SherbyNet - /page/wikibook/entry.mu

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</style><div role="note" class="hatnote navigation-not-searchable"><span>Not to be confused with <a href="Dell" title="Dell">Dell</a>.</span> <span>For other uses, see <a href="Del_(disambiguation)" class="mw-disambig" title="Del (disambiguation)">Del (disambiguation)</a>.</span></div>

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<p><b>Del</b>, or <b>nabla</b>, is an <a href="Operator_(mathematics)" title="Operator (mathematics)">operator</a> used in mathematics (particularly in <a href="Vector_calculus" title="Vector calculus">vector calculus</a>) as a <a href="Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</a> <a href="Differential_operator" title="Differential operator">differential operator</a>, usually represented by <b>∇</b> (the <i><a href="Nabla_symbol" title="Nabla symbol">nabla symbol</a></i>). When applied to a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> defined on a <a href="Dimension_(mathematics)" class="mw-redirect" title="Dimension (mathematics)">one-dimensional</a> domain, it denotes the standard <a href="Derivative" title="Derivative">derivative</a> of the function as defined in <a href="Calculus" title="Calculus">calculus</a>. When applied to a <i>field</i> (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the <a href="Gradient" title="Gradient">gradient</a> or (locally) steepest slope of a <a href="Scalar_field" title="Scalar field">scalar field</a> (or sometimes of a <a href="Vector_field" title="Vector field">vector field</a>, as in the <a href="Navier%E2%80%93Stokes_equations#Interpretation_as_v·(∇v)" title="Navier–Stokes equations">Navier–Stokes equations</a>); the <a href="Divergence" title="Divergence">divergence</a> of a vector field; or the <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a> (rotation) of a vector field.

</p><p>Del is a very convenient <a href="Mathematical_notation" title="Mathematical notation">mathematical notation</a> for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be <a href="Formal_calculation" title="Formal calculation">formally</a> defined as a vector operator whose components are the corresponding <a href="Partial_derivative" title="Partial derivative">partial derivative</a> operators. As a vector operator, it can act on scalar and vector fields in three different ways, giving rise to three different differential operations: first, it can act on scalar fields by a formal scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formal <a href="Dot_product" title="Dot product">dot product</a>—to give a scalar field called the divergence; and lastly, it can act on vector fields by a formal <a href="Cross_product" title="Cross product">cross product</a>—to give a vector field called the curl. These formal products do not necessarily <a href="Commutative_operation" class="mw-redirect" title="Commutative operation">commute</a> with other operators or products. These three uses are summarized as:

</p>

<ul><li>Gradient: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {grad} f=\nabla f}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {grad} f=\nabla f}</annotation>

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</math></span><img src="./18a38180a41e74e85359ef2155baaab943962707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.508ex; height:2.509ex;" alt="{\displaystyle \operatorname {grad} f=\nabla f}" loading="lazy"></span></li>

<li>Divergence: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} \mathbf {v} =\nabla \cdot \mathbf {v} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {div} \mathbf {v} =\nabla \cdot \mathbf {v} }</annotation>

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</math></span><img src="./16d79512c67d6fbe11afebadcce100506e856a54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.089ex; height:2.176ex;" alt="{\displaystyle \operatorname {div} \mathbf {v} =\nabla \cdot \mathbf {v} }" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {curl} \mathbf {v} =\nabla \times \mathbf {v} }</annotation>

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</math></span><img src="./fcf0c080491729864206a5aaf9fdd2fe2a0a05a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.967ex; height:2.176ex;" alt="{\displaystyle \operatorname {curl} \mathbf {v} =\nabla \times \mathbf {v} }" loading="lazy"></span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>In the <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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</math></span><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> with coordinates <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\dots ,x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{1},\dots ,x_{n})}</annotation>

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</math></span><img src="./56204ada1ecf9dd5c6beddfdfd0f341cd69ff632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\dots ,x_{n})}" loading="lazy"></span> and <a href="Standard_basis" title="Standard basis">standard basis</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 \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}}</annotation>

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</math></span><img src="./62f7ce281ad2a6a82a622d5d7ef611a3b18ae2b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.226ex; height:2.843ex;" alt="{\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}}" loading="lazy"></span>, del is a vector operator whose <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\dots ,x_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{1},\dots ,x_{n}}</annotation>

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</math></span><img src="./e5afdbc2d248d8fa9ba2c4f5188d946a0537e753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\dots ,x_{n}}" loading="lazy"></span> components are the <a href="Partial_derivative" title="Partial derivative">partial derivative</a> operators <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial \over \partial x_{1}},\dots ,{\partial \over \partial x_{n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\partial \over \partial x_{1}},\dots ,{\partial \over \partial x_{n}}}</annotation>

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</math></span><img src="./961f4aed90424f512618a1f1a25b7797205d094e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.419ex; height:5.843ex;" alt="{\displaystyle {\partial \over \partial x_{1}},\dots ,{\partial \over \partial x_{n}}}" loading="lazy"></span>; that is,

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<annotation encoding="application/x-tex">{\displaystyle \nabla =\sum _{i=1}^{n}\mathbf {e} _{i}{\partial \over \partial x_{i}}=\left({\partial \over \partial x_{1}},\ldots ,{\partial \over \partial x_{n}}\right)}</annotation>

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</math></span><img src="./3cbe31df1ef0f70f083d4137a993bf3f960a2ad3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.023ex; height:6.843ex;" alt="{\displaystyle \nabla =\sum _{i=1}^{n}\mathbf {e} _{i}{\partial \over \partial x_{i}}=\left({\partial \over \partial x_{1}},\ldots ,{\partial \over \partial x_{n}}\right)}" loading="lazy"></span></dd></dl>

<p>where the expression in parentheses is a row vector. In <a href="Three-dimensional" class="mw-redirect" title="Three-dimensional">three-dimensional</a> Cartesian coordinate system <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math></span><img src="./f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"></span> with coordinates <span class="mwe-math-element mwe-math-element-inline"><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,y,z)}">

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<annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation>

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</math></span><img src="./22a8c93372e8f8b6e24d523bd5545aed3430baf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}" loading="lazy"></span> and standard basis or unit vectors of 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 \{\mathbf {e} _{x},\mathbf {e} _{y},\mathbf {e} _{z}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{\mathbf {e} _{x},\mathbf {e} _{y},\mathbf {e} _{z}\}}</annotation>

