Vector identities

A few helpful vector identities:

Triple Products

<math>\vec{A} \times (\vec{B} \times \vec{C}) = \vec{B}(\vec{A} \cdot \vec{C}) – \vec{C}(\vec{A} \cdot \vec{B})<math>
<math> \vec{A}\cdot(\vec{B}\times \vec{C}) = \vec{B}\cdot(\vec{C}\times \vec{A}) = \vec{C}\cdot(\vec{A}\times \vec{B})<math>

<math>\vec{\nabla} (fg) = f(\vec{\nabla} \cdot g) + g(\vec{\nabla} \cdot f)<math>
<math>\vec{\nabla}(\vec{A} \cdot \vec{B}) = \vec{A} \times (\vec{\nabla} \times \vec{B})+\vec{B} \times (\vec{\nabla} \times \vec{A})+(\vec{A} \cdot \vec{\nabla})\vec{B}+(\vec{B} \cdot \vec{\nabla})\vec{A} <math>
<math>\vec{\nabla} \cdot (f\vec{A})=f(\vec{\nabla} \cdot \vec{A})+\vec{A} \cdot (\vec{\nabla} f) <math>
<math>\vec{\nabla} \cdot (\vec{A} \times \vec{B})=\vec{B} \cdot (\vec{\nabla} \times \vec{A})-\vec{A} \cdot (\vec{\nabla} \times \vec{B}) <math>
<math>\nabla\times (\vec{A}\times\vec{B})= (\vec{B}\cdot\nabla) \vec{A}-(\vec{A}\cdot\nabla)\vec{B} + \vec{A} (\nabla\cdot\vec{B}) – \vec{B}(\nabla\cdot\vec{A})<math>
<math>\nabla\times (f\vec{A})=f(\nabla\times\vec{A})-\vec{A}\times(\nabla f)<math>

<math> \vec{\nabla} \cdot \left( f \vec{\nabla} f \right) = f \vec{\nabla} \cdot \left( \vec{\nabla} f \right) + \left( \vec{\nabla} f \cdot \vec{\nabla} f \right) = f \nabla^2 f + \left( \vec{\nabla} f \right)^2 <math>
therefore
<math> f \nabla^2 f = \vec{\nabla} \cdot \left( f \vec{\nabla} f \right) – \left( \vec{\nabla} f \right)^2<math>

<math>\int (\vec{\nabla} \cdot \vec{A}) \,dv = \oint_{S} \vec{A} \cdot d\vec{a}<math>

Stokes's Theorem:

<math> \oint_{C} \vec{A} \cdot d\vec{l}= \oint_{S} (\nabla \times \vec{A})\cdot \vec{n} \,da<math>