Vector calculus identities

The following identities are important in vector calculus:

Operator notations

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

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

Curl

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

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.^[1]^[2]

A less general but similar idea is used in geometric algebra where the so-called Hestenes overdot notation is employed.^[3] 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

For scalar fields $\psi$ and $\phi$ , vector fields $\mathbf {A}$ and $\mathbf {B}$ , and cartesian functions $f$ and $g$ :

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

where $J A$ denotes the Jacobian of $A$ . For more details, refer to these notes ^[4]

As a special case, when $A = B$ ,

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

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{aligned}\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{aligned}}

Second derivatives

Curl of the gradient

The curl of the gradient of any continuously 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 the divergence of its gradient:

\nabla ^{2}\psi =\nabla \cdot (\nabla \psi )

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)
$\mathbf {A} \times \left(\mathbf {B} \times \mathbf {C} \right)=\left(\mathbf {A} \times \mathbf {B} \right)\times \mathbf {C} \ +\ \mathbf {B} \times \left(\mathbf {A} \times \mathbf {C} \right)$ (Jacobi identity)
$\mathbf {A} \times \left(\mathbf {B} \times \mathbf {C} \right)\ +\ \mathbf {C} \times \left(\mathbf {A} \times \mathbf {B} \right)\ +\ \mathbf {B} \times \left(\mathbf {C} \times \mathbf {A} \right)=0$ (Jacobi identity)
$\left(\mathbf {A} \times \mathbf {B} \right)\cdot \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\left(\mathbf {B} \cdot \mathbf {D} \right)-\left(\mathbf {B} \cdot \mathbf {C} \right)\left(\mathbf {A} \cdot \mathbf {D} \right)$
$\left(\mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {C} \right)\right)\mathbf {D} =\left(\mathbf {A} \cdot \mathbf {D} \right)\left(\mathbf {B} \times \mathbf {C} \right)+\left(\mathbf {B} \cdot \mathbf {D} \right)\left(\mathbf {C} \times \mathbf {A} \right)+\left(\mathbf {C} \cdot \mathbf {D} \right)\left(\mathbf {A} \times \mathbf {B} \right)$
$\left(\mathbf {A} \times \mathbf {B} \right)\times \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {D} \right)\right)\mathbf {C} -\left(\mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {C} \right)\right)\mathbf {D}$

Differentiation

Gradient

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

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

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

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 three-dimensional volume with a corresponding two-dimensional boundary S = ∂V (a closed surface):

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

$\ointclockwise$

{\scriptstyle \partial S}

\mathbf {A} \cdot {\rm {d}}{\boldsymbol {\ell }}=-

$\ointctrclockwise$

{\scriptstyle \partial S}

\mathbf {A} \cdot {\rm {d}}{\boldsymbol {\ell }}.

References

↑ Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lectures on Physics. Addison-Wesley. Vol II, p. 27–4. ISBN 0-8053-9049-9.
↑ Kholmetskii, A. L.; Missevitch, O. V. (2005). "The Faraday induction law in relativity theory" (PDF). p. 4. arXiv:physics/0504223.
↑ Doran, C.; Lasenby, A. (2003). Geometric algebra for physicists. Cambridge University Press. p. 169. ISBN 978-0-521-71595-9.
↑ Kelly, P. (2013). "Chapter 1.14 Tensor Calculus 1: Tensor Fields" (PDF). Mechanics Lecture Notes Part III: Foundations of Continuum Mechanics. University of Auckland. Retrieved 7 December 2017.

Vector calculus identities

Operator notations

Gradient

Divergence

Curl

Laplacian

Special notations

Properties

Distributive properties

Product rule for the gradient

Product of a scalar and a vector

Quotient rule

Chain rule

Vector dot product

Vector cross product

Second derivatives

Curl of the gradient

Divergence of the curl

Divergence of the gradient

Curl of the curl

Summary of important identities

Addition and multiplication

Differentiation

Gradient

Divergence

Curl

Second derivatives

Third derivatives

Integration

Surface–volume integrals

Curve–surface integrals

See also

References

Further reading