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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Geometric calculus</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Matrix_calculus" title="Matrix calculus">matrix calculus</a> or <a href="Vector_calculus" title="Vector calculus">vector calculus</a>.</div>
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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}">
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<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation>
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</math></span><img src="./17d063dc86a53a2efb1fe86f4a5d47d498652766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}" loading="lazy"></span></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;">
<ul><li><a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul>
<div class="hlist">
<ul><li><a href="Limit_of_a_function" title="Limit of a function">Limits</a></li>
<li><a href="Continuous_function" title="Continuous function">Continuity</a></li></ul>
</div><div class="hlist">
<ul><li><a href="Rolle's_theorem" title="Rolle's theorem">Rolle's theorem</a></li>
<li><a href="Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li>
<li><a href="Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul>
</div></td></tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><span style="font-size:120%"><a href="Differential_calculus" title="Differential calculus">Differential</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">
Definitions</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Derivative" title="Derivative">Derivative</a> (<a href="Generalizations_of_the_derivative" title="Generalizations of the derivative">generalizations</a>)</li>
<li><a href="Differential_(mathematics)" title="Differential (mathematics)">Differential</a>
<ul><li><a href="Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li>
<li><a href="Differential_of_a_function" title="Differential of a function">of a function</a></li>
<li><a href="Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td>
</tr><tr><th class="sidebar-heading">
Concepts</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li>
<li><a href="Second_derivative" title="Second derivative">Second derivative</a></li>
<li><a href="Implicit_function" title="Implicit function">Implicit differentiation</a></li>
<li><a href="Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li>
<li><a href="Related_rates" title="Related rates">Related rates</a></li>
<li><a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li>
<li><a href="Product_rule" title="Product rule">Product</a></li>
<li><a href="Chain_rule" title="Chain rule">Chain</a></li>
<li><a href="Power_rule" title="Power rule">Power</a></li>
<li><a href="Quotient_rule" title="Quotient rule">Quotient</a></li>
<li><a href="L'H%C3%B4pital's_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li>
<li><a href="Inverse_function_rule" title="Inverse function rule">Inverse</a></li>
<li><a href="General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li>
<li><a href="Fa%C3%A0_di_Bruno's_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li>
<li><a href="Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Integral" title="Integral">Integral</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">
<ul><li><a href="Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li>
<li><a href="Integral_transform" title="Integral transform">Integral transform</a></li>
<li><a href="Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
Definitions</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Antiderivative" title="Antiderivative">Antiderivative</a></li>
<li><a href="Integral" title="Integral">Integral</a> (<a href="Improper_integral" title="Improper integral">improper</a>)</li>
<li><a href="Riemann_integral" title="Riemann integral">Riemann integral</a></li>
<li><a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li>
<li><a href="Contour_integration" title="Contour integration">Contour integration</a></li>
<li><a href="Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
Integration by</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Integration_by_parts" title="Integration by parts">Parts</a></li>
<li><a href="Disc_integration" title="Disc integration">Discs</a></li>
<li><a href="Shell_integration" title="Shell integration">Cylindrical shells</a></li>
<li><a href="Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="Euler_substitution" title="Euler substitution">Euler</a>)</li>
<li><a href="Integration_using_Euler's_formula" title="Integration using Euler's formula">Euler's formula</a></li>
<li><a href="Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li>
<li><a href="Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li>
<li><a href="Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li>
<li><a href="Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li>
<li><a href="Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Series_(mathematics)" title="Series (mathematics)">Series</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">
<ul><li><a href="Geometric_series" title="Geometric series">Geometric</a> (<a href="Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li>
<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li>
<li><a href="Alternating_series" title="Alternating series">Alternating</a></li>
<li><a href="Power_series" title="Power series">Power</a></li>
<li><a href="Binomial_series" title="Binomial series">Binomial</a></li>
<li><a href="Taylor_series" title="Taylor series">Taylor</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li>
<li><a href="Ratio_test" title="Ratio test">Ratio</a></li>
<li><a href="Root_test" title="Root test">Root</a></li>
<li><a href="Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li>
<li><a href="Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li>
<li><br><a href="Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li>
<li><a href="Alternating_series_test" title="Alternating series test">Alternating series</a></li>
<li><a href="Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li>
<li><a href="Dirichlet's_test" title="Dirichlet's test">Dirichlet</a></li>
<li><a href="Abel's_test" title="Abel's test">Abel</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Vector_calculus" title="Vector calculus">Vector</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">
<ul><li><a href="Gradient" title="Gradient">Gradient</a></li>
<li><a href="Divergence" title="Divergence">Divergence</a></li>
<li><a href="Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li>
<li><a href="Laplace_operator" title="Laplace operator">Laplacian</a></li>
<li><a href="Directional_derivative" title="Directional derivative">Directional derivative</a></li>
<li><a href="Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
Theorems</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Gradient_theorem" title="Gradient theorem">Gradient</a></li>
<li><a href="Green's_theorem" title="Green's theorem">Green's</a></li>
<li><a href="Stokes'_theorem" title="Stokes' theorem">Stokes'</a></li>
<li><a href="Divergence_theorem" title="Divergence theorem">Divergence</a></li>
<li><a href="Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li>
<li><a href="Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Multivariable_calculus" title="Multivariable calculus">Multivariable</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">
Formalisms</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Matrix_calculus" title="Matrix calculus">Matrix</a></li>
<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li>
<li><a href="Exterior_derivative" title="Exterior derivative">Exterior</a></li>
</ul></td>
</tr><tr><th class="sidebar-heading">
Definitions</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>
<li><a href="Multiple_integral" title="Multiple integral">Multiple integral</a></li>
<li><a href="Line_integral" title="Line integral">Line integral</a></li>
<li><a href="Surface_integral" title="Surface integral">Surface integral</a></li>
<li><a href="Volume_integral" title="Volume integral">Volume integral</a></li>
<li><a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li>
<li><a href="Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td>
</tr></tbody></table></div></div></td>
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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Advanced</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <b>geometric calculus</b> extends <a href="Geometric_algebra" title="Geometric algebra">geometric algebra</a> to include <a href="Derivative" title="Derivative">differentiation</a> and <a href="Integral" title="Integral">integration</a>. The formalism is powerful and can be shown to reproduce other mathematical theories including <a href="Vector_calculus" title="Vector calculus">vector calculus</a>, <a href="Differential_geometry" title="Differential geometry">differential geometry</a>, and <a href="Differential_form" title="Differential form">differential forms</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Differentiation">Differentiation</h2></div>
<p>With a geometric algebra given, let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b}</annotation>
</semantics>
</math></span><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> be <a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> and let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> be a <a href="Multivector" title="Multivector">multivector</a>-valued function of a vector. The <a href="Directional_derivative" title="Directional derivative">directional derivative</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> along <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b}</annotation>
</semantics>
</math></span><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> is defined as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\nabla _{b}F)(a)=\lim _{\epsilon \rightarrow 0}{\frac {F(a+\epsilon b)-F(a)}{\epsilon }},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ϵ<!-- ϵ --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>+</mo>
<mi>ϵ<!-- ϵ --></mi>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϵ<!-- ϵ --></mi>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\nabla _{b}F)(a)=\lim _{\epsilon \rightarrow 0}{\frac {F(a+\epsilon b)-F(a)}{\epsilon }},}</annotation>
</semantics>
</math></span><img src="./38ae7692fd8d7fd39d577f310216f7f11f791a0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.843ex; height:5.843ex;" alt="{\displaystyle (\nabla _{b}F)(a)=\lim _{\epsilon \rightarrow 0}{\frac {F(a+\epsilon b)-F(a)}{\epsilon }},}" loading="lazy"></span></dd></dl>
<p>provided that the limit exists for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b}</annotation>
</semantics>
</math></span><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span>, where the limit is taken for scalar <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϵ<!-- ϵ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation>
</semantics>
</math></span><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"></span>. This is similar to the usual definition of a directional derivative but extends it to functions that are not necessarily scalar-valued.
