View source for Vector calculus identities

{{See also|Vector algebra relations}}
{{More footnotes|date=August 2017}}
{{Calculus |Vector}}
The following are important [[Identity (mathematics)|identities]] involving derivatives and integrals in [[vector calculus]].

==Operator notations==

===Gradient===
{{main|Gradient}}

For a function <math>f(x,y,z)</math> in three-dimensional [[Cartesian coordinate system|Cartesian coordinate]] variables, the gradient is the vector field:

:<math>\operatorname{grad}(f) \ =\ \nabla f \ =\  \tfrac{\partial f}{\partial x} \mathbf{i} + \tfrac{\partial f}{\partial y}  \mathbf{j} + \tfrac{\partial f}{\partial z} \mathbf{k}
\ =\ 
(\tfrac{\partial f}{\partial x}, \tfrac{\partial f}{\partial y}, \tfrac{\partial f}{\partial z})
</math>

where '''i''', '''j''', '''k''' are the [[standard basis|standard]] [[unit vector]]s for the ''x'', ''y'', ''z''-axes. More generally, for a function of ''n'' variables <math>\psi(x_1,\ldots,x_n)</math>, also called a [[scalar (mathematics)|scalar]] field, the gradient is the [[vector field]]: <blockquote><math>\nabla\psi = (\tfrac{\partial \psi}{\partial x_1},\ldots,\tfrac{\partial \psi}{\partial x_n}).</math></blockquote>For a vector field <math>\mathbf{A}= (A_1,\ldots,A_n)</math> written as a 1 × ''n'' row vector, also called a tensor field of order 1, the gradient or [[covariant derivative]] is the ''n × n'' [[Jacobian matrix and determinant|Jacobian matrix]]:<blockquote><math>\nabla \!\mathbf{A}=\mathbf{J}_{\mathbf{A}} = \left(\frac{\partial A_i}{\partial x_j}\right)_{\!ij}.</math></blockquote>For a [[tensor field]] <math>\mathbf{A}</math> of any order ''k,'' the gradient <math>\operatorname{grad}(\mathbf{A}) = \nabla\!\mathbf{A} </math> is a tensor field of order ''k+''1.

===Divergence===
{{main|Divergence}}

In Cartesian coordinates, the divergence of a [[continuously differentiable]] [[vector field]] <math>\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}</math> is the scalar-valued function:

:<math>\operatorname{div} \mathbf{F} \ =\  \nabla\cdot\mathbf{F} \ =\ ( \tfrac{\partial}{\partial x}, \tfrac{\partial}{\partial y}, \tfrac{\partial}{\partial z} ) \cdot (F_x,F_y,F_z) \ =\  \tfrac{\partial F_x}{\partial x}+\tfrac{\partial F_y}{\partial y}+\tfrac{\partial F_z}{\partial z}.</math>

The divergence of a [[tensor field]] <math>\mathbf{A}</math>of non-zero order ''k'' is written as <math>\operatorname{div}(\mathbf{A}) = \nabla \cdot \mathbf{A}</math>, a [[Tensor contraction|contraction]] to a tensor field of order ''k''–1. Specifically, the divergence of a vector is a scalar. The divergence of a higher order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity,

:<math>\nabla \cdot (\mathbf{B} \otimes \hat{\mathbf{A}}) = \hat{\mathbf{A}}(\nabla \cdot \mathbf{B})+(\mathbf{B}\cdot \nabla) \hat{\mathbf{A}}</math>

where <math> \mathbf{B}\cdot\nabla </math> is the [[directional derivative]] in the direction of <math> \mathbf{B} </math> multiplied by its magnitude. Specifically, for the outer product of two vectors,

:<math>\nabla \cdot \left(\mathbf{b} \mathbf{a}^\mathrm{T}\right) = \mathbf{a}(\nabla \cdot \mathbf{b})+(\mathbf{b}\cdot \nabla) \mathbf{a}.</math>

===Curl===
{{main|Curl (mathematics)}}

In Cartesian coordinates, for <math>\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}</math> the curl is the vector field:

