Change of Variable in multiple Integrals

[ahhppull]

Homework Statement

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.

Homework Equations

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.

 

[SammyS]

Homework Statement

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.

Homework Equations

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.

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.​

 

[ahhppull]

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.​

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

 

[haruspex]

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

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

 

[ahhppull]

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

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)

 

[SammyS]

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)

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.