</p><p>Next, choose a set of <a href="Basis_vector" class="mw-redirect" title="Basis vector">basis vectors</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{i}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{e_{i}\}}</annotation>
</semantics>
</math></span><img src="./d45dea23899c892d30b142d73ed0fa19233ee4a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.208ex; height:2.843ex;" alt="{\displaystyle \{e_{i}\}}" loading="lazy"></span> and consider the operators, denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial _{i}}</annotation>
</semantics>
</math></span><img src="./9b6be63a3ddb914c608027404afea96933df7c79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.034ex; height:2.509ex;" alt="{\displaystyle \partial _{i}}" loading="lazy"></span>, that perform directional derivatives in the directions of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation>
</semantics>
</math></span><img src="./ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"></span>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{i}:F\mapsto (x\mapsto (\nabla _{e_{i}}F)(x)).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>:</mo>
<mi>F</mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msub>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial _{i}:F\mapsto (x\mapsto (\nabla _{e_{i}}F)(x)).}</annotation>
</semantics>
</math></span><img src="./04a69b74bfd3010386678d8514e3b4b6656bd46b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.974ex; height:3.009ex;" alt="{\displaystyle \partial _{i}:F\mapsto (x\mapsto (\nabla _{e_{i}}F)(x)).}" loading="lazy"></span></dd></dl>
<p>Then, using the <a href="Einstein_summation_notation" class="mw-redirect" title="Einstein summation notation">Einstein summation notation</a>, consider the operator:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i}\partial _{i},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{i}\partial _{i},}</annotation>
</semantics>
</math></span><img src="./099ffcb91263643be504aafda14b518bc4619326.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.564ex; height:3.009ex;" alt="{\displaystyle e^{i}\partial _{i},}" loading="lazy"></span></dd></dl>
<p>which means
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\mapsto e^{i}\partial _{i}F,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mi>F</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F\mapsto e^{i}\partial _{i}F,}</annotation>
</semantics>
</math></span><img src="./94907c7dac82c41d874eff976d6208dc05fa3b08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.66ex; height:3.009ex;" alt="{\displaystyle F\mapsto e^{i}\partial _{i}F,}" loading="lazy"></span></dd></dl>
<p>where the geometric product is applied after the directional derivative. More verbosely:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\mapsto (x\mapsto e^{i}(\nabla _{e_{i}}F)(x)).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msub>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F\mapsto (x\mapsto e^{i}(\nabla _{e_{i}}F)(x)).}</annotation>
</semantics>
</math></span><img src="./b5626e46a01c89d8874545290f03d768796688e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.886ex; height:3.343ex;" alt="{\displaystyle F\mapsto (x\mapsto e^{i}(\nabla _{e_{i}}F)(x)).}" loading="lazy"></span></dd></dl>
<p>This operator is independent of the choice of frame, and can thus be used to define what in geometric calculus is called the <i>vector derivative</i>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla =e^{i}\partial _{i}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla =e^{i}\partial _{i}.}</annotation>
</semantics>
</math></span><img src="./a96754c45a587124bb7f1491f145e6505f6fc55a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.598ex; height:3.009ex;" alt="{\displaystyle \nabla =e^{i}\partial _{i}.}" loading="lazy"></span></dd></dl>
<p>This is similar to the usual definition of the <a href="Gradient" title="Gradient">gradient</a>, but it, too, extends to functions that are not necessarily scalar-valued.
</p><p>The directional derivative is linear regarding its direction, that is:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{\alpha a+\beta b}=\alpha \nabla _{a}+\beta \nabla _{b}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mi>a</mi>
<mo>+</mo>
<mi>β<!-- β --></mi>
<mi>b</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>α<!-- α --></mi>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>β<!-- β --></mi>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla _{\alpha a+\beta b}=\alpha \nabla _{a}+\beta \nabla _{b}.}</annotation>
</semantics>
</math></span><img src="./3e202c5455c495780160fdffa97da93b0a0bedff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.332ex; height:2.843ex;" alt="{\displaystyle \nabla _{\alpha a+\beta b}=\alpha \nabla _{a}+\beta \nabla _{b}.}" loading="lazy"></span></dd></dl>
<p>From this follows that the directional derivative is the inner product of its direction by the vector derivative. All needs to be observed is that the direction <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> can be written <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(a\cdot e^{i})e_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=(a\cdot e^{i})e_{i}}</annotation>
</semantics>
</math></span><img src="./7bdd2a20374f16914502c8f46bbce43d8f3d037b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.813ex; height:3.176ex;" alt="{\displaystyle a=(a\cdot e^{i})e_{i}}" loading="lazy"></span>, so that:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{a}=\nabla _{(a\cdot e^{i})e_{i}}=(a\cdot e^{i})\nabla _{e_{i}}=a\cdot (e^{i}\nabla _{e_{i}})=a\cdot \nabla .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo>=</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla _{a}=\nabla _{(a\cdot e^{i})e_{i}}=(a\cdot e^{i})\nabla _{e_{i}}=a\cdot (e^{i}\nabla _{e_{i}})=a\cdot \nabla .}</annotation>
</semantics>
</math></span><img src="./222bc7c5de99686900d55cf11f4b73d00b6d7403.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:48.801ex; height:3.509ex;" alt="{\displaystyle \nabla _{a}=\nabla _{(a\cdot e^{i})e_{i}}=(a\cdot e^{i})\nabla _{e_{i}}=a\cdot (e^{i}\nabla _{e_{i}})=a\cdot \nabla .}" loading="lazy"></span></dd></dl>
<p>For this reason, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{a}F(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla _{a}F(x)}</annotation>
</semantics>
</math></span><img src="./778c451c134e3bc07df9aa0b3b55970b89efe7a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.918ex; height:2.843ex;" alt="{\displaystyle \nabla _{a}F(x)}" loading="lazy"></span> is often noted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot \nabla F(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\cdot \nabla F(x)}</annotation>
</semantics>
</math></span><img src="./38465ffedc81d50c3b85a6581e917858bd8b4f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.725ex; height:2.843ex;" alt="{\displaystyle a\cdot \nabla F(x)}" loading="lazy"></span>.
</p><p>The standard <a href="Order_of_operations" title="Order of operations">order of operations</a> for the vector derivative is that it acts only on the function closest to its immediate right. Given two functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G}</annotation>
</semantics>
</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span>, then for example we have
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla FG=(\nabla F)G.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>F</mi>
<mi>G</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mi>G</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla FG=(\nabla F)G.}</annotation>
</semantics>
</math></span><img src="./2bb6a70cb331164b5877737eb40fa17690b398b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.561ex; height:2.843ex;" alt="{\displaystyle \nabla FG=(\nabla F)G.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Product_rule">Product rule</h3></div>
<p>Although the partial derivative exhibits a <a href="Product_rule" title="Product rule">product rule</a>, the vector derivative only partially inherits this property. Consider two functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G}</annotation>
</semantics>
</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nabla (FG)&=e^{i}\partial _{i}(FG)\\&=e^{i}((\partial _{i}F)G+F(\partial _{i}G))\\&=e^{i}(\partial _{i}F)G+e^{i}F(\partial _{i}G).\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo stretchy="false">(</mo>
<mi>F</mi>
<mi>G</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>F</mi>
<mi>G</mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mi>G</mi>
<mo>+</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mi>G</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mi>G</mi>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mi>F</mi>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mi>G</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nabla (FG)&=e^{i}\partial _{i}(FG)\\&=e^{i}((\partial _{i}F)G+F(\partial _{i}G))\\&=e^{i}(\partial _{i}F)G+e^{i}F(\partial _{i}G).\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./215337f804d07bf436322ea8f2c138f4ae7a0058.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:33.238ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}\nabla (FG)&=e^{i}\partial _{i}(FG)\\&=e^{i}((\partial _{i}F)G+F(\partial _{i}G))\\&=e^{i}(\partial _{i}F)G+e^{i}F(\partial _{i}G).\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Since the geometric product is not <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i}F\neq Fe^{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mi>F</mi>
<mo>≠<!-- ≠ --></mo>
<mi>F</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{i}F\neq Fe^{i}}</annotation>
</semantics>
</math></span><img src="./3fd916db9bd3b5480356f17741403d1b3d8035fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.346ex; height:3.176ex;" alt="{\displaystyle e^{i}F\neq Fe^{i}}" loading="lazy"></span> in general, we need a new notation to proceed. A solution is to adopt the <i><a href="Overdot" class="mw-redirect" title="Overdot">overdot</a> notation</i>, in which the scope of a vector derivative with an overdot is the multivector-valued function sharing the same overdot. In this case, if we define
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\nabla }}F{\dot {G}}=e^{i}F(\partial _{i}G),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>G</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mi>F</mi>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mi>G</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\dot {\nabla }}F{\dot {G}}=e^{i}F(\partial _{i}G),}</annotation>
</semantics>
