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Math Help - Change of Variables...final tomorrow!

  1. #1
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    Exclamation Change of Variables...final tomorrow!

    Use the transformation:
    x = (u+v)/2
    y = (v−u)/2
    double integral bounded by R of: (x − y)(x + y)dxdy

    where R is the region bounded by the lines y = x + 2, y = x, y = −x + 2 and
    y = −x + 4.


    Again...its the bounds of the integral that are killing me. Thanks ahead of time my book is just terrible.
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  2. #2
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    Note from your transformation that u=x-y and v=x+y. Now, note that our region is a rectangle at 45 in the xy axes, and out bounding lines can be rewritten as: x-y=-2, x-y=0, x+y=2, and x+y=4, so we have bounds in uv of u=-2, u=0, v=2 and v=4. Does that help?

    --Kevin C.
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  3. #3
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    how did you find out what u and v were equal to
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  4. #4
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    I found u and v by the same methods one uses to solve a system of equations;
    since you have:
    x=\frac{u+v}{2}=\tfrac{1}{2}u+\tfrac{1}{2}v
    y=\frac{v-u}{2}=-\tfrac{1}{2}u+\tfrac{1}{2}v
    adding the two equations together gives:
    x+y=\tfrac{1}{2}u+\tfrac{1}{2}v-\tfrac{1}{2}u+\tfrac{1}{2}v=v
    and subtracting the second equation from the first gives:
    x-y=\tfrac{1}{2}u+\tfrac{1}{2}v-(-\tfrac{1}{2}u+\tfrac{1}{2}v)=u
    Does that make sense?

    --Kevin C.
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