Vector calculus identities

The following are important identities involving derivatives and integrals in vector calculus.

Operator notation

Gradient

For a function $f(x, y, z)$ in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
$\operatorname(f) = \nabla f = \begin \frac,\ \frac,\ \frac \end f = \frac \mathbf + \frac \mathbf + \frac \mathbf$
where i, j, k are the standard unit vectors for the ''x'', ''y'', ''z''-axes. More generally, for a function of ''n'' variables $\psi(x_1, \ldots, x_n)$, also called a scalar field, the gradient is the vector field:
$\nabla\psi = \begin\frac, \ldots,\ \frac \end\psi = \frac \mathbf_1 + \dots + \frac\mathbf_n .$
where $\mathbf_$ are orthogonal unit vectors in arbitrary directions.

As the name implies, the gradient is proportional to and points in the direction of the function's most rapid (positive) change.

For a vector field $\mathbf = \left(A_1, \ldots, A_n\right)$ 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:
$\mathbf_ = (\nabla \!\mathbf)^\mathrm = \left(\frac\right)_.$

For a tensor field $\mathbf$ of any order ''k'', the gradient $\operatorname(\mathbf) = (\nabla\!\mathbf)^\mathrm$ is a tensor field of order ''k'' + 1.

Divergence

In Cartesian coordinates, the divergence of a continuously differentiable vector field $\mathbf = F_x\mathbf + F_y\mathbf + F_z\mathbf$ is the scalar-valued function:
$\operatorname\mathbf = \nabla\cdot\mathbf = \begin\frac,\ \frac,\ \frac\end \cdot \beginF_,\ F_,\ F_\end = \frac + \frac + \frac.$

As the name implies the divergence is a measure of how much vectors are diverging.

The divergence of a tensor field $\mathbf$ of non-zero order ''k'' is written as $\operatorname(\mathbf) = \nabla \cdot \mathbf$, a 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,
$\nabla \cdot \left(\mathbf \otimes \hat\right) = \hat (\nabla \cdot \mathbf) + (\mathbf \cdot \nabla) \hat$
where $\mathbf \cdot \nabla$ is the directional derivative in the direction of $\mathbf$ multiplied by its magnitude. Specifically, for the outer product of two vectors,
$\nabla \cdot \left(\mathbf \mathbf^\mathsf\right) = \mathbf\left(\nabla \cdot \mathbf\right) + \left(\mathbf \cdot \nabla\right) \mathbf.$

Curl

In Cartesian coordinates, for $\mathbf = F_x\mathbf + F_y\mathbf + F_z\mathbf$ the curl is the vector field:

\begin\operatorname\mathbf &=& \nabla \times \mathbf =
\begin\frac,\ \frac,\ \frac\end \times \beginF_,\ F_,\ F_\end
=
\begin
\mathbf & \mathbf & \mathbf \\
\frac & \frac & \frac \\
F_x & F_y & F_z
\end \\ em &=&
\left(\frac - \frac\right) \mathbf +
\left(\frac - \frac\right) \mathbf +
\left(\frac - \frac\right) \mathbf
\end
where i, j, and k are the unit vectors for the ''x''-, ''y''-, and ''z''-axes, respectively.

As the name implies the curl is a measure of how much nearby vectors tend in a circular direction.

In Einstein notation, the vector field $\mathbf = \begin F_1 & F_2 & F_3 \end$ has curl given by:
$\nabla \times \mathbf = \varepsilon^\mathbf_i \frac$
where $\varepsilon$ = ±1 or 0 is the Levi-Civita parity symbol.

Laplacian

In Cartesian coordinates, the Laplacian of a function $f(x,y,z)$ is
$\Delta f = \nabla^2\! f = (\nabla \cdot \nabla) f = \frac + \frac + \frac.$

The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.

For a tensor field, $\mathbf$, the Laplacian is generally written as:
$\Delta\mathbf = \nabla^2\! \mathbf = (\nabla \cdot \nabla) \mathbf$
and is a tensor field of the same order.

When the Laplacian is equal to 0, the function is called a harmonic function. That is,
$\Delta f = 0$

Special notations

In ''Feynman subscript notation'',
$\nabla_\mathbf\! \left( \mathbf \right) = \mathbf \! \left( \nabla \mathbf \right) + \left( \mathbf \nabla \right) \mathbf$
where the notation ∇_B means the subscripted gradient operates on only the factor B.

Less general but similar is the ''Hestenes'' ''overdot notation'' in geometric algebra. The above identity is then expressed as:
$\dot \left( \mathbf \dot \right) = \mathbf \! \left( \nabla \mathbf \right) + \left( \mathbf \nabla \right) \mathbf$
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 $\psi$, $\phi$ and vector fields $\mathbf$, $\mathbf$, we have the following derivative identities.

