Verifying Solution of 3-D Laplace Eq. u=1/(x^2+y^2+z^2)^2

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Homework Help Overview

The discussion revolves around verifying whether the function u=1/(x^2 + y^2 + z^2)^2 is a solution to the 3-dimensional Laplace equation, specifically examining the conditions under which the sum of the second partial derivatives equals zero.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the computation of second derivatives and question the conditions under which their sum can equal zero. There is a discussion about potential misinterpretations of the function's form and the implications of the derivatives being equal and positive.

Discussion Status

Participants are actively engaging in calculating the second derivatives and discussing their findings. Some have provided alternative expressions for the derivatives, while others are questioning the assumptions about the positivity of the individual terms. There is no explicit consensus yet on the correctness of the derivatives or the overall solution.

Contextual Notes

There is confusion regarding the correct form of the function, with multiple participants suggesting different exponents. The discussion also highlights the need to clarify the behavior of the derivatives and their contributions to the sum.

fk378
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Homework Statement


Verify that the function u=1/(x^2 + y^2 + z^2)^2 is a solution of the 3-dimensional Laplace equation uxx+uyy+uzz=0



The Attempt at a Solution


I know how to solve the partial derivatives, so I know that uxx=uyy=uzz for this problem. How can their sum equal 0?
 
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are you sure its not ^3/2?
 
lzkelley said:
are you sure its not ^3/2?

You mean ^(1/2), yes? 1/r is the Green's function for the Laplace equation.
 
Ah, yes, the function should read:
u=1/(x^2 + y^2 + z^2)^(1/2)

Can you explain how the sum of the partial derivatives should equal zero, if their individual expressions are equal and positive?
 
fk378 said:
Ah, yes, the function should read:
u=1/(x^2 + y^2 + z^2)^(1/2)

Can you explain how the sum of the partial derivatives should equal zero, if their individual expressions are equal and positive?

You said you know you to find the second derivatives. Then do it. The individual expression aren't 'equal and positive'. Tell me what is the second derivative u_xx? It has two terms which cancel when summed over x,y and z.
 
For u_xx I'm getting 3(x^2 + y^2 + z^2)^-(5/2)
 
That's not what I get. I get an 'x' in the numerator after the first derivative coming from the chain rule. When I apply the quotient rule to that to get the second derivative I get two terms.
 
Ok, for my u_xx I now get -(x^2 + y^2 + z^2)^(-3/2) + 3(x^2) (x^2 + y^2 + z^2)^(-5/2)
 
Right. Now sum over x, y and z.
 

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