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Vector Identities

(Definition)

It is difficult to get anywhere in physics without a firm understanding of vectors and their common operations. Here, we will give vector identities as a reference. Basic terminology to keep straight.

Operation	Symbol
Gradient	$\nabla f$
Laplacian	$\nabla^2$
divergence	$\nabla \cdot$
curl	$\nabla \times$

Vector Magnitude

$A = \left \vert \mathbf{A} \right \vert = \sqrt{{A_x}^2 + {A_y}^2 + {A_z}^2 }$
$A = \sqrt{\mathbf{A} \cdot \mathbf{A}}$

scalar product (Dot Product)

$\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z$
$\mathbf{A} \cdot \mathbf{B} = \left \vert \mathbf{A} \right \vert \left \vert \mathbf{B} \right \vert \cos \theta$

vector product (Cross Product)

$\mathbf{A} \times \mathbf{B} = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{... ...\right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$

It can be easier to remember with determinant formulation

$\mathbf{A} \times \mathbf{B} = \left\vert \begin{matrix} \mathbf{\hat{i}} & \ma... ...\right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$

Vector Triple Product, aka. BAC CAB

$\mathbf{A} \times \left ( \mathbf{B} \times \mathbf{C} \right ) = \mathbf{B} \l... ...t \mathbf{C} \right ) - \mathbf{C} \left ( \mathbf{A} \cdot \mathbf{B} \right)$

scalar Triple Product

$\mathbf{A} \cdot \left ( \mathbf{B} \times \mathbf{C} \right ) = \mathbf{B} \cd... ...f{A} \right ) = \mathbf{C} \cdot \left ( \mathbf{A} \times \mathbf{B} \right )$

Gradient

$\nabla f = \frac{\partial f}{\partial x} \mathbf{\hat{i}} + \frac{\partial f}{\partial y} \mathbf{\hat{j}} + \frac{\partial f}{\partial z} \mathbf{\hat{k}}$

Gradient identities

$\nabla \left ( f + g \right ) = \nabla f + \nabla g$
$\nabla \left ( \alpha f \right ) = \alpha \nabla f$
$\nabla \left ( f \, g \right ) = f \nabla g + g \nabla f$
$\nabla \left ( f/g \right ) = \frac{\left ( g \nabla f - f \nabla g \right )}{g^2}$

Divergence

$\nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z}$

Divergence of the cross product

$\nabla \cdot \left ( \mathbf{A} \times \mathbf{B} \right ) = \mathbf{B} \cdot \... ...athbf{A} \right ) - \mathbf{A} \cdot \left ( \nabla \times \mathbf{B} \right )$