# Homework Help: Change of Variable in multiple Integrals

1. Nov 24, 2012

### ahhppull

1. The problem statement, all variables and given/known data

Let D be the triangular region in the xy-plane with the vertices (1, 2), (3, 6), and (7, 4).
Consider the transformation T : x = 3u − 2v, y = u + v.

(a) Find the vertices of the triangle in the uv-plane whose image under the transformation T is the triangle D.

(b) Find the Jacobian of the transformation T.

2. Relevant equations

3. The attempt at a solution
I think I got the answers, just checking to make sure.

For a, I got the vertices; (-1,3),(-3,9) and (13,11).

For b, I got 5.

2. Nov 24, 2012

### SammyS

Staff Emeritus
You have (a) wrong.
If (x,y) = (1, 2), what are u & v ?

etc.

You found that if (u,v) = (1, 2) , then (x,y) = (-1,3) , etc. But this is not what's being asked. ​

3. Nov 24, 2012

### ahhppull

I don't understand. How would I do this then?

4. Nov 24, 2012

### haruspex

At the point (x,y) = (1,2), if x = 3u − 2v and y = u + v what are u and v?

5. Nov 24, 2012

### ahhppull

So I set 1 = 3u -2v and 2 = u+v. Then, I do 2-v=u and substitute u into the first equation?

I get (1,1)

6. Nov 24, 2012

### SammyS

Staff Emeritus
Yes.

You can also solve the set of equations:
x = 3u − 2v

y = u + v​
for u and v, and then plug in the set of (x,y) pairs to get the set of (u,v) pairs.