Vector Field Helper

Which coordinate system is vector field

V

defined in?

Let

V

be a vector field defined in Cartesian coordinates (

x

y

, and

z

). Let

V_{x}

V_{y}

, and

V_{z}

denote the

x

y

, and

z

vector components of

V

, respectively. If a vector component is equal to zero or some other scalar constant (e.g.,

V_{x} = 0

), then it is a function of no variables.

		Zero Terms
Which vector components is $V$ defined for?	$x$ $y$ $z$	$V_{x} = 0$	$V_{y} = 0$	$V_{z} = 0$
The component $V_{x}$ is a function of which (if any) variables:	$x$ $y$ $z$	$\frac{\partial V_{x}}{\partial x} = 0$	$\frac{\partial V_{x}}{\partial y} = 0$	$\frac{\partial V_{x}}{\partial z} = 0$
The component $V_{y}$ is a function of which (if any) variables:	$x$ $y$ $z$	$\frac{\partial V_{y}}{\partial x} = 0$	$\frac{\partial V_{y}}{\partial y} = 0$	$\frac{\partial V_{y}}{\partial z} = 0$
The component $V_{z}$ is a function of which (if any) variables:	$x$ $y$ $z$	$\frac{\partial V_{z}}{\partial x} = 0$	$\frac{\partial V_{z}}{\partial y} = 0$	$\frac{\partial V_{z}}{\partial z} = 0$
Show Zero Terms

The vectors $x$ , $y$ , $z$ denote the unit vectors in the $x$ , $y$ , and $z$ directions respectively.

In the equations below, zero terms are shown in BLUE.

Divergence of the Vector Field

\nabla \cdot V =

\frac{\partial V_{x}}{\partial x}

+

\frac{\partial V_{y}}{\partial y}

+

\frac{\partial V_{z}}{\partial z}

0

\nabla \times V =

x

(

\frac{\partial V_{z}}{\partial y}

-

\frac{\partial V_{y}}{\partial z}

)

+

y

(

\frac{\partial V_{x}}{\partial z}

-

\frac{\partial V_{z}}{\partial x}

)

+

z

(

\frac{\partial V_{y}}{\partial x}

-

\frac{\partial V_{x}}{\partial y}

)

0

Designed and Programmed by Aaron Gaba - April 2022

Updated in December 2022