Change of variables in integration.

[vineel49]

Homework Statement

The original integral is
$$\left[\int_0^{\infty} {\int_0^{\infty} {F(x + y,x - y) \cdot dx \cdot dy} } \right]$$

What should be the limits of the integrals. (position represented by '?' symbol)
$$\left[\int_?^? {\int_?^? {F(u,v) \cdot (\frac{1}{2})du \cdot dv} } \right]$$ .

When x+y is substituted by 'u' and x-y is substituted by 'v'

Homework Equations

use x+y= u , x-y=v, I am confused about the limits of the integrals

The Attempt at a Solution

dx*dy = 0.5 * du * dv - This I got by using Jacobian matrix.
I need help in deciding the limits of the integrals.

My approach:
$$\left[\int_0^{\infty} {\int_{\left| v \right|}^{\infty} {F(u,v) \cdot (\frac{1}{2})du \cdot dv} } \right]$$

Is this correct?

 

[tiny-tim]

hi vineel49!

use x+y= u , x-y=v, I am confused about the limits of the integrals

try drawing it …

that's simply rotating the axes by 45°, isn't it?

(and scaling up or down by the Jacobian)

so if one of the new limits is 0 to ∞, the other must be … (not 0) to … ?