Answer
See proof
Work Step by Step
Step 1: \[ \vec{F}(x,y,z) = f(x,y)\hat{i} + g(x,y)\hat{j} + 0\hat{k} \] Step 2: \[ \text{curl} \, \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ f & g & 0 \end{vmatrix} = \left(\frac{\partial f}{\partial x} - \frac{\partial g}{\partial y}\right)\hat{k} \] Then, \[ \iint \vec{\nabla} \times \vec{F} \cdot \hat{k} \, dA = \iint \left(\frac{\partial f}{\partial x} - \frac{\partial g}{\partial y}\right) \, dA \] Using Green's Theorem: \[ \iint_C \vec{F} \cdot d\vec{r} = \iint_R \vec{\nabla} \times \vec{F} \cdot \hat{k} \, dA = \iint_R \left(\frac{\partial f}{\partial x} - \frac{\partial g}{\partial y}\right) \, dA \]
