I'm reading Introduction to Fourier Optics – J. Goodman and got to this statements which is referred to as Green's Theorem:
Let $U(P)$ and $G(P)$ be any two complex-valued functions of position, and let $S$ be a closed surface surrounding a volume $V$. If $U$, $G$, and their first and second partial derivatives are single-valued and continuous within and on $S$, then we have $$\iiint_V \Bigr(U \nabla^2 G – G \nabla^2 U\Bigr) dv = \iint_S \Bigr(U\frac{\partial G}{\partial n} -G\frac{\partial U}{\partial n}\Bigr)ds$$ where $\frac{\partial}{\partial n}$ signifies a partial derivative in the outward normal direction at each point on $S$.
I remembered to have learned about Green's Theorem in the mathematics courses but did not recall this form. I looked it up and got to what I remembered, that is a 2-dimensional case given by $$\iint_S \Bigr(\frac{\partial P}{\partial x} – \frac{\partial Q}{\partial y}\Bigr)dxdy=\int_C Qdx+ Pdy$$ which could be considered a particular case of Stokes' Theorem.
Are the two formulas related (which means I missed the link, in which case, can someone suggest some directions), or are they different formulas and the author in the book misfortunately used "Green's Theorem"?
Best Answer
I searched further and after some luck with the terms used in the search engine I got to what are called Green's Identities. The formula I had trouble with is the second identity.
Now that I found the "correct" name of it, I could look for links to the Stokes theorem and got to this material, so if someone has a similar problem, you can start from these.