Vector calculus identities

Gradient

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

grad ⁡ ( f ) = ∇ f = ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k

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

The gradient of a tensor field, A , of order n, is generally written as

grad ⁡ ( A ) = ∇ A

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

grad ⁡ ( ψ ) = ∇ ψ

is a vector field.

Divergence

In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field F = F x i + F y j + F z k is defined as the scalar-valued function:

div F = ∇ ⋅ F = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) ⋅ ( F x , F y , F z ) = ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z .

The divergence of a tensor field, A , of non-zero order n, is generally written as

div ⁡ ( A ) = ∇ ⋅ 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,

∇ ⋅ ( B ⊗ A ^ ) = A ^ ( ∇ ⋅ B ) + ( B ⋅ ∇ ) A ^

where B ⋅ ∇ is the directional derivative in the direction of B multiplied by its magnitude. Specifically, for the outer product of two vectors,

∇ ⋅ ( a b T ) = b ( ∇ ⋅ a ) + ( a ⋅ ∇ ) b   .

Curl

In Cartesian coordinates, for F = F x i + F y j + F z k :

curl( F ) = ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z | ∇ × F = ( ∂ F z ∂ y − ∂ F y ∂ z ) i + ( ∂ F x ∂ z − ∂ F z ∂ x ) j + ( ∂ F y ∂ x − ∂ F x ∂ y ) k

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

For a 3-dimensional vector field v , curl is also a 3-dimensional vector field, generally written as:

∇ × v

or in Einstein notation as:

ε i j k ∂ v k ∂ x j

where ε is the Levi-Civita symbol.

Laplacian

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

Δ f = ∇ 2 f = ( ∇ ⋅ ∇ ) f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 .

For a tensor field, A , the laplacian is generally written as:

Δ A = ∇ 2 A = ( ∇ ⋅ ∇ ) A

and is a tensor field of the same order.

Special notations

In Feynman subscript notation,

∇ B ( A ⋅ B ) = A × ( ∇ × B ) + ( A ⋅ ∇ ) 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:

∇ ˙ ( A ⋅ B ˙ ) = A × ( ∇ × B ) + ( A ⋅ ∇ ) 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.

Distributive properties

∇ ( ψ + ϕ ) = ∇ ψ + ∇ ϕ ∇ ⋅ ( A + B ) = ∇ ⋅ A + ∇ ⋅ B ∇ × ( A + B ) = ∇ × A + ∇ × B

Product rule for the gradient

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

∇ ( ψ ϕ ) = ϕ ∇ ψ + ψ ∇ ϕ

Product of a scalar and a vector

∇ ⋅ ( ψ A ) = ψ ( ∇ ⋅ A ) + A ⋅ ( ∇ ψ ) ∇ × ( ψ A ) = ψ ( ∇ × A ) + ( ∇ ψ ) × A

Quotient rule

∇ ( f g ) = g ∇ f − ( ∇ g ) f g 2 ∇ ⋅ ( A g ) = g ∇ ⋅ A − ( ∇ g ) ⋅ A g 2 ∇ × ( A g ) = g ∇ × A − ( ∇ g ) × A g 2

Chain rule

∇ ( f ∘ g ) = ( f ′ ∘ g ) ∇ g ∇ ( f ∘ A ) = ( ∇ f ∘ A ) ∇ A ∇ ⋅ ( A ∘ f ) = ( A ′ ∘ f ) ⋅ ∇ f ∇ × ( A ∘ f ) = − ( A ′ ∘ f ) × ∇ f

Vector dot product

∇ ( A ⋅ B ) = J A T B + J B T A = ( A ⋅ ∇ ) B + ( B ⋅ ∇ ) A + A × ( ∇ × B ) + B × ( ∇ × A )   .

where J_A denotes the Jacobian of A.

Alternatively, using Feynman subscript notation,

∇ ( A ⋅ B ) = ∇ A ( A ⋅ B ) + ∇ B ( A ⋅ B )   .

As a special case, when A = B,

1 2 ∇ ( A ⋅ A ) = J A T A = ( A ⋅ ∇ ) A + A × ( ∇ × A )   .

Vector cross product

∇ ⋅ ( A × B ) = ( ∇ × A ) ⋅ B − A ⋅ ( ∇ × B ) ∇ × ( A × B ) = A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) + ( B ⋅ ∇ ) A − ( A ⋅ ∇ ) B = ( ∇ ⋅ B + B ⋅ ∇ ) A − ( ∇ ⋅ A + A ⋅ ∇ ) B = ∇ ⋅ ( B A T ) − ∇ ⋅ ( A B T ) = ∇ ⋅ ( B A T − A B T )

Curl of the gradient

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

∇ × ( ∇ ϕ ) = 0

Divergence of the curl

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

∇ ⋅ ( ∇ × A ) = 0

Divergence of the gradient

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

∇ 2 ψ = ∇ ⋅ ( ∇ ψ )

Note that the result is a scalar quantity.