:<math>\operatorname{curl}\mathbf F\ =\ \nabla \times \mathbf{F} \ =\  \left|\begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{matrix}\right|
\ =\ 
\left(\tfrac{\partial F_z}{\partial y}{-} \tfrac{\partial F_y}{\partial z}\right)\! \mathbf{i} + \left(\tfrac{\partial F_x}{\partial z} {-} \tfrac{\partial F_z}{\partial x}\right)\! \mathbf{j} + \left(\tfrac{\partial F_y}{\partial x}{-}\tfrac{\partial F_x}{\partial y}\right)\!\mathbf{k}</math>

where '''i''', '''j''', and '''k''' are the [[unit vector]]s for the ''x''-, ''y''-, and ''z''-axes, respectively. In [[Einstein notation]], the vector field <math>\mathbf{F}=(F_1,F_2,F_3)</math>has curl given by:

:<math> \nabla\times\mathbf{F} \ =\ \varepsilon^{ijk} \frac {\partial F_k}{\partial x^j}</math>

where <math>\varepsilon=\plusmn 1</math> or '''0''' is the [[Levi-Civita symbol|Levi-Civita parity symbol]].

===Laplacian===
{{main|Laplace operator}}

In '''[[Cartesian coordinates]]''', the Laplacian of a function <math>f(x,y,z)</math> is

:<math>\Delta f = \nabla^2\! f = (\nabla \cdot \nabla) f = \frac{\partial^2\! f}{\partial x^2} + \frac{\partial^2\! f}{\partial y^2} + \frac{\partial^2\! f}{\partial z^2}.</math>

For a [[tensor field]], <math> \mathbf{A} </math>, the Laplacian is generally written as:

:<math>\Delta\mathbf{A} = \nabla^2\! \mathbf{A} = (\nabla \cdot \nabla) \mathbf{A}</math>

and is a tensor field of the same order.

===Special notations===
In ''Feynman subscript notation'',

:<math> \nabla_\mathbf{B}\! \left( \mathbf{A {\cdot} B} \right)\ =\ \mathbf{A} {\times}\! \left( \nabla {\times} \mathbf{B} \right) \,+\, \left( \mathbf{A} {\cdot} \nabla \right) \mathbf{B} </math>

where the notation ∇<sub>'''B'''</sub>  means the subscripted gradient operates on only the factor '''B'''.<ref name=Feynman>{{cite book |first1=R. P. |last1=Feynman |first2=R. B. |last2=Leighton |first3=M. |last3=Sands |title=The Feynman Lectures on Physics |publisher = Addison-Wesley |year=1964 |isbn=0-8053-9049-9 |nopp= true |pages= Vol II, p. 27–4}}</ref><ref name=Missevitch>{{cite arXiv |eprint=physics/0504223 |first1=A. L. |last1=Kholmetskii |first2=O. V. |last2=Missevitch |title=The Faraday induction law in relativity theory |year=2005 |url=https://arxiv.org/ftp/physics/papers/0504/0504223.pdf |page=4 }}</ref>

Less general but similar is the ''Hestenes'' ''overdot notation'' in [[geometric algebra]].<ref name=Doran>{{cite book |first1=C. |last1=Doran |authorlink1=Chris J. L. Doran |first2=A. |last2=Lasenby |title=Geometric algebra for physicists |year=2003 |publisher=Cambridge University Press |page=169 |isbn=978-0-521-71595-9}}</ref> The above identity is then expressed as:

:<math> \dot{\nabla} \left( \mathbf{A} {\cdot} \dot{\mathbf{B}} \right)\ =\ \mathbf{A} {\times}\! \left( \nabla {\times} \mathbf{B} \right)\,+\, \left( \mathbf{A} {\cdot} \nabla \right) \mathbf{B} </math>

where overdots define the scope of the vector derivative. The dotted vector, in this case '''B''', is differentiated, while the (undotted) '''A''' is held constant.

For the remainder of this article, Feynman subscript notation will be used where appropriate.

==First derivative identities==
For scalar fields <math>\psi</math>, <math>\phi</math> and vector fields <math>\mathbf{A}</math>, <math>\mathbf{B}</math>, we have the following derivative identities.

===Distributive properties===

:<math> \nabla ( \psi + \phi ) \ =\ \nabla \psi + \nabla \phi </math>
:<math> \nabla( \mathbf{A} + \mathbf{B} ) \ =\ \nabla \!\mathbf{A} + \nabla \mathbf{B} </math>
:<math> \nabla \,{\cdot}\, ( \mathbf{A} + \mathbf{B} ) \ =\ \nabla {\cdot} \mathbf{A} + \nabla {\cdot} \mathbf{B} </math>
:<math> \nabla {\times} ( \mathbf{A} + \mathbf{B} )\ =\ \nabla {\times} \mathbf{A} + \nabla {\times} \mathbf{B} </math>

===Product rule ===
We have the following generalizations of the [[product rule]] in single variable [[calculus]].