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</math></span><img src="./84589fe1107dc80af0233db9847cf54e28e7c3bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.292ex; height:3.009ex;" alt="{\displaystyle \{\mathbf {e} _{x},\mathbf {e} _{y},\mathbf {e} _{z}\}}" loading="lazy"></span>, del is written as:

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<annotation encoding="application/x-tex">{\displaystyle \nabla =\mathbf {e} _{x}{\partial \over \partial x}+\mathbf {e} _{y}{\partial \over \partial y}+\mathbf {e} _{z}{\partial \over \partial z}=\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)}</annotation>

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</math></span><img src="./f76d82c11031984f27bec14e8f32c6e70943afc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.273ex; height:6.176ex;" alt="{\displaystyle \nabla =\mathbf {e} _{x}{\partial \over \partial x}+\mathbf {e} _{y}{\partial \over \partial y}+\mathbf {e} _{z}{\partial \over \partial z}=\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)}" loading="lazy"></span></dd></dl>

<p>As a vector operator, del naturally acts on scalar fields via <a href="Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>, and naturally acts on vector fields via dot products and cross products.

</p><p>More specifically, for any scalar field <span class="mwe-math-element mwe-math-element-inline"><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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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =(F_{x},F_{y},F_{z})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {e} _{i}{\partial \over \partial x_{i}}\right)f:={\partial \over \partial x_{i}}(\mathbf {e} _{i}f)={\partial f \over \partial x_{i}}\mathbf {e} _{i}}</annotation>

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</math></span><img src="./c942822cf5a86927327fabc581dad5dcbc012c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.944ex; height:6.176ex;" alt="{\displaystyle \left(\mathbf {e} _{i}{\partial \over \partial x_{i}}\right)f:={\partial \over \partial x_{i}}(\mathbf {e} _{i}f)={\partial f \over \partial x_{i}}\mathbf {e} _{i}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {e} _{i}{\partial \over \partial x_{i}}\right)\cdot \mathbf {F} :={\partial \over \partial x_{i}}(\mathbf {e} _{i}\cdot \mathbf {F} )={\partial F_{i} \over \partial x_{i}}}</annotation>

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</math></span><img src="./bc5498c348aa2e11b2ae44174fd7c5776161a6a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.863ex; height:6.176ex;" alt="{\displaystyle \left(\mathbf {e} _{i}{\partial \over \partial x_{i}}\right)\cdot \mathbf {F} :={\partial \over \partial x_{i}}(\mathbf {e} _{i}\cdot \mathbf {F} )={\partial F_{i} \over \partial x_{i}}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {e} _{x}{\partial \over \partial x}\right)\times \mathbf {F} :={\partial \over \partial x}(\mathbf {e} _{x}\times \mathbf {F} )={\partial \over \partial x}(0,-F_{z},F_{y})}</annotation>

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</math></span><img src="./15dc5b1d667b9c4a83ebaf6266578f917ad96c68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.255ex; height:6.176ex;" alt="{\displaystyle \left(\mathbf {e} _{x}{\partial \over \partial x}\right)\times \mathbf {F} :={\partial \over \partial x}(\mathbf {e} _{x}\times \mathbf {F} )={\partial \over \partial x}(0,-F_{z},F_{y})}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {e} _{y}{\partial \over \partial y}\right)\times \mathbf {F} :={\partial \over \partial y}(\mathbf {e} _{y}\times \mathbf {F} )={\partial \over \partial y}(F_{z},0,-F_{x})}</annotation>

</semantics>

</math></span><img src="./bfe5d25e5a03170713d7a3a6ad2e94717a52441d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.61ex; height:6.176ex;" alt="{\displaystyle \left(\mathbf {e} _{y}{\partial \over \partial y}\right)\times \mathbf {F} :={\partial \over \partial y}(\mathbf {e} _{y}\times \mathbf {F} )={\partial \over \partial y}(F_{z},0,-F_{x})}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {e} _{z}{\partial \over \partial z}\right)\times \mathbf {F} :={\partial \over \partial z}(\mathbf {e} _{z}\times \mathbf {F} )={\partial \over \partial z}(-F_{y},F_{x},0),}</annotation>

</semantics>

</math></span><img src="./fb5ef3acce0f0cac598231229cbdc82a72667571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.007ex; height:6.176ex;" alt="{\displaystyle \left(\mathbf {e} _{z}{\partial \over \partial z}\right)\times \mathbf {F} :={\partial \over \partial z}(\mathbf {e} _{z}\times \mathbf {F} )={\partial \over \partial z}(-F_{y},F_{x},0),}" loading="lazy"></span></dd></dl>

<p>then using the above definition 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 \nabla }">

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<annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \nabla f=\left(\mathbf {e} _{x}{\partial \over \partial x}\right)f+\left(\mathbf {e} _{y}{\partial \over \partial y}\right)f+\left(\mathbf {e} _{z}{\partial \over \partial z}\right)f={\partial f \over \partial x}\mathbf {e} _{x}+{\partial f \over \partial y}\mathbf {e} _{y}+{\partial f \over \partial z}\mathbf {e} _{z}}</annotation>

</semantics>

</math></span><img src="./4afa95ffc300ed9ef7ab33ccd5f3a2826b39fb84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:70.217ex; height:6.176ex;" alt="{\displaystyle \nabla f=\left(\mathbf {e} _{x}{\partial \over \partial x}\right)f+\left(\mathbf {e} _{y}{\partial \over \partial y}\right)f+\left(\mathbf {e} _{z}{\partial \over \partial z}\right)f={\partial f \over \partial x}\mathbf {e} _{x}+{\partial f \over \partial y}\mathbf {e} _{y}+{\partial f \over \partial z}\mathbf {e} _{z}}" loading="lazy"></span></dd></dl>

<p>and

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<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {F} =\left(\mathbf {e} _{x}{\partial \over \partial x}\cdot \mathbf {F} \right)+\left(\mathbf {e} _{y}{\partial \over \partial y}\cdot \mathbf {F} \right)+\left(\mathbf {e} _{z}{\partial \over \partial z}\cdot \mathbf {F} \right)={\partial F_{x} \over \partial x}+{\partial F_{y} \over \partial y}+{\partial F_{z} \over \partial z}}</annotation>

