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<title>Multivariable calculus</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Multivariable calculus</span></span>
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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>&nbsp;(<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>&nbsp;(<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>&nbsp;(<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>&nbsp;(<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 mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"></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>
<li><a href="Geometric_calculus" title="Geometric calculus">Geometric</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>
</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%">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">
<ul><li><a href="Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li>
<li><a href="Generalized_function" title="Generalized function">Generalized functions</a></li>
<li><a href="Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content">
<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%">Specialized</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;">
<ul><li><a href="Fractional_calculus" title="Fractional calculus">Fractional</a></li>
<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin</a></li>
<li><a href="Stochastic_calculus" title="Stochastic calculus">Stochastic</a></li>
<li><a href="Calculus_of_variations" title="Calculus of variations">Variations</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<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%">Miscellanea</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;">
<ul><li><a href="Precalculus" title="Precalculus">Precalculus</a></li>
<li><a href="History_of_calculus" title="History of calculus">History</a></li>
<li><a href="Glossary_of_calculus" title="Glossary of calculus">Glossary</a></li>
<li><a href="List_of_calculus_topics" title="List of calculus topics">List of topics</a></li>
<li><a href="Integration_Bee" title="Integration Bee">Integration Bee</a></li>
<li><a href="Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li>
<li><a href="Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li></ul></div></div></td>
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<p><b>Multivariable calculus</b> (also known as <b>multivariate calculus</b>) is the extension of <a href="Calculus" title="Calculus">calculus</a> in one <a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a> to calculus with <a href="Function_of_several_real_variables" title="Function of several real variables">functions of several variables</a>: the <a href="Differential_calculus" title="Differential calculus">differentiation</a> and <a href="Integral" title="Integral">integration</a> of functions involving multiple variables (<i><a href="Multivariate_(mathematics)" class="mw-redirect" title="Multivariate (mathematics)">multivariate</a></i>), rather than just one.<sup id="cite_ref-CourantJohn1999_1-0" class="reference"><a href="#cite_note-CourantJohn1999-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Multivariable calculus may be thought of as an elementary part of <a href="Calculus_on_Euclidean_space" title="Calculus on Euclidean space">calculus on Euclidean space</a>. The special case of calculus in three dimensional space is often called <i><a href="Vector_calculus" title="Vector calculus">vector calculus</a></i>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2></div>
<p>In single-variable calculus, operations like differentiation and integration are made to functions of a single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the <a href="Domain_of_a_function" title="Domain of a function">domain</a> is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces:
</p>
<ol><li>There are infinite ways to approach a single point in higher dimensions, as opposed to two (from the positive and negative direction) in 1D;</li>
<li>There are multiple extended objects associated with the dimension; for example, for a 1D function, it must be represented as a curve on the 2D <a href="Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a>, but a function with two variables is a surface in 3D, while curves can also live in 3D space.</li></ol>
<p>The consequence of the first difference is the difference in the definition of the limits and continuity. Directional <a href="Limit_of_a_function" title="Limit of a function">limits</a> and <a href="Directional_derivative" title="Directional derivative">derivatives</a> define the limit and differential along a 1D parametrized curve, reducing the problem to the 1D case. Further higher-dimensional objects can be constructed from these operators.
</p><p>The consequence of the second difference is the existence of multiple types of integration, including <a href="Line_integral" title="Line integral">line integrals</a>, <a href="Surface_integral" title="Surface integral">surface integrals</a> and <a href="Volume_integral" title="Volume integral">volume integrals</a>. Due to the non-uniqueness of these integrals, an <a href="Antiderivative" title="Antiderivative">antiderivative</a> or <a href="Indefinite_integral" class="mw-redirect" title="Indefinite integral">indefinite integral</a> cannot be properly defined.
</p>
<div class="mw-heading mw-heading2"><h2 id="Limits">Limits</h2></div>
<p>A study of <a href="Limit_of_a_function" title="Limit of a function">limits</a> and <a href="Continuous_function" title="Continuous function">continuity</a> in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions.
</p><p>A limit along a path may be defined by considering a parametrised path <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 s(t):\mathbb {R} \to \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
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</math></span><img src="./f3bb769f922389fa74ff0f7fcad21b56ee1c3fbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.865ex; height:2.843ex;" alt="{\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}}" loading="lazy"></span> in n-dimensional Euclidean space. Any 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({\overrightarrow {x}}):\mathbb {R} ^{n}\to \mathbb {R} ^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle f({\overrightarrow {x}}):\mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation>
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</math></span><img src="./c5f5c568373fc6ea408b247abdebd8499fbd8a8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.277ex; height:3.509ex;" alt="{\displaystyle f({\overrightarrow {x}}):\mathbb {R} ^{n}\to \mathbb {R} ^{m}}" loading="lazy"></span> can then be projected on the path as a 1D 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(s(t))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle f(s(t))}</annotation>
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</math></span><img src="./ff6a11382517dd787e45a6ed8b3b74a92704a755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.827ex; height:2.843ex;" alt="{\displaystyle f(s(t))}" loading="lazy"></span>. The limit 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}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> to the point <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 s(t_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle s(t_{0})}</annotation>
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</math></span><img src="./a8d7837fe830c733cc02d209bffb7c2d654a23c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.794ex; height:2.843ex;" alt="{\displaystyle s(t_{0})}" loading="lazy"></span> along the path <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 s(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
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<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>
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</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span> can hence be defined as
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</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \lim _{{\overrightarrow {x}}\to s(t_{0})}f({\overrightarrow {x}})=\lim _{t\to t_{0}}f(s(t))}">
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<annotation encoding="application/x-tex">{\displaystyle \lim _{{\overrightarrow {x}}\to s(t_{0})}f({\overrightarrow {x}})=\lim _{t\to t_{0}}f(s(t))}</annotation>
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</math></span><img src="./a8fed5a8600fd164605823f7d6fd4f23829c2056.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.665ex; height:6.009ex;" alt="{\displaystyle \lim _{{\overrightarrow {x}}\to s(t_{0})}f({\overrightarrow {x}})=\lim _{t\to t_{0}}f(s(t))}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>
<p>Note that the value of this limit can be dependent on the form 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 s(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>
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</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span>, i.e. the path chosen, not just the point which the limit approaches.<sup id="cite_ref-CourantJohn1999_1-1" class="reference"><a href="#cite_note-CourantJohn1999-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 19–22">: 19–22 </span></sup> For example, consider the function
