Vector calculus identities

See also: Vector algebra relations

The following identities are important in vector calculus:

Operator notations

Gradient

Main article: Gradient

In the three-dimensional Cartesian coordinate system, the gradient of some function $f(x,y,z)$ is given by:

$\operatorname{grad}(f) = \nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}$

where i, j, k are the standard unit vectors.

The gradient of a tensor field, $\mathbf{A}$, of order n, is generally written as

$\operatorname{grad}(\mathbf{A}) = \nabla \mathbf{A}$

and is a tensor field of order n + 1. In particular, if the tensor field has order 0 (i.e. a scalar), $\psi$, the resulting gradient,

$\operatorname{grad}(\psi) = \nabla \psi$

is a vector field.

Divergence

Main article: Divergence

In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field $\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}$ is defined as the scalar-valued function:

$\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot (F_x,F_y,F_z) = \frac{\partial F_x}{\partial x} +\frac{\partial F_y}{\partial y} +\frac{\partial F_z}{\partial z}.$

The divergence of a tensor field, $\mathbf{A}$, of non-zero order n, is generally written as

$\operatorname{div}(\mathbf{A}) = \nabla \cdot \mathbf{A}$

and is a contraction to a tensor field of order n − 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, thereby allowing the use of the identity,

$\nabla \cdot (\mathbf{B} \otimes \hat{\mathbf{A}}) = \hat{\mathbf{A}}(\nabla \cdot \mathbf{B})+(\mathbf{B}\cdot \nabla) \hat{\mathbf{A}}$

where $\mathbf{B}\cdot\nabla$ is the directional derivative in the direction of $\mathbf{B}$ multiplied by its magnitude. Specifically, for the outer product of two vectors,

$\nabla \cdot (\mathbf{a} \mathbf{b}^\mathrm{T}) = \mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \mathbf{b} \ .$

Curl

Main article: Curl (mathematics)

In Cartesian coordinates, for $\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}$:

curl($\mathbf{F}$) = $\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ F_x & F_y & F_z \end{vmatrix}$

$\nabla \times \mathbf{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \mathbf{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{k}$

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.

For a 3-dimensional vector field $\mathbf{v}$, curl is also a 3-dimensional vector field, generally written as:

$\nabla \times \mathbf{v}$

or in Einstein notation as:

$\varepsilon^{ijk} \frac {\partial v_k}{\partial x^j}$

where ε is the Levi-Civita symbol.

Laplacian

Main article: Laplace operator

In Cartesian coordinates, the Laplacian of a function $f(x,y,z)$ is

$\Delta f = \nabla^2 f = (\nabla \cdot \nabla) f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.$

For a tensor field, $\mathbf{A}$, the laplacian is generally written as:

$\Delta\mathbf{A} = \nabla^2 \mathbf{A} = (\nabla \cdot \nabla) \mathbf{A}$

and is a tensor field of the same order.

Special notations

In Feynman subscript notation,

$\nabla_\mathbf{B} \left( \mathbf{A \cdot B} \right) = \mathbf{A} \times \left( \nabla \times \mathbf{B} \right) + \left( \mathbf{A} \cdot \nabla \right) \mathbf{B}$

where the notation ∇_B means the subscripted gradient operates on only the factor B.

A less general but similar idea is used in geometric algebra where the so-called Hestenes overdot notation is employed. The above identity is then expressed as:

$\dot{\nabla} \left( \mathbf{A} \cdot \dot{\mathbf{B}} \right) = \mathbf{A} \times \left( \nabla \times \mathbf{B} \right) + \left( \mathbf{A} \cdot \nabla \right) \mathbf{B}$

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.

Properties

Distributive properties

$\nabla ( \psi + \phi ) = \nabla \psi + \nabla \phi$

$\nabla \cdot ( \mathbf{A} + \mathbf{B} ) = \nabla \cdot \mathbf{A} + \nabla \cdot \mathbf{B}$

$\nabla \times ( \mathbf{A} + \mathbf{B} ) = \nabla \times \mathbf{A} + \nabla \times \mathbf{B}$

Product rule for the gradient

The gradient of the product of two scalar fields $\psi$ and $\phi$ follows the same form as the product rule in single variable calculus.

