Change of Variable in multiple Integrals

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Homework Help Overview

The discussion revolves around a transformation of a triangular region in the xy-plane defined by specific vertices and the corresponding transformation to the uv-plane. Participants are tasked with finding the vertices of the transformed triangle and the Jacobian of the transformation.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the transformation equations and the process of finding the vertices in the uv-plane. There is an attempt to verify the correctness of the calculated vertices and the Jacobian. Questions arise regarding the proper interpretation of the transformation and the relationship between the (x,y) and (u,v) coordinates.

Discussion Status

Some participants express uncertainty about the correctness of their answers and seek clarification on the transformation process. There is an ongoing exploration of how to derive the values of u and v from given (x,y) coordinates, with guidance being offered on solving the equations.

Contextual Notes

Participants are working under the constraints of a homework assignment, which may limit the information they can use or the methods they can apply. There is a focus on ensuring that the transformation is applied correctly to find the appropriate vertices and Jacobian.

ahhppull
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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.
 
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ahhppull said:

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.​
 
SammyS said:
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?
 
ahhppull said:
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?
 
haruspex said:
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)
 
ahhppull said:
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.
 

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