The exercise asks
Use Green's Theorem in the form of equation 13 to prove Green's First Identity: $$ \iint\limits_D f \nabla^2 g \, dA = \oint_C f(\nabla g) \cdot \mathbf{n} \, ds - \iint\limits_D \nabla f \cdot \nabla g \, dA $$ where $D$ and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of $f$ and $g$ exist and are continuous.
Equation 13 is $$ \oint_C \mathbf{F} \cdot \mathbf{n} \, ds = \iint\limits_D \text{div} \; \mathbf{F}(x,y) \, dA $$
One approach I tried was adding the right-most integral to the left side, rewriting $f \nabla^2 g + \nabla f \cdot \nabla g$ as $\nabla \cdot (f \nabla g)$. I did this using the math below: $$ f \nabla^2 g + \nabla f \cdot \nabla g = f\frac{\partial^2 g}{\partial x^2} + f\frac{\partial^2 g}{\partial y^2} + f\frac{\partial^2 g}{\partial z^2} + \frac{\partial f}{\partial x} \frac{\partial g}{\partial x} +\frac{\partial f}{\partial y} \frac{\partial g}{\partial y} + \frac{\partial f}{\partial z} \frac{\partial g}{\partial z} \\ = f\frac{\partial^2 g}{\partial x^2} + \frac{\partial f}{\partial x} \frac{\partial g}{\partial x} + f\frac{\partial^2 g}{\partial y^2} + \frac{\partial f}{\partial y} \frac{\partial g}{\partial y} + f\frac{\partial^2 g}{\partial z^2} + \frac{\partial f}{\partial z} \frac{\partial g}{\partial z} \\ = \frac{\partial}{\partial x} \left[ f \frac{\partial g}{\partial x} \right] + \frac{\partial}{\partial y} \left[ f \frac{\partial g}{\partial y} \right] + \frac{\partial}{\partial z} \left[ f \frac{\partial g}{\partial z} \right] \\ = \nabla \cdot (f \nabla g) $$
From here my goal was to show that $ \iint_D \nabla \cdot (f \nabla g) \, dA = \oint_C f(\nabla g) \cdot \mathbf{n} \, ds = \iint_D \text{div} \; f(\nabla g) \, dA$, but then I got stuck here. Is there a way to continue using this approach or is there another, possibly more elegant way that this can be done?
Also, my highest level of math knowledge is just vector calculus (to be more specific, curl and divergence because that's what I'm studying at the moment), so please don't somehow find a way to explain the answer in terms of differential geometry or something, because it's not going to be useful. (This has happened numerous times)