$\nabla (\psi \, \phi) = \phi \,\nabla \psi + \psi \,\nabla \phi$

Product of a scalar and a vector

$\nabla \cdot ( \psi \mathbf{A} ) = \psi ( \nabla \cdot \mathbf{A}) + \mathbf{A}\cdot (\nabla \psi)$

$\nabla \times ( \psi \mathbf{A} ) = \psi ( \nabla \times \mathbf{A}) + ( \nabla \psi ) \times \mathbf{A}$

Quotient rule

$\nabla\left(\frac{f}{g}\right) = \frac{g\nabla f - (\nabla g)f}{g^2}$

$\nabla \cdot \left(\frac{\mathbf{A}}{g}\right) = \frac{g \nabla \cdot \mathbf{A} - (\nabla g) \cdot \mathbf{A}}{g^2}$

$\nabla \times \left(\frac{\mathbf{A}}{g}\right) = \frac{g \nabla \times \mathbf{A} - (\nabla g) \times \mathbf{A}}{g^2}$

Chain rule

$\nabla(f \circ g) = (f' \circ g) \nabla g$

$\nabla(f \circ \mathbf{A}) = (\nabla f \circ \mathbf{A}) \nabla \mathbf{A}$

$\nabla \cdot (\mathbf{A} \circ f) = (\mathbf{A}' \circ f) \cdot \nabla f$

$\nabla \times (\mathbf{A} \circ f) = -(\mathbf{A}' \circ f) \times \nabla f$

Vector dot product

$\begin{align} \nabla(\mathbf{A} \cdot \mathbf{B}) &= \mathbf{J}^\mathrm{T}_\mathbf{A} \mathbf{B} + \mathbf{J}^\mathrm{T}_\mathbf{B} \mathbf{A} \\ &= (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} + \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) \ . \end{align}$

where J_A denotes the Jacobian of A.

Alternatively, using Feynman subscript notation,

$\nabla(\mathbf{A} \cdot \mathbf{B}) = \nabla_\mathbf{A}(\mathbf{A} \cdot \mathbf{B}) + \nabla_\mathbf{B} (\mathbf{A} \cdot \mathbf{B}) \ .$

As a special case, when A = B,

$\begin{align} \frac{1}{2} \nabla \left( \mathbf{A}\cdot\mathbf{A} \right) &= \mathbf{J}^\mathrm{T}_\mathbf{A} \mathbf A \\ &= (\mathbf{A} \cdot \nabla) \mathbf{A} + \mathbf{A} \times (\nabla \times \mathbf{A}) \ . \end{align}$

Vector cross product

$\nabla \cdot (\mathbf{A} \times \mathbf{B}) = (\nabla \times \mathbf{A}) \cdot \mathbf{B} - \mathbf{A} \cdot (\nabla \times \mathbf{B})$

$\begin{align}\nabla \times (\mathbf{A} \times \mathbf{B}) &= \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} \\ &= (\nabla \cdot \mathbf{B} + \mathbf{B} \cdot \nabla)\mathbf{A} -(\nabla \cdot \mathbf{A} + \mathbf{A} \cdot \nabla )\mathbf{B} \\ &= \nabla \cdot (\mathbf{B} \mathbf{A}^\mathrm{T}) - \nabla \cdot (\mathbf{A} \mathbf{B}^\mathrm{T}) \\ &= \nabla \cdot (\mathbf{B} \mathbf{A}^\mathrm{T} - \mathbf{A} \mathbf{B}^\mathrm{T}) \end{align}$

Second derivatives

Curl of the gradient

The curl of the gradient of any twice-differentiable scalar field $\ \phi$ is always the zero vector:

$\nabla \times ( \nabla \phi ) = \mathbf{0}$

Divergence of the curl

The divergence of the curl of any vector field A is always zero:

$\nabla \cdot ( \nabla \times \mathbf{A} ) = 0$

Divergence of the gradient

The Laplacian of a scalar field is defined as the divergence of the gradient:

$\nabla^2 \psi = \nabla \cdot (\nabla \psi)$

Note that the result is a scalar quantity.

