Seemingly simple multiple integral substitution

  • #1

Homework Statement



This problem seems fine but has me stumped for some reason...

For the variables

u=x2−y2
v= x y

Under the condition u>0
We're simply asked to determine x(u,v) and y(u,v) so that we can eventually use them to solve a multiple integral by substitution.


Homework Equations





The Attempt at a Solution



I've tried a number of different methods - for example, trigonometric and hyperbolic substitutions for x and y, various linear and quadratic combinations of u and v, etc. - and for some reason I can't find a simple way of expressing x and y in terms of u and v. I noticed that if you let z=x+yi then R(z2)=u and I(z2)=2v, but I'm not sure how this helps. I know there's something really simple I must be overlooking, but any help would be greatly appreciated.

I can actually solve the integral they gave us - which is pretty easy - just by finding the jacobian d(u,v)/d(x,y) and using the fact that this is the inverse of the Jacobian d(x,y)/d(u,v) without ever explicitly determining the function in terms of u and v - doing it this way, nice factors in the integrand cancel out, which I'm guessing was the point of the question - but I still can't find x (u,v) and y (u,v).
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
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Homework Statement



This problem seems fine but has me stumped for some reason...

For the variables

u=x2−y2
v= x y

Under the condition u>0
We're simply asked to determine x(u,v) and y(u,v) so that we can eventually use them to solve a multiple integral by substitution.


Homework Equations





The Attempt at a Solution



I've tried a number of different methods - for example, trigonometric and hyperbolic substitutions for x and y, various linear and quadratic combinations of u and v, etc. - and for some reason I can't find a simple way of expressing x and y in terms of u and v. I noticed that if you let z=x+yi then R(z2)=u and I(z2)=2v, but I'm not sure how this helps. I know there's something really simple I must be overlooking, but any help would be greatly appreciated.

I can actually solve the integral they gave us - which is pretty easy - just by finding the jacobian d(u,v)/d(x,y) and using the fact that this is the inverse of the Jacobian d(x,y)/d(u,v) without ever explicitly determining the function in terms of u and v - doing it this way, nice factors in the integrand cancel out, which I'm guessing was the point of the question - but I still can't find x (u,v) and y (u,v).
Authors sometimes put problems like that in their homework problems. Your specific problem worked without you ever having to solve explicitly for x and y because the problem was cooked up to work that way. So you are correct; that was the point of the question. Don't waste your time trying to work it the hard way.
 
  • #3
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,317
1,007

Homework Statement



This problem seems fine but has me stumped for some reason...

For the variables

u=x2−y2
v= x y

Under the condition u>0
We're simply asked to determine x(u,v) and y(u,v) so that we can eventually use them to solve a multiple integral by substitution.


Homework Equations





The Attempt at a Solution



I've tried a number of different methods - for example, trigonometric and hyperbolic substitutions for x and y, various linear and quadratic combinations of u and v, etc. - and for some reason I can't find a simple way of expressing x and y in terms of u and v. I noticed that if you let z=x+yi then R(z2)=u and I(z2)=2v, but I'm not sure how this helps. I know there's something really simple I must be overlooking, but any help would be greatly appreciated.

I can actually solve the integral they gave us - which is pretty easy - just by finding the jacobian d(u,v)/d(x,y) and using the fact that this is the inverse of the Jacobian d(x,y)/d(u,v) without ever explicitly determining the function in terms of u and v - doing it this way, nice factors in the integrand cancel out, which I'm guessing was the point of the question - but I still can't find x (u,v) and y (u,v).
Solve v = xy for y (or for x if you rather).
Then plug that into the equation for u .

Multiplying by x2 gives a quadratic in x2 .

Solve that for x2 .
 
  • #4
BiGyElLoWhAt
Gold Member
1,570
116
Either substitution or elimination, whichever you prefer...
 
  • #5
Sorry LC Kurtz and others - maybe I didn't make it clear - expressing x and y in terms of u and v was a part of the problem that we were explicitly asked to do - which seemed strange because it wasn't necessary, under normal circumstances I'd tackle the integral straight away

The quadratic in x^2 method seems like it will give me what I want (and explains the restriction u > 0 which will probably force us to take the positive root)

Just a little irritating to have to do things the long way around - spent way too long on some ultimately unnecessary algebra

Thanks for all your help
 
  • #6
BiGyElLoWhAt
Gold Member
1,570
116
Yea, it is unnecessary for this problem, but there will be times when it's not. Look at it as practice for those times.
 

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