:<math> \nabla (\psi \phi)\ =\ \phi\, \nabla \psi  \,+\, \psi\, \nabla \phi </math>
:<math> \nabla ( \psi \mathbf{A} )
\ =\ (\nabla \psi)^{\mathbf{T}} \mathbf{A} \,+\, \psi\, \nabla\! \mathbf{A}
\ =\ \nabla \psi \otimes \mathbf{A} \,+\, \psi\, \nabla\!\mathbf{A}
 </math>   
:<math> \nabla \,{\cdot}\, ( \psi \mathbf{A} ) \ =\ \psi\, \nabla {\cdot} \mathbf{A} \,+\,  (\nabla \psi){\cdot}\mathbf{A} </math>
:<math> \nabla {\times} ( \psi \mathbf{A} ) \ =\ \psi\, \nabla {\times} \mathbf{A} \,+\, ( \nabla \psi ) {\times} \mathbf{A} </math>

In the second formula, the transposed gradient <math>(\nabla \psi)^{\mathbf{T}}</math> is an ''n'' × 1 column vector, <math>\mathbf{A}</math> is a 1 × ''n'' row vector, and their product is an  ''n × n''  matrix: this may also be considered as the [[tensor product]] of two vectors, or of a covector and a vector''.'' 

===Quotient rule===

:<math> \nabla\!\left(\frac{\psi}{\phi}\right)\ =\  \frac{\phi\,\nabla \psi - (\nabla\!\phi)\psi}{\phi^2}</math>
:<math> \nabla \,{\cdot}\! \left(\frac{\mathbf{A}}{\phi}\right) \ =\ \frac{\phi\, \nabla{\cdot}\mathbf{A} - (\nabla\! \phi){\cdot}\mathbf{A}}{\phi^2}</math>
:<math> \nabla \,{\times}\! \left(\frac{\mathbf{A}}{\phi}\right) \ =\ \frac{\phi\, \nabla{\times}\mathbf{A} - (\nabla\! \phi){\times} \mathbf{A}}{\phi^2}</math>

===Chain rule===
Let <math>f(x)
</math> be a one-variable function from scalars to scalars,  <math>\mathbf{r}(t) = (r_1(t),\ldots,r_n(t))
</math> a [[Parametrization (geometry)|parametrized]] curve, and <math>F:\mathbb{R}^n\to\mathbb{R}
</math> a function from vectors to scalars. We have the following special cases of the multi-variable [[chain rule]].

:<math>\nabla(f \circ F) \,=\, (f'\! \circ F)\, \nabla\! F</math>
:<math>(F\circ \mathbf{r})'  = (\nabla\!F\circ \mathbf{r})\cdot \mathbf{r}'
</math>
:<math>\nabla(F \circ \mathbf{A}) \ =\ (\nabla\! F \circ \mathbf{A})\, \nabla\! \mathbf{A}</math>

For a [[Coordinate system|coordinate parametrization]] <math>\Phi:\mathbb{R}^n \to \mathbb{R}^n
</math> we have:

:<math>\nabla \cdot (\mathbf{A} \circ \Phi) = \mathrm{tr}\!\left(\,(\nabla\!\mathbf{A} \circ \Phi) \, \mathbf{J}_\Phi\,\right)</math>

Here we take the [[Trace (linear algebra)|trace]] of the product of two ''n × n'' matrices: the gradient of '''A''' and the Jacobian of Φ.

===Dot product rule===

:<math>\begin{align}
\nabla(\mathbf{A} \cdot \mathbf{B}) & \ =\ (\mathbf{A} {\cdot} \nabla)\mathbf{B} \,+\, (\mathbf{B}\, {\cdot} \nabla)\mathbf{A} \,+\, \mathbf{A} {\times} (\nabla {\times} \mathbf{B}) \,+\, \mathbf{B} {\times} (\nabla {\times} \mathbf{A}) \\
&\ =\   \mathbf{A}\,\mathbf{J}_\mathbf{B} + \mathbf{B}\,\mathbf{J}_\mathbf{A}   
\ =\ \mathbf{A}\,\nabla\mathbf{B}  + \mathbf{B}\,\nabla\!\mathbf{A}
\end{align}</math>

where <math>\mathbf{J}_{\mathbf{A}} = \nabla \!\mathbf{A}=(\partial A_i/\partial x_j)_{ij}</math> denotes the [[Jacobian matrix and determinant|Jacobian matrix]] of the vector field <math>\mathbf{A} = (A_1,\ldots,A_n)</math>.

