Why does Green's theorem fail for this exact differential form $\int Mdx + Ndy$

Question

Why does Green's theorem fail for this exact differential form $\int {Mdx + Ndy} = 0$ since it's an exact differential and M and N both are not functions of z ( the case for which i want to ask the question). Green says this $$\int \int {\partial N/\partial X - \partial M/\partial Y}dA$$ which is equal to zero as this is an exact differential. If i do the line integral in some path to some point to some point then it is equal to the change of function between the final and initial points which may not be zero. So my question is, why does not Gauss' theorem become true for this?

Answer 1

A note on terminology: The theorem you refer to as Gauss's theorem is more commonly called Green's theorem. Gauss's theorem often refers to the divergence theorem (which is a generalazation of green's theorem, and thus both may be reffered to as Gauss's theorem).

Now, let us state Green's theorem (by wikipedia):

Let $C$ be a positively oriented, piecewise smooth, simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $M$ and $N$ are functions of $(x, y)$ defined on an open region containing $D$ and have continuous partial derivatives there, then $$ \int_C M\,dx + N\, dy = \iint_D \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right) \, dx\, dy $$ where the path of integration along $C$ is anticlockwise.

Note that the theorem requires the curve to be closed. Thus, you may not apply the theorem for curves that have different end points.