</math></span><img src="./f4ac4039824ab1c45fc76716911bbeb1cc6c15e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.543ex; height:3.343ex;" alt="{\displaystyle {\dot {\nabla }}F{\dot {G}}=e^{i}F(\partial _{i}G),}" loading="lazy"></span></dd></dl>
<p>then the product rule for the vector derivative is
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla (FG)=\nabla FG+{\dot {\nabla }}F{\dot {G}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo stretchy="false">(</mo>
<mi>F</mi>
<mi>G</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>F</mi>
<mi>G</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>G</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla (FG)=\nabla FG+{\dot {\nabla }}F{\dot {G}}.}</annotation>
</semantics>
</math></span><img src="./0ad307b8fbb6421dc38f44f00d00fa95451181d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.905ex; height:3.343ex;" alt="{\displaystyle \nabla (FG)=\nabla FG+{\dot {\nabla }}F{\dot {G}}.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Interior_and_exterior_derivative">Interior and exterior derivative</h3></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> be an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>-grade multivector. Then we can define an additional pair of operators, the interior and exterior derivatives,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot F=\langle \nabla F\rangle _{r-1}=e^{i}\cdot \partial _{i}F,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>F</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>F</mi>
<msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mi>F</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot F=\langle \nabla F\rangle _{r-1}=e^{i}\cdot \partial _{i}F,}</annotation>
</semantics>
</math></span><img src="./c3d248056ae07024be2c7f0e7a01484b0c1b2dc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.097ex; height:3.176ex;" alt="{\displaystyle \nabla \cdot F=\langle \nabla F\rangle _{r-1}=e^{i}\cdot \partial _{i}F,}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \wedge F=\langle \nabla F\rangle _{r+1}=e^{i}\wedge \partial _{i}F.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>∧<!-- ∧ --></mo>
<mi>F</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>F</mi>
<msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mi>F</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla \wedge F=\langle \nabla F\rangle _{r+1}=e^{i}\wedge \partial _{i}F.}</annotation>
</semantics>
</math></span><img src="./c0321f3d2a4d42f07dcf7ba7732cecf9eae40693.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.904ex; height:3.176ex;" alt="{\displaystyle \nabla \wedge F=\langle \nabla F\rangle _{r+1}=e^{i}\wedge \partial _{i}F.}" loading="lazy"></span></dd></dl>
<p>In particular, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> is grade 1 (vector-valued function), then we can write
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla F=\nabla \cdot F+\nabla \wedge F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>F</mi>
<mo>=</mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>F</mi>
<mo>+</mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>∧<!-- ∧ --></mo>
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla F=\nabla \cdot F+\nabla \wedge F}</annotation>
</semantics>
</math></span><img src="./82870582fd3129b3857583aa3fc23db8b061444e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.231ex; height:2.343ex;" alt="{\displaystyle \nabla F=\nabla \cdot F+\nabla \wedge F}" loading="lazy"></span></dd></dl>
<p>and identify the <a href="Divergence" title="Divergence">divergence</a> and <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a> as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot F=\operatorname {div} F,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>F</mi>
<mo>=</mo>
<mi>div</mi>
<mo><!-- --></mo>
<mi>F</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot F=\operatorname {div} F,}</annotation>
</semantics>
</math></span><img src="./974d766a4555708d4b0a114a47febb1749674825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.396ex; height:2.509ex;" alt="{\displaystyle \nabla \cdot F=\operatorname {div} F,}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \wedge F=I\,\operatorname {curl} F.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>∧<!-- ∧ --></mo>
<mi>F</mi>
<mo>=</mo>
<mi>I</mi>
<mspace width="thinmathspace"></mspace>
<mi>curl</mi>
<mo><!-- --></mo>
<mi>F</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla \wedge F=I\,\operatorname {curl} F.}</annotation>
</semantics>
</math></span><img src="./c3d12e72efa70a877990888074ff3e79afdc1dfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.962ex; height:2.176ex;" alt="{\displaystyle \nabla \wedge F=I\,\operatorname {curl} F.}" loading="lazy"></span></dd></dl>
<p>Unlike the vector derivative, neither the interior derivative operator nor the exterior derivative operator is invertible.
</p>
<div class="mw-heading mw-heading3"><h3 id="Multivector_derivative">Multivector derivative</h3></div>
<p>The derivative with respect to a vector as discussed above can be generalized to a derivative with respect to a general multivector, called the <b>multivector derivative</b>.
</p><p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> be a multivector-valued function of a multivector. The directional derivative of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> in the direction <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> are multivectors, is defined as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A*\partial _{X}F(X)=\lim _{\epsilon \to 0}{\frac {F(X+\epsilon A)-F(X)}{\epsilon }}\ ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∗<!-- ∗ --></mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ϵ<!-- ϵ --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>+</mo>
<mi>ϵ<!-- ϵ --></mi>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϵ<!-- ϵ --></mi>
</mfrac>
</mrow>
<mtext> </mtext>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A*\partial _{X}F(X)=\lim _{\epsilon \to 0}{\frac {F(X+\epsilon A)-F(X)}{\epsilon }}\ ,}</annotation>
</semantics>
</math></span><img src="./8fe3f4ec49f43d316bd2dc3bc4f6388a7a4237e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.542ex; height:5.843ex;" alt="{\displaystyle A*\partial _{X}F(X)=\lim _{\epsilon \to 0}{\frac {F(X+\epsilon A)-F(X)}{\epsilon }}\ ,}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A*B=\langle AB\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∗<!-- ∗ --></mo>
<mi>B</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>A</mi>
<mi>B</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A*B=\langle AB\rangle }</annotation>
</semantics>
</math></span><img src="./df3ac9a0b50969aa6332a790b3b2a703b0dcfa2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.117ex; height:2.843ex;" alt="{\displaystyle A*B=\langle AB\rangle }" loading="lazy"></span> is the <a href="Geometric_algebra#Extensions_of_the_inner_and_exterior_products" title="Geometric algebra">scalar product</a>. With <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{i}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{e_{i}\}}</annotation>
</semantics>
</math></span><img src="./d45dea23899c892d30b142d73ed0fa19233ee4a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.208ex; height:2.843ex;" alt="{\displaystyle \{e_{i}\}}" loading="lazy"></span> a vector basis and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e^{i}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{e^{i}\}}</annotation>
</semantics>
</math></span><img src="./58c5a4b8c73f70d664b9860baf59f7b1f228d003.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.208ex; height:3.176ex;" alt="{\displaystyle \{e^{i}\}}" loading="lazy"></span> the corresponding <a href="Geometric_algebra#Dual_basis" title="Geometric algebra">dual basis</a>, the multivector derivative is defined in terms of the directional derivative as<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial X}}=\partial _{X}=\sum _{i<\dots <j}e^{i}\wedge \cdots \wedge e^{j}(e_{j}\wedge \cdots \wedge e_{i})*\partial _{X}\ .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>X</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo><</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo><</mo>
<mi>j</mi>
</mrow>
</munder>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∧<!-- ∧ --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>∗<!-- ∗ --></mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mtext> </mtext>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial X}}=\partial _{X}=\sum _{i<\dots <j}e^{i}\wedge \cdots \wedge e^{j}(e_{j}\wedge \cdots \wedge e_{i})*\partial _{X}\ .}</annotation>
</semantics>
</math></span><img src="./70622c22bb66ef56965330f6056936803fa58665.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:50.94ex; height:6.843ex;" alt="{\displaystyle {\frac {\partial }{\partial X}}=\partial _{X}=\sum _{i<\dots <j}e^{i}\wedge \cdots \wedge e^{j}(e_{j}\wedge \cdots \wedge e_{i})*\partial _{X}\ .}" loading="lazy"></span></dd></dl>
<p>This equation is just expressing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{X}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial _{X}}</annotation>
</semantics>
</math></span><img src="./bcceacf0fe653ba62ef9521ea1f81c744e137c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.867ex; height:2.509ex;" alt="{\displaystyle \partial _{X}}" loading="lazy"></span> in terms of components in a reciprocal basis of blades, as discussed in the article section <a href="Geometric_algebra#Dual_basis" title="Geometric algebra">Geometric algebra#Dual basis</a>.
</p><p>A key property of the multivector derivative is that
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{X}\langle XA\rangle =P_{X}(A)\ ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>X</mi>
<mi>A</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>=</mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mtext> </mtext>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial _{X}\langle XA\rangle =P_{X}(A)\ ,}</annotation>
</semantics>
</math></span><img src="./16f71af155500967b9d1c0ce3ede65e6527088bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.402ex; height:2.843ex;" alt="{\displaystyle \partial _{X}\langle XA\rangle =P_{X}(A)\ ,}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{X}(A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P_{X}(A)}</annotation>
</semantics>
</math></span><img src="./6a4a94a84c701844934ac4037a46c5ddeddde982.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.677ex; height:2.843ex;" alt="{\displaystyle P_{X}(A)}" loading="lazy"></span> is the projection of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> onto the grades contained in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>.