</semantics>

</math></span><img src="./0edd1cf558e999e50d7f97b821c87db0ec489def.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:74.31ex; height:6.343ex;" alt="{\displaystyle \nabla \cdot \mathbf {F} =\left(\mathbf {e} _{x}{\partial \over \partial x}\cdot \mathbf {F} \right)+\left(\mathbf {e} _{y}{\partial \over \partial y}\cdot \mathbf {F} \right)+\left(\mathbf {e} _{z}{\partial \over \partial z}\cdot \mathbf {F} \right)={\partial F_{x} \over \partial x}+{\partial F_{y} \over \partial y}+{\partial F_{z} \over \partial z}}" loading="lazy"></span></dd></dl>

<p>and

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nabla \times \mathbf {F} &=\left(\mathbf {e} _{x}{\partial \over \partial x}\times \mathbf {F} \right)+\left(\mathbf {e} _{y}{\partial \over \partial y}\times \mathbf {F} \right)+\left(\mathbf {e} _{z}{\partial \over \partial z}\times \mathbf {F} \right)\\&={\partial \over \partial x}(0,-F_{z},F_{y})+{\partial \over \partial y}(F_{z},0,-F_{x})+{\partial \over \partial z}(-F_{y},F_{x},0)\\&=\left({\partial F_{z} \over \partial y}-{\partial F_{y} \over \partial z}\right)\mathbf {e} _{x}+\left({\partial F_{x} \over \partial z}-{\partial F_{z} \over \partial x}\right)\mathbf {e} _{y}+\left({\partial F_{y} \over \partial x}-{\partial F_{x} \over \partial y}\right)\mathbf {e} _{z}\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./020532e2cae21dcf7b2a8c0d0b6bc1b5c5ed4f47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.497ex; margin-bottom: -0.175ex; width:71.174ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\nabla \times \mathbf {F} &=\left(\mathbf {e} _{x}{\partial \over \partial x}\times \mathbf {F} \right)+\left(\mathbf {e} _{y}{\partial \over \partial y}\times \mathbf {F} \right)+\left(\mathbf {e} _{z}{\partial \over \partial z}\times \mathbf {F} \right)\\&={\partial \over \partial x}(0,-F_{z},F_{y})+{\partial \over \partial y}(F_{z},0,-F_{x})+{\partial \over \partial z}(-F_{y},F_{x},0)\\&=\left({\partial F_{z} \over \partial y}-{\partial F_{y} \over \partial z}\right)\mathbf {e} _{x}+\left({\partial F_{x} \over \partial z}-{\partial F_{z} \over \partial x}\right)\mathbf {e} _{y}+\left({\partial F_{y} \over \partial x}-{\partial F_{x} \over \partial y}\right)\mathbf {e} _{z}\end{aligned}}}" loading="lazy"></span></dd></dl>

<dl><dd><b>Example:</b></dd>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(x,y,z)=x+y+z}</annotation>

</semantics>

</math></span><img src="./77639b27d3bcbcaf53fd6e06e3e2b8a4269991a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.082ex; height:2.843ex;" alt="{\displaystyle f(x,y,z)=x+y+z}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \nabla f=\mathbf {e} _{x}{\partial f \over \partial x}+\mathbf {e} _{y}{\partial f \over \partial y}+\mathbf {e} _{z}{\partial f \over \partial z}=\left(1,1,1\right)}</annotation>

</semantics>

</math></span><img src="./d85c2aa6dde1c4cb2c2416f429c4dd3e66f213fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.705ex; height:6.176ex;" alt="{\displaystyle \nabla f=\mathbf {e} _{x}{\partial f \over \partial x}+\mathbf {e} _{y}{\partial f \over \partial y}+\mathbf {e} _{z}{\partial f \over \partial z}=\left(1,1,1\right)}" loading="lazy"></span></dd>

<p>Del can also be expressed in other coordinate systems, see for example <a href="Del_in_cylindrical_and_spherical_coordinates" title="Del in cylindrical and spherical coordinates">del in cylindrical and spherical coordinates</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Notational_uses">Notational uses</h2></div>

<p>Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the <a href="Gradient" title="Gradient">gradient</a>, <a href="Divergence" title="Divergence">divergence</a>, <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a>, <a href="Directional_derivative" title="Directional derivative">directional derivative</a>, and <a href="Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Gradient">Gradient</h3></div>

<p>The vector derivative of a <a href="Scalar_field" title="Scalar field">scalar field</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 f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {grad} f={\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}=\nabla f}</annotation>

</semantics>

</math></span><img src="./a06554fe16b9ac73b42d64542888e5be4ed28a15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.646ex; height:6.176ex;" alt="{\displaystyle \operatorname {grad} f={\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}=\nabla f}" loading="lazy"></span></dd></dl>

<p>It always points in the <a href="Direction_(geometry)" title="Direction (geometry)">direction</a> of greatest increase 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}">

</mstyle>

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<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>, and it has a <a href="Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> equal to the maximum rate of increase at the point—just like a standard derivative. In particular, if a hill is defined as a <a href="Height_function" title="Height function">height function</a> over a plane <span class="mwe-math-element mwe-math-element-inline"><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(x,y)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h(x,y)}</annotation>

</semantics>

</math></span><img src="./7235c6799c7d4112231c9941bd428fe6a4111fe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.667ex; height:2.843ex;" alt="{\displaystyle h(x,y)}" loading="lazy"></span>, the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.

</p><p>In particular, this notation is powerful because the gradient <a href="Product_rule" title="Product rule">product rule</a> looks very similar to the 1d-derivative case:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla (fg)=f\nabla g+g\nabla f}</annotation>

</semantics>

</math></span><img src="./7d27f6e61a59a516822d5fab9c8adbf3b4fe2612.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.74ex; height:2.843ex;" alt="{\displaystyle \nabla (fg)=f\nabla g+g\nabla f}" loading="lazy"></span></dd></dl>

<p>However, the rules for <a href="Dot_product" title="Dot product">dot products</a> do not turn out to be simple, as illustrated by:

</p>

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<annotation encoding="application/x-tex">{\displaystyle \nabla (\mathbf {u} \cdot \mathbf {v} )=(\mathbf {u} \cdot \nabla )\mathbf {v} +(\mathbf {v} \cdot \nabla )\mathbf {u} +\mathbf {u} \times (\nabla \times \mathbf {v} )+\mathbf {v} \times (\nabla \times \mathbf {u} )}</annotation>