</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(x,y)={\frac {x^{2}y}{x^{4}+y^{2}}}.}">
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<annotation encoding="application/x-tex">{\displaystyle f(x,y)={\frac {x^{2}y}{x^{4}+y^{2}}}.}</annotation>
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</math></span><img src="./24ff8e3d0552fd0adfd937bbdb840d42b6856ec6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.627ex; height:6.343ex;" alt="{\displaystyle f(x,y)={\frac {x^{2}y}{x^{4}+y^{2}}}.}" loading="lazy"></span></dd></dl>
<p>If the point <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 (0,0)}">
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<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation>
</semantics>
</math></span><img src="./5d630d3e781a53b0a3559ae7e5b45f9479a3141a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}" loading="lazy"></span> is approached through the line <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 y=kx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mi>k</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=kx}</annotation>
</semantics>
</math></span><img src="./a0ce87b0aaacaad25a36d2f52dc1fae2ead92689.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.795ex; height:2.509ex;" alt="{\displaystyle y=kx}" loading="lazy"></span>, or in parametric form:
</p>

<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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(t)=t,\,y(t)=kt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>t</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>k</mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x(t)=t,\,y(t)=kt}</annotation>
</semantics>
</math></span><img src="./c6a84461d010af3cee578968c6135b89fb8fd109.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.291ex; height:2.843ex;" alt="{\displaystyle x(t)=t,\,y(t)=kt}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table>
<p>Then the limit along the path will be:
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \lim _{t\to 0}f(x(t),y(t))=\lim _{t\to 0}{\frac {kt^{3}}{t^{4}+k^{2}t^{2}}}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>k</mi>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lim _{t\to 0}f(x(t),y(t))=\lim _{t\to 0}{\frac {kt^{3}}{t^{4}+k^{2}t^{2}}}=0}</annotation>
</semantics>
</math></span><img src="./758df685fa89a82c8edd440263171dda1834e5ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.227ex; height:6.176ex;" alt="{\displaystyle \lim _{t\to 0}f(x(t),y(t))=\lim _{t\to 0}{\frac {kt^{3}}{t^{4}+k^{2}t^{2}}}=0}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table>
<p>On the other hand, if the path <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 y=\pm x^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mo>±<!-- ± --></mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=\pm x^{2}}</annotation>
</semantics>
</math></span><img src="./9008c88b968b21ab02d6e69852327af8f3fad95c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.446ex; height:3.009ex;" alt="{\displaystyle y=\pm x^{2}}" loading="lazy"></span> (or parametrically, <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(t)=t,\,y(t)=\pm t^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>t</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>±<!-- ± --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x(t)=t,\,y(t)=\pm t^{2}}</annotation>
</semantics>
</math></span><img src="./0aff2c966de817d5c177e6ffddc9153ceed78f68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.943ex; height:3.176ex;" alt="{\displaystyle x(t)=t,\,y(t)=\pm t^{2}}" loading="lazy"></span>) is chosen, then the limit becomes:
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \lim _{t\to 0}f(x(t),y(t))=\lim _{t\to 0}{\frac {\pm t^{4}}{t^{4}+t^{4}}}=\pm {\frac {1}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>±<!-- ± --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo>±<!-- ± --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lim _{t\to 0}f(x(t),y(t))=\lim _{t\to 0}{\frac {\pm t^{4}}{t^{4}+t^{4}}}=\pm {\frac {1}{2}}}</annotation>
</semantics>
</math></span><img src="./716ab7acf1e97454e9ac851f6bb1a85e6a49afd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.606ex; height:6.176ex;" alt="{\displaystyle \lim _{t\to 0}f(x(t),y(t))=\lim _{t\to 0}{\frac {\pm t^{4}}{t^{4}+t^{4}}}=\pm {\frac {1}{2}}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span></td></tr></tbody></table>
<p>Since taking different paths towards the same point yields different values, a general limit at the point <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 (0,0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation>
</semantics>
</math></span><img src="./5d630d3e781a53b0a3559ae7e5b45f9479a3141a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}" loading="lazy"></span> cannot be defined for the function.
</p><p>A general limit can be defined if the limits to a point along all possible paths converge to the same value, i.e. we say for a 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:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation>
</semantics>
</math></span><img src="./aad78382c3d23bcb4051b3148f1a23b1d0ba52e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.079ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}" loading="lazy"></span> that the limit 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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> to some point <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_{0}\in \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}\in \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./e5c5d325c966538b6db03962c2bcf80fe26af8b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.121ex; height:2.676ex;" alt="{\displaystyle x_{0}\in \mathbb {R} ^{n}}" loading="lazy"></span> is L, if and only if
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \lim _{t\to t_{0}}f(s(t))=L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</munder>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lim _{t\to t_{0}}f(s(t))=L}</annotation>
</semantics>
</math></span><img src="./ce45ab260480f7cefba1d785d3cb61dc7d7ca619.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.558ex; height:4.343ex;" alt="{\displaystyle \lim _{t\to t_{0}}f(s(t))=L}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span></td></tr></tbody></table>
<p>for all continuous 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 s(t):\mathbb {R} \to \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./f3bb769f922389fa74ff0f7fcad21b56ee1c3fbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.865ex; height:2.843ex;" alt="{\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}}" loading="lazy"></span> such that <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 s(t_{0})=x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t_{0})=x_{0}}</annotation>
</semantics>
</math></span><img src="./80815cd762cf86f1fee45b08a64885a3959024a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.276ex; height:2.843ex;" alt="{\displaystyle s(t_{0})=x_{0}}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Continuity">Continuity</h3></div>
<p>From the concept of limit along a path, we can then derive the definition for multivariate continuity in the same manner, that is: we say for a 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:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation>
</semantics>
</math></span><img src="./aad78382c3d23bcb4051b3148f1a23b1d0ba52e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.079ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}" loading="lazy"></span> that <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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is continuous at the point <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_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span>, if and only if
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \lim _{t\to t_{0}}f(s(t))=f(x_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</munder>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lim _{t\to t_{0}}f(s(t))=f(x_{0})}</annotation>
</semantics>
</math></span><img src="./e921b1ecba9eb47ee76d784dbb1cc2b82cc9cbd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.447ex; height:4.343ex;" alt="{\displaystyle \lim _{t\to t_{0}}f(s(t))=f(x_{0})}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span></td></tr></tbody></table>
<p>for all continuous 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 s(t):\mathbb {R} \to \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./f3bb769f922389fa74ff0f7fcad21b56ee1c3fbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.865ex; height:2.843ex;" alt="{\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}}" loading="lazy"></span> such that <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 s(t_{0})=x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t_{0})=x_{0}}</annotation>
</semantics>
</math></span><img src="./80815cd762cf86f1fee45b08a64885a3959024a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.276ex; height:2.843ex;" alt="{\displaystyle s(t_{0})=x_{0}}" loading="lazy"></span>.
</p><p>As with limits, being continuous along <i>one</i> path <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 s(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>
</semantics>
</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span> does not imply multivariate continuity.