</p><p>The multivector derivative finds applications in <a href="Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian field theory</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Integration">Integration</h2></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{1},\ldots ,e_{n}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{e_{1},\ldots ,e_{n}\}}</annotation>
</semantics>
</math></span><img src="./014c2407d0cf40698e1a1ba829d2622919d89ede.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.943ex; height:2.843ex;" alt="{\displaystyle \{e_{1},\ldots ,e_{n}\}}" loading="lazy"></span> be a set of basis vectors that span an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensional vector space. From geometric algebra, we interpret the <a href="Pseudoscalar" title="Pseudoscalar">pseudoscalar</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}\wedge e_{2}\wedge \cdots \wedge e_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{1}\wedge e_{2}\wedge \cdots \wedge e_{n}}</annotation>
</semantics>
</math></span><img src="./46c010c96b20bb36cc5ba75356c97b237eab2c3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.048ex; height:2.343ex;" alt="{\displaystyle e_{1}\wedge e_{2}\wedge \cdots \wedge e_{n}}" loading="lazy"></span> to be the <a href="Signed_volume" class="mw-redirect" title="Signed volume">signed volume</a> of the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-<a href="Parallelepiped#Parallelotope" title="Parallelepiped">parallelotope</a> subtended by these basis vectors. If the basis vectors are <a href="Orthonormal" class="mw-redirect" title="Orthonormal">orthonormal</a>, then this is the unit pseudoscalar.
</p><p>More generally, we may restrict ourselves to a subset of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> of the basis vectors, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq k\leq n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<mi>k</mi>
<mo>≤<!-- ≤ --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1\leq k\leq n}</annotation>
</semantics>
</math></span><img src="./78ec00bc2eb99b403bee93def3c12ae87f1e3c3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle 1\leq k\leq n}" loading="lazy"></span>, to treat the length, area, or other general <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-volume of a subspace in the overall <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensional vector space. We denote these selected basis vectors by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{i_{1}},\ldots ,e_{i_{k}}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{e_{i_{1}},\ldots ,e_{i_{k}}\}}</annotation>
</semantics>
</math></span><img src="./a4c87fb84f73202803a87d98fe43cf9775a4e8b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.961ex; height:3.009ex;" alt="{\displaystyle \{e_{i_{1}},\ldots ,e_{i_{k}}\}}" loading="lazy"></span>. A general <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-volume of the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-parallelotope subtended by these basis vectors is the grade <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> multivector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}}</annotation>
</semantics>
</math></span><img src="./510f51896e7d8e37d60291da9ee9e3b2cdabe5b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.643ex; height:2.676ex;" alt="{\displaystyle e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}}" loading="lazy"></span>.
</p><p>Even more generally, we may consider a new set of vectors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x^{i_{1}}e_{i_{1}},\ldots ,x^{i_{k}}e_{i_{k}}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msup>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msup>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{x^{i_{1}}e_{i_{1}},\ldots ,x^{i_{k}}e_{i_{k}}\}}</annotation>
</semantics>
</math></span><img src="./2dad4fa2984ffd22df4a81e047aab34e5d6f84ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.91ex; height:3.343ex;" alt="{\displaystyle \{x^{i_{1}}e_{i_{1}},\ldots ,x^{i_{k}}e_{i_{k}}\}}" loading="lazy"></span> proportional to the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> basis vectors, where each of the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x^{i_{j}}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
</msup>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{x^{i_{j}}\}}</annotation>
</semantics>
</math></span><img src="./a8a4327ee9dcd8e05486d777608ac8a121511a7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.169ex; height:3.176ex;" alt="{\displaystyle \{x^{i_{j}}\}}" loading="lazy"></span> is a component that scales one of the basis vectors. We are free to choose components as infinitesimally small as we wish as long as they remain nonzero. Since the outer product of these terms can be interpreted as a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-volume, a natural way to define a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> is
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}d^{k}X&=\left(dx^{i_{1}}e_{i_{1}}\right)\wedge \left(dx^{i_{2}}e_{i_{2}}\right)\wedge \cdots \wedge \left(dx^{i_{k}}e_{i_{k}}\right)\\&=\left(e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}\right)dx^{i_{1}}dx^{i_{2}}\cdots dx^{i_{k}}.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mi>X</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msup>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>∧<!-- ∧ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msup>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
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<mrow>
<mo>(</mo>
<mrow>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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</msub>
</mrow>
</msup>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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<mtr>
<mtd></mtd>
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<mi></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
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<msub>
<mi>e</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
<msup>
<mi>x</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</mrow>
</msup>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msup>
<mo>⋯<!-- ⋯ --></mo>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msup>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}d^{k}X&=\left(dx^{i_{1}}e_{i_{1}}\right)\wedge \left(dx^{i_{2}}e_{i_{2}}\right)\wedge \cdots \wedge \left(dx^{i_{k}}e_{i_{k}}\right)\\&=\left(e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}\right)dx^{i_{1}}dx^{i_{2}}\cdots dx^{i_{k}}.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./f5b00a0284b68b9e610dbdb7a9e71a5e7791f966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.655ex; margin-bottom: -0.183ex; width:46ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}d^{k}X&=\left(dx^{i_{1}}e_{i_{1}}\right)\wedge \left(dx^{i_{2}}e_{i_{2}}\right)\wedge \cdots \wedge \left(dx^{i_{k}}e_{i_{k}}\right)\\&=\left(e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}\right)dx^{i_{1}}dx^{i_{2}}\cdots dx^{i_{k}}.\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>The measure is therefore always proportional to the unit pseudoscalar of a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-dimensional subspace of the vector space. Compare the <a href="Riemannian_volume_form" class="mw-redirect" title="Riemannian volume form">Riemannian volume form</a> in the theory of differential forms. The integral is taken with respect to this measure:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}F(x)\,d^{k}X=\int _{V}F(x)\left(e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}\right)dx^{i_{1}}dx^{i_{2}}\cdots dx^{i_{k}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>V</mi>
</mrow>
</msub>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mi>X</mi>
<mo>=</mo>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>V</mi>
</mrow>
</msub>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msup>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msup>
<mo>⋯<!-- ⋯ --></mo>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{V}F(x)\,d^{k}X=\int _{V}F(x)\left(e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}\right)dx^{i_{1}}dx^{i_{2}}\cdots dx^{i_{k}}.}</annotation>
</semantics>
</math></span><img src="./f33badafa74e2d089826f93b8c324a421326ba79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:62.133ex; height:5.676ex;" alt="{\displaystyle \int _{V}F(x)\,d^{k}X=\int _{V}F(x)\left(e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}\right)dx^{i_{1}}dx^{i_{2}}\cdots dx^{i_{k}}.}" loading="lazy"></span></dd></dl>
<p>More formally, consider some directed volume <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> of the subspace. We may divide this volume into a sum of <a href="Simplices" class="mw-redirect" title="Simplices">simplices</a>. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{i}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{x_{i}\}}</annotation>
</semantics>
</math></span><img src="./c9d90fd628e1497151325d23b808b6d0296e7701.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.454ex; height:2.843ex;" alt="{\displaystyle \{x_{i}\}}" loading="lazy"></span> be the coordinates of the vertices. At each vertex we assign a measure <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta U_{i}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta U_{i}(x)}</annotation>
</semantics>
</math></span><img src="./0790285493fc7f88216392e6b60ea8cef387c1cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.462ex; height:2.843ex;" alt="{\displaystyle \Delta U_{i}(x)}" loading="lazy"></span> as the average measure of the simplices sharing the vertex. Then the integral of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation>
</semantics>
</math></span><img src="./71a82805d469cdfa7856c11d6ee756acd1dc7174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}" loading="lazy"></span> with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U(x)}</annotation>
</semantics>
</math></span><img src="./6d626d3a1e65c94535c811c73fa83389cfb76683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.922ex; height:2.843ex;" alt="{\displaystyle U(x)}" loading="lazy"></span> over this volume is obtained in the limit of finer partitioning of the volume into smaller simplices:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}F\,dU=\lim _{n\rightarrow \infty }\sum _{i=1}^{n}F(x_{i})\,\Delta U_{i}(x_{i}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>V</mi>
</mrow>
</msub>
<mi>F</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>U</mi>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
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<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munder>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mn>1</mn>
</mrow>
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<mi>F</mi>
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</mrow>
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<mi>U</mi>
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<mi>i</mi>
</mrow>
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<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{V}F\,dU=\lim _{n\rightarrow \infty }\sum _{i=1}^{n}F(x_{i})\,\Delta U_{i}(x_{i}).}</annotation>
</semantics>
</math></span><img src="./7f7e9744342c13a7961154849951cb6d6addb50f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.777ex; height:6.843ex;" alt="{\displaystyle \int _{V}F\,dU=\lim _{n\rightarrow \infty }\sum _{i=1}^{n}F(x_{i})\,\Delta U_{i}(x_{i}).}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Fundamental_theorem_of_geometric_calculus">Fundamental theorem of geometric calculus</h3></div>
<p>The reason for defining the vector derivative and integral as above is that they allow a strong generalization of <a href="Stokes'_theorem" title="Stokes' theorem">Stokes' theorem</a>. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}(A;x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">L</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>;</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}(A;x)}</annotation>
</semantics>
