An open source physics library

create new user


Username:
Password:

forget your password?

Main Menu

Sections

Encyclopedia

Papers

Books

lectures

Talkback

Polls

Forums

Bug Reports

Feedback

Downloads

Snapshots

Information

Legalese

About

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

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