Vector algebra application

This note will contain some basic relations from vector algebra and their application in vector calculus

Double cross product #

$$ \begin{align*} \vec a \times (\vec b \times \vec c) = (\vec a \cdot \vec c)\vec b-(\vec a \cdot \vec b)\vec c \end{align*} $$

Curl of Curl of a vector field #

This is not a proof using the Grassmann identity, but just a way to remember it.

$$ \begin{align*} \vec \nabla \times (\vec \nabla \times \vec v) = \vec \nabla (\vec \nabla \cdot \vec v) - \vec \nabla ^{2}\vec v \end{align*} $$

Curl of curl is equal to gradient of divergence minus laplacian

Scalar triple product #

$$ \begin{align*} \vec a \cdot (\vec b \times \vec c)=\vec b \cdot (\vec c \times \vec a) = \vec c \cdot (\vec a \times \vec b)\\ \text{[a b c]=[b c a]=[c a b]} \end{align*} $$

zero when operating twice #

Divergence of Curl #

$$ \nabla \cdot (\nabla \times F) =0 $$

Curl of gradient #

$$ \begin{align*} \nabla \times (\nabla f) = \vec 0 \end{align*} $$