</p><p>Continuity in each argument not being sufficient for multivariate continuity can also be seen from the following example.<sup id="cite_ref-CourantJohn1999_1-2" class="reference"><a href="#cite_note-CourantJohn1999-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 17–19">: 17–19 </span></sup> For example, for a real-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:\mathbb {R} ^{2}\to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./a1c262f451d196f94214e2b8856b462c2d209306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.24ex; height:3.009ex;" alt="{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }" loading="lazy"></span> with two real-valued parameters, <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,y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x,y)}</annotation>
</semantics>
</math></span><img src="./29473ed0c4e838ac9dbe074535e507166c0e9101.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle f(x,y)}" loading="lazy"></span>, continuity 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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> 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="./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> for fixed <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 y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> and continuity 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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> 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 y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> for fixed <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> does not imply continuity 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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>.
</p><p>Consider
</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(x,y)={\begin{cases}{\frac {y}{x}}-y&amp;{\text{if}}\quad 0\leq y<x\leq 1\\{\frac {x}{y}}-x&amp;{\text{if}}\quad 0\leq x<y\leq 1\\1-x&amp;{\text{if}}\quad 0<x=y\\0&amp;{\text{everywhere else}}.\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>y</mi>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if</mtext>
</mrow>
<mspace width="1em"></mspace>
<mn>0</mn>
<mo>≤<!-- ≤ --></mo>
<mi>y</mi>
<mo>&lt;</mo>
<mi>x</mi>
<mo>≤<!-- ≤ --></mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mi>y</mi>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if</mtext>
</mrow>
<mspace width="1em"></mspace>
<mn>0</mn>
<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
<mo>&lt;</mo>
<mi>y</mi>
<mo>≤<!-- ≤ --></mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if</mtext>
</mrow>
<mspace width="1em"></mspace>
<mn>0</mn>
<mo>&lt;</mo>
<mi>x</mi>
<mo>=</mo>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>everywhere else</mtext>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x,y)={\begin{cases}{\frac {y}{x}}-y&amp;{\text{if}}\quad 0\leq y&lt;x\leq 1\\{\frac {x}{y}}-x&amp;{\text{if}}\quad 0\leq x&lt;y\leq 1\\1-x&amp;{\text{if}}\quad 0&lt;x=y\\0&amp;{\text{everywhere else}}.\end{cases}}}</annotation>
</semantics>
</math></span><img src="./002a06de158e3bdbc3ecd6abbab7b24507b40f87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:38.579ex; height:12.509ex;" alt="{\displaystyle f(x,y)={\begin{cases}{\frac {y}{x}}-y&amp;{\text{if}}\quad 0\leq y<x\leq 1\\{\frac {x}{y}}-x&amp;{\text{if}}\quad 0\leq x<y\leq 1\\1-x&amp;{\text{if}}\quad 0<x=y\\0&amp;{\text{everywhere else}}.\end{cases}}}" loading="lazy"></span></dd></dl>
<p>It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle <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 (0,1)\times (0,1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (0,1)\times (0,1)}</annotation>
</semantics>
</math></span><img src="./3d78ba14cbb06e9b9e05ca044e25479a0104311e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.177ex; height:2.843ex;" alt="{\displaystyle (0,1)\times (0,1)}" loading="lazy"></span>. Furthermore, the functions defined for constant <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> 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 y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" 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 0\leq a\leq 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>≤<!-- ≤ --></mo>
<mi>a</mi>
<mo>≤<!-- ≤ --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0\leq a\leq 1}</annotation>
</semantics>
</math></span><img src="./4267954a08c5bed68b666ee33dc530a6c9462f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.752ex; height:2.343ex;" alt="{\displaystyle 0\leq a\leq 1}" loading="lazy"></span> 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 g_{a}(x)=f(x,a)\quad }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mspace width="1em"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{a}(x)=f(x,a)\quad }</annotation>
</semantics>
</math></span><img src="./632bd718e14516805227102094212e9b885ad3f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.452ex; height:2.843ex;" alt="{\displaystyle g_{a}(x)=f(x,a)\quad }" 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 \quad h_{a}(y)=f(a,y)\quad }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="1em"></mspace>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mspace width="1em"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \quad h_{a}(y)=f(a,y)\quad }</annotation>
</semantics>
</math></span><img src="./3052c76f412153fdc2a57f7a4893691be5a864fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.656ex; height:2.843ex;" alt="{\displaystyle \quad h_{a}(y)=f(a,y)\quad }" loading="lazy"></span></dd></dl>
<p>are continuous. Specifically,
</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_{0}(x)=f(x,0)=h_{0}(0,y)=f(0,y)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{0}(x)=f(x,0)=h_{0}(0,y)=f(0,y)=0}</annotation>
</semantics>
</math></span><img src="./030dfc69aef812f50c4f10c73d0b146c56c469c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.467ex; height:2.843ex;" alt="{\displaystyle g_{0}(x)=f(x,0)=h_{0}(0,y)=f(0,y)=0}" loading="lazy"></span> for all <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>. Therefore, <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(0,0)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(0,0)=0}</annotation>
</semantics>
</math></span><img src="./73eda9d9d0146876d534fe2d9d228990825ec759.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.708ex; height:2.843ex;" alt="{\displaystyle f(0,0)=0}" loading="lazy"></span> and moreover, along the coordinate axes, <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 \lim _{x\to 0}f(x,0)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0}f(x,0)=0}</annotation>
</semantics>
</math></span><img src="./c063218a04b0d260a3cc8ecd9b7f9aa35ca9998f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.667ex; height:4.009ex;" alt="{\displaystyle \lim _{x\to 0}f(x,0)=0}" 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 \lim _{y\to 0}f(0,y)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lim _{y\to 0}f(0,y)=0}</annotation>
</semantics>
</math></span><img src="./d1e2ade0be5d47fdbb3fba8ee6d384eb2c932b52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.37ex; height:4.343ex;" alt="{\displaystyle \lim _{y\to 0}f(0,y)=0}" loading="lazy"></span>. Therefore the function is continuous along both individual arguments.</dd></dl>
<p>However, consider the parametric path <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(t)=t,\,y(t)=t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>t</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x(t)=t,\,y(t)=t}</annotation>
</semantics>
</math></span><img src="./7990fe49bf4fa86ee87a7d3bf1ab368cace30a8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.08ex; height:2.843ex;" alt="{\displaystyle x(t)=t,\,y(t)=t}" loading="lazy"></span>. The parametric function becomes
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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(t),y(t))={\begin{cases}1-t&amp;{\text{if}}\quad t>0\\0&amp;{\text{everywhere else}}.\end{cases}}}">
<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>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>t</mi>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if</mtext>
</mrow>
<mspace width="1em"></mspace>
<mi>t</mi>
<mo>&gt;</mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>everywhere else</mtext>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x(t),y(t))={\begin{cases}1-t&amp;{\text{if}}\quad t&gt;0\\0&amp;{\text{everywhere else}}.\end{cases}}}</annotation>
</semantics>
</math></span><img src="./0a2b0465d090da7ff26a5f171c114b9652c61d98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.897ex; height:6.176ex;" alt="{\displaystyle f(x(t),y(t))={\begin{cases}1-t&amp;{\text{if}}\quad t>0\\0&amp;{\text{everywhere else}}.\end{cases}}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span></td></tr></tbody></table>
<p>Therefore,
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \lim _{t\to 0^{+}}f(x(t),y(t))=1\neq f(0,0)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mn>0</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
</mrow>
</munder>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>≠<!-- ≠ --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lim _{t\to 0^{+}}f(x(t),y(t))=1\neq f(0,0)=0}</annotation>
</semantics>
</math></span><img src="./d7f9ea2cb2e8e69a8701f169f792fb35ac66d9a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.62ex; height:4.343ex;" alt="{\displaystyle \lim _{t\to 0^{+}}f(x(t),y(t))=1\neq f(0,0)=0}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span></td></tr></tbody></table>
<p>It is hence clear that the function is not multivariate continuous, despite being continuous in both coordinates.
