Solve the Problem $\mathrm { M } = \frac { \mathrm { y } } { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } }$

Essay

Solve the problem.
-Let $\mathrm { M } = \frac { \mathrm { y } } { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } }$ and $\mathrm { N } = \frac { - \mathrm { x } } { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } }$ . Show that $\int _ { \mathrm { C } } \mathrm { Mdx } + \mathrm { Ndy } z \int _ { \mathrm { R } } \left( \frac { \partial \mathrm { N } } { \partial \mathrm { x } } - \frac { \partial \mathrm { M } } { \partial \mathrm { y } } \right) \mathrm { dx } \mathrm { dy }$ , where $\mathrm { R }$ is the region bounded by the unit circle $C$ centered at the origin. Why is Green's Theorem failing in this case?