Curl Formula in Different Coordinate Systems | The Right Gate

# Curl formula in different coordinate systems.

Curl of a vector field is one of the basic operations in the study of Electromagnetics. This article discusses the representation of the curl formula in different coordinate systems viz. Cartesian, Cylindrical and Spherical.

## What is the curl of a vector field?

Technically, the curl of a vector field is a vector whose magnitude is the maximum circulation of the given field per unit area and direction is normal to the area when it is oriented for maximum circulation.

In simple words, the curl of a vector field at any point gives us an idea about the whirling nature of the field at that point. More is the curl, more will be the whirling nature at that point. It also indicates the direction along which the whirling is maximum at that point. Check out the following article for the detailed definition of the Curl.

## Curl Formula in different Coordinate Systems

Before quoting the curl formula in different coordinate systems viz. Cartesian, Cylindrical and Spherical, have a look at an intuitive proof for the same.

#### Curl Formula in Cartesian Coordinate System

Let the vector field is A whose curl operation is to be calculated. Then A would have the standard form as follows – $\overrightarrow A=A_x{\widehat a}_x+A_y{\widehat a}_y+A_z{\widehat a}_z$. Then curl is defined as follows: –

#### Curl Formula in Cylindrical Coordinate System

If A is the vector field whose curl operation is to be calculated, then for cylindrical coordinates, it would have the standard form as follows – $\overrightarrow A=A_\rho\widehat a\rho+A\phi{\widehat a}_\phi+A_z{\widehat a}_z$ Then curl is defined as follows: –

#### Curl Formula in Spherical Coordinate System

For spherical coordinates, A would have the standard form as follows – $\overrightarrow A=A_r{\widehat a}_r+A_\theta{\widehat a}_\theta+A_\phi{\widehat a}_\phi$ Then curl is defined as follows: –

Suggested Community: Electromagnetics for GATE & ESE