</math></span><img src="./f591f35b9bf0a4095d47b925aab09a9df23e9c24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.176ex; height:2.843ex;" alt="{\displaystyle {\mathsf {L}}(A;x)}" loading="lazy"></span> be a multivector-valued function of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>-grade input <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> and general position <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>, linear in its first argument. Then the fundamental theorem of geometric calculus relates the integral of a derivative over the volume <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> to the integral over its boundary:
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}{\dot {\mathsf {L}}}\left({\dot {\nabla }}dX;x\right)=\oint _{\partial V}{\mathsf {L}}(dS;x).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>V</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">L</mi>
</mrow>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mi>d</mi>
<mi>X</mi>
<mo>;</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mo>∮<!-- ∮ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>V</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">L</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>d</mi>
<mi>S</mi>
<mo>;</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{V}{\dot {\mathsf {L}}}\left({\dot {\nabla }}dX;x\right)=\oint _{\partial V}{\mathsf {L}}(dS;x).}</annotation>
</semantics>
</math></span><img src="./096f1d78c2fc5bfbc38dea5b49926d5783d3af3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.449ex; height:5.676ex;" alt="{\displaystyle \int _{V}{\dot {\mathsf {L}}}\left({\dot {\nabla }}dX;x\right)=\oint _{\partial V}{\mathsf {L}}(dS;x).}" loading="lazy"></span>
</p><p>As an example, let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}(A;x)=\langle F(x)AI^{-1}\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">L</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>;</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>A</mi>
<msup>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}(A;x)=\langle F(x)AI^{-1}\rangle }</annotation>
</semantics>
</math></span><img src="./cb166b1b53fc7a145581a7fff320dfc541b73661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.256ex; height:3.176ex;" alt="{\displaystyle {\mathsf {L}}(A;x)=\langle F(x)AI^{-1}\rangle }" loading="lazy"></span> for a vector-valued function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation>
</semantics>
</math></span><img src="./71a82805d469cdfa7856c11d6ee756acd1dc7174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}" loading="lazy"></span> and a (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>
</semantics>
</math></span><img src="./fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span>)-grade multivector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>. We find that
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{V}{\dot {\mathsf {L}}}\left({\dot {\nabla }}dX;x\right)&=\int _{V}\langle {\dot {F}}(x){\dot {\nabla }}\,dX\,I^{-1}\rangle \\&=\int _{V}\langle {\dot {F}}(x){\dot {\nabla }}\,|dX|\rangle \\&=\int _{V}\nabla \cdot F(x)\,|dX|.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>V</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">L</mi>
</mrow>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mi>d</mi>
<mi>X</mi>
<mo>;</mo>
<mi>x</mi>
</mrow>
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</mrow>
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<mi></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>V</mi>
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</msub>
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<mrow class="MJX-TeXAtom-ORD">
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<mover>
<mi>F</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
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</mover>
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</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>X</mi>
<mspace width="thinmathspace"></mspace>
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<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
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</msup>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
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<msub>
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<mrow class="MJX-TeXAtom-ORD">
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</msub>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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<mover>
<mi>F</mi>
<mo>˙<!-- ˙ --></mo>
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</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
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<mo stretchy="false">|</mo>
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</mtd>
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<mtr>
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<mi></mi>
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<mrow class="MJX-TeXAtom-ORD">
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</mrow>
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<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
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<mo stretchy="false">|</mo>
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</mtd>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{V}{\dot {\mathsf {L}}}\left({\dot {\nabla }}dX;x\right)&=\int _{V}\langle {\dot {F}}(x){\dot {\nabla }}\,dX\,I^{-1}\rangle \\&=\int _{V}\langle {\dot {F}}(x){\dot {\nabla }}\,|dX|\rangle \\&=\int _{V}\nabla \cdot F(x)\,|dX|.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./f823ba07272921312b5f691e33cead3a09c9f633.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:37.295ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}\int _{V}{\dot {\mathsf {L}}}\left({\dot {\nabla }}dX;x\right)&=\int _{V}\langle {\dot {F}}(x){\dot {\nabla }}\,dX\,I^{-1}\rangle \\&=\int _{V}\langle {\dot {F}}(x){\dot {\nabla }}\,|dX|\rangle \\&=\int _{V}\nabla \cdot F(x)\,|dX|.\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Likewise,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\oint _{\partial V}{\mathsf {L}}(dS;x)&=\oint _{\partial V}\langle F(x)\,dS\,I^{-1}\rangle \\&=\oint _{\partial V}\langle F(x){\hat {n}}\,|dS|\rangle \\&=\oint _{\partial V}F(x)\cdot {\hat {n}}\,|dS|.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>V</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">L</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>d</mi>
<mi>S</mi>
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<mi>x</mi>
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<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
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<mn>1</mn>
</mrow>
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</mtd>
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<mtr>
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<mtr>
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<mi></mi>
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<mi>F</mi>
<mo stretchy="false">(</mo>
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</mtd>
</mtr>
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</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\oint _{\partial V}{\mathsf {L}}(dS;x)&=\oint _{\partial V}\langle F(x)\,dS\,I^{-1}\rangle \\&=\oint _{\partial V}\langle F(x){\hat {n}}\,|dS|\rangle \\&=\oint _{\partial V}F(x)\cdot {\hat {n}}\,|dS|.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./57bd18d20b73b5eee39c3e2886a4db911355b12e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:33.554ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}\oint _{\partial V}{\mathsf {L}}(dS;x)&=\oint _{\partial V}\langle F(x)\,dS\,I^{-1}\rangle \\&=\oint _{\partial V}\langle F(x){\hat {n}}\,|dS|\rangle \\&=\oint _{\partial V}F(x)\cdot {\hat {n}}\,|dS|.\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Thus we recover the <a href="Divergence_theorem" title="Divergence theorem">divergence theorem</a>,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}\nabla \cdot F(x)\,|dX|=\oint _{\partial V}F(x)\cdot {\hat {n}}\,|dS|.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>V</mi>
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</msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
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<mo stretchy="false">|</mo>
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<mi>d</mi>
<mi>X</mi>
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<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<msub>
<mo>∮<!-- ∮ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>V</mi>
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<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">^<!-- ^ --></mo>
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<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mi>d</mi>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{V}\nabla \cdot F(x)\,|dX|=\oint _{\partial V}F(x)\cdot {\hat {n}}\,|dS|.}</annotation>
</semantics>
</math></span><img src="./6b3a1844f2b3aa829aca829b97d17a70ce6a1c96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.749ex; height:5.676ex;" alt="{\displaystyle \int _{V}\nabla \cdot F(x)\,|dX|=\oint _{\partial V}F(x)\cdot {\hat {n}}\,|dS|.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Covariant_derivative">Covariant derivative</h2></div>
<p>A sufficiently smooth <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-surface in an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensional space is deemed a <a href="Manifold" title="Manifold">manifold</a>. To each point on the manifold, we may attach a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-blade <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> that is tangent to the manifold. Locally, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> acts as a pseudoscalar of the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-dimensional space. This blade defines a <a href="Geometric_algebra#Projection_and_rejection" title="Geometric algebra">projection</a> of vectors onto the manifold:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}_{B}(A)=(A\cdot B^{-1})B.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mi>B</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}_{B}(A)=(A\cdot B^{-1})B.}</annotation>
</semantics>
</math></span><img src="./b0355fbb1bb22299fcb2707bab3d72bb136c5f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.487ex; height:3.176ex;" alt="{\displaystyle {\mathcal {P}}_{B}(A)=(A\cdot B^{-1})B.}" loading="lazy"></span></dd></dl>
<p>Just as the vector derivative <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation>
</semantics>
</math></span><img src="./a3d0e93b78c50237f9ea83d027e4ebbdaef354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" loading="lazy"></span> is defined over the entire <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensional space, we may wish to define an <i>intrinsic derivative</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial }</annotation>
</semantics>
</math></span><img src="./62b4e7c1cedb9564609aefd2aa2309972f455c24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.318ex; height:2.176ex;" alt="{\displaystyle \partial }" loading="lazy"></span>, locally defined on the manifold:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial F={\mathcal {P}}_{B}(\nabla )F.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>F</mi>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo stretchy="false">)</mo>
<mi>F</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial F={\mathcal {P}}_{B}(\nabla )F.}</annotation>
</semantics>
</math></span><img src="./e6e521fed5531ade3d9505c076576c42920067ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.387ex; height:2.843ex;" alt="{\displaystyle \partial F={\mathcal {P}}_{B}(\nabla )F.}" loading="lazy"></span></dd></dl>
<p>(Note: The right hand side of the above may not lie in the tangent space to the manifold. Therefore, it is not the same as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}_{B}(\nabla F)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>F</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}_{B}(\nabla F)}</annotation>
</semantics>
</math></span><img src="./939b05f730609bcd795d2fa84b23072c35f451e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.583ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}_{B}(\nabla F)}" loading="lazy"></span>, which necessarily does lie in the tangent space.)