Alternatively, using Feynman subscript notation,

:<math> \nabla(\mathbf{A} \cdot \mathbf{B}) = \nabla_\mathbf{A}(\mathbf{A}  \cdot \mathbf{B}) +  \nabla_\mathbf{B} (\mathbf{A} \cdot \mathbf{B}) \ . </math>

See these notes.<ref>{{cite book |last1=Kelly |first1=P. |year=2013 |title=Mechanics Lecture Notes Part III: Foundations of Continuum Mechanics |url=http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/ |publisher=University of Auckland |chapter=Chapter 1.14 Tensor Calculus 1: Tensor Fields |chapter-url=http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf |accessdate=7 December 2017}}</ref> 

As a special case, when {{math|'''A''' {{=}} '''B'''}},

:<math>\tfrac{1}{2} \nabla \left( \mathbf{A}\cdot\mathbf{A} \right)
\ =\ (\mathbf{A} \cdot \nabla) \mathbf{A} + \mathbf{A} \times (\nabla \times \mathbf{A}) 
\ =\  \mathbf A \,\mathbf{J}_\mathbf{A}
\ =\ \mathbf{A} \,\nabla\! \mathbf{A}  . </math>

The generalization of the dot product formula to Riemannian manifolds is a defining property of a [[Riemannian connection]], which differentiates a vector field to give a vector-valued [[Differential form|1-form]].
===Cross product rule===

:<math> \nabla \cdot (\mathbf{A} \times \mathbf{B}) = (\nabla \times \mathbf{A}) \cdot \mathbf{B} - \mathbf{A} \cdot (\nabla \times \mathbf{B}) </math>

:<math> \begin{align}
\nabla \times (\mathbf{A} \times \mathbf{B}) &= \mathbf{A}(\nabla \cdot \mathbf{B}) - \mathbf{B}(\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} \\
&= (\nabla \cdot \mathbf{B}  + \mathbf{B} \cdot \nabla)\mathbf{A} -(\nabla \cdot \mathbf{A} + \mathbf{A} \cdot \nabla )\mathbf{B} \\
&= \nabla \cdot (\mathbf{B} \mathbf{A}^\mathrm{T}) - \nabla \cdot (\mathbf{A} \mathbf{B}^\mathrm{T}) \\
&= \nabla \cdot (\mathbf{B} \mathbf{A}^\mathrm{T} - \mathbf{A} \mathbf{B}^\mathrm{T}) 
\end{align} </math>

==Second derivative identities==

===Curl of gradient is zero===

The [[Curl (mathematics)|curl]] of the [[gradient]] of ''any'' continuously twice-differentiable [[scalar field]] <math>\ \phi </math> is always the [[zero vector]]:

:<math>\nabla \times ( \nabla \phi )  = \mathbf{0}</math>

===Divergence of curl is zero===
The [[divergence]] of the curl of ''any'' [[vector field]] '''A''' is always zero:
:<math>\nabla \cdot ( \nabla \times \mathbf{A} ) = 0 </math>

The above two vanishing properties are a special case of the vanishing of the square of the [[exterior derivative]] in the [[De Rham cohomology|De Rham]] [[chain complex]].

===Divergence of gradient===
The [[Laplacian]] of a scalar field is the divergence of its gradient:
:<math> \nabla^2 \psi = \nabla \cdot (\nabla \psi) </math>
The result is a scalar quantity.

===Curl of curl===
:<math> \nabla \times \left( \nabla \times \mathbf{A} \right) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A}</math>

Here ∇<sup>2</sup> is the [[vector Laplacian]] operating on the vector field '''A'''.

