Suitable change of variables for this triple integral?

[LHS]

Homework Statement

[PLAIN]http://img542.imageshack.us/img542/5600/unledsn.png

Homework Equations

The Attempt at a Solution

The first part is fine, just struggling to find a change of variables that'll help, tried spherical due to the x^2+y^2+z^2, didn't help enormously
Thanks!

(from sheet 3, not take home test http://www.maths.ox.ac.uk/courses/course/12489/material)

 

[Wingeer]

Hm. This looks really convoluted. Have you had any linear algebra? More spesific; Conic sections?

 

[LHS]

Hmm.. yes I have covered some of that, but nothing I can see that'll help in this situation.
the question seems to be wanting you to use a Jacobian etc.. maybe involving the first part

 

[Wingeer]

I am not sure as you are integrating over the whole domain of R3.

 

[LHS]

-(x+y-z)^2-(x-y+z)^2-(-x+y+z)^2

Then let r=x+y-z s=... seems to work! Thanks though!

 

[Wingeer]

Aah! A very nice observation! I knew there would be some algebra involved.

 

[LHS]

Thanks :) It is annoying when they set questions where there just seems to be some arbitrary factorization which you can spend hours on and learn relatively little from. Have a good day!

 

[Ray Vickson]

Thanks :) It is annoying when they set questions where there just seems to be some arbitrary factorization which you can spend hours on and learn relatively little from. Have a good day!

Actually, the questioner is asking you to "discover" a well-known and completely
standard method of dealing with such problems. After all your struggles, you now
know a valuable tool: finding and using the canonical form of a quadratic.

RGV