</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> is a vector tangent to the manifold, then indeed both the vector derivative and intrinsic derivative give the same directional derivative:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot \partial F=a\cdot \nabla F.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>F</mi>
<mo>=</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>F</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\cdot \partial F=a\cdot \nabla F.}</annotation>
</semantics>
</math></span><img src="./24b55963357c28fb46beff6ee8adec66afef62aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.299ex; height:2.176ex;" alt="{\displaystyle a\cdot \partial F=a\cdot \nabla F.}" loading="lazy"></span></dd></dl>
<p>Although this operation is perfectly valid, it is not always useful because <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial F}</annotation>
</semantics>
</math></span><img src="./014dfffa1461a32be4e057eee19c864a3b7841c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.059ex; height:2.176ex;" alt="{\displaystyle \partial F}" loading="lazy"></span> itself is not necessarily on the manifold. Therefore, we define the <i>covariant derivative</i> to be the forced projection of the intrinsic derivative back onto the manifold:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot DF={\mathcal {P}}_{B}(a\cdot \partial F)={\mathcal {P}}_{B}(a\cdot {\mathcal {P}}_{B}(\nabla )F).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>D</mi>
<mi>F</mi>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo stretchy="false">)</mo>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\cdot DF={\mathcal {P}}_{B}(a\cdot \partial F)={\mathcal {P}}_{B}(a\cdot {\mathcal {P}}_{B}(\nabla )F).}</annotation>
</semantics>
</math></span><img src="./3d914d3e64afd8b88d937aabe19319adc4252900.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.691ex; height:2.843ex;" alt="{\displaystyle a\cdot DF={\mathcal {P}}_{B}(a\cdot \partial F)={\mathcal {P}}_{B}(a\cdot {\mathcal {P}}_{B}(\nabla )F).}" loading="lazy"></span></dd></dl>
<p>Since any general multivector can be expressed as a sum of a projection and a rejection, in this case
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot \partial F={\mathcal {P}}_{B}(a\cdot \partial F)+{\mathcal {P}}_{B}^{\perp }(a\cdot \partial F),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>F</mi>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\cdot \partial F={\mathcal {P}}_{B}(a\cdot \partial F)+{\mathcal {P}}_{B}^{\perp }(a\cdot \partial F),}</annotation>
</semantics>
</math></span><img src="./6729aac0a8f827779874c71b77f8978a99e4b2ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.445ex; height:3.176ex;" alt="{\displaystyle a\cdot \partial F={\mathcal {P}}_{B}(a\cdot \partial F)+{\mathcal {P}}_{B}^{\perp }(a\cdot \partial F),}" loading="lazy"></span></dd></dl>
<p>we introduce a new function, the <a href="Shape_tensor" class="mw-redirect" title="Shape tensor">shape tensor</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {S}}(a)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">S</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathsf {S}}(a)}</annotation>
</semantics>
</math></span><img src="./eda42d4f00bffa531be257314f9396930505216d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.332ex; height:2.843ex;" alt="{\displaystyle {\mathsf {S}}(a)}" loading="lazy"></span>, which satisfies
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\times {\mathsf {S}}(a)={\mathcal {P}}_{B}^{\perp }(a\cdot \partial F),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">S</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F\times {\mathsf {S}}(a)={\mathcal {P}}_{B}^{\perp }(a\cdot \partial F),}</annotation>
</semantics>
</math></span><img src="./8e4ce7c2f2cba9b8e1c8b9d26b710b57fea9fdf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.675ex; height:3.176ex;" alt="{\displaystyle F\times {\mathsf {S}}(a)={\mathcal {P}}_{B}^{\perp }(a\cdot \partial F),}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>×<!-- × --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \times }</annotation>
</semantics>
</math></span><img src="./0ffafff1ad26cbe49045f19a67ce532116a32703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }" loading="lazy"></span> is the <a href="Commutator" title="Commutator">commutator product</a>. In a local coordinate basis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{i}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{e_{i}\}}</annotation>
</semantics>
</math></span><img src="./d45dea23899c892d30b142d73ed0fa19233ee4a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.208ex; height:2.843ex;" alt="{\displaystyle \{e_{i}\}}" loading="lazy"></span> spanning the tangent surface, the shape tensor is given by
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {S}}(a)=e^{i}\wedge {\mathcal {P}}_{B}^{\perp }(a\cdot \partial e_{i}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">S</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo>∧<!-- ∧ --></mo>
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathsf {S}}(a)=e^{i}\wedge {\mathcal {P}}_{B}^{\perp }(a\cdot \partial e_{i}).}</annotation>
</semantics>
</math></span><img src="./b2128eed5f5c60f8fe14684acf24dd8540291204.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.702ex; height:3.176ex;" alt="{\displaystyle {\mathsf {S}}(a)=e^{i}\wedge {\mathcal {P}}_{B}^{\perp }(a\cdot \partial e_{i}).}" loading="lazy"></span></dd></dl>
<p>Importantly, on a general manifold, the covariant derivative does not commute. In particular, the <a href="Commutator" title="Commutator">commutator</a> is related to the shape tensor by
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a\cdot D,\,b\cdot D]F=-({\mathsf {S}}(a)\times {\mathsf {S}}(b))\times F.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>D</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>D</mi>
<mo stretchy="false">]</mo>
<mi>F</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">S</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">S</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mi>F</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [a\cdot D,\,b\cdot D]F=-({\mathsf {S}}(a)\times {\mathsf {S}}(b))\times F.}</annotation>
</semantics>
</math></span><img src="./6c457b5b98d9984ddf7e83417307b47560b27170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.105ex; height:2.843ex;" alt="{\displaystyle [a\cdot D,\,b\cdot D]F=-({\mathsf {S}}(a)\times {\mathsf {S}}(b))\times F.}" loading="lazy"></span></dd></dl>
<p>Clearly the term <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {S}}(a)\times {\mathsf {S}}(b)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">S</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">S</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathsf {S}}(a)\times {\mathsf {S}}(b)}</annotation>
</semantics>
</math></span><img src="./eefd4137db13e158bafbe321efb0c9fa4327d0b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.271ex; height:2.843ex;" alt="{\displaystyle {\mathsf {S}}(a)\times {\mathsf {S}}(b)}" loading="lazy"></span> is of interest. However it, like the intrinsic derivative, is not necessarily on the manifold. Therefore, we can define the <a href="Riemann_tensor" class="mw-redirect" title="Riemann tensor">Riemann tensor</a> to be the projection back onto the manifold:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {R}}(a\wedge b)=-{\mathcal {P}}_{B}({\mathsf {S}}(a)\times {\mathsf {S}}(b)).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">R</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>∧<!-- ∧ --></mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">S</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">S</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathsf {R}}(a\wedge b)=-{\mathcal {P}}_{B}({\mathsf {S}}(a)\times {\mathsf {S}}(b)).}</annotation>
</semantics>
</math></span><img src="./04c8d16d3e50d6bced71c6eb49618e31b184b530.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.852ex; height:2.843ex;" alt="{\displaystyle {\mathsf {R}}(a\wedge b)=-{\mathcal {P}}_{B}({\mathsf {S}}(a)\times {\mathsf {S}}(b)).}" loading="lazy"></span></dd></dl>
<p>Lastly, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> is of grade <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>, then we can define interior and exterior covariant derivatives as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\cdot F=\langle DF\rangle _{r-1},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>F</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>D</mi>
<mi>F</mi>
<msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D\cdot F=\langle DF\rangle _{r-1},}</annotation>
</semantics>
</math></span><img src="./0f9b1b9568ee0db8f807ce70f59e138551a19e6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.638ex; height:2.843ex;" alt="{\displaystyle D\cdot F=\langle DF\rangle _{r-1},}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\wedge F=\langle DF\rangle _{r+1},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo>∧<!-- ∧ --></mo>
<mi>F</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>D</mi>
<mi>F</mi>
<msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D\wedge F=\langle DF\rangle _{r+1},}</annotation>
</semantics>
</math></span><img src="./a400c34a6f93af681a1f3254c5bd5e1b0524e3fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.542ex; height:2.843ex;" alt="{\displaystyle D\wedge F=\langle DF\rangle _{r+1},}" loading="lazy"></span></dd></dl>
<p>and likewise for the intrinsic derivative.
