Question

Determine an example of a vector field that would yield a positive value for a line integral around a circle that is traversed once clockwise for any nonzero radius and explain how you know your vector field is correct.

Answer 1

Let us assume a general vector field denoted by the following function

Here assume that u and v have continous partial derivatives.

Let C be a circle of radius r around which we want to find the line integral of above vector field.

According to Green's Theorem,

If C is a positively oriented, piecewise smooth, simple closed curve in a plane, and and R is the region bounded by it . If u and v are functions of (x,y) defined on an open region containing R and have continuous partial derivatives there, then

where dA is the area element of the region enclosed by the Curve C

In our case, C is a smooth circle with non-zero radius r

Line integral of vector field F over the Circle C can be denoted as

(condition that the line integral is positive)

as we are moving clockwise , the limit on theta is from 0 to -2 pi

For any vector field which satisfies above condition will have a positive value of line integral around a circle.

Let us take an example:

Here, which satisfies (1)