Integration by parts of non-parametric partial derivative

Question

Let $A(x,y,z)$ be some smooth, but wiggly function on $R^3$. Let $B(x,y,z) = \frac{\partial A}{\partial z}$. Let $x,y \in\Gamma$, which is a simple 2D surface. In my case, it is a planar triangle. I need to calculate the following integral

I = $\iint_\Gamma B(x,y,0)dxdy$

I will be calculating this integral numerically. I don't want to take derivatives of $A$, because numerically, $A$ is much better behaved within $\Gamma$ than $B$, and hence the integral of $B$ will converge much slower than the integral for $A$. It would be great to apply a Green's theorem to convert the above integral into an integral of $A$ over $\Gamma$ and its perimeter $\partial \Gamma$.

Do you think it is possible and how?

Answer 1

Green's theorem says that $\int_{\partial A} P(x,y)dx+ Q(x,y)dy= \int_A \left(\frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}\right)dxdy$

Here you have $\int_A B(x, y, 0)dA$. In order to write that as $\int_A \left(\frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}\right)dxdy$ you will need to find function P and Q such that $\frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}= B(x, y, 0)$. Whether or not you can do that depends upon B and your creativity.