Vector calculus identities

The following identities are important in vector calculus:

Single operators (summary)

This section explicitly lists what some symbols mean for clarity.

Divergence

Divergence of a vector field

For a vector field $\mathbf {v}$ , divergence is generally written as

\operatorname {div} (\mathbf {v} )=\nabla \cdot \mathbf {v}

and is a scalar field.

Divergence of a tensor

For a second order tensor ${\stackrel {\mathbf {\mathfrak {T}} }{}}$ , divergence is generally written as

\operatorname {div} (\mathbf {\mathfrak {T}} )=\nabla \cdot \mathbf {\mathfrak {T}}

and is a vector.

More generally speaking, the divergence of a tensor of order n is a contraction to a tensor of order n-1.

Curl

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

\operatorname {curl} (\mathbf {v} )=\nabla \times \mathbf {v}

and is a vector field.

Gradient

Gradient of a vector field

For a vector field $\mathbf {v}$ , gradient is generally written as:

\operatorname {grad} (\mathbf {v} )=\nabla \mathbf {v}

and is a tensor.

Gradient of a scalar field

For a scalar field, $\psi$ , the gradient is generally written as

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

and is a vector field.

Combinations of multiple operators

Curl of the gradient

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

\nabla \times (\nabla \phi )={\vec {0}}

One way to establish this identity (and most of the others listed in this article) is to use three-dimensional Cartesian coordinates. According to the article on curl,

\nabla \times \nabla \phi ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\\\{\partial _{x}}&{\partial _{y}}&{\partial _{z}}\\\\\partial _{x}\phi &\partial _{y}\phi &\partial _{z}\phi \end{vmatrix}}\ ,

where the right hand side is a determinant, and i, j, k are unit vectors pointing in the positive axes directions, and ∂_x = ∂ / ∂ x etc. For example, the x-component of the above equation is:

\mathbf {i} \left(\partial _{y}\partial _{z}-\partial _{z}\partial _{y}\right)\phi ={\vec {0}}\ ,

where the left-hand side equals zero due to the equality of mixed partial derivatives.

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

Properties

Distributive property

\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}

Vector dot product

\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} )

In simpler form, using Feynman subscript notation:

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

where the notation ∇_A means the subscripted gradient operates on only the factor A.^[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:

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

where overdots define the scope of the vector derivative. In the first term it is only the first (dotted) factor that is differentiated, while the second is held constant. Likewise, in the second term it is the second (dotted) factor that is differentiated, and the first is held constant.

As a special case, when A = B:

{\frac {1}{2}}\nabla \left(\mathbf {A} \cdot \mathbf {A} \right)=\mathbf {A} \times (\nabla \times \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {A} .

Vector cross product

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

\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}

\mathbf {A\ \times } \left(\mathbf {\nabla \times B} \right)=\nabla _{B}\left(\mathbf {A\cdot B} \right)-\left(\mathbf {A\cdot \nabla } \right)\mathbf {B} \ ,

where the Feynman subscript notation ∇_B means the subscripted gradient operates on only the factor B.^[1]^[2] In overdot notation, explained above:^[3]

\mathbf {A\ \times } \left(\mathbf {\nabla \times B} \right)={\dot {\nabla }}\left(\mathbf {A\cdot } {\dot {\mathbf {B} }}\right)-\left(\mathbf {A\cdot \nabla } \right)\mathbf {B} \ .

^[4]

Product of a scalar and a vector

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

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

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

Summary of all 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)
$\mathbf {\left(A\times B\right)\cdot } \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\left(\mathbf {B} \cdot \mathbf {D} \right)-\left(\mathbf {B} \cdot \mathbf {C} \right)\left(\mathbf {A} \cdot \mathbf {D} \right)$
$\mathbf {\left(A\times B\right)\times } \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \mathbf {B\times D} \right)\mathbf {C} -\left(\mathbf {A} \cdot \mathbf {B\times C} \right)\mathbf {D}$

Differentiation

Addition

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

Gradient of products

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

Divergence of products

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

Curl of products

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

Second derivatives

$\nabla \cdot (\nabla \times \mathbf {A} )=0$
$\nabla \times (\nabla \psi )=0$
$\nabla \cdot (\nabla \psi )=\nabla ^{2}\psi$ (scalar Laplacian)
$\nabla \left(\nabla \cdot \mathbf {A} \right)-\nabla \times \nabla \times \mathbf {A} =\nabla ^{2}\mathbf {A}$ (vector Laplacian)
$\psi \nabla ^{2}\phi -\phi \nabla ^{2}\psi =\nabla \cdot \left(\psi \nabla \phi -\phi \nabla \psi \right)$

Integration

$\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \mathbf {A} \cdot d\mathbf {s} =\iiint \limits _{V}\left(\nabla \cdot \mathbf {A} \right)dv$ (Divergence theorem)
$\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset {\psi }d\mathbf {s} =\iiint \limits _{V}\nabla \psi \,dv$
$\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \left({\hat {\mathbf {n} }}\times \mathbf {A} \right)ds=\iiint \limits _{V}\left(\nabla \times \mathbf {A} \right)dv$
$\iiint \limits _{V}\left(\psi \nabla ^{2}\varphi +\nabla \varphi \cdot \nabla \psi \right)dv=\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \psi \left(\nabla \varphi \cdot {\hat {\mathbf {n} }}\right)ds$ (Green's first identity)

$\iiint \limits _{V}\left(\psi \nabla ^{2}\varphi -\varphi \nabla ^{2}\psi \right)dv=\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \left[\left(\psi \nabla \varphi -\varphi \nabla \psi \right)\cdot {\hat {\mathbf {n} }}\right]ds=\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \left[\psi {\frac {\partial \varphi }{\partial n}}-\varphi {\frac {\partial \psi }{\partial n}}\right]ds$ (Green's second identity)

$\oint \limits _{C}\mathbf {A} \cdot d\mathbf {l} =\iint \limits _{S}\left(\nabla \times \mathbf {A} \right)\cdot d\mathbf {s}$ (Stokes' theorem)

$\oint \limits _{C}\psi d\mathbf {l} =\iint \limits _{S}\left({\hat {\mathbf {n} }}\times \nabla \psi \right)ds$

Notes and references

^ ^a ^b Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lecture on Physics. Addison-Wesley. Vol II, p. 27–4. ISBN 0805390499. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)
^ ^a ^b Kholmetskii, A. L.; Missevitch, O. V. (2005). "The Faraday induction law in relativity theory". p. 4. arXiv:physics/0504223. {{cite arXiv}}: |class= ignored (help)
^ ^a ^b Doran, C.; Lasenby, A. (2003). Geometric algebra for physicists. Cambridge University Press. p. 169. ISBN 978-0-521-71595-9.
^ Adams, Robert A.; Essex, Christopher (2008). Calculus: Several Variables (7th ed.). Toronto: Pearson Canada. p. 897. ISBN 0201798026.