</semantics>

</math></span><img src="./f7eebaef43ae4007156eb7bc6ab979b29f698840.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.226ex; height:2.843ex;" alt="{\displaystyle \nabla (\mathbf {u} \cdot \mathbf {v} )=(\mathbf {u} \cdot \nabla )\mathbf {v} +(\mathbf {v} \cdot \nabla )\mathbf {u} +\mathbf {u} \times (\nabla \times \mathbf {v} )+\mathbf {v} \times (\nabla \times \mathbf {u} )}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Divergence">Divergence</h3></div>

<p>The <a href="Divergence" title="Divergence">divergence</a> of a <a href="Vector_field" title="Vector field">vector field</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}}</annotation>

</semantics>

</math></span><img src="./10b4e06cd52dcd7bfc0c41a823495529c055c220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.257ex; height:3.009ex;" alt="{\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}}" loading="lazy"></span> is a <a href="Scalar_field" title="Scalar field">scalar field</a> that can be represented as:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {div} \mathbf {v} ={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}=\nabla \cdot \mathbf {v} }</annotation>

</semantics>

</math></span><img src="./eefd05ad90ca9c5663be1f53b0e4a7804598c485.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.938ex; height:6.343ex;" alt="{\displaystyle \operatorname {div} \mathbf {v} ={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}=\nabla \cdot \mathbf {v} }" loading="lazy"></span></dd></dl>

<p>The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point.

</p><p>The power of the del notation is shown by the following product rule:

</p>

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<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot (f\mathbf {v} )=(\nabla f)\cdot \mathbf {v} +f(\nabla \cdot \mathbf {v} )}</annotation>

</semantics>

</math></span><img src="./fcf11ad2dd2a7fb966d6cb88a0be6094bc0452a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.28ex; height:2.843ex;" alt="{\displaystyle \nabla \cdot (f\mathbf {v} )=(\nabla f)\cdot \mathbf {v} +f(\nabla \cdot \mathbf {v} )}" loading="lazy"></span></dd></dl>

<p>The formula for the <a href="Vector_product" class="mw-redirect" title="Vector product">vector product</a> is slightly less intuitive, because this product is not commutative:

</p>

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<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot (\mathbf {u} \times \mathbf {v} )=(\nabla \times \mathbf {u} )\cdot \mathbf {v} -\mathbf {u} \cdot (\nabla \times \mathbf {v} )}</annotation>

</semantics>

</math></span><img src="./4e16989946642f58f29d7dba97f3948ee0ac6a79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.422ex; height:2.843ex;" alt="{\displaystyle \nabla \cdot (\mathbf {u} \times \mathbf {v} )=(\nabla \times \mathbf {u} )\cdot \mathbf {v} -\mathbf {u} \cdot (\nabla \times \mathbf {v} )}" loading="lazy"></span></dd></dl>

<p>The <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a> of a vector field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}}</annotation>

</semantics>

</math></span><img src="./10b4e06cd52dcd7bfc0c41a823495529c055c220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.257ex; height:3.009ex;" alt="{\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}}" loading="lazy"></span> is a <a href="Vector_field" title="Vector field">vector</a> function that can be represented as:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {curl} \mathbf {v} =\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right){\hat {\mathbf {x} }}+\left({\partial v_{x} \over \partial z}-{\partial v_{z} \over \partial x}\right){\hat {\mathbf {y} }}+\left({\partial v_{y} \over \partial x}-{\partial v_{x} \over \partial y}\right){\hat {\mathbf {z} }}=\nabla \times \mathbf {v} }</annotation>

</semantics>

</math></span><img src="./059cf5650923c41ba44fccd43d5030d76cc38efe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:73.84ex; height:6.343ex;" alt="{\displaystyle \operatorname {curl} \mathbf {v} =\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right){\hat {\mathbf {x} }}+\left({\partial v_{x} \over \partial z}-{\partial v_{z} \over \partial x}\right){\hat {\mathbf {y} }}+\left({\partial v_{y} \over \partial x}-{\partial v_{x} \over \partial y}\right){\hat {\mathbf {z} }}=\nabla \times \mathbf {v} }" loading="lazy"></span></dd></dl>

<p>The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centered at that point.

</p><p>The vector product operation can be visualized as a pseudo-<a href="Determinant" title="Determinant">determinant</a>:

</p>

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {v} =\left|{\begin{matrix}{\hat {\mathbf {x} }}&{\hat {\mathbf {y} }}&{\hat {\mathbf {z} }}\\[2pt]{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\[2pt]v_{x}&v_{y}&v_{z}\end{matrix}}\right|}</annotation>

</semantics>

</math></span><img src="./2c22add6c547b447848574525afdfe35578f1f62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:23.807ex; height:11.843ex;" alt="{\displaystyle \nabla \times \mathbf {v} =\left|{\begin{matrix}{\hat {\mathbf {x} }}&{\hat {\mathbf {y} }}&{\hat {\mathbf {z} }}\\[2pt]{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\[2pt]v_{x}&v_{y}&v_{z}\end{matrix}}\right|}" loading="lazy"></span></dd></dl>

<p>Again the power of the notation is shown by the product rule:

</p>

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times (f\mathbf {v} )=(\nabla f)\times \mathbf {v} +f(\nabla \times \mathbf {v} )}</annotation>

</semantics>

</math></span><img src="./a00d526a83adcd232599b48a3f4b484f54ccab18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.764ex; height:2.843ex;" alt="{\displaystyle \nabla \times (f\mathbf {v} )=(\nabla f)\times \mathbf {v} +f(\nabla \times \mathbf {v} )}" loading="lazy"></span></dd></dl>

<p>The rule for the vector product does not turn out to be simple:

</p>

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times (\mathbf {u} \times \mathbf {v} )=\mathbf {u} \,(\nabla \cdot \mathbf {v} )-\mathbf {v} \,(\nabla \cdot \mathbf {u} )+(\mathbf {v} \cdot \nabla )\,\mathbf {u} -(\mathbf {u} \cdot \nabla )\,\mathbf {v} }</annotation>

</semantics>

</math></span><img src="./c89a8cf9a1de32fd5d9f464d53054c51d4d88b77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.772ex; height:2.843ex;" alt="{\displaystyle \nabla \times (\mathbf {u} \times \mathbf {v} )=\mathbf {u} \,(\nabla \cdot \mathbf {v} )-\mathbf {v} \,(\nabla \cdot \mathbf {u} )+(\mathbf {v} \cdot \nabla )\,\mathbf {u} -(\mathbf {u} \cdot \nabla )\,\mathbf {v} }" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Directional_derivative">Directional derivative</h3></div>