</p>
<div class="mw-heading mw-heading2"><h2 id="Relation_to_differential_geometry">Relation to differential geometry</h2></div>
<p>On a manifold, locally we may assign a tangent surface spanned by a set of basis vectors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{i}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{e_{i}\}}</annotation>
</semantics>
</math></span><img src="./d45dea23899c892d30b142d73ed0fa19233ee4a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.208ex; height:2.843ex;" alt="{\displaystyle \{e_{i}\}}" loading="lazy"></span>. We can associate the components of a <a href="Metric_tensor" title="Metric tensor">metric tensor</a>, the <a href="Christoffel_symbols" title="Christoffel symbols">Christoffel symbols</a>, and the <a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a> as follows:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}=e_{i}\cdot e_{j},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{ij}=e_{i}\cdot e_{j},}</annotation>
</semantics>
</math></span><img src="./aa36db7c71828331cbf0e1ad0af4568f9d3a28c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.887ex; height:2.343ex;" alt="{\displaystyle g_{ij}=e_{i}\cdot e_{j},}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{ij}^{k}=(e_{i}\cdot De_{j})\cdot e^{k},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msubsup>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi>D</mi>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma _{ij}^{k}=(e_{i}\cdot De_{j})\cdot e^{k},}</annotation>
</semantics>
</math></span><img src="./528538f9509db4146384f262fa38a484e301d33e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.816ex; height:3.676ex;" alt="{\displaystyle \Gamma _{ij}^{k}=(e_{i}\cdot De_{j})\cdot e^{k},}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{ijkl}=({\mathsf {R}}(e_{i}\wedge e_{j})\cdot e_{k})\cdot e_{l}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
<mi>l</mi>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">R</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{ijkl}=({\mathsf {R}}(e_{i}\wedge e_{j})\cdot e_{k})\cdot e_{l}.}</annotation>
</semantics>
</math></span><img src="./8e06acbb800a9a3e9e5c62aa5a8defffbde8c5fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.249ex; height:3.009ex;" alt="{\displaystyle R_{ijkl}=({\mathsf {R}}(e_{i}\wedge e_{j})\cdot e_{k})\cdot e_{l}.}" loading="lazy"></span></dd></dl>
<p>These relations embed the theory of differential geometry within geometric calculus.
</p>
<div class="mw-heading mw-heading2"><h2 id="Relation_to_differential_forms">Relation to differential forms</h2></div>
<p>In a <a href="Local_coordinate_system" class="mw-redirect" title="Local coordinate system">local coordinate system</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1},\ldots ,x^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x^{1},\ldots ,x^{n}}</annotation>
</semantics>
</math></span><img src="./d7ba4766c77e384fcf3fa5a11063d993cd794355.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:3.009ex;" alt="{\displaystyle x^{1},\ldots ,x^{n}}" loading="lazy"></span>), the coordinate differentials <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle dx^{1}}</annotation>
</semantics>
</math></span><img src="./c90b31cc2db56a2b2e1ed1251c91bbc3d4faf9ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.6ex; height:2.676ex;" alt="{\displaystyle dx^{1}}" loading="lazy"></span>, ..., <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle dx^{n}}</annotation>
</semantics>
</math></span><img src="./3a786b4fe5c88ebb490d1d895c5ec27976fa1c8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.764ex; height:2.343ex;" alt="{\displaystyle dx^{n}}" loading="lazy"></span> form a basic set of one-forms within the <a href="Coordinate_chart" class="mw-redirect" title="Coordinate chart">coordinate chart</a>. Given a <a href="Multi-index" class="mw-redirect" title="Multi-index">multi-index</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=(i_{1},\ldots ,i_{k})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I=(i_{1},\ldots ,i_{k})}</annotation>
</semantics>
</math></span><img src="./c2ece80b6570bb8dd4a0d1fb11440d41be9f001f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.006ex; height:2.843ex;" alt="{\displaystyle I=(i_{1},\ldots ,i_{k})}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i_{p}\leq n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1\leq i_{p}\leq n}</annotation>
</semantics>
</math></span><img src="./3936b324a12b0d29c0709778a21e55a386364284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.616ex; height:2.843ex;" alt="{\displaystyle 1\leq i_{p}\leq n}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<mi>p</mi>
<mo>≤<!-- ≤ --></mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq k}</annotation>
</semantics>
</math></span><img src="./b068c6969a1fda37f002fa9006eb5d633fd287a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.74ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq k}" loading="lazy"></span>, we can define a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-form
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =f_{I}\,dx^{I}=f_{i_{1}i_{2}\cdots i_{k}}\,dx^{i_{1}}\wedge dx^{i_{2}}\wedge \cdots \wedge dx^{i_{k}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>⋯<!-- ⋯ --></mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msup>
<mo>∧<!-- ∧ --></mo>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msup>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∧<!-- ∧ --></mo>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega =f_{I}\,dx^{I}=f_{i_{1}i_{2}\cdots i_{k}}\,dx^{i_{1}}\wedge dx^{i_{2}}\wedge \cdots \wedge dx^{i_{k}}.}</annotation>
</semantics>
</math></span><img src="./7cb08313ff9462a44422ab706c88aa8de96c3806.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.422ex; height:3.343ex;" alt="{\displaystyle \omega =f_{I}\,dx^{I}=f_{i_{1}i_{2}\cdots i_{k}}\,dx^{i_{1}}\wedge dx^{i_{2}}\wedge \cdots \wedge dx^{i_{k}}.}" loading="lazy"></span></dd></dl>
<p>We can alternatively introduce a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-grade multivector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=f_{i_{1}i_{2}\cdots i_{k}}e^{i_{1}}\wedge e^{i_{2}}\wedge \cdots \wedge e^{i_{k}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>⋯<!-- ⋯ --></mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msub>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msup>
<mo>∧<!-- ∧ --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msup>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∧<!-- ∧ --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A=f_{i_{1}i_{2}\cdots i_{k}}e^{i_{1}}\wedge e^{i_{2}}\wedge \cdots \wedge e^{i_{k}}}</annotation>
</semantics>
</math></span><img src="./18ef43e58ff8742d4df35bba56ce23c98ef4df6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.007ex; height:3.343ex;" alt="{\displaystyle A=f_{i_{1}i_{2}\cdots i_{k}}e^{i_{1}}\wedge e^{i_{2}}\wedge \cdots \wedge e^{i_{k}}}" loading="lazy"></span></dd></dl>
<p>and a measure
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}d^{k}X&=\left(dx^{i_{1}}e_{i_{1}}\right)\wedge \left(dx^{i_{2}}e_{i_{2}}\right)\wedge \cdots \wedge \left(dx^{i_{k}}e_{i_{k}}\right)\\&=\left(e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}\right)dx^{i_{1}}dx^{i_{2}}\cdots dx^{i_{k}}.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mi>X</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msup>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>∧<!-- ∧ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msup>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∧<!-- ∧ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msup>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msup>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
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<mo>⋯<!-- ⋯ --></mo>
<mi>d</mi>
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</mtd>
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</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}d^{k}X&=\left(dx^{i_{1}}e_{i_{1}}\right)\wedge \left(dx^{i_{2}}e_{i_{2}}\right)\wedge \cdots \wedge \left(dx^{i_{k}}e_{i_{k}}\right)\\&=\left(e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}\right)dx^{i_{1}}dx^{i_{2}}\cdots dx^{i_{k}}.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./f5b00a0284b68b9e610dbdb7a9e71a5e7791f966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.655ex; margin-bottom: -0.183ex; width:46ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}d^{k}X&=\left(dx^{i_{1}}e_{i_{1}}\right)\wedge \left(dx^{i_{2}}e_{i_{2}}\right)\wedge \cdots \wedge \left(dx^{i_{k}}e_{i_{k}}\right)\\&=\left(e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}\right)dx^{i_{1}}dx^{i_{2}}\cdots dx^{i_{k}}.\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Apart from a subtle difference in meaning for the exterior product with respect to differential forms versus the exterior product with respect to vectors (in the former the <i>increments</i> are covectors, whereas in the latter they represent scalars), we see the correspondences of the differential form
