Definitions

# List of vector identities

## Triple products

• $vec\left\{A\right\} times \left(vec\left\{B\right\} times vec\left\{C\right\}\right) = \left(vec\left\{C\right\} times vec\left\{B\right\}\right) times vec\left\{A\right\} = vec\left\{B\right\}\left(vec\left\{A\right\} cdot vec\left\{C\right\}\right) - vec\left\{C\right\}\left(vec\left\{A\right\} cdot vec\left\{B\right\}\right)$
• $vec\left\{A\right\}cdot\left(vec\left\{B\right\}times vec\left\{C\right\}\right) = vec\left\{B\right\}cdot\left(vec\left\{C\right\}times vec\left\{A\right\}\right) = vec\left\{C\right\}cdot\left(vec\left\{A\right\}times vec\left\{B\right\}\right)$

these can be proved by giving arbitrary components to $A,$ $B,$ and $C,$

$vec\left\{A\right\} = \left(a_x, a_y, a_z\right)$

$vec\left\{B\right\} = \left(b_x, b_y, b_z\right)$

$vec\left\{C\right\} = \left(c_x, c_y, c_z\right)$

then finding the values of each statement, such as $vec\left\{B\right\}times vec\left\{C\right\}$, in terms of the generic components will show that both sides of the equation are equal.

## Other products

• $\left(vec\left\{A\right\} times vec\left\{B\right\}\right) cdot \left(vec\left\{A\right\} times vec\left\{B\right\}\right) = A^2 B^2 - \left(vec\left\{A\right\} cdot vec\left\{B\right\}\right)^2 = vec\left\{B\right\} cdot vec\left\{A\right\} times \left(vec\left\{B\right\} times vec\left\{A\right\}\right)$ (note: this is another form of Lagrange's formula).

## Product rules

• $vec\left\{nabla\right\} \left(fg\right) = f\left(vec\left\{nabla\right\}g\right) + g\left(vec\left\{nabla\right\} f\right)$
• $vec\left\{nabla\right\}\left(vec\left\{A\right\} cdot vec\left\{B\right\}\right) = vec\left\{A\right\} times \left(vec\left\{nabla\right\} times vec\left\{B\right\}\right)+vec\left\{B\right\} times \left(vec\left\{nabla\right\} times vec\left\{A\right\}\right)+\left(vec\left\{A\right\} cdot vec\left\{nabla\right\}\right)vec\left\{B\right\}+\left(vec\left\{B\right\} cdot vec\left\{nabla\right\}\right)vec\left\{A\right\}$
• $vec\left\{nabla\right\} cdot \left(fvec\left\{A\right\}\right)=f\left(vec\left\{nabla\right\} cdot vec\left\{A\right\}\right)+vec\left\{A\right\} cdot \left(vec\left\{nabla\right\} f\right)$
• $vec\left\{nabla\right\} cdot \left(vec\left\{A\right\} times vec\left\{B\right\}\right)=vec\left\{B\right\} cdot \left(vec\left\{nabla\right\} times vec\left\{A\right\}\right)-vec\left\{A\right\} cdot \left(vec\left\{nabla\right\} times vec\left\{B\right\}\right)$
• $nablatimes \left(vec\left\{A\right\}timesvec\left\{B\right\}\right)= \left(vec\left\{B\right\}cdotnabla\right) vec\left\{A\right\}-\left(vec\left\{A\right\}cdotnabla\right)vec\left\{B\right\} + vec\left\{A\right\} \left(nablacdotvec\left\{B\right\}\right) - vec\left\{B\right\}\left(nablacdotvec\left\{A\right\}\right)$
• $nablatimes \left(vec\left\{A\right\}timesvec\left\{B\right\}\right)= vec\left\{A\right\} times \left(nablatimesvec\left\{B\right\}\right) - vec\left\{B\right\} times \left(nablatimesvec\left\{A\right\}\right) - \left(vec\left\{A\right\}timesnabla\right) times vec\left\{B\right\} + \left(vec\left\{B\right\}timesnabla\right) times vec\left\{A\right\}$
• $nablatimes \left(fvec\left\{A\right\}\right)=f\left(nablatimesvec\left\{A\right\}\right)+\left(nabla f\right)timesvec\left\{A\right\}$

## Green's first identity

• $vec\left\{nabla\right\} cdot left\left(f vec\left\{nabla\right\} f right\right) = f vec\left\{nabla\right\} cdot left\left(vec\left\{nabla\right\} f right\right) + left\left(vec\left\{nabla\right\} f cdot vec\left\{nabla\right\} f right\right) = f nabla^2 f + left\left(vec\left\{nabla\right\} f cdot vec\left\{nabla\right\} f right\right)$
therefore
$f nabla^2 f = vec\left\{nabla\right\} cdot left\left(f vec\left\{nabla\right\} f right\right) - left\left(vec\left\{nabla\right\} f cdot vec\left\{nabla\right\} f right\right)$

## Fundamental theorems

Divergence theorem

• $int_\left\{V\right\} \left(vec\left\{nabla\right\} cdot vec\left\{A\right\}\right) ,dv = oint_\left\{S\right\} vec\left\{A\right\} cdot dvec\left\{a\right\}$

• $int_\left\{S\right\} \left(nabla times vec\left\{A\right\}\right) cdot dvec\left\{a\right\} = oint_\left\{C\right\} vec\left\{A\right\} cdot dvec\left\{l\right\}$