Vector calculus identities

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The following identities are important in vector calculus:

[edit] Single Operators (Summary)

This section explicitly lists what some symbols mean for clarity.

[edit] Divergence

[edit] Divergence of a Vector Field

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

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

and is a scalar field.

[edit] Divergence of a Tensor

For a Tensor $\mathbf{A}$ , Divergence is generally written as

$\nabla \cdot \mathbf{A} = div(\mathbf{A})$

and is a vector.

[edit] Curl

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

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

and is a vector field.

[edit] Gradient

[edit] Gradient of a Vector Field

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

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

and is a tensor

[edit] Gradient of a Scalar Field

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

$\nabla \psi = grad(\psi)$

and is a vector field.

[edit] Combinations of multiple operators

[edit] Curl of the gradient

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

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

[edit] Divergence of the curl

The divergence of the curl of any vector field $\ \mathbf{A}$ is always zero:

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

[edit] 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.

[edit] Curl of the curl

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

[edit] Properties

[edit] Distributive property

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

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

[edit] Vector dot product

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

[edit] Vector cross product

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

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