Finding the Origin of an Integral Identity | Numerical PDEs Homework

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Discussion Overview

The discussion revolves around the origin of an integral identity encountered in a numerical PDEs homework context. Participants explore the relationship between the integral of \(v^2\) over a region and the integral of \(v^2\Delta \phi\), where \(\phi\) is defined as \(\frac{1}{2d}|x|^2\). The scope includes theoretical aspects related to analysis and its application in solving PDEs.

Discussion Character

  • Homework-related
  • Exploratory

Main Points Raised

  • One participant questions the origin of the integral identity, suggesting it may relate to analysis, despite lacking formal training in that area.
  • Another participant expresses confusion about the identity and its derivation, noting their background in calculus, differential equations, and linear algebra.
  • A later reply proposes a potential resolution by relating the absolute value function to a radial coordinate \(r\) and asserting that the Laplacian of \(r^2\) simplifies to a constant, leading to the conclusion that the Laplacian of \(\phi\) equals one.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the derivation of the integral identity, and while one participant suggests a possible explanation, no consensus is reached on the validity of that explanation.

Contextual Notes

Limitations include the participants' varying levels of familiarity with analysis, which may affect their understanding of the identity's derivation. The discussion does not resolve the mathematical steps involved in the identity.

Brian T
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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
 
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Brian T said:
I haven't taken an analysis class
Brian T said:
As I was working on my numerical PDEs homework
I don't understand this.
 
Krylov said:
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
 
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
 

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