<p>The <a href="Directional_derivative" title="Directional derivative">directional derivative</a> of a scalar field <span class="mwe-math-element mwe-math-element-inline"><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,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x,y,z)}</annotation>

</semantics>

</math></span><img src="./f5d48dce2c4341575269f1709237a2e18923237a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.729ex; height:2.843ex;" alt="{\displaystyle f(x,y,z)}" loading="lazy"></span> in the direction

</mrow>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} (x,y,z)=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}}</annotation>

</semantics>

</math></span><img src="./062fcc09f1c251589149a25909e1fa249587320e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.452ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} (x,y,z)=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}}" loading="lazy"></span> is defined as:

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<annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} \cdot \nabla )f=\lim _{h\to 0}{\frac {f(x+a_{x}h,y+a_{y}h,z+a_{z}h)-f(x,y,z)}{h}}.}</annotation>

</semantics>

</math></span><img src="./5a28c4ebfd48336df992c1cc223a91d420bea838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:56.132ex; height:5.843ex;" alt="{\displaystyle (\mathbf {a} \cdot \nabla )f=\lim _{h\to 0}{\frac {f(x+a_{x}h,y+a_{y}h,z+a_{z}h)-f(x,y,z)}{h}}.}" loading="lazy"></span></dd></dl>

<p>Which is equal to the following when the gradient exists

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \operatorname {grad} f=a_{x}{\partial f \over \partial x}+a_{y}{\partial f \over \partial y}+a_{z}{\partial f \over \partial z}=\mathbf {a} \cdot (\nabla f)}</annotation>

</semantics>

</math></span><img src="./4b76b1bcec0bb257c2f0b2d4119e48cb217c78e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.316ex; height:6.176ex;" alt="{\displaystyle \mathbf {a} \cdot \operatorname {grad} f=a_{x}{\partial f \over \partial x}+a_{y}{\partial f \over \partial y}+a_{z}{\partial f \over \partial z}=\mathbf {a} \cdot (\nabla f)}" loading="lazy"></span></dd></dl>

<p>This gives the rate of change of a field <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation>

</semantics>

</math></span><img src="./1a957216653a9ee0d0133dcefd13fb75e36b8b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }" loading="lazy"></span>, scaled by the magnitude 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 \mathbf {a} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation>

</semantics>

</math></span><img src="./1a957216653a9ee0d0133dcefd13fb75e36b8b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }" loading="lazy"></span>. In operator notation, the element in parentheses can be considered a single coherent unit; <a href="Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a> uses this convention extensively, terming it the <a href="Convective_derivative" class="mw-redirect" title="Convective derivative">convective derivative</a>—the "moving" derivative of the fluid.

</p><p>Note 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 (\mathbf {a} \cdot \nabla )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} \cdot \nabla )}</annotation>

</semantics>

</math></span><img src="./f359b483581016360ef91d2b1b1e8a193e10baac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.724ex; height:2.843ex;" alt="{\displaystyle (\mathbf {a} \cdot \nabla )}" loading="lazy"></span> is an operator that maps scalars to scalars. It can be extended to act on a vector field by applying the operator component-wise to each component of the vector.

</p>

<div class="mw-heading mw-heading3"><h3 id="Laplacian">Laplacian</h3></div>

<p>The <a href="Laplace_operator" title="Laplace operator">Laplace operator</a> is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:

</p>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \Delta ={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}=\nabla \cdot \nabla =\nabla ^{2}}</annotation>

</semantics>

</math></span><img src="./f07578e67b23dba5c9583d405fd975e85348b696.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.659ex; height:6.343ex;" alt="{\displaystyle \Delta ={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}=\nabla \cdot \nabla =\nabla ^{2}}" loading="lazy"></span></dd></dl>

<p>and the definition for more general coordinate systems is given in <a href="Vector_Laplacian" class="mw-redirect" title="Vector Laplacian">vector Laplacian</a>.

</p><p>The Laplacian is ubiquitous throughout modern <a href="Mathematical_physics" title="Mathematical physics">mathematical physics</a>, appearing for example in <a href="Laplace's_equation" title="Laplace's equation">Laplace's equation</a>, <a href="Poisson's_equation" title="Poisson's equation">Poisson's equation</a>, the <a href="Heat_equation" title="Heat equation">heat equation</a>, the <a href="Wave_equation" title="Wave equation">wave equation</a>, and the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Hessian_matrix">Hessian matrix</h3></div>

<p>While <span class="mwe-math-element mwe-math-element-inline"><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 ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}}</annotation>

</semantics>

</math></span><img src="./f4be87ad083e5ead48d92b0c82f2d4e719cb34a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="{\displaystyle \nabla ^{2}}" loading="lazy"></span> usually represents the <a href="Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a>, sometimes <span class="mwe-math-element mwe-math-element-inline"><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 ^{2}}">

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}}</annotation>

</semantics>

</math></span><img src="./f4be87ad083e5ead48d92b0c82f2d4e719cb34a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="{\displaystyle \nabla ^{2}}" loading="lazy"></span> also represents the <a href="Hessian_matrix" title="Hessian matrix">Hessian matrix</a>. The former refers to the inner product 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 \nabla }">

</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>, while the latter refers to the <a href="Dyadic_product" class="mw-redirect" title="Dyadic product">dyadic product</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 \nabla }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation>

</semantics>

</p>

<msup>

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</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}=\nabla \cdot \nabla ^{T}}</annotation>

</semantics>

</math></span><img src="./66bb7115488b7c9662e1df1ade5dc9165f50fa71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.029ex; height:2.676ex;" alt="{\displaystyle \nabla ^{2}=\nabla \cdot \nabla ^{T}}" loading="lazy"></span>.</dd></dl>

<p>So whether <span class="mwe-math-element mwe-math-element-inline"><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 ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}}</annotation>

</semantics>

</p>

<div class="mw-heading mw-heading3"><h3 id="Tensor_derivative">Tensor derivative</h3></div>

<p>Del can also be applied to a vector field with the result being a <a href="Tensor" title="Tensor">tensor</a>. The <a href="Tensor_derivative" class="mw-redirect" title="Tensor derivative">tensor derivative</a> of a vector field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation>

</semantics>

</math></span><img src="./35c1866e359fbfd2e0f606c725ba5cc37a5195d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }" loading="lazy"></span> (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply 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 \nabla \otimes \mathbf {v} }">

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<annotation encoding="application/x-tex">{\displaystyle \nabla \otimes \mathbf {v} }</annotation>

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</math></span><img src="./f19c83421802bfd7c50332c34ff4bee23dbdf217.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.187ex; height:2.343ex;" alt="{\displaystyle \nabla \otimes \mathbf {v} }" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation>

</semantics>

</math></span><img src="./de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"></span> represents the <a href="Dyadic_product" class="mw-redirect" title="Dyadic product">dyadic product</a>. This quantity is equivalent to the transpose of the <a href="Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a> of the vector field with respect to space. The divergence of the vector field can then be expressed as the <a href="Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of this matrix.

