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}) = 
abla \cdot \mathbf{v}

and is a scalar field.

Divergence of a tensor

For a tensor

\mathbf{A}

, divergence is generally written as

:

\operatorname{div}(\mathbf{A}) = 
abla \cdot \mathbf{A}

and is a vector.

Curl

For a vector field

\mathbf{v}

, curl is generally written as

:

\operatorname{curl}(\mathbf{v}) = 
abla 	imes \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}) = 
abla \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) = 
abla \psi

and is a vector field.

COMBINATIONS OF MULTIPLE OPERATORS

Curl of the gradient

The Curl of the Gradient of ''any'' Scalar Field

\ \psi

is always zero:

:

abla 	imes ( 
abla \psi )  = 0

Divergence of the curl

The Divergence of the curl of ''any'' Vector Field

\ \mathbf{A}

is always zero:
:

abla \cdot ( 
abla 	imes \mathbf{A} ) = 0

Divergence of the gradient

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

abla \cdot (
abla \psi) = 
abla^2 \psi

Note that the result is a scalar quantity.

Curl of the curl

:

abla 	imes 
abla 	imes \mathbf{A} = 
abla(
abla \cdot \mathbf{A}) - 
abla^{2}\mathbf{A}

PROPERTIES

Distributive property

:

abla \cdot ( \mathbf{A} + \mathbf{B} ) = 
abla \cdot \mathbf{A} + 
abla \cdot \mathbf{B}

abla 	imes ( \mathbf{A} + \mathbf{B} ) = 
abla 	imes \mathbf{A} + 
abla 	imes \mathbf{B}

Vector dot product

:

abla(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot 
abla)\mathbf{B} + (\mathbf{B} \cdot 
abla)\mathbf{A} + \mathbf{A} 	imes (
abla 	imes \mathbf{B}) + \mathbf{B} 	imes (
abla 	imes \mathbf{A})

Vector cross product

:

abla \cdot (\mathbf{A} 	imes \mathbf{B}) = \mathbf{B} \cdot 
abla 	imes \mathbf{A} - \mathbf{A} \cdot 
abla 	imes \mathbf{B}

abla 	imes (\mathbf{A} 	imes \mathbf{B}) = \mathbf{A} (
abla \cdot \mathbf{B}) - \mathbf{B} (
abla \cdot \mathbf{A}) + (\mathbf{B} \cdot 
abla) \mathbf{A} - (\mathbf{A} \cdot 
abla) \mathbf{B}

Product of a scalar and a vector

:

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

abla 	imes (\psi\mathbf{A}) = \psi
abla 	imes \mathbf{A} - \mathbf{A} 	imes 
abla\psi

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 .
:

abla (\psi \, \phi) = \phi \,
abla \psi  + \psi \,
abla \phi

MORE IDENTITIES

:

rac{1}{2} 
abla A^2 = \mathbf{A} 	imes (
abla 	imes \mathbf{A}) + (\mathbf{A} \cdot 
abla) \mathbf{A}

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Vector Calculus Identities