Stoke's Theorem

Question

Verify Stoke's Theorem for ->F=(xy+x^2)i + (x^2+y^2)j over the square bounded by the lines x = -1, x = 1, y = -1 and y = 1.

What is the final answer?

Answer 1

The final answer is zero. This is most easily seen by working out that the curl of F is

( 2x - y) k. Since the normal to the square surface is k, the surface integral of the curl dot k is

simply the surface integral ∫∫ (2x -y.) dx dy or ∫∫ 2x dx dy - ∫∫ y dx dy. Both of these latter two evaluate to zero.

The contour integral around the perimeter of the square is the sum of four line integrals. The contribution of the top and bottom edges cancel as do the contributions of the left and right edges. . This also results in zero.