Curl of the curl

$\nabla \times \left( \nabla \times \mathbf{A} \right) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A}$

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

Summary of important identities

Addition and multiplication

• $\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}$
• $\mathbf{A}\cdot\mathbf{B}=\mathbf{B}\cdot\mathbf{A}$
• $\mathbf{A}\times\mathbf{B}=\mathbf{-B}\times\mathbf{A}$
• $\left(\mathbf{A}+\mathbf{B}\right)\cdot\mathbf{C}=\mathbf{A}\cdot\mathbf{C}+\mathbf{B}\cdot\mathbf{C}$
• $\left(\mathbf{A}+\mathbf{B}\right)\times\mathbf{C}=\mathbf{A}\times\mathbf{C}+\mathbf{B}\times\mathbf{C}$
• $\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)=\mathbf{B}\cdot\left(\mathbf{C}\times\mathbf{A}\right)=\mathbf{C}\cdot\left(\mathbf{A}\times\mathbf{B}\right)$ (scalar triple product)
• $\mathbf{A}\times\left(\mathbf{B}\times\mathbf{C}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right)\mathbf{B}-\left(\mathbf{A}\cdot\mathbf{B}\right)\mathbf{C}$ (vector triple product)
• $\left(\mathbf{A}\times\mathbf{B}\right)\times\mathbf{C}=\left(\mathbf{A}\cdot\mathbf{C}\right)\mathbf{B}-\left(\mathbf{B}\cdot\mathbf{C}\right)\mathbf{A}$ (vector triple product)
• $\left(\mathbf{A}\times\mathbf{B}\right)\cdot\left(\mathbf{C}\times\mathbf{D}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right)\left(\mathbf{B}\cdot\mathbf{D}\right)-\left(\mathbf{B}\cdot\mathbf{C}\right)\left(\mathbf{A}\cdot\mathbf{D}\right)$
• $\left(\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)\right)\mathbf{D}=\left(\mathbf{A}\cdot\mathbf{D}\right)\left(\mathbf{B}\times\mathbf{C}\right)+\left(\mathbf{B}\cdot\mathbf{D}\right)\left(\mathbf{C}\times\mathbf{A}\right)+\left(\mathbf{C}\cdot\mathbf{D}\right)\left(\mathbf{A}\times\mathbf{B}\right)$
• $\left(\mathbf{A}\times\mathbf{B}\right)\times\left(\mathbf{C}\times\mathbf{D}\right) =\left(\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{D}\right)\right)\mathbf{C}-\left(\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)\right)\mathbf{D}$

Differentiation

Gradient

• $\nabla(\psi+\phi)=\nabla\psi+\nabla\phi$
• $\nabla (\psi \, \phi) = \phi \,\nabla \psi + \psi \,\nabla \phi$
• $\nabla\left(\mathbf{A}\cdot\mathbf{B}\right)=\left(\mathbf{A}\cdot\nabla\right)\mathbf{B}+\left(\mathbf{B}\cdot\nabla\right)\mathbf{A}+\mathbf{A}\times\left(\nabla\times\mathbf{B}\right)+\mathbf{B}\times\left(\nabla\times\mathbf{A}\right)$

Divergence

• $\nabla\cdot(\mathbf{A}+\mathbf{B})=\nabla\cdot\mathbf{A}+\nabla\cdot\mathbf{B}$
• $\nabla\cdot\left(\psi\mathbf{A}\right)=\psi\nabla\cdot\mathbf{A}+\mathbf{A}\cdot\nabla \psi$
• $\nabla\cdot\left(\mathbf{A}\times\mathbf{B}\right)=\mathbf{B}\cdot (\nabla\times\mathbf{A})-\mathbf{A}\cdot(\nabla\times\mathbf{B})$

Curl

• $\nabla\times(\mathbf{A}+\mathbf{B})=\nabla\times\mathbf{A}+\nabla\times\mathbf{B}$
• $\nabla\times\left(\psi\mathbf{A}\right)=\psi\nabla\times\mathbf{A}+\nabla\psi\times\mathbf{A}$
• $\nabla\times\left(\mathbf{A}\times\mathbf{B}\right)=\mathbf{A}\left(\nabla\cdot\mathbf{B}\right)-\mathbf{B}\left(\nabla\cdot\mathbf{A}\right)+\left(\mathbf{B}\cdot\nabla\right)\mathbf{A}-\left(\mathbf{A}\cdot\nabla\right)\mathbf{B}$

Second derivatives

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.