</p><p>For a small displacement <span class="mwe-math-element mwe-math-element-inline"><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 \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta \mathbf {r} }</annotation>

</semantics>

</math></span><img src="./316af5a88d44e3de8269a3bfc89367414edeb9a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.151ex; height:2.343ex;" alt="{\displaystyle \delta \mathbf {r} }" loading="lazy"></span>, the change in the vector field is given by:

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<annotation encoding="application/x-tex">{\displaystyle \delta \mathbf {v} =(\nabla \otimes \mathbf {v} )^{T}\cdot \delta \mathbf {r} }</annotation>

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</math></span><img src="./f2fa911ddd72e96e203e4840d1ecd894e67d6f7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.774ex; height:3.176ex;" alt="{\displaystyle \delta \mathbf {v} =(\nabla \otimes \mathbf {v} )^{T}\cdot \delta \mathbf {r} }" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Product_rules">Product rules</h2></div>

<p>For <a href="Vector_calculus" title="Vector calculus">vector calculus</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nabla (fg)&=f\nabla g+g\nabla f\\\nabla (\mathbf {u} \cdot \mathbf {v} )&=\mathbf {u} \times (\nabla \times \mathbf {v} )+\mathbf {v} \times (\nabla \times \mathbf {u} )+(\mathbf {u} \cdot \nabla )\mathbf {v} +(\mathbf {v} \cdot \nabla )\mathbf {u} \\\nabla \cdot (f\mathbf {v} )&=f(\nabla \cdot \mathbf {v} )+\mathbf {v} \cdot (\nabla f)\\\nabla \cdot (\mathbf {u} \times \mathbf {v} )&=\mathbf {v} \cdot (\nabla \times \mathbf {u} )-\mathbf {u} \cdot (\nabla \times \mathbf {v} )\\\nabla \times (f\mathbf {v} )&=(\nabla f)\times \mathbf {v} +f(\nabla \times \mathbf {v} )\\\nabla \times (\mathbf {u} \times \mathbf {v} )&=\mathbf {u} \,(\nabla \cdot \mathbf {v} )-\mathbf {v} \,(\nabla \cdot \mathbf {u} )+(\mathbf {v} \cdot \nabla )\,\mathbf {u} -(\mathbf {u} \cdot \nabla )\,\mathbf {v} \end{aligned}}}</annotation>

</semantics>

</math></span><img src="./a6cc7515bdc1acdc88b00be5906ae2a6ab7f143d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:65.979ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\nabla (fg)&=f\nabla g+g\nabla f\\\nabla (\mathbf {u} \cdot \mathbf {v} )&=\mathbf {u} \times (\nabla \times \mathbf {v} )+\mathbf {v} \times (\nabla \times \mathbf {u} )+(\mathbf {u} \cdot \nabla )\mathbf {v} +(\mathbf {v} \cdot \nabla )\mathbf {u} \\\nabla \cdot (f\mathbf {v} )&=f(\nabla \cdot \mathbf {v} )+\mathbf {v} \cdot (\nabla f)\\\nabla \cdot (\mathbf {u} \times \mathbf {v} )&=\mathbf {v} \cdot (\nabla \times \mathbf {u} )-\mathbf {u} \cdot (\nabla \times \mathbf {v} )\\\nabla \times (f\mathbf {v} )&=(\nabla f)\times \mathbf {v} +f(\nabla \times \mathbf {v} )\\\nabla \times (\mathbf {u} \times \mathbf {v} )&=\mathbf {u} \,(\nabla \cdot \mathbf {v} )-\mathbf {v} \,(\nabla \cdot \mathbf {u} )+(\mathbf {v} \cdot \nabla )\,\mathbf {u} -(\mathbf {u} \cdot \nabla )\,\mathbf {v} \end{aligned}}}" loading="lazy"></span></dd></dl>

<p>For <a href="Matrix_calculus" title="Matrix calculus">matrix calculus</a> (for which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \cdot \mathbf {v} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \cdot \mathbf {v} }</annotation>

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</math></span><img src="./b3161cc5faca46f8bd209de2beb24159a50bc66e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.575ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} \cdot \mathbf {v} }" loading="lazy"></span> can be written <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} ^{\text{T}}\mathbf {v} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ^{\text{T}}\mathbf {v} }</annotation>

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</math></span><img src="./5a7fe855c58f7ac464e607d7369f971019f80c8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.315ex; height:2.676ex;" alt="{\displaystyle \mathbf {u} ^{\text{T}}\mathbf {v} }" loading="lazy"></span>):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(\mathbf {A} \nabla \right)^{\text{T}}\mathbf {u} &=\nabla ^{\text{T}}\left(\mathbf {A} ^{\text{T}}\mathbf {u} \right)-\left(\nabla ^{\text{T}}\mathbf {A} ^{\text{T}}\right)\mathbf {u} \end{aligned}}}</annotation>

</semantics>

</math></span><img src="./45a43f3cdda201d3a119e98e6253f063d517f6e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:36.95ex; height:3.509ex;" alt="{\displaystyle {\begin{aligned}\left(\mathbf {A} \nabla \right)^{\text{T}}\mathbf {u} &=\nabla ^{\text{T}}\left(\mathbf {A} ^{\text{T}}\mathbf {u} \right)-\left(\nabla ^{\text{T}}\mathbf {A} ^{\text{T}}\right)\mathbf {u} \end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Another relation of interest (see e.g. <i><a href="Euler_equations" class="mw-redirect" title="Euler equations">Euler equations</a></i>) is the following, 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 \mathbf {u} \otimes \mathbf {v} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \otimes \mathbf {v} }</annotation>