==Summary of important identities==


===Differentiation===

====Gradient====

*<math>\nabla(\psi+\phi)=\nabla\psi+\nabla\phi </math>
*<math>\nabla(\psi \phi) = \phi\nabla \psi  + \psi \nabla \phi </math>
*<math>\nabla(\psi \mathbf{A} ) = \nabla \psi \otimes \mathbf{A} + \psi \nabla \mathbf{A}</math>
*<math>\nabla(\mathbf{A}\cdot\mathbf{B})= (\mathbf{A}\cdot\nabla)\mathbf{B} + (\mathbf{B}\cdot\nabla)\mathbf{A}+ \mathbf{A} \times (\nabla \times \mathbf{B} )+ \mathbf{B}\times\left(\nabla\times\mathbf{A}\right)</math>

====Divergence====

* <math> \nabla\cdot(\mathbf{A}+\mathbf{B})= \nabla\cdot\mathbf{A}+\nabla\cdot\mathbf{B} </math>

*<math> \nabla\cdot\left(\psi\mathbf{A}\right)= \psi\nabla\cdot\mathbf{A}+\mathbf{A}\cdot\nabla \psi</math>
*<math> \nabla\cdot\left(\mathbf{A}\times\mathbf{B}\right)= (\nabla\times\mathbf{A})\cdot \mathbf{B}-(\nabla\times\mathbf{B})\cdot \mathbf{A}</math>

====Curl====

*<math>\nabla\times(\mathbf{A}+\mathbf{B})=\nabla\times\mathbf{A}+\nabla\times\mathbf{B} </math>
*<math>\nabla\times\left(\psi\mathbf{A}\right)=\psi\,(\nabla\times\mathbf{A})+\nabla\psi\times\mathbf{A}</math>
*<math>\nabla\times\left(\psi\nabla\phi\right)= \nabla \psi \times \nabla \phi</math>
*<math>\nabla\times\left(\mathbf{A}\times\mathbf{B}\right)= \mathbf{A}\left(\nabla\cdot\mathbf{B}\right)-\mathbf{B} \left( \nabla\cdot\mathbf{A}\right)+\left(\mathbf{B}\cdot\nabla\right)\mathbf{A}- \left(\mathbf{A}\cdot\nabla\right)\mathbf{B} </math>

====Second derivatives====
[[File:DCG chart.svg|right|thumb|300px|DCG chart:

A simple chart depicting all rules pertaining to second derivatives.
D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively.

Arrows indicate existence of second derivatives. Blue circle in the middle represents curl of curl, whereas the other two red circles(dashed) mean that DD and GG do not exist.
]]

*<math>\nabla\cdot(\nabla\times\mathbf{A})=0 </math>
*<math>\nabla\times(\nabla\psi)= \mathbf{0} </math>
*<math>\nabla\cdot(\nabla\psi)=\nabla^{2}\psi </math>        ([[Laplace operator|scalar Laplacian]])
*<math>\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla^{2}\mathbf{A}</math>   ([[vector Laplacian]])
*<math>\nabla\cdot(\phi\nabla\psi)=\phi\nabla^{2}\psi + \nabla\phi\cdot\nabla\psi </math> 
*<math>\psi\nabla^2\phi-\phi\nabla^2\psi= \nabla\cdot\left(\psi\nabla\phi-\phi\nabla\psi\right)</math>
*<math>\nabla^2(\phi\psi)=\phi\nabla^2\psi + 2(\nabla\phi)\cdot(\nabla\psi) + (\nabla^2\phi)\psi</math>
*<math>\nabla^2(\psi\mathbf{A})=\mathbf{A}\nabla^2\psi+2(\nabla\psi\cdot\nabla)\mathbf{A}+\psi\nabla^2\mathbf{A}</math>
*<math>\nabla^2(\mathbf{A}\cdot\mathbf{B})\ =\ \mathbf{A}{\cdot}\nabla^2\mathbf{B} \,-\, \mathbf{B}{\cdot}\nabla^2\!\mathbf{A}\, +\, 2\nabla {\cdot}\, ((\mathbf{B}\cdot\nabla)\mathbf{A} \,+\, \mathbf{B}{\times}(\nabla{\times}\mathbf{A}))</math> ([[Green's identities|Green's vector identity]])

====Third derivatives====

*<math>\nabla^{2}(\nabla\psi) = \nabla(\nabla\cdot(\nabla\psi)) = \nabla(\nabla^{2}\psi)</math>
*<math>\nabla^{2}(\nabla\cdot\mathbf{A}) = \nabla\cdot(\nabla(\nabla\cdot\mathbf{A})) =\nabla\cdot(\nabla^{2}\mathbf{A})</math>
*<math>\nabla^{2}(\nabla\times\mathbf{A}) = -\nabla\times(\nabla\times(\nabla\times\mathbf{A})) = \nabla\times(\nabla^{2}\mathbf{A})</math>

===Integration===
Below, the curly symbol ∂ means "[[boundary (topology)|boundary of]]".

