Finding the Origin of an Integral Identity | Numerical PDEs Homework

[Brian T]

Hey all,
As I was working on my numerical PDEs homework, an identity came up which we used to solve a problem. I was able to answer the question, but my question here is where does the identity come from (I figured it has something to do with analysis) ?

The identity is
The integral of $$v^2$$ over some region D is equivalent to the integral of $$v^2\Delta \phi$$ over D, where $$\phi = \frac{1}{2d}|x|^2$$

I haven't taken an analysis class so not too sure where this comes from

 

[S.G. Janssens]

I haven't taken an analysis class

As I was working on my numerical PDEs homework

I don't understand this.

 

[Brian T]

I don't understand this.

The reqs for the class are the standard calc/DE/lin. alg sequence, and I've also taken the PDE theory class so I thought it was worth a shot. I've been managing to learn the basic aspects of analysis and some of the major theorems along the way, but just not too sure where the identity comes from

 

[Brian T]

Actually, I think I figured it out. Abs(x) is just r, and the laplacian of r^2 is just a constant (2d), so that laplacian of phi is one, and hence the identity