Exact Differentials: Proving Existence of u(x,y) in Connected Open Region

[STEMucator]

Homework Statement

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 u_x = M and u_y = N )

Homework Equations

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

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 M_y = N_x in R, then the differential is exact.

Any push in the right direction would be great.