</semantics>

</math></span><img src="./f5d579ce37084592ce1baaaddbd0f50592934783.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.737ex; height:2.176ex;" alt="{\displaystyle \mathbf {u} \otimes \mathbf {v} }" loading="lazy"></span> is the <a href="Outer_product" title="Outer product">outer product</a> tensor:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {u} \otimes \mathbf {v} )=(\nabla \cdot \mathbf {u} )\mathbf {v} +(\mathbf {u} \cdot \nabla )\mathbf {v} \end{aligned}}}</annotation>

</semantics>

</math></span><img src="./399413288555112b0775b3c30c708897db6839fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.492ex; height:2.843ex;" alt="{\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {u} \otimes \mathbf {v} )=(\nabla \cdot \mathbf {u} )\mathbf {v} +(\mathbf {u} \cdot \nabla )\mathbf {v} \end{aligned}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Second_derivatives">Second derivatives</h2></div>

<p>When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field <i>f</i> or a vector field <i><b>v</b></i>; the use of the scalar <a href="Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a> and <a href="Vector_Laplacian" class="mw-redirect" title="Vector Laplacian">vector Laplacian</a> gives two more:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {div} (\operatorname {grad} f)&=\nabla \cdot (\nabla f)=\nabla ^{2}f\\\operatorname {curl} (\operatorname {grad} f)&=\nabla \times (\nabla f)\\\operatorname {grad} (\operatorname {div} \mathbf {v} )&=\nabla (\nabla \cdot \mathbf {v} )\\\operatorname {div} (\operatorname {curl} \mathbf {v} )&=\nabla \cdot (\nabla \times \mathbf {v} )\\\operatorname {curl} (\operatorname {curl} \mathbf {v} )&=\nabla \times (\nabla \times \mathbf {v} )\\\Delta f&=\nabla ^{2}f\\\Delta \mathbf {v} &=\nabla ^{2}\mathbf {v} \end{aligned}}}</annotation>

</semantics>

</math></span><img src="./94af07d422714768a170cdc8c379c959ad6c07a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:31.743ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {div} (\operatorname {grad} f)&=\nabla \cdot (\nabla f)=\nabla ^{2}f\\\operatorname {curl} (\operatorname {grad} f)&=\nabla \times (\nabla f)\\\operatorname {grad} (\operatorname {div} \mathbf {v} )&=\nabla (\nabla \cdot \mathbf {v} )\\\operatorname {div} (\operatorname {curl} \mathbf {v} )&=\nabla \cdot (\nabla \times \mathbf {v} )\\\operatorname {curl} (\operatorname {curl} \mathbf {v} )&=\nabla \times (\nabla \times \mathbf {v} )\\\Delta f&=\nabla ^{2}f\\\Delta \mathbf {v} &=\nabla ^{2}\mathbf {v} \end{aligned}}}" loading="lazy"></span></dd></dl>

<p>These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation>

</semantics>

</math></span><img src="./971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy"></span> in most cases), two of them are always zero:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {curl} (\operatorname {grad} f)&=\nabla \times (\nabla f)=0\\\operatorname {div} (\operatorname {curl} \mathbf {v} )&=\nabla \cdot (\nabla \times \mathbf {v} )=0\end{aligned}}}</annotation>

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</math></span><img src="./bdcdec8546fe66be7f6247ff963ce6d7996be6a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.61ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {curl} (\operatorname {grad} f)&=\nabla \times (\nabla f)=0\\\operatorname {div} (\operatorname {curl} \mathbf {v} )&=\nabla \cdot (\nabla \times \mathbf {v} )=0\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Two of them are always equal:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {div} (\operatorname {grad} f)=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f}</annotation>

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</math></span><img src="./8f3a762325091088033a0380cd92adb712ec6cb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.588ex; height:3.176ex;" alt="{\displaystyle \operatorname {div} (\operatorname {grad} f)=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f}" loading="lazy"></span></dd></dl>

<p>The 3 remaining vector derivatives are related by the equation:

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times \left(\nabla \times \mathbf {v} \right)=\nabla (\nabla \cdot \mathbf {v} )-\nabla ^{2}\mathbf {v} }</annotation>

</semantics>

</math></span><img src="./c4f949de15a0864a5765bfec3362d9b4dd00a072.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.884ex; height:3.176ex;" alt="{\displaystyle \nabla \times \left(\nabla \times \mathbf {v} \right)=\nabla (\nabla \cdot \mathbf {v} )-\nabla ^{2}\mathbf {v} }" loading="lazy"></span></dd></dl>

<p>And one of them can even be expressed with the tensor product, if the functions are well-behaved:

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<annotation encoding="application/x-tex">{\displaystyle \nabla (\nabla \cdot \mathbf {v} )=\nabla \cdot (\mathbf {v} \otimes \nabla )}</annotation>

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</math></span><img src="./ebf5e8a47989462bb99bfbf8afea47f8d433a078.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.481ex; height:2.843ex;" alt="{\displaystyle \nabla (\nabla \cdot \mathbf {v} )=\nabla \cdot (\mathbf {v} \otimes \nabla )}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Precautions">Precautions</h2></div>

<p>Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold if the del symbol is replaced by any other vector. This is part of the value to be gained in notationally representing this operator as a vector.

</p><p>Though one can often replace del with a vector and obtain a vector identity, making those identities mnemonic, the reverse is <i>not</i> necessarily reliable, because del does not commute in general.

