Answer
See proof.
Work Step by Step
Step 1: \[ \vec{F}(x,y,z) = f(x,y)\hat{i} + g(x,y)\hat{j} + 0\hat{k} \] And, \[ \vec{\nabla} \cdot \vec{F} = \frac{\partial f}{\partial y} = \frac{\partial g}{\partial x} \] Step 2: From the given figure, the boundary of the region \(R\) is \(\partial R = C_1 - C_2\), then \[ \iint_{\partial R} \vec{F} \cdot d\vec{r} - \iint_{C_2} \vec{F} \cdot d\vec{r} = \iint_{C_1 - C_2} \vec{F} \cdot d\vec{r} = \iint_{\partial R} \vec{F} \cdot d\vec{r}, \] using Green's Theorem. \[ = \iint_R (\frac{\partial g}{\partial x} - \frac{\partial f}{\partial y}) \, dA \quad \text{since } \frac{\partial f}{\partial y} = \frac{\partial g}{\partial x} = 0. \] Hence, \[ \iint_{C_1} \vec{F} \cdot d\vec{r} = \iint_{C_2} \vec{F} \cdot d\vec{r} \]