</p>
<div class="mw-heading mw-heading3"><h3 id="Theorems_regarding_multivariate_limits_and_continuity">Theorems regarding multivariate limits and continuity</h3></div>
<ul><li>All properties of linearity and superposition from single-variable calculus carry over to multivariate calculus.</li>
<li><b>Composition</b>: 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:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation>
</semantics>
</math></span><img src="./aad78382c3d23bcb4051b3148f1a23b1d0ba52e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.079ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}" 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:\mathbb {R} ^{m}\to \mathbb {R} ^{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} ^{m}\to \mathbb {R} ^{p}}</annotation>
</semantics>
</math></span><img src="./9796e0c0d52eae6b5eac392da285a51848aceb8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.758ex; height:2.676ex;" alt="{\displaystyle g:\mathbb {R} ^{m}\to \mathbb {R} ^{p}}" loading="lazy"></span> are both multivariate continuous functions at the points <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_{0}\in \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}\in \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./e5c5d325c966538b6db03962c2bcf80fe26af8b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.121ex; height:2.676ex;" alt="{\displaystyle x_{0}\in \mathbb {R} ^{n}}" 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 f(x_{0})\in \mathbb {R} ^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x_{0})\in \mathbb {R} ^{m}}</annotation>
</semantics>
</math></span><img src="./9e9c426e7c9aab35581eec3bf369568709e8f9b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.666ex; height:2.843ex;" alt="{\displaystyle f(x_{0})\in \mathbb {R} ^{m}}" loading="lazy"></span> respectively, then <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\circ f:\mathbb {R} ^{n}\to \mathbb {R} ^{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>∘<!-- ∘ --></mo>
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g\circ f:\mathbb {R} ^{n}\to \mathbb {R} ^{p}}</annotation>
</semantics>
</math></span><img src="./834a45c86c58385c44cebd6730258bc5e8148aca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.774ex; height:2.676ex;" alt="{\displaystyle g\circ f:\mathbb {R} ^{n}\to \mathbb {R} ^{p}}" loading="lazy"></span> is also a multivariate continuous function at the point <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_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span>.</li>
<li><b>Multiplication</b>: 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:\mathbb {R} ^{n}\to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>
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<mrow class="MJX-TeXAtom-ORD">
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<mi>g</mi>
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<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>
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</math></span><img src="./6032f72d4d3deedc8f4a9e49617e46b95bfc8e6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.242ex; height:2.676ex;" alt="{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"></span> are both continuous functions at the point <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_{0}\in \mathbb {R} ^{n}}">
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<annotation encoding="application/x-tex">{\displaystyle x_{0}\in \mathbb {R} ^{n}}</annotation>
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<mrow class="MJX-TeXAtom-ORD">
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<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle fg:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>
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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
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</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" 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 f/g:\mathbb {R} ^{n}\to \mathbb {R} }">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle f/g:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
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</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span> provided that <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(x_{0})\neq 0}">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle g(x_{0})\neq 0}</annotation>
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<li>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:\mathbb {R} ^{n}\to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>
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</math></span><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"></span> is a continuous function at point <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_{0}\in \mathbb {R} ^{n}}">
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<mrow class="MJX-TeXAtom-ORD">
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<msub>
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<mo>∈<!-- ∈ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle x_{0}\in \mathbb {R} ^{n}}</annotation>
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<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<annotation encoding="application/x-tex">{\displaystyle |f|}</annotation>
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</math></span><img src="./940e58aa437f429f47fd743a819e41fedaa1ff9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.572ex; height:2.843ex;" alt="{\displaystyle |f|}" loading="lazy"></span> is also continuous at the same point.</li>
<li>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:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation>
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<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle x_{0}\in \mathbb {R} ^{n}}</annotation>
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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is multivariate continuous 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 x_{0}}">
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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
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<tbody><tr>
<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof</div>
</th></tr>
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<td style="color:inherit; border: solid 1px Silver; padding: 0.6em; background: var(--background-color-base, #fff);">
<p>From the Lipschitz continuity condition 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 f}">
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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> we have
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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(s(t))-f(s(t_{0}))|\leq K|s(t)-s(t_{0})|}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
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<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mo>≤<!-- ≤ --></mo>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
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<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle |f(s(t))-f(s(t_{0}))|\leq K|s(t)-s(t_{0})|}</annotation>
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</math></span><img src="./3c55b1e809600e54258087e23a88fb92e468f696.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.674ex; height:2.843ex;" alt="{\displaystyle |f(s(t))-f(s(t_{0}))|\leq K|s(t)-s(t_{0})|}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_8" class="reference nourlexpansion" style="font-weight:bold;">8</span></td></tr></tbody></table>
<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 K}">
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<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
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</math></span><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> is the Lipschitz constant. Note also that, 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 s(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
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<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>
</semantics>
</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span> is continuous 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 t_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>
</semantics>
</math></span><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"></span>, for every <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 >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>δ<!-- δ --></mi>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta &gt;0}</annotation>
</semantics>
</math></span><img src="./595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"></span> there exists 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 \epsilon >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϵ<!-- ϵ --></mi>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \epsilon &gt;0}</annotation>
</semantics>
</math></span><img src="./568095ad3924314374a5ab68fae17343661f2a71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}" loading="lazy"></span> such that <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 |s(t)-s(t_{0})|<\delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&lt;</mo>
<mi>δ<!-- δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |s(t)-s(t_{0})|&lt;\delta }</annotation>
</semantics>
</math></span><img src="./2a71efa9d2d30a9df26500f31bd9270501156aee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.814ex; height:2.843ex;" alt="{\displaystyle |s(t)-s(t_{0})|<\delta }" 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 \forall |t-t_{0}|<\epsilon }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>t</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&lt;</mo>
<mi>ϵ<!-- ϵ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \forall |t-t_{0}|&lt;\epsilon }</annotation>
</semantics>
</math></span><img src="./4cd947d7bc7faddc48d83d72809359a9d4cd5b22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.203ex; height:2.843ex;" alt="{\displaystyle \forall |t-t_{0}|<\epsilon }" loading="lazy"></span>.