• $\nabla\cdot(\nabla\times\mathbf{A})=0$
• $\nabla\times(\nabla\psi)= \mathbf{0}$
• $\nabla\cdot(\nabla\psi)=\nabla^{2}\psi$ (scalar Laplacian)
• $\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla^{2}\mathbf{A}$ (vector Laplacian)
• $\nabla\cdot(\phi\nabla\psi)=\phi\nabla^{2}\psi + \nabla\phi\cdot\nabla\psi$
• $\psi\nabla^2\phi-\phi\nabla^2\psi= \nabla\cdot\left(\psi\nabla\phi-\phi\nabla\psi\right)$
• $\nabla^2(\phi\psi)=\phi\nabla^2\psi+2\nabla\phi\cdot\nabla\psi+\psi\nabla^2\phi$
• $\nabla^2(\psi\mathbf{A})=\mathbf{A}\nabla^2\psi+2(\nabla\psi\cdot\nabla)\mathbf{A}+\psi\nabla^2\mathbf{A}$
• $\nabla^2(\mathbf{A}\cdot\mathbf{B})= \mathbf{A}\cdot\nabla^2\mathbf{B} - \mathbf{B}\cdot\nabla^2\mathbf{A} + 2\nabla\cdot((\mathbf{B}\cdot\nabla)\mathbf{A} + \mathbf{B}\times\nabla\times\mathbf{A})$ (Green's vector identity)

Third derivatives

• $\nabla^{2}(\nabla\psi) = \nabla(\nabla\cdot(\nabla\psi)) = \nabla(\nabla^{2}\psi)$
• $\nabla^{2}(\nabla\cdot\mathbf{A}) = \nabla\cdot(\nabla(\nabla\cdot\mathbf{A})) =\nabla\cdot(\nabla^{2}\mathbf{A})$
• $\nabla^{2}(\nabla\times\mathbf{A}) = -\nabla\times(\nabla\times(\nabla\times\mathbf{A})) = \nabla\times(\nabla^{2}\mathbf{A})$

Integration

Below, the curly symbol ∂ means "boundary of".

Surface–volume integrals

In the following surface–volume integral theorems, V denotes a 3d volume with a corresponding 2d boundary S = ∂V (a closed surface):

• $\scriptstyle \partial V$ $\mathbf{A}\cdot d\mathbf{S}=\iiint_V \left(\nabla \cdot \mathbf{A}\right)dV$ (Divergence theorem)
• $\scriptstyle \partial V$ $\psi d \mathbf{S} = \iiint_V \nabla \psi\, dV$
• $\scriptstyle \partial V$ $\left(\hat{\mathbf{n}}\times\mathbf{A}\right)dS=\iiint _{V}\left(\nabla\times\mathbf{A}\right)dV$
• $\scriptstyle \partial V$ $\psi\left(\nabla\varphi\cdot\hat{\mathbf{n}}\right)dS = \iiint _{V}\left(\psi\nabla^{2}\varphi+\nabla\varphi\cdot\nabla\psi\right)dV$ (Green's first identity)
• $\scriptstyle \partial V$ $\left[\left(\psi\nabla\varphi-\varphi\nabla\psi\right)\cdot\hat{\mathbf{n}}\right]dS=\,\!$ $\scriptstyle \partial V$ $\left[\psi\frac{\partial\varphi}{\partial n}-\varphi\frac{\partial\psi}{\partial n}\right]dS$ $\displaystyle=\iiint_{V}\left(\psi\nabla^{2}\varphi-\varphi\nabla^{2}\psi\right)dV\,\!$ (Green's second identity)

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):

• $\oint_{\partial S}\mathbf{A}\cdot d\boldsymbol{\ell}=\iint_{S}\left(\nabla\times\mathbf{A}\right)\cdot d\mathbf{s}$   (Stokes' theorem)
• $\oint_{\partial S}\psi d\boldsymbol{\ell}=\iint_{S}\left(\hat{\mathbf{n}}\times\nabla\psi\right)dS$

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):

${\scriptstyle \partial S}$ $\mathbf{A}\cdot{\rm d}\boldsymbol{\ell}=-$ ${\scriptstyle \partial S}$ $\mathbf{A}\cdot{\rm d}\boldsymbol{\ell}.$

See also

• Exterior derivative
• Vector calculus
• Del in cylindrical and spherical coordinates
• Comparison of vector algebra and geometric algebra