- #1

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## 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 :

[itex]\int_C Fdp[/itex]

depends on only the endpoints of C ( that is, any two curves from P

_{1}to P

_{2}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.