Integrationg over exp with two variables

[cutesteph]

Homework Statement

f(x,y) = exp(-x^2 +xy -y^2)

transform with
x =(1/sqrt(2)) *(u – v), y = (1/sqrt(2))* (u + v) .

Homework Equations

Jacobian

The Attempt at a Solution

Jacobian = 1

f(u,v) = exp(-(u^2)/2 -(3v^2/2)

double integral f(u,v) du dv

the bounds would be x > 0 => ( u-v) >0 => u > v
and x < ∞ => u < ∞

v > 0 to v < ∞

I am lost on what to do next. If anyone can be as kind as to help, I would greatly appreciate it!

 

[tiny-tim]

hi there cutesteph!

(try using the X² button just above the Reply box )

what are your limits for x and y ?

i'll assume they're both from 0 to ∞

draw the region (in x,y), and mark a grid of lines of equal u and v

u goes from 0 to ∞

for each value of u, where does v go from and to? ​

 

[cutesteph]

So the limits v from 0 to infinity and u from -v to v.

∫0 to∞ exp(-u²/2)∫-u to u exp(-3v²/2) dv du

 

[tiny-tim]

So the limits v from 0 to infinity and u from -v to v.

isn't it the other way round?

 

[Ray Vickson]

Homework Statement

f(x,y) = exp(-x^2 +xy -y^2)

transform with
x =(1/sqrt(2)) *(u – v), y = (1/sqrt(2))* (u + v) .

Homework Equations

Jacobian

The Attempt at a Solution

Jacobian = 1

f(u,v) = exp(-(u^2)/2 -(3v^2/2)

double integral f(u,v) du dv

the bounds would be x > 0 => ( u-v) >0 => u > v
and x < ∞ => u < ∞

v > 0 to v < ∞

I am lost on what to do next. If anyone can be as kind as to help, I would greatly appreciate it!

You never actually answered the question about the limits on x and y, and without your answer I cannot possibly tell what are the limits on u and v. However, you can determine the latter for yourself by noting that
$$u = \frac{x+y}{\sqrt{2}}, \; v = \frac{y-x}{\sqrt{2}}$$
If you know the ranges of x and y you can figure out the ranges on u and v.