Solve the Problem $\int _ { C } \frac { \partial f } { \partial y } d x - \frac { \partial f } { \partial x } d y = 0$

question 125

Essay

Solve the problem.
-Assuming all the necessary derivatives exist, show that if $\int _ { C } \frac { \partial f } { \partial y } d x - \frac { \partial f } { \partial x } d y = 0$ for all closed curves C to which Green's Theorem applies, then $f$ satisfies the Laplace equation $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } = 0$ for all regions bounded by closed curves $\mathrm { C }$ to which Green's Theorem applies.

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