# 12.3: Vector Identities


## Algebraic Identities

\begin{align} \mathbf { A } \cdot ( \mathbf { B } \times \mathbf { C } ) &= \mathbf { B } \cdot ( \mathbf { C } \times \mathbf { A } ) = \mathbf { C } \cdot ( \mathbf { A } \times \mathbf { B } ) \\[5pt] \mathbf { A } \times ( \mathbf { B } \times \mathbf { C } ) &= \mathbf { B } ( \mathbf { A } \cdot \mathbf { C } ) - \mathbf { C } ( \mathbf { A } \cdot \mathbf { B } ) \end{align} \nonumber

## Identities Involving Differential Operators

\begin{align} \nabla \cdot ( \nabla \times \mathbf { A } ) &= 0\\[5pt] \nabla \times ( \nabla f ) &= 0\\[5pt] \nabla \times ( f \mathbf { A } ) &= f ( \nabla \times \mathbf { A } ) + ( \nabla f ) \times \mathbf { A }\\[5pt] \nabla \cdot ( \mathbf { A } \times \mathbf { B } ) &= \mathbf { B } \cdot ( \nabla \times \mathbf { A } ) - \mathbf { A } \cdot ( \nabla \times \mathbf { B } )\\[5pt] \nabla \cdot ( \nabla f ) &= \nabla ^ { 2 } f\\[5pt] \nabla \times \nabla \times \mathbf { A } &= \nabla ( \nabla \cdot \mathbf { A } ) - \nabla ^ { 2 } \mathbf { A }\\[5pt] \nabla ^ { 2 } \mathbf { A }& = \nabla ( \nabla \cdot \mathbf { A } ) - \nabla \times ( \nabla \times \mathbf { A } )\end{align} \nonumber

## Divergence Theorem

Given a closed surface $${\mathcal S}$$ enclosing a contiguous volume $${\mathcal V}$$, $\int_{\mathcal V} \left( \nabla \cdot {\bf A} \right) dv = \oint_{\mathcal S} {\bf A}\cdot d{\bf s} \nonumber$ where the surface normal $$d{\bf s}$$ is pointing out of the volume.

## Stokes’ Theorem

Given a closed curve $${\mathcal C}$$ bounding a contiguous surface $${\mathcal S}$$, $\int_{\mathcal S} \left( \nabla \times {\bf A} \right) \cdot d{\bf s} = \oint_{\mathcal C} {\bf A}\cdot d{\bf l} \nonumber$ where the direction of the surface normal $$d{\bf s}$$ is related to the direction of integration along $${\mathcal C}$$ by the “right hand rule.”

This page titled 12.3: Vector Identities is shared under a CC BY-SA license and was authored, remixed, and/or curated by Steven W. Ellingson (Virginia Tech Libraries' Open Education Initiative) .