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \cong A^{\dagger }\cdot d^{k}X=A\cdot \left(d^{k}X\right)^{\dagger },}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \omega \cong A^{\dagger }\cdot d^{k}X=A\cdot \left(d^{k}X\right)^{\dagger },}</annotation>
</semantics>
</math></span><img src="./9b1f3e8c810ed8297efbc6244211312cd19372a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.762ex; height:3.843ex;" alt="{\displaystyle \omega \cong A^{\dagger }\cdot d^{k}X=A\cdot \left(d^{k}X\right)^{\dagger },}" loading="lazy"></span></dd></dl>
<p>its derivative
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\omega \cong (D\wedge A)^{\dagger }\cdot d^{k+1}X=(D\wedge A)\cdot \left(d^{k+1}X\right)^{\dagger },}">
<semantics>
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<mi>d</mi>
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<annotation encoding="application/x-tex">{\displaystyle d\omega \cong (D\wedge A)^{\dagger }\cdot d^{k+1}X=(D\wedge A)\cdot \left(d^{k+1}X\right)^{\dagger },}</annotation>
</semantics>
</math></span><img src="./e77f48a2d78c2953472a1e19c131bf130eb19f88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.811ex; height:3.843ex;" alt="{\displaystyle d\omega \cong (D\wedge A)^{\dagger }\cdot d^{k+1}X=(D\wedge A)\cdot \left(d^{k+1}X\right)^{\dagger },}" loading="lazy"></span></dd></dl>
<p>and its <a href="Hodge_dual" class="mw-redirect" title="Hodge dual">Hodge dual</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \star \omega \cong (I^{-1}A)^{\dagger }\cdot d^{k}X,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>⋆<!-- ⋆ --></mo>
<mi>ω<!-- ω --></mi>
<mo>≅<!-- ≅ --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mi>A</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>†<!-- † --></mo>
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<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mi>X</mi>
<mo>,</mo>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \star \omega \cong (I^{-1}A)^{\dagger }\cdot d^{k}X,}</annotation>
</semantics>
</math></span><img src="./ebc933d1724b4fbb68430fd06486266c79fa3ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.383ex; height:3.176ex;" alt="{\displaystyle \star \omega \cong (I^{-1}A)^{\dagger }\cdot d^{k}X,}" loading="lazy"></span></dd></dl>
<p>embed the theory of differential forms within geometric calculus.
</p>
<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>
<p>Following is a diagram summarizing the history of geometric calculus.
</p>
<div style="clear:both;" class=""></div>
<div class="mw-heading mw-heading2"><h2 id="References_and_further_reading">References and further reading</h2></div>
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</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="David_Hestenes" title="David Hestenes">David Hestenes</a>, Garrett Sobczyk: Clifford Algebra to Geometric Calculus, a Unified Language for mathematics and Physics (Dordrecht/Boston:G.Reidel Publ.Co., 1984, <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>90-277-2561-6</bdi></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFDoranLasenby2007" class="citation book cs1">Doran, Chris; Lasenby, Anthony (2007). <i>Geometric Algebra for Physicists</i>. Cambridge University press. p. 395. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-71595-9</bdi>.</cite></span>
</li>
</ol></div></div>
<ul><li><cite id="CITEREFMacdonald2012" class="citation book cs1">Macdonald, Alan (2012). <a rel="nofollow" class="external text" href="http://faculty.luther.edu/~macdonal/vagc/"><i>Vector and Geometric Calculus</i></a>. Charleston: CreateSpace. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781480132450</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/829395829">829395829</a>.</cite></li></ul>
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<li><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a> / <a href="Stochastic_calculus" title="Stochastic calculus">analysis</a></li>
<li><a href="Functional_integration" title="Functional integration">Path integral</a></li>
<li><a href="Malliavin_calculus" title="Malliavin calculus">Stochastic variational calculus</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_physics" title="Mathematical physics">Mathematical<br>physics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a>
<ul><li><a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a></li>
<li><a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a></li></ul></li>
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<ul><li><a href="Classical_field_theory" title="Classical field theory">Classical</a></li>
<li><a href="Conformal_field_theory" title="Conformal field theory">Conformal</a></li>
<li><a href="Effective_field_theory" title="Effective field theory">Effective</a></li>
<li><a href="Gauge_theory" title="Gauge theory">Gauge</a></li>
<li><a href="Quantum_field_theory" title="Quantum field theory">Quantum</a></li>
<li><a href="Statistical_field_theory" title="Statistical field theory">Statistical</a></li>
<li><a href="Topological_field_theory" class="mw-redirect" title="Topological field theory">Topological</a></li></ul></li>
<li><a href="Perturbation_theory" title="Perturbation theory">Perturbation theory</a>
<ul><li><a href="Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">in quantum mechanics</a></li></ul></li>
<li><a href="Potential_theory" title="Potential theory">Potential theory</a></li>
<li><a href="String_theory" title="String theory">String theory</a>
<ul><li><a href="Bosonic_string_theory" title="Bosonic string theory">Bosonic</a></li>
<li><a href="Topological_string_theory" title="Topological string theory">Topological</a></li></ul></li>
<li><a href="Supersymmetry" title="Supersymmetry">Supersymmetry</a>
<ul><li><a href="Supersymmetric_quantum_mechanics" title="Supersymmetric quantum mechanics">Supersymmetric quantum mechanics</a></li>
<li><a href="Supersymmetric_theory_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">Supersymmetric theory of stochastic dynamics</a></li></ul></li></ul>
</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Algebraicstructures48" scope="row" class="navbox-group" style="width:1%"><a href="Algebraic_structure" title="Algebraic structure">Algebraic<br>structures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li>
<li><a href="Path_integral_formulation" title="Path integral formulation">Feynman integral</a></li>
<li><a href="Poisson_algebra" title="Poisson algebra">Poisson algebra</a></li>
<li><a href="Quantum_group" title="Quantum group">Quantum group</a></li>
<li><a href="Renormalization_group" title="Renormalization group">Renormalization group</a></li>
<li><a href="Particle_physics_and_representation_theory" title="Particle physics and representation theory">Representation theory</a></li>
<li><a href="Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li>
<li><a href="Superalgebra" title="Superalgebra">Superalgebra</a></li>
<li><a href="Supersymmetry_algebra" title="Supersymmetry algebra">Supersymmetry algebra</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Decision_theory" title="Decision theory">Decision sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Game_theory" title="Game theory">Game theory</a></li>
<li><a href="Operations_research" title="Operations research">Operations research</a></li>
<li><a href="Mathematical_optimization" title="Mathematical optimization">Optimization</a></li>
<li><a href="Social_choice_theory" title="Social choice theory">Social choice theory</a></li>
<li><a href="Statistics" title="Statistics">Statistics</a></li>
<li><a href="Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li>
<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li>
<li><a href="Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li>
<li><a href="Mathematical_psychology" title="Mathematical psychology">Psychology</a></li>
<li><a href="Mathematical_sociology" title="Mathematical sociology">Sociology</a></li>
<li>"<a href="The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a>"</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Mathematics" title="Mathematics">Mathematics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>
<ul><li><a href="Japan_Society_for_Industrial_and_Applied_Mathematics" title="Japan Society for Industrial and Applied Mathematics">Japan Society for Industrial and Applied Mathematics</a></li></ul></li>
<li><a href="Soci%C3%A9t%C3%A9_de_Math%C3%A9matiques_Appliqu%C3%A9es_et_Industrielles" title="Société de Mathématiques Appliquées et Industrielles">Société de Mathématiques Appliquées et Industrielles</a></li>
<li><a href="International_Council_for_Industrial_and_Applied_Mathematics" title="International Council for Industrial and Applied Mathematics">International Council for Industrial and Applied Mathematics</a></li>
<li>European Community on Computational Methods in Applied Sciences</li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><b>Category</b></li>
<li><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a> / <a href="Topic_outline_of_mathematics" class="mw-redirect" title="Topic outline of mathematics">outline</a> / <a href="List_of_mathematics_topics" class="mw-redirect" title="List of mathematics topics">topics list</a></li></ul>
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