﻿ Curl

## Curl

 Figure 1
The curl is a vector. It is a measure of the circulation of a vector field. The component of the curl of the vector H in the z direction normal to the surface is defined as
 (1)
where , is the area of the rectangular loop lying in the xy plane shown in Figure 1.

Since the distances are small, when y changes to , the value of the x component of H changes to . Integrating H around the rectangular loop shown in Figure 1 in the counter-clockwise direction,

 (2)
Dividing Equation 2 by gives the z component of the curl of H .
 (3)
The x and y components of the curl are found in a similar manner.
 (4)
The curl can also be expressed as a determinant.
 (5)
Equation 6 is the integral form of Faraday's law.
 (6)
Replacing H by E in Equation 1. Plug Equation 6 into Equation 1. Then consider a surface to be divided into many small surfaces. Apply Equation 1 to each of the small surfaces and sum over all the small surfaces. This results in the following differential form for Faraday's law,
 (7)

Also, since

 (8)
It follows that
 (9)
Equation 9 is the differential form of Ampere's law.