Vector Identities

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}} & \m...
...\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} \...
...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} \c...
...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 ) $

Divergence of the curl

$ \nabla \cdot \left ( \nabla \times \mathbf{A} \right ) = 0 $

Laplacian Identities

$ \nabla \times \left ( \nabla \times \mathbf{A} \right ) = \nabla \left ( \nabla \cdot \mathbf{A} \right ) - \nabla^2 \mathbf{A} $



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by bloftin.