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Vector calculus identities

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 .

Divergence of a tensor

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

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

and is a vector.

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.

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

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

and is a tensor.

For a scalar field, ψ, the gradient is generally written as

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

and is a vector.

Combinations of multiple operators

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

$\nabla \times ( \nabla \phi ) = 0$

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,

$\nabla \times \nabla \phi = \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \\ { \partial_x } & { \partial_y } & { \partial_z } \\ \\ \partial_x \phi & \partial_y \phi & \partial_z \phi \end{bmatrix} \ ,$

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:

$\mathbf{i} \left( \partial_y \partial_z - \partial_z \partial_y \right) \phi = 0 \ ,$

where the left-hand side evaluates as zero assuming the order of differentiation is immaterial.

Divergence of the curl

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

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

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

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

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]

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

Product of a scalar and a vector

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

The gradient of the product of two scalar fields ψ and φ follows the same form as the Product rule in single variable Calculus.

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

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