# Showing the equivalence of two integrals

1. Oct 12, 2011

### aftershock

1. The problem statement, all variables and given/known data

Show for the z-component of the curl of a vector function that the integral off this over an infinitesimal rectangle in the x-y plane is equal to the contour integral of the original vector function around the perimeter.

2. Relevant equations

Short of explaining how to do line and surface integrals I'm not sure there's relevant equations.

3. The attempt at a solution

I wrote the vector function as M$\hat{i}$ +N$\hat{j}$ +P$\hat{k}$

Where M,N, and P are functions of both x and y (z is held constant since we're on the x-y plane.

I get 4 separate integrals, starting from the bottom and going counter clockwise call the 4 terms

M + N +M' + N' (primes here have nothing to do with derivatives)

When I take the z-component of the curl and do a double integral dy dx I get only the two terms M +N

Do M' + N' just somehow cancel out?