Vector calculus identities

See also: Vector algebra relations

The following identities are important in vector calculus:

Single operators (summary)

This section explicitly lists what some symbols mean for clarity.

Divergence

Main article: Divergence

Divergence of a tensor, ${\stackrel {\mathbf {\mathfrak {T}} }{}}$, of non-zero order n, is generally written as

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

and is a contraction to a tensor of order n-1. Specifically, the divergence of a vector is a scalar. The divergence of a higher order tensor may be found by decomposing the tensor into a sum of outer products, thereby allowing the use of the identity,

$\nabla \cdot (\mathbf {a} \otimes {\hat {\mathbf {\mathfrak {T}} }})={\hat {\mathbf {\mathfrak {T}} }}(\nabla \cdot \mathbf {a} )+(\mathbf {a} \cdot \nabla ){\hat {\mathbf {\mathfrak {T}} }}$

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

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

Curl

Main article: Curl (mathematics)

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

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

and is also a 3-dimensional vector field.

Gradient

Main article: 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 second-order 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.

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:

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

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. In overdot notation, explained above:

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

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)

Differentiation

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.

Gradient

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

Divergence

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

Curl

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

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)

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

See also

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

Further reading

• Balanis, Constantine A. Advanced Engineering Electromagnetics. ISBN 0471621943.
• Schey, H. M. (1997). Div Grad Curl and all that: An informal text on vector calculus. W. W. Norton & Company. ISBN 0-393-96997-5.
• Griffiths, David J. (1999). Introduction to Electrodynamics. Prentice Hall. ISBN 0-13-805326-X.