====Surface–volume integrals====
In the following surface–volume integral theorems, ''V'' denotes a three-dimensional volume with a corresponding two-dimensional [[Boundary (topology)|boundary]] ''S'' = ∂''V'' (a [[closed surface]]):

*{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\mathbf{A}\cdot d\mathbf{S}\ =\ \iiint_V \left(\nabla \cdot \mathbf{A}\right)dV </math>}} ([[Divergence theorem]])
*{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\psi\, d \mathbf{S} \ =\ \iiint_V \nabla \psi\, dV</math>}}
*{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\mathbf{A}\times d\mathbf{S} \ =\ -\iiint _{V}\nabla\times\mathbf{A} \,dV</math>}}
*{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math> \psi  \nabla\!\varphi \cdot d\mathbf{S} \ =\ \iiint _{V} (\psi\, \nabla^{2}\!\varphi\,+\,\nabla\!\varphi\,{\cdot}\,\nabla\!\psi)\,dV </math>}} ([[Green's first identity]])
*{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\left(\psi\,\nabla\!\varphi-\varphi\,\nabla\!\psi\right)\cdot d\mathbf{S}
\ =\ 
</math>{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\left(\psi\frac{\partial\varphi}{\partial n}-\varphi\frac{\partial\psi}{\partial n}\right)dS</math>}} <math>\displaystyle
\ =\ 
\iiint_{V}\left(\psi\,\nabla^{2}\!\varphi-\varphi\,\nabla^{2}\!\psi\right)\,dV</math>}} ([[Green's second identity]])
*{{Oiint|preintegral=<math>\iiint_V  \mathbf{A} \cdot \nabla \psi \, dV \ =\ </math>|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math> \psi \mathbf{A} \cdot d\mathbf{S} - \iiint_V \psi\, \nabla {\cdot} \mathbf{A}\,  dV</math>}}   ([[Integration by parts]])
*<math> \iiint_V  \psi\,\nabla{\cdot} \mathbf A \,dV \ =\ \iint_{\partial V} \psi \mathbf{A}\cdot d\mathbf S - \iiint_V \nabla\psi\, \cdot\mathbf{A}\,dV</math>  ([[Integration by parts]])<br />

====Curve–surface integrals====
In the following curve–surface integral theorems, ''S'' denotes a 2d open surface with a corresponding 1d boundary ''C'' = ∂''S'' (a [[closed curve]]):

*<math> \oint_{\!\!\!\!\partial S}\mathbf{A}\cdot d\boldsymbol{\ell}\ =\ \iint_{S}\left(\nabla\times\mathbf{A}\right)\cdot d\mathbf{S} </math>  ([[Stokes' theorem]])
*<math> \oint_{\!\!\!\!\partial S}\psi\, d\boldsymbol{\ell}\ =\ -\iint_{S}\nabla\psi\times d\mathbf{S} </math>

Integration around a closed curve in the [[clockwise]] sense is the negative of the same line integral in the counterclockwise sense (analogous to interchanging the limits in a [[definite integral]]):

:{{intorient|
| preintegral = {{intorient|
| preintegral =
|symbol=oint
| intsubscpt = <math>{\scriptstyle \partial S}</math>
| integrand = <math>\mathbf{A}\cdot{\rm d}\boldsymbol{\ell}=-</math>
}}
|symbol=ointctr
| intsubscpt = <math>{\scriptstyle \partial S}</math>
| integrand = <math>\mathbf{A}\cdot{\rm d}\boldsymbol{\ell}.</math>
}}

==See also==
* [[Exterior calculus identities]]
* [[Exterior derivative]]
* [[Vector calculus]]
* [[Del in cylindrical and spherical coordinates]]
* [[Comparison of vector algebra and geometric algebra]]

==References==
{{reflist}}

==Further reading==
{{Refbegin}}
* {{cite book | title = Advanced Engineering Electromagnetics | first = Constantine A. | last = Balanis | isbn = 0-471-62194-3 }}
* {{cite book | first = H. M. | last = Schey | title = Div Grad Curl and all that:  An informal text on vector calculus | publisher=W. W. Norton & Company | year= 1997 | isbn = 0-393-96997-5 }}
* {{cite book | first = David J. | last = Griffiths | title = Introduction to Electrodynamics | publisher = Prentice Hall | year = 1999 | isbn = 0-13-805326-X | url-access = registration | url = https://archive.org/details/introductiontoel00grif_0 }}
{{Refend}}

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