Distributive properties

:$\begin \nabla ( \psi + \phi ) &= \nabla \psi + \nabla \phi \\ \nabla ( \mathbf + \mathbf ) &= \nabla \mathbf + \nabla \mathbf \\ \nabla \cdot ( \mathbf + \mathbf ) &= \nabla \mathbf + \nabla \cdot \mathbf \\ \nabla \times ( \mathbf + \mathbf ) &= \nabla \times \mathbf + \nabla \times \mathbf \end$

Product rule for multiplication by a scalar

We have the following generalizations of the product rule in single variable calculus.

:$\begin \nabla ( \psi \phi ) &= \phi\, \nabla \psi + \psi\, \nabla \phi \\ \nabla ( \psi \mathbf ) &= (\nabla \psi) \mathbf^ + \psi \nabla \mathbf \ =\ \nabla \psi \otimes \mathbf + \psi\, \nabla \mathbf \\ \nabla \cdot ( \psi \mathbf ) &= \psi\, \nabla \mathbf + ( \nabla \psi ) \, \mathbf \\ \nabla ( \psi \mathbf ) &= \psi\, \nabla \mathbf + ( \nabla \psi ) \mathbf \\ \nabla^(f g) &= f\,\nabla^g + 2\,\nabla\! f\cdot\!\nabla g+g\, \nabla^f \end$

In the second formula, the transposed gradient $(\nabla \psi)^$ is an ''n'' × 1 column vector, $\mathbf$ is a 1 × ''n'' row vector, and their product is an ''n × n'' matrix (or more precisely, a dyad); This may also be considered as the tensor product $\otimes$ of two vectors, or of a covector and a vector''.''

Quotient rule for division by a scalar

:\begin
\nabla\left(\frac\right)
&= \frac \\ em \nabla\left(\frac\right)
&= \frac \\ em \nabla \cdot \left(\frac\right)
&= \frac \\ em \nabla \times \left(\frac\right)
&= \frac
\end

Chain rule

Let $f(x)$ be a one-variable function from scalars to scalars, $\mathbf(t) = (r_1(t),\ldots,r_n(t))$ a parametrized curve, and $F:\mathbb^n\to\mathbb$ a function from vectors to scalars. We have the following special cases of the multi-variable chain rule.

:$\begin \nabla(f \circ F) &= \left(f' \circ F\right)\, \nabla F \\ (F \circ \mathbf)' &= (\nabla F \circ \mathbf) \cdot \mathbf' \\ \nabla(F \circ \mathbf) &= (\nabla F \circ \mathbf)\, \nabla \mathbf \\ \nabla \times (\mathbf r \circ F) &= \nabla F \times (\mathbf r' \circ F) \end$

For a coordinate parametrization $\Phi:\mathbb^n \to \mathbb^n$ we have:

:$\nabla \cdot (\mathbf \circ \Phi) = \mathrm \left((\nabla\mathbf \circ \Phi) \mathbf_\Phi\right)$

Here we take the trace of the product of two ''n × n'' matrices: the gradient of A and the Jacobian of $\Phi$.

Dot product rule

:$\begin \nabla(\mathbf \cdot \mathbf) &\ =\ (\mathbf \cdot \nabla)\mathbf \,+\, (\mathbf \cdot \nabla)\mathbf \,+\, \mathbf (\nabla \mathbf) \,+\, \mathbf (\nabla \mathbf) \\ &\ =\ \mathbf\cdot\mathbf_\mathbf + \mathbf\cdot\mathbf_\mathbf \ =\ (\nabla\mathbf)\cdot \mathbf \,+\, (\nabla\mathbf) \cdot\mathbf \end$

where $\mathbf_ = (\nabla \!\mathbf)^\mathrm = (\partial A_i/\partial x_j)_$ denotes the Jacobian matrix of the vector field $\mathbf = (A_1,\ldots,A_n)$.

Alternatively, using Feynman subscript notation,

:$\nabla(\mathbf \cdot \mathbf) = \nabla_\mathbf(\mathbf \cdot \mathbf) + \nabla_\mathbf (\mathbf \cdot \mathbf) \ .$

See these notes.

As a special case, when ,

:$\tfrac \nabla \left( \mathbf \cdot \mathbf \right) \ =\ \mathbf \cdot \mathbf_\mathbf \ =\ (\nabla \mathbf)\cdot \mathbf\ =\ (\mathbf \nabla) \mathbf \,+\, \mathbf (\nabla \mathbf) \ =\ A \nabla (A) .$

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 1-form.

