We can get a physical feel for what the circulation integral represents if
we look at two situations: one where there is uniform flow, and one where
the flow is rotating.
Here it is clear that the circulation is 0, as there is no flow around the
path.
Here the circulation is greater than 0, since the flow clearly has a
component along the path. By convention, the circulation is calculated
around the path in counterclockwise direction. Flow in a clockwise
direction would yield a negative circulation.
The Flux Integral
The flux integral measures flow across a path C.The velocity of a fluid perpendicular to a path is F(dot)n, where n is the unit normal to the path C.
The flux integral is 
Recall from before that T = ( dx/dt i +
dy/dt j ) / ||v|| = v / ||v||
n = T x k = ( ( dx/dt i + dy/dt j ) x
k ) / ||v||
= ( dy/dt i - dx/dt j ) / ||v||
Alternatively, n can be found by inspection. For example, Given a
semicircle as below, we can easily pick out the normal vectors
indicated.
An alternative method of computing the flux integral is as follows:
Pictorially the flux can be looked at in the following way:
With fluid leaving the area enclosed by the path, the flux is > 0.
With fluid entering the area enclosed, the flux < 0
If fluid flow is uniform, as much fluid enters the area as exits, and the
flux is 0.
Example:
Find the Circulation and Flux for F = xi + yj and C is
a semicircle:
We can do the integral in two parts - over C1, the
circular portion of the path, and over C2, the flat
part of the curve along the x-axis.
Paramatrizing the Curves:
C1: r1(t) = a cos t i
+ a sin t j; 0<t<Pi
C2: r2(t) = ati; -1<t<1
On C1: T1 = v(t) / ||v(t)|| = - a sin ti + a cos t
j / ||v||
n1 = a cos t i + a sin t j /
||v||
On C2: T2 = i
n2 = -j
Note that the circulation was zero, as expected for a radial field - there
is no flow around the path, and the flux was positive, as expected for a
field expanding outward from the area enclosed by the path.