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 counterclockwise 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.