Chapter 15.4, Problem 26ES

Suppose that $\text{F}\left(x,y,\right)=f\left(x,y\right)\text{i}+g\left(x,y\right)\text{j}$ is a vector field on the $xy\text{-plane}$ and that f and g have continuous first partial derivatives with $f_y=g_x$ everywhere. Use Green’s Theorem to explain why ${\displaystyle {\int }_{C_1}\text{F}\cdot d\text{r}={\displaystyle {\int }_{C_2}\text{F}\cdot d\text{r}}}$ where $C_1\text{ and }{C}_2$ are the oriented curves in the accompanying figure. [Note: Compare this result with Theorems 15.3.2 and 15.3.3.]