- #1

lus1450

- 40

- 1

## Homework Statement

Let ##H## be the parallelogram in ##\mathbb{R}^2## whose vertices are ##(1,1), (3,2), (4,5), (2,4).## Find the affine map ##T## which sense ##(0,0)## to ##(1,1), (1,0)## to ##(3,2), (0,1)## to ##(2,4)##. Show that ##J_T=5## (the Jacobian). Use ##T## to convert the integral

$$

\alpha = \int_H e^{x-y}dxdy

$$

to an integral over ##I^2## (unit square) and thus compute ##\alpha##.

## Homework Equations

## The Attempt at a Solution

So I'm not sure why I'm having some trouble with this. I've done everything up to solving the integral. The affine map I found is ##T(x,y) = (1,1) + (2x+y,x+3y)##, and I've checked that it is correct. However, in my mind, it makes sense that in changing the integral to be over the square, I take ##T^{-1}(H) = I^2##, that is, obtaining:

$$

\alpha = \int_H e^{x-y}dxdy = \int_{I^2} f(T^{-1}(x,y))|J_{T^{-1}}|dxdy

$$

However, the integral is supposed to be ##f(T(x,y))|J_T|##, not of ##T^{-1}##. Can someone explain why this is so? I think I'm just forgetting something from lower division calculus, as it's been a while. It made sense back then, but for some reason this problem is screwing me up. Thanks in advance.