</p><p>Hence, for every <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 \alpha >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha &gt;0}</annotation>
</semantics>
</math></span><img src="./edd4f784b6e8bb68fa774213ceacbab2d97825dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}" loading="lazy"></span>, choose <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 ={\frac {\alpha }{K}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>δ<!-- δ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mi>K</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta ={\frac {\alpha }{K}}}</annotation>
</semantics>
</math></span><img src="./6f229a0b14ed84b3c69d7360c4629122cd50994e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.049ex; height:4.676ex;" alt="{\displaystyle \delta ={\frac {\alpha }{K}}}" loading="lazy"></span>; there exists 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 \epsilon >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϵ<!-- ϵ --></mi>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \epsilon &gt;0}</annotation>
</semantics>
</math></span><img src="./568095ad3924314374a5ab68fae17343661f2a71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}" loading="lazy"></span> such that 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 t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> satisfying <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 |t-t_{0}|<\epsilon }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>t</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&lt;</mo>
<mi>ϵ<!-- ϵ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |t-t_{0}|&lt;\epsilon }</annotation>
</semantics>
</math></span><img src="./52f0e636a9410e98857e8833093717ef2c71591f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.91ex; height:2.843ex;" alt="{\displaystyle |t-t_{0}|<\epsilon }" 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 |s(t)-s(t_{0})|<\delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&lt;</mo>
<mi>δ<!-- δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |s(t)-s(t_{0})|&lt;\delta }</annotation>
</semantics>
</math></span><img src="./2a71efa9d2d30a9df26500f31bd9270501156aee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.814ex; height:2.843ex;" alt="{\displaystyle |s(t)-s(t_{0})|<\delta }" 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 |f(s(t))-f(s(t_{0}))|\leq K|s(t)-s(t_{0})|<K\delta =\alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≤<!-- ≤ --></mo>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&lt;</mo>
<mi>K</mi>
<mi>δ<!-- δ --></mi>
<mo>=</mo>
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |f(s(t))-f(s(t_{0}))|\leq K|s(t)-s(t_{0})|&lt;K\delta =\alpha }</annotation>
</semantics>
</math></span><img src="./2357eff2eed809c8e10c6ba3b41dd45dca502c9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.473ex; height:2.843ex;" alt="{\displaystyle |f(s(t))-f(s(t_{0}))|\leq K|s(t)-s(t_{0})|<K\delta =\alpha }" loading="lazy"></span>. Hence <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 \lim _{t\to t_{0}}f(s(t))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</munder>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lim _{t\to t_{0}}f(s(t))}</annotation>
</semantics>
</math></span><img src="./11ebd47bda981ada87a4675803a1e183d2b512bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.876ex; height:4.343ex;" alt="{\displaystyle \lim _{t\to t_{0}}f(s(t))}" loading="lazy"></span> converges 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 f(s(t_{0}))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(s(t_{0}))}</annotation>
</semantics>
</math></span><img src="./6a9bf282a2a61cb5f6e72612cad45bbfa92e26a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.881ex; height:2.843ex;" alt="{\displaystyle f(s(t_{0}))}" loading="lazy"></span> regardless of the precise form 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 s(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>
</semantics>
</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span>.
</p>
</td></tr></tbody></table></div>
<div class="mw-heading mw-heading2"><h2 id="Differentiation">Differentiation</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1236090951">
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/* end https://en.wikipedia.org/ */
</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Partial_derivative" title="Partial derivative">Partial derivative</a> and <a href="Directional_derivative" title="Directional derivative">Directional derivative</a></div>
<div class="mw-heading mw-heading3"><h3 id="Directional_derivative">Directional derivative</h3></div>
<p>The derivative of a single-variable function is defined as
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 {df}{dx}}=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>f</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>h</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>h</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>h</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {df}{dx}}=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}}</annotation>
</semantics>
</math></span><img src="./fe1245295a482e859e37d518299e8d4a6387fbda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.97ex; height:5.843ex;" alt="{\displaystyle {\frac {df}{dx}}=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_9" class="reference nourlexpansion" style="font-weight:bold;">9</span></td></tr></tbody></table>
<p>Using the extension of limits discussed above, one can then extend the definition of the derivative to a scalar-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:\mathbb {R} ^{n}\to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"></span> along some path <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 s(t):\mathbb {R} \to \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./f3bb769f922389fa74ff0f7fcad21b56ee1c3fbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.865ex; height:2.843ex;" alt="{\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}}" loading="lazy"></span>:
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \left.{\frac {df}{dx}}\right|_{s(t),t=t_{0}}=\lim _{h\to 0}{\frac {f(s(t_{0}+h))-f(s(t_{0}))}{|s(t_{0}+h)-s(t_{0})|}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>f</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>t</mi>
<mo>=</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>h</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>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left.{\frac {df}{dx}}\right|_{s(t),t=t_{0}}=\lim _{h\to 0}{\frac {f(s(t_{0}+h))-f(s(t_{0}))}{|s(t_{0}+h)-s(t_{0})|}}}</annotation>
</semantics>
</math></span><img src="./6fbeeddab7f50d7c55db955330004c3a426efd3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.176ex; height:6.676ex;" alt="{\displaystyle \left.{\frac {df}{dx}}\right|_{s(t),t=t_{0}}=\lim _{h\to 0}{\frac {f(s(t_{0}+h))-f(s(t_{0}))}{|s(t_{0}+h)-s(t_{0})|}}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_10" class="reference nourlexpansion" style="font-weight:bold;">10</span></td></tr></tbody></table>
<p>Unlike limits, for which the value depends on the exact form of the path <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 s(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>
</semantics>
</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span>, it can be shown that the derivative along the path depends only on the tangent vector of the path 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 s(t_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t_{0})}</annotation>
</semantics>
</math></span><img src="./a8d7837fe830c733cc02d209bffb7c2d654a23c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.794ex; height:2.843ex;" alt="{\displaystyle s(t_{0})}" loading="lazy"></span>, i.e. <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 s'(t_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s'(t_{0})}</annotation>
</semantics>
</math></span><img src="./7dbd43d6980b546a8e86b80e1cf9be90e8f99bfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.478ex; height:3.009ex;" alt="{\displaystyle s'(t_{0})}" loading="lazy"></span>, provided that <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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is <a href="Lipschitz_continuous" class="mw-redirect" title="Lipschitz continuous">Lipschitz continuous</a> 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 s(t_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t_{0})}</annotation>
</semantics>
</math></span><img src="./a8d7837fe830c733cc02d209bffb7c2d654a23c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.794ex; height:2.843ex;" alt="{\displaystyle s(t_{0})}" loading="lazy"></span>, and that the limit exits for at least one such path.