Cross product rule

:\begin
\nabla \cdot (\mathbf \times \mathbf)
&\ =\ (\nabla \mathbf) \cdot \mathbf
\,-\, \mathbf \cdot (\nabla \mathbf) \\ pt \nabla \times (\mathbf \times \mathbf)
&\ =\ \mathbf(\nabla \mathbf) \,-\, \mathbf(\nabla \mathbf)
\,+\, (\mathbf \nabla) \mathbf \,-\, (\mathbf \nabla) \mathbf \\ pt &\ =\ (\nabla \, \mathbf \,+\, \mathbf\, \nabla)\mathbf
\,-\, (\nabla \mathbf \,+\, \mathbf \nabla) \mathbf \\ pt &\ =\ \nabla \left(\mathbf \mathbf^\mathrm\right)
\,-\, \nabla \left(\mathbf \mathbf^\mathrm\right) \\ pt &\ =\ \nabla \left(\mathbf \mathbf^\mathrm \,-\, \mathbf \mathbf^\mathrm\right) \\
\mathbf \times (\nabla \times \mathbf)
&\ =\ \nabla_(\mathbf \mathbf) \,-\, (\mathbf \nabla) \mathbf \\ pt &\ =\ \mathbf \cdot \mathbf_\mathbf \,-\, (\mathbf \nabla) \mathbf =\ (\nabla\mathbf)\cdot\mathbf \,-\, (\mathbf \nabla) \mathbf \\ pt &\ =\ \mathbf \cdot (\mathbf_\mathbf \,-\, \mathbf_\mathbf^\mathrm)\\ pt (\mathbf \times \nabla) \times \mathbf
&\ =\ (\nabla\mathbf) \cdot \mathbf \,-\, \mathbf (\nabla \mathbf)\\
&\ =\ \mathbf \times (\nabla \times \mathbf) \,+\, (\mathbf \nabla) \mathbf \,-\, \mathbf (\nabla \mathbf)
\end

Note that the matrix $\mathbf_\mathbf \,-\, \mathbf_\mathbf^\mathrm$ is antisymmetric.

Second derivative identities

Divergence of curl is zero

The divergence of the curl of ''any'' vector field A is always zero:
$\nabla \cdot ( \nabla \times \mathbf ) = 0$

This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.

Divergence of gradient is Laplacian

The Laplacian of a scalar field is the divergence of its gradient:
$\Delta \psi = \nabla^2 \psi = \nabla \cdot (\nabla \psi)$
The result is a scalar quantity.

Divergence of divergence is not defined

Divergence of a vector field A is a scalar, and you cannot take the divergence of a scalar quantity. Therefore:
$\nabla \cdot (\nabla \cdot \mathbf) \text$

Curl of gradient is zero

The curl of the gradient of ''any'' continuously twice-differentiable scalar field $\varphi$ (i.e., differentiability class $C^2$) is always the zero vector:
$\nabla \times ( \nabla \varphi ) = \mathbf$

It can be easily proved by expressing $\nabla \times ( \nabla \varphi )$ in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). This result is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.

Curl of curl

$\nabla \times \left( \nabla \times \mathbf \right) \ =\ \nabla(\nabla \mathbf) \,-\, \nabla^\mathbf$

Here ∇² is the vector Laplacian operating on the vector field A.

Curl of divergence is not defined

The divergence of a vector field A is a scalar, and you cannot take curl of a scalar quantity. Therefore
$\nabla \times (\nabla \cdot \mathbf) \text$

A mnemonic

The figure to the right is a mnemonic for some of these identities. The abbreviations used are:
* D: divergence,
* C: curl,
* G: gradient,
* L: Laplacian,
* CC: curl of curl.

Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist.

Summary of important identities

Differentiation

Gradient

*$\nabla(\psi+\phi)=\nabla\psi+\nabla\phi$
*$\nabla(\psi \phi) = \phi\nabla \psi + \psi \nabla \phi$
*$\nabla(\psi \mathbf ) = \nabla \psi \otimes \mathbf + \psi \nabla \mathbf$
*$\nabla(\mathbf \cdot \mathbf) = (\mathbf \cdot \nabla)\mathbf + (\mathbf \cdot \nabla)\mathbf + \mathbf \times (\nabla \times \mathbf) + \mathbf \times (\nabla \times \mathbf)$

Divergence

*$\nabla\cdot(\mathbf+\mathbf)= \nabla\cdot\mathbf+\nabla\cdot\mathbf$
*$\nabla\cdot\left(\psi\mathbf\right)= \psi\nabla\cdot\mathbf+\mathbf\cdot\nabla \psi$
*$\nabla\cdot\left(\mathbf\times\mathbf\right)= (\nabla\times\mathbf)\cdot \mathbf-(\nabla\times\mathbf)\cdot \mathbf$

