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Change of Variable in multiple Integrals

  1. Nov 24, 2012 #1
    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. jcsd
  3. Nov 24, 2012 #2

    SammyS

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    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. ​
     
  4. Nov 24, 2012 #3
    I don't understand. How would I do this then?
     
  5. Nov 24, 2012 #4

    haruspex

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    At the point (x,y) = (1,2), if x = 3u − 2v and y = u + v what are u and v?
     
  6. Nov 24, 2012 #5
    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)
     
  7. Nov 24, 2012 #6

    SammyS

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    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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