</p>

<div style="margin-left:0">
<table class="mw-collapsible mw-archivedtalk" style="color:inherit; background: transparent; text-align: left; border: 1px solid Silver; margin: 0.2em auto auto; width:100%; clear: both; padding: 1px;">

<tbody><tr>
<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof</div>
</th></tr>
<tr>
<td style="color:inherit; border: solid 1px Silver; padding: 0.6em; background: var(--background-color-base, #fff);">
<p>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 s(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation>
</semantics>
</math></span><img src="./c484de351ba40ccb9a5ad522c29c1aac5686c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}" loading="lazy"></span> continuous up to the first derivative (this statement is well defined 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 s}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
</semantics>
</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> is a function of one variable), we can write the <a href="Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</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 s}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
</semantics>
</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> around <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 t_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>
</semantics>
</math></span><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"></span> using <a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a> to construct the remainder:
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 s(t)=s(t_{0})+s'(\tau )(t-t_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>s</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>τ<!-- τ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(t)=s(t_{0})+s'(\tau )(t-t_{0})}</annotation>
</semantics>
</math></span><img src="./a349c82b45c2496a74210897d0c81548317e5c71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.641ex; height:3.009ex;" alt="{\displaystyle s(t)=s(t_{0})+s'(\tau )(t-t_{0})}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_11" class="reference nourlexpansion" style="font-weight:bold;">11</span></td></tr></tbody></table>
<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 \tau \in [t_{0},t]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>τ<!-- τ --></mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau \in [t_{0},t]}</annotation>
</semantics>
</math></span><img src="./58ac6f55b4d2583f6bf34d88a41fb91a7c353b2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.104ex; height:2.843ex;" alt="{\displaystyle \tau \in [t_{0},t]}" loading="lazy"></span>.
</p><p>Substituting this into <b><a href="#math_10">10</a></b>,
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \left.{\frac {df}{dx}}\right|_{s(t),t=t_{0}}=\lim _{h\to 0}{\frac {f(s(t_{0})+s'(\tau )h)-f(s(t_{0}))}{|s'(\tau )h|}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>f</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>t</mi>
<mo>=</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>h</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>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>s</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>τ<!-- τ --></mi>
<mo stretchy="false">)</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>s</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>τ<!-- τ --></mi>
<mo stretchy="false">)</mo>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left.{\frac {df}{dx}}\right|_{s(t),t=t_{0}}=\lim _{h\to 0}{\frac {f(s(t_{0})+s'(\tau )h)-f(s(t_{0}))}{|s'(\tau )h|}}}</annotation>
</semantics>
</math></span><img src="./1b9a88955a7c64e3fec507c952307dbe66da7ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:45.963ex; height:6.676ex;" alt="{\displaystyle \left.{\frac {df}{dx}}\right|_{s(t),t=t_{0}}=\lim _{h\to 0}{\frac {f(s(t_{0})+s'(\tau )h)-f(s(t_{0}))}{|s'(\tau )h|}}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_12" class="reference nourlexpansion" style="font-weight:bold;">12</span></td></tr></tbody></table>
<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 \tau (h)\in [t_{0},t_{0}+h]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>τ<!-- τ --></mi>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>h</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau (h)\in [t_{0},t_{0}+h]}</annotation>
</semantics>
</math></span><img src="./f260b8a858b2aa351ce0b7d519c2efa451910223.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.486ex; height:2.843ex;" alt="{\displaystyle \tau (h)\in [t_{0},t_{0}+h]}" loading="lazy"></span>.
</p><p>Lipschitz continuity gives us <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)-f(y)|\leq K|x-y|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≤<!-- ≤ --></mo>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |f(x)-f(y)|\leq K|x-y|}</annotation>
</semantics>
</math></span><img src="./e46de0c9666270531699a90ec3feec3ccc828b0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.579ex; height:2.843ex;" alt="{\displaystyle |f(x)-f(y)|\leq K|x-y|}" loading="lazy"></span> for some finite <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="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" 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 \forall x,y\in \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \forall x,y\in \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./30b95c283b7a844fa1a04bf92aa2b5c448d46be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.549ex; height:2.676ex;" alt="{\displaystyle \forall x,y\in \mathbb {R} ^{n}}" loading="lazy"></span>. It follows that <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+O(h))-f(x)|\sim O(h)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>∼<!-- ∼ --></mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |f(x+O(h))-f(x)|\sim O(h)}</annotation>
</semantics>
</math></span><img src="./261b5aad532a9a9d544e904e54f2de095bcb0af8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.751ex; height:2.843ex;" alt="{\displaystyle |f(x+O(h))-f(x)|\sim O(h)}" loading="lazy"></span>.
</p><p>Note also that given the continuity 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 s'(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s'(t)}</annotation>
</semantics>
</math></span><img src="./e9fbde1038511c9270c270eb6aa61af29db613e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.424ex; height:3.009ex;" alt="{\displaystyle s'(t)}" 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 s'(\tau )=s'(t_{0})+O(h)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>τ<!-- τ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>s</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s'(\tau )=s'(t_{0})+O(h)}</annotation>
</semantics>
</math></span><img src="./19c5d4d5142de946798c4abfe446a1e286175cc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.125ex; height:3.009ex;" alt="{\displaystyle s'(\tau )=s'(t_{0})+O(h)}" loading="lazy"></span> 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 h\to 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h\to 0}</annotation>
</semantics>
</math></span><img src="./3f15c1e6800f6e3fca9fec99720fc67a13780215.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.116ex; height:2.176ex;" alt="{\displaystyle h\to 0}" loading="lazy"></span>.
</p><p>Substituting these two conditions into <b><a href="#math_12">12</a></b>,
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \left.{\frac {df}{dx}}\right|_{s(t),t=t_{0}}=\lim _{h\to 0}{\frac {f(s(t_{0})+s'(t_{0})h)-f(s(t_{0}))+O(h^{2})}{|s'(t_{0})h|+O(h^{2})}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>f</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>t</mi>
<mo>=</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>h</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>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>s</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>s</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>+</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left.{\frac {df}{dx}}\right|_{s(t),t=t_{0}}=\lim _{h\to 0}{\frac {f(s(t_{0})+s'(t_{0})h)-f(s(t_{0}))+O(h^{2})}{|s'(t_{0})h|+O(h^{2})}}}</annotation>
</semantics>
</math></span><img src="./acd7e9fbaa83db6321dab2869ef6db2c4263b788.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:55.471ex; height:6.843ex;" alt="{\displaystyle \left.{\frac {df}{dx}}\right|_{s(t),t=t_{0}}=\lim _{h\to 0}{\frac {f(s(t_{0})+s'(t_{0})h)-f(s(t_{0}))+O(h^{2})}{|s'(t_{0})h|+O(h^{2})}}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_13" class="reference nourlexpansion" style="font-weight:bold;">13</span></td></tr></tbody></table>
<p>whose limit depends only on <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 s'(t_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s'(t_{0})}</annotation>
</semantics>
</math></span><img src="./7dbd43d6980b546a8e86b80e1cf9be90e8f99bfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.478ex; height:3.009ex;" alt="{\displaystyle s'(t_{0})}" loading="lazy"></span> as the dominant term.