Curl of the curl

∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A

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

Addition and multiplication

A + B = B + A

A ⋅ B = B ⋅ A

A × B = − B × A

( A + B ) ⋅ C = A ⋅ C + B ⋅ C

( A + B ) × C = A × C + B × C

A ⋅ ( B × C ) = B ⋅ ( C × A ) = C ⋅ ( A × B ) (scalar triple product)

A × ( B × C ) = ( A ⋅ C ) B − ( A ⋅ B ) C (vector triple product)

( A × B ) × C = ( A ⋅ C ) B − ( B ⋅ C ) A (vector triple product)

( A × B ) ⋅ ( C × D ) = ( A ⋅ C ) ( B ⋅ D ) − ( B ⋅ C ) ( A ⋅ D )

( A ⋅ ( B × C ) ) D = ( A ⋅ D ) ( B × C ) + ( B ⋅ D ) ( C × A ) + ( C ⋅ D ) ( A × B )

( A × B ) × ( C × D ) = ( A ⋅ ( B × D ) ) C − ( A ⋅ ( B × C ) ) D

Gradient

∇ ( ψ + ϕ ) = ∇ ψ + ∇ ϕ

∇ ( ψ ϕ ) = ϕ ∇ ψ + ψ ∇ ϕ

∇ ( A ⋅ B ) = ( A ⋅ ∇ ) B + ( B ⋅ ∇ ) A + A × ( ∇ × B ) + B × ( ∇ × A )

Divergence

∇ ⋅ ( A + B ) = ∇ ⋅ A + ∇ ⋅ B

∇ ⋅ ( ψ A ) = ψ ∇ ⋅ A + A ⋅ ∇ ψ

∇ ⋅ ( A × B ) = B ⋅ ( ∇ × A ) − A ⋅ ( ∇ × B )

Curl

∇ × ( A + B ) = ∇ × A + ∇ × B

∇ × ( ψ A ) = ψ ∇ × A + ∇ ψ × A

∇ × ( A × B ) = A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) + ( B ⋅ ∇ ) A − ( A ⋅ ∇ ) B

Second derivatives

∇ ⋅ ( ∇ × A ) = 0

∇ × ( ∇ ψ ) = 0

∇ ⋅ ( ∇ ψ ) = ∇ 2 ψ (scalar Laplacian)

∇ ( ∇ ⋅ A ) − ∇ × ( ∇ × A ) = ∇ 2 A (vector Laplacian)

∇ ⋅ ( ϕ ∇ ψ ) = ϕ ∇ 2 ψ + ∇ ϕ ⋅ ∇ ψ

ψ ∇ 2 ϕ − ϕ ∇ 2 ψ = ∇ ⋅ ( ψ ∇ ϕ − ϕ ∇ ψ )

∇ 2 ( ϕ ψ ) = ϕ ∇ 2 ψ + 2 ∇ ϕ ⋅ ∇ ψ + ψ ∇ 2 ϕ

∇ 2 ( ψ A ) = A ∇ 2 ψ + 2 ( ∇ ψ ⋅ ∇ ) A + ψ ∇ 2 A

∇ 2 ( A ⋅ B ) = A ⋅ ∇ 2 B − B ⋅ ∇ 2 A + 2 ∇ ⋅ ( ( B ⋅ ∇ ) A + B × ∇ × A ) (Green's vector identity)

Third derivatives

∇ 2 ( ∇ ψ ) = ∇ ( ∇ ⋅ ( ∇ ψ ) ) = ∇ ( ∇ 2 ψ )

∇ 2 ( ∇ ⋅ A ) = ∇ ⋅ ( ∇ ( ∇ ⋅ A ) ) = ∇ ⋅ ( ∇ 2 A )

∇ 2 ( ∇ × A ) = − ∇ × ( ∇ × ( ∇ × A ) ) = ∇ × ( ∇ 2 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):

∂ V A ⋅ d S = ∭ V ( ∇ ⋅ A ) d V (Divergence theorem)

∂ V ψ d S = ∭ V ∇ ψ d V

∂ V ( n ^ × A ) d S = ∭ V ( ∇ × A ) d V

∂ V ψ ( ∇ φ ⋅ n ^ ) d S = ∭ V ( ψ ∇ 2 φ + ∇ φ ⋅ ∇ ψ ) d V (Green's first identity)

∂ V [ ( ψ ∇ φ − φ ∇ ψ ) ⋅ n ^ ] d S = ∂ V [ ψ ∂ φ ∂ n − φ ∂ ψ ∂ n ] d S = ∭ V ( ψ ∇ 2 φ − φ ∇ 2 ψ ) d V (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):

∮ ∂ S A ⋅ d ℓ = ∬ S ( ∇ × A ) ⋅ d s   (Stokes' theorem)

∮ ∂ S ψ d ℓ = ∬ S ( n ^ × ∇ ψ ) d S

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):

∂ S A ⋅ d ℓ = − ∂ S A ⋅ d ℓ .