</p><p>A counterexample that demonstrates the divergence (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {v} }">

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<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {v} }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \cdot \nabla }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\mathbf {u} \cdot \mathbf {v} )f&\equiv (\mathbf {v} \cdot \mathbf {u} )f\\(\nabla \cdot \mathbf {v} )f&=\left({\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}\right)f={\frac {\partial v_{x}}{\partial x}}f+{\frac {\partial v_{y}}{\partial y}}f+{\frac {\partial v_{z}}{\partial z}}f\\(\mathbf {v} \cdot \nabla )f&=\left(v_{x}{\frac {\partial }{\partial x}}+v_{y}{\frac {\partial }{\partial y}}+v_{z}{\frac {\partial }{\partial z}}\right)f=v_{x}{\frac {\partial f}{\partial x}}+v_{y}{\frac {\partial f}{\partial y}}+v_{z}{\frac {\partial f}{\partial z}}\\\Rightarrow (\nabla \cdot \mathbf {v} )f&\neq (\mathbf {v} \cdot \nabla )f\\\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./75513abf8ce9fbf6a803ab3106ec423b7def562c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:68.078ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}(\mathbf {u} \cdot \mathbf {v} )f&\equiv (\mathbf {v} \cdot \mathbf {u} )f\\(\nabla \cdot \mathbf {v} )f&=\left({\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}\right)f={\frac {\partial v_{x}}{\partial x}}f+{\frac {\partial v_{y}}{\partial y}}f+{\frac {\partial v_{z}}{\partial z}}f\\(\mathbf {v} \cdot \nabla )f&=\left(v_{x}{\frac {\partial }{\partial x}}+v_{y}{\frac {\partial }{\partial y}}+v_{z}{\frac {\partial }{\partial z}}\right)f=v_{x}{\frac {\partial f}{\partial x}}+v_{y}{\frac {\partial f}{\partial y}}+v_{z}{\frac {\partial f}{\partial z}}\\\Rightarrow (\nabla \cdot \mathbf {v} )f&\neq (\mathbf {v} \cdot \nabla )f\\\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>A counterexample that relies on del's differential properties:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\nabla x)\times (\nabla y)&=\left(\mathbf {e} _{x}{\frac {\partial x}{\partial x}}+\mathbf {e} _{y}{\frac {\partial x}{\partial y}}+\mathbf {e} _{z}{\frac {\partial x}{\partial z}}\right)\times \left(\mathbf {e} _{x}{\frac {\partial y}{\partial x}}+\mathbf {e} _{y}{\frac {\partial y}{\partial y}}+\mathbf {e} _{z}{\frac {\partial y}{\partial z}}\right)\\&=(\mathbf {e} _{x}\cdot 1+\mathbf {e} _{y}\cdot 0+\mathbf {e} _{z}\cdot 0)\times (\mathbf {e} _{x}\cdot 0+\mathbf {e} _{y}\cdot 1+\mathbf {e} _{z}\cdot 0)\\&=\mathbf {e} _{x}\times \mathbf {e} _{y}\\&=\mathbf {e} _{z}\\(\mathbf {u} x)\times (\mathbf {u} y)&=xy(\mathbf {u} \times \mathbf {u} )\\&=xy\mathbf {0} \\&=\mathbf {0} \end{aligned}}}</annotation>

</semantics>

</math></span><img src="./53468dc139c3fdd4a23c4d6ab35d001d8b60263a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.838ex; width:72.063ex; height:24.843ex;" alt="{\displaystyle {\begin{aligned}(\nabla x)\times (\nabla y)&=\left(\mathbf {e} _{x}{\frac {\partial x}{\partial x}}+\mathbf {e} _{y}{\frac {\partial x}{\partial y}}+\mathbf {e} _{z}{\frac {\partial x}{\partial z}}\right)\times \left(\mathbf {e} _{x}{\frac {\partial y}{\partial x}}+\mathbf {e} _{y}{\frac {\partial y}{\partial y}}+\mathbf {e} _{z}{\frac {\partial y}{\partial z}}\right)\\&=(\mathbf {e} _{x}\cdot 1+\mathbf {e} _{y}\cdot 0+\mathbf {e} _{z}\cdot 0)\times (\mathbf {e} _{x}\cdot 0+\mathbf {e} _{y}\cdot 1+\mathbf {e} _{z}\cdot 0)\\&=\mathbf {e} _{x}\times \mathbf {e} _{y}\\&=\mathbf {e} _{z}\\(\mathbf {u} x)\times (\mathbf {u} y)&=xy(\mathbf {u} \times \mathbf {u} )\\&=xy\mathbf {0} \\&=\mathbf {0} \end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Central to these distinctions is the fact that del is not simply a vector; it is a <a href="Vector_operator" title="Vector operator">vector operator</a>. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function.

</p><p>For that reason, identities involving del must be derived with care, using both vector identities and <i>differentiation</i> identities such as the product rule.

</p>

<ul><li><a href="Del_in_cylindrical_and_spherical_coordinates" title="Del in cylindrical and spherical coordinates">Del in cylindrical and spherical coordinates</a></li>

<li><a href="Notation_for_differentiation" title="Notation for differentiation">Notation for differentiation</a></li>

<li><a href="Vector_calculus_identities" title="Vector calculus identities">Vector calculus identities</a></li>

<li><a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a></li>

<li><a href="Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li>

<li><a href="Table_of_mathematical_symbols" class="mw-redirect" title="Table of mathematical symbols">Table of mathematical symbols</a></li>

<li><a href="Quabla_operator" class="mw-redirect" title="Quabla operator">Quabla operator</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><a href="Willard_Gibbs" class="mw-redirect" title="Willard Gibbs">Willard Gibbs</a> & <a href="Edwin_Bidwell_Wilson" title="Edwin Bidwell Wilson">Edwin Bidwell Wilson</a> (1901) <a href="Vector_Analysis" title="Vector Analysis">Vector Analysis</a>, <a href="Yale_University_Press" title="Yale University Press">Yale University Press</a>, 1960: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>.</li>

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</style><cite id="CITEREFSchey1997" class="citation book cs1">Schey, H. M. (1997). <i>Div, Grad, Curl, and All That: An Informal Text on Vector Calculus</i>. New York: Norton. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-393-96997-5</bdi>.</cite></li>

<li><cite id="CITEREFMiller" class="citation web cs1">Miller, Jeff. <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/calculus.html">"Earliest Uses of Symbols of Calculus"</a>.</cite></li>

<li><cite id="CITEREFArnold_Neumaier1998" class="citation web cs1">Arnold Neumaier (January 26, 1998). Cleve Moler (ed.). <a rel="nofollow" class="external text" href="http://www.netlib.org/na-digest-html/98/v98n03.html#2">"History of Nabla"</a>. NA Digest, Volume 98, Issue 03. netlib.org.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFTai1994" class="citation report cs1">Tai, Chen-To (1994). A survey of the improper use of ∇ in vector analysis (Report). Radiation Laboratory, University of Michigan. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027.42%2F7869">2027.42/7869</a></span>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-01" href="https://en.wikipedia.org/wiki/?title=Del&oldid=1303696798">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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