</p>
</td></tr></tbody></table></div>
<p>It is therefore possible to generate the definition of the directional derivative as follows: The directional derivative of a scalar-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:\mathbb {R} ^{n}\to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"></span> along the unit vector <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 {\hat {\mathbf {u}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {u}}}}</annotation>
</semantics>
</math></span><img src="./c9ba9882d416bf92d99b9023e6f4026af2e55a5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {u}}}}" loading="lazy"></span> at some point <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_{0}\in \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}\in \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./e5c5d325c966538b6db03962c2bcf80fe26af8b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.121ex; height:2.676ex;" alt="{\displaystyle x_{0}\in \mathbb {R} ^{n}}" loading="lazy"></span> is
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 _{\hat {\mathbf {u}}}f(x_{0})=\lim _{t\to 0}{\frac {f(x_{0}+{\hat {\mathbf {u}}}t)-f(x_{0})}{t}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mi>t</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=\lim _{t\to 0}{\frac {f(x_{0}+{\hat {\mathbf {u}}}t)-f(x_{0})}{t}}}</annotation>
</semantics>
</math></span><img src="./26de8a9bfaf752fe27f348b3c33e85fe1e4addef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.191ex; height:5.843ex;" alt="{\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=\lim _{t\to 0}{\frac {f(x_{0}+{\hat {\mathbf {u}}}t)-f(x_{0})}{t}}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_14" class="reference nourlexpansion" style="font-weight:bold;">14</span></td></tr></tbody></table>
<p>or, when expressed in terms of ordinary differentiation,
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 _{\hat {\mathbf {u}}}f(x_{0})=\left.{\frac {df(x_{0}+{\hat {\mathbf {u}}}t)}{dt}}\right|_{t=0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=\left.{\frac {df(x_{0}+{\hat {\mathbf {u}}}t)}{dt}}\right|_{t=0}}</annotation>
</semantics>
</math></span><img src="./94948b92d0a1b0923e64bb898bb326452257db50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.051ex; height:6.509ex;" alt="{\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=\left.{\frac {df(x_{0}+{\hat {\mathbf {u}}}t)}{dt}}\right|_{t=0}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_15" class="reference nourlexpansion" style="font-weight:bold;">15</span></td></tr></tbody></table>
<p>which is a well defined expression 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 f(x_{0}+{\hat {\mathbf {u}}}t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x_{0}+{\hat {\mathbf {u}}}t)}</annotation>
</semantics>
</math></span><img src="./df8448f68f361d6a733d093fa34c90ef675cf5de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.637ex; height:2.843ex;" alt="{\displaystyle f(x_{0}+{\hat {\mathbf {u}}}t)}" loading="lazy"></span> is a scalar function with one variable 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 t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>.
</p><p>It is not possible to define a unique scalar derivative without a direction; it is clear for example that <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 _{\hat {\mathbf {u}}}f(x_{0})=-\nabla _{-{\hat {\mathbf {u}}}}f(x_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=-\nabla _{-{\hat {\mathbf {u}}}}f(x_{0})}</annotation>
</semantics>
</math></span><img src="./4496335a3d234d706b61a4e82a5f1c1696405ca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.566ex; height:3.176ex;" alt="{\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=-\nabla _{-{\hat {\mathbf {u}}}}f(x_{0})}" loading="lazy"></span>. It is also possible for directional derivatives to exist for some directions but not for others.
</p>
<div class="mw-heading mw-heading3"><h3 id="Partial_derivative">Partial derivative</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Partial_derivative" title="Partial derivative">Partial derivative</a></div>
<p>The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a <a href="Derivative" title="Derivative">derivative</a> with respect to one variable with all other variables held constant.<sup id="cite_ref-CourantJohn1999_1-3" class="reference"><a href="#cite_note-CourantJohn1999-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 26ff">: 26ff </span></sup>
</p><p>A partial derivative may be thought of as the directional derivative of the function along a coordinate axis.
</p><p>Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In <a href="Vector_calculus" title="Vector calculus">vector calculus</a>, the <a href="Del" title="Del">del</a> operator (<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 used to define the concepts of <a href="Gradient" title="Gradient">gradient</a>, <a href="Divergence" title="Divergence">divergence</a>, and <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a> in terms of partial derivatives. A matrix of partial derivatives, the <b><a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></b> matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. The derivative can thus be understood as a <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> which directly varies from point to point in the domain of the function.
</p><p><a href="Differential_equations" class="mw-redirect" title="Differential equations">Differential equations</a> containing partial derivatives are called <a href="Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a> or PDEs. These equations are generally more difficult to solve than <a href="Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary differential equations</a>, which contain derivatives with respect to only one variable.<sup id="cite_ref-CourantJohn1999_1-4" class="reference"><a href="#cite_note-CourantJohn1999-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 654ff">: 654ff </span></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Multiple_integration">Multiple integration</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multiple_integral" title="Multiple integral">Multiple integral</a></div>
<p>The multiple integral extends the concept of the <a href="Integral" title="Integral">integral</a> to functions of any number of variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. <a href="Fubini's_theorem" title="Fubini's theorem">Fubini's theorem</a> guarantees that a multiple integral may be evaluated as a <i>repeated integral</i> or <i>iterated integral</i> as long as the integrand is continuous throughout the domain of integration.<sup id="cite_ref-CourantJohn1999_1-5" class="reference"><a href="#cite_note-CourantJohn1999-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 367ff">: 367ff </span></sup>
</p><p>The <a href="Surface_integral" title="Surface integral">surface integral</a> and the <a href="Line_integral" title="Line integral">line integral</a> are used to integrate over curved <a href="Manifold" title="Manifold">manifolds</a> such as <a href="Surface_(mathematics)" title="Surface (mathematics)">surfaces</a> and <a href="Curve" title="Curve">curves</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Fundamental_theorem_of_calculus_in_multiple_dimensions">Fundamental theorem of calculus in multiple dimensions</h3></div>
<p>In single-variable calculus, the <a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a> establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus:<sup id="cite_ref-CourantJohn1999_1-6" class="reference"><a href="#cite_note-CourantJohn1999-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 543ff">: 543ff </span></sup>
</p>
<ul><li><a href="Gradient_theorem" title="Gradient theorem">Gradient theorem</a></li>
<li><a href="Stokes'_theorem#Special_cases" title="Stokes' theorem">Stokes' theorem</a></li>
<li><a href="Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li>
<li><a href="Green's_theorem" title="Green's theorem">Green's theorem</a>.</li></ul>
<p>In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized <a href="Generalized_Stokes_theorem" title="Generalized Stokes theorem">Stokes' theorem</a>, which applies to the integration of <a href="Differential_forms" class="mw-redirect" title="Differential forms">differential forms</a> over <a href="Differentiable_manifold" title="Differentiable manifold">manifolds</a>.<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>
<div class="mw-heading mw-heading2"><h2 id="Applications_and_uses">Applications and uses</h2></div>
<p>Techniques of multivariable calculus are used to study many objects of interest in the material world. In particular,
</p>
<table class="wikitable" style="text-align:center">

<tbody><tr>
<th></th>
<th></th>
<th>Type of functions</th>
<th>Applicable techniques
</th></tr>
<tr>
<th><a href="Curve" title="Curve">Curves</a>
</th>
<td><span typeof="mw:File"></span></td>
<td><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:\mathbb {R} \to \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./5de4ca000609cbf1e9d28f447c384ad967980da1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} ^{n}}" loading="lazy"></span> <br> 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 n>1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>&gt;</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n&gt;1}</annotation>
</semantics>
</math></span><img src="./ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}" loading="lazy"></span></td>
<td>Lengths of curves, <a href="Line_integral" title="Line integral">line integrals</a>, and <a href="Curvature" title="Curvature">curvature</a>.
