# Jacobian Transformations

Can someone help me with the following?
I am supposed to evaluate
∫∫ e^(x+y)dA where the area of integration is given by the inequality |x|+|y|≤1.
So, suppose I do one of these Jacobians, and I set u = |x| and v = |y|, so wouldn’t the equation have to satisfy the inequality u+v≤1, and u≥0, v≥0? So, would wouldn’t the Jacobian be 1? But clearly I’m doing something wrong here, so any hints would be greatly appreciated!! Thanks!!

Related Calculus and Beyond Homework Help News on Phys.org
Dick
Homework Helper
The function in a change of variable should one-to-one (as your example shows - it simply omits regions of the integration domain). Differentiability is also handy. Absolute value is neither.

I'm not sure if I understand what you mean. Can you clarify more or point out where I'm going wrong with this? Thanks.

Dick
Homework Helper
I thought I did? You can SEE where it's going wrong, right? The formula for change of variables involving Jacobians has premises. You can't blindly use any old functions. |-1|=|1|=1. It's not one-to-one. Hence you can't blindly use it in change of variable. Jacobian isn't even defined at (0,0).

arildno
Homework Helper
Gold Member
Dearly Missed
Note that absolute values are non-negative, so for example, |x|<=1, that is $-1\leq{x}\leq{1}$

An equivalent inequality to the one you have been given is:
$$|x|-1\leq{y}\leq{1}-|x|$$
In the region $x\geq{0}$, this translates to:
$$x-1\leq{y}\leq{1-x}, 0\leq{x}\leq{1}$$
Make a similar translation for negative x's!

Last edited:
Thanks, that is very helpful. So, can I just ask, since |x|+|y| ≤1, so this means that |x|≤1-|y|. |y| - 1 ≤ |x| ≤ 1-|y|. So this means that y – 1 ≤ x ≤ 1-y, where 0 ≤ y ≤ 1. But I don’t see why I can’t just use those parameters to integrate? So, could I do ∫∫ e^(x+y)dxdy where y – 1 ≤ x ≤ 1-y and where 0 ≤ y ≤ 1?

arildno
Homework Helper
Gold Member
Dearly Missed
Of course you could use y rather than x as the "outer" variable.

Dick
Homework Helper
Yes. That's exactly what you should do.

arildno
Homework Helper
Gold Member
Dearly Missed
Yes. That's exactly what you should do.
Eeh?

Dick