# Exact differentials

1. Jan 29, 2013

### Zondrina

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

Let R be a connected open region ( in the plane ). Suppose that F = (M,N) is a vector function defined on R and is such that for any ( piecewise smooth ) curve C in R :

$\int_C Fdp$

depends on only the endpoints of C ( that is, any two curves from P1 to P2 in R give the same value for the integral).

Prove that there exists a function u(x,y) defined on R such that ∇u = F.

( i.e ux = M and uy = N )

2. Relevant equations

Err I think this may have to do with simply connected regions?

3. The attempt at a solution

I'm not quite sure where to start with this one? I'm having trouble seeing how the info provided leads to what I need.

I think it has to do with if R is a simply connected open region and Mdx + Ndy is such that My = Nx in R, then the differential is exact.

Any push in the right direction would be great.