</td></tr>
<tr>
<th><a href="Surface_(mathematics)" title="Surface (mathematics)">Surfaces</a>
</th>
<td><span typeof="mw:File"></span></td>
<td><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:\mathbb {R} ^{2}\to \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./ae8e8f3264524547597748653089bbf61fadf2e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.459ex; height:3.009ex;" alt="{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{n}}" loading="lazy"></span> <br> 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 n>2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>&gt;</mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n&gt;2}</annotation>
</semantics>
</math></span><img src="./44e71ac55b9fbf1e9f341b946cda63d61d3ef2cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>2}" loading="lazy"></span></td>
<td><a href="Area" title="Area">Areas</a> of surfaces, <a href="Surface_integral" title="Surface integral">surface integrals</a>, <a href="Flux" title="Flux">flux</a> through surfaces, and curvature.
</td></tr>
<tr>
<th><a href="Scalar_fields" class="mw-redirect" title="Scalar fields">Scalar fields</a>
</th>
<td><span typeof="mw:File"></span></td>
<td><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:\mathbb {R} ^{n}\to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"></span></td>
<td>Maxima and minima, <a href="Lagrange_multipliers" class="mw-redirect" title="Lagrange multipliers">Lagrange multipliers</a>, <a href="Directional_derivative" title="Directional derivative">directional derivatives</a>, <a href="Level_set" title="Level set">level sets</a>.
</td></tr>
<tr>
<th><a href="Vector_fields" class="mw-redirect" title="Vector fields">Vector fields</a>
</th>
<td><span typeof="mw:File"></span></td>
<td><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:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./58fa8a698b05433f8431190a957e0f413cda8e7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.079ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}" loading="lazy"></span></td>
<td>Any of the operations of <a href="Vector_calculus" title="Vector calculus">vector calculus</a> including <a href="Gradient" title="Gradient">gradient</a>, <a href="Divergence" title="Divergence">divergence</a>, and <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a>.
</td></tr></tbody></table>
<p>Multivariable calculus can be applied to analyze <a href="Deterministic_system" title="Deterministic system">deterministic systems</a> that have multiple <a href="Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a>. Functions with <a href="Independent_variable" class="mw-redirect" title="Independent variable">independent variables</a> corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the <a href="System_dynamics" title="System dynamics">system dynamics</a>.
</p><p>Multivariate calculus is used in the <a href="Optimal_control" title="Optimal control">optimal control</a> of <a href="Continuous_time" class="mw-redirect" title="Continuous time">continuous time</a> <a href="Dynamic_systems" class="mw-redirect" title="Dynamic systems">dynamic systems</a>. It is used in <a href="Regression_analysis" title="Regression analysis">regression analysis</a> to derive formulas for estimating relationships among various sets of <a href="Empirical_data" class="mw-redirect" title="Empirical data">empirical data</a>.
</p><p>Multivariable calculus is used in many fields of <a href="Natural_science" title="Natural science">natural</a> and <a href="Social_science" title="Social science">social science</a> and <a href="Engineering" title="Engineering">engineering</a> to model and study high-dimensional systems that exhibit deterministic behavior. In <a href="Economics" title="Economics">economics</a>, for example, <a href="Consumer_choice" title="Consumer choice">consumer choice</a> over a variety of goods, and <a href="Profit_maximization" title="Profit maximization">producer choice</a> over various inputs to use and outputs to produce, are modeled with multivariate calculus.
</p><p>Non-deterministic, or <a href="Stochastic_process" title="Stochastic process">stochastic</a> systems can be studied using a different kind of mathematics, such as <a href="Stochastic_calculus" title="Stochastic calculus">stochastic calculus</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="List_of_multivariable_calculus_topics" title="List of multivariable calculus topics">List of multivariable calculus topics</a></li>
<li><a href="Multivariate_statistics" title="Multivariate statistics">Multivariate statistics</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-CourantJohn1999-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-CourantJohn1999_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-CourantJohn1999_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-CourantJohn1999_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-CourantJohn1999_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-CourantJohn1999_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-CourantJohn1999_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-CourantJohn1999_1-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFRichard_CourantFritz_John1999" class="citation book cs1">Richard Courant; Fritz John (14 December 1999). <i>Introduction to Calculus and Analysis Volume II/2</i>. Springer Science &amp; Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-3-540-66570-0</bdi>.</cite></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="CITEREFSpivak1965" class="citation book cs1">Spivak, Michael (1965). <a rel="nofollow" class="external text" href="https://archive.org/details/SpivakM.CalculusOnManifolds_201703"><i>Calculus on Manifolds</i></a>. New York: W. A. Benjamin, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>9780805390216</bdi>.</cite></span>
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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Multivariate_calculus" class="extiw external" title="commons:Category:Multivariate calculus">Multivariate calculus</a></span>.</div></div>
</div>
<ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=cw6pHhjhKmk">UC Berkeley video lectures on Multivariable Calculus, Fall 2009, Professor Edward Frenkel</a></li>
<li><a rel="nofollow" class="external text" href="https://www.youtube.com/playlist?list=PL4C4C8A7D06566F38">MIT video lectures on Multivariable Calculus, Fall 2007</a></li>
<li><a rel="nofollow" class="external text" href="http://www.math.gatech.edu/~cain/notes/calculus.html"><i>Multivariable Calculus</i></a>: A free online textbook by George Cain and James Herod</li>
<li><a rel="nofollow" class="external text" href="https://math.etsu.edu/Multicalc/"><i>Multivariable Calculus Online</i></a>: A free online textbook by Jeff Knisley</li>
<li><a rel="nofollow" class="external text" href="http://www.ecs.umass.edu/mie/faculty/perot/mie440/Multivariable%20Calculus.pdf"><i>Multivariable Calculus – A Very Quick Review</i></a>, Prof. Blair Perot, University of Massachusetts Amherst</li>
<li><a rel="nofollow" class="external text" href="http://www.stat.rice.edu/~dobelman/notes_papers/math/calculus.MV.pdf"><i>Multivariable Calculus</i></a>, Online text by Dr. Jerry Shurman</li></ul>
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<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>
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