Curl

*$\nabla\times(\mathbf+\mathbf)=\nabla\times\mathbf+\nabla\times\mathbf$
*$\nabla\times\left(\psi\mathbf\right)=\psi\,(\nabla\times\mathbf)-(\mathbf\times\nabla)\psi=\psi\,(\nabla\times\mathbf)+(\nabla\psi)\times\mathbf$
*$\nabla\times\left(\psi\nabla\phi\right)= \nabla \psi \times \nabla \phi$
*$\nabla\times\left(\mathbf\times\mathbf\right)= \mathbf\left(\nabla\cdot\mathbf\right)-\mathbf \left( \nabla\cdot\mathbf\right)+\left(\mathbf\cdot\nabla\right)\mathbf- \left(\mathbf\cdot\nabla\right)\mathbf$

Vector dot Del Operator

*$(\mathbf \cdot \nabla)\mathbf = \frac\bigg[\nabla(\mathbf \cdot \mathbf) - \nabla\times(\mathbf \times \mathbf) - \mathbf\times(\nabla \times \mathbf) - \mathbf\times(\nabla \times \mathbf) - \mathbf(\nabla \cdot \mathbf) + \mathbf(\nabla \cdot\mathbf)\bigg]$

*$(\mathbf \cdot \nabla)\mathbf = \frac\nabla , \mathbf, ^2-\mathbf\times(\nabla\times\mathbf) = \frac\nabla , \mathbf, ^2 + (\nabla\times\mathbf)\times \mathbf$

Second derivatives

*$\nabla \cdot (\nabla \times \mathbf) = 0$
*$\nabla \times (\nabla\psi) = \mathbf$
*$\nabla \cdot (\nabla\psi) = \nabla^2\psi$ ( scalar Laplacian)
*$\nabla\left(\nabla \cdot \mathbf\right) - \nabla \times \left(\nabla \times \mathbf\right) = \nabla^2\mathbf$ (vector Laplacian)
*$\nabla \cdot (\phi\nabla\psi) = \phi\nabla^2\psi + \nabla\phi \cdot \nabla\psi$
*$\psi\nabla^2\phi - \phi\nabla^2\psi = \nabla \cdot \left(\psi\nabla\phi - \phi\nabla\psi\right)$
*$\nabla^2(\phi\psi) = \phi\nabla^2\psi + 2(\nabla\phi) \cdot(\nabla\psi) + \left(\nabla^2\phi\right)\psi$
*$\nabla^2(\psi\mathbf) = \mathbf\nabla^2\psi + 2(\nabla\psi \cdot \nabla)\mathbf + \psi\nabla^2\mathbf$
*$\nabla^2(\mathbf \cdot \mathbf) = \mathbf \cdot \nabla^2\mathbf - \mathbf \cdot \nabla^2\!\mathbf + 2\nabla \cdot ((\mathbf \cdot \nabla)\mathbf + \mathbf \times (\nabla \times \mathbf))$ ( Green's vector identity)

Third derivatives

*$\nabla^2(\nabla\psi) = \nabla(\nabla \cdot (\nabla\psi)) = \nabla\left(\nabla^2\psi\right)$
*$\nabla^2(\nabla \cdot \mathbf) = \nabla \cdot (\nabla(\nabla \cdot \mathbf)) = \nabla \cdot \left(\nabla^2\mathbf\right)$
*$\nabla^(\nabla\times\mathbf) = -\nabla \times (\nabla \times (\nabla \times \mathbf)) = \nabla \times \left(\nabla^2\mathbf\right)$

Integration

Below, the curly symbol ∂ means " boundary of" a surface or solid.

Surface–volume integrals

In the following surface–volume integral theorems, ''V'' denotes a three-dimensional volume with a corresponding two-dimensional boundary ''S'' = ∂''V'' (a closed surface):

*
* (divergence theorem)
*
* ( Green's first identity)
* (Green's second identity)
* (integration by parts)
* (integration by parts)
* (integration by parts)

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):

*$\oint_\mathbf\cdot d\boldsymbol\ =\ \iint_\left(\nabla \times \mathbf\right)\cdot d\mathbf$ (Stokes' theorem)
*$\oint_\psi\, d\boldsymbol\ =\ -\iint_ \nabla\psi \times d\mathbf$

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):
:

Endpoint-curve integrals

In the following endpoint–curve integral theorems, ''P'' denotes a 1d open path with signed 0d boundary points $\mathbf-\mathbf = \partial P$ and integration along ''P'' is from $\mathbf$ to $\mathbf$:

*$\psi, _ = \psi(\mathbf)-\psi(\mathbf) = \int_ \nabla\psi\cdot d\boldsymbol$ (Gradient theorem).

See also

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References

Further reading

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{{Refend

Mathematical identities
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