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Math Help - Change of variables in 3 dimensions

  1. #1
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    Change of variables in 3 dimensions

    (First post, hoorah!)


    Hi there - I'm revising for my exams in a few weeks and decided to look at some of the older past papers for my uni course, but they tend to deviate off the syllabus a fair bit so I haven't been taught how to approach a problem like this systematically, I'm using mostly guesswork and logic...


    (i) 'The domain S in the (x,y) plane is bounded by y=x, y=ax (0 \leq a \leq 1) and xy^2=1 (x,y \geq 0). Find a transformation
    u=f(x,y) v=g(x,y)
    such that S is transformed into a rectangle in the (u,v) plane.'

    I figured I may as well aim to transform into the unit square if I could, since it's only a short step translating from a rectangle in the (u,v) plane to the unit square. The only problem which struck me here was the fact a rectangle needs 4 'bounds' whereas the given area in the (x,y) plane has only 3 bounds. Through, as I said, mostly fairly unconfident guesswork, I came up with u=\frac{xy^2(y-ax)}{\frac{1}{\sqrt{x}}-ax}, v=\frac{xy^2(y-x)}{\frac{1}{\sqrt{x}}-x} - would this work? It gives u=1 and v=1 along the xy^2=1 boundary and 0 along the y=x and y=ax boundaries, so would that be okay? In general, what's a good method for approaching this sort of question systematically?

    (ii) 'Evaluate \int_D \frac{y^2z^2}{x} \, dx \, dy \, dz, where D is the region bounded by

    y=x, y=zx, xy^2=1 (x,y \geq 0)

    and the planes z=0, z=1.'

    Obviously a change of variables is the way they want you to go, and I'm happy with the actual process of changing variables, the Jacobian, the volume integral etc, but how do I actually work out -what- variables to change to? It's not like it's a symmetrical area which points towards spherical polars or anything like that, so I'm not sure how to proceed!

    Thanks very much for the help!
    Last edited by mathmos6; May 11th 2009 at 12:48 PM.
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  2. #2
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    For (i), I would write the inequalities as a\leqslant\frac yx\leqslant1 and 0\leqslant xy^2\leqslant1. So if u(x,y) = \frac xy and v(x,y) = xy^2 then S corresponds to the rectangle a\leqslant u\leqslant1,\ 0\leqslant v\leqslant1 in the (u,v)-plane.

    Then uv = y^3 and v/u^2 = x^3, from which y^2/x = (uv)^{2/3}(v/u^2)^{-1/3} = u^{4/3}v^{1/3}. So for (ii) you just have to integrate u^{4/3}v^{1/3}z^2 (multiplied by a suitable Jacobian determinant) over the region z\leqslant u\leqslant1,\ 0\leqslant v\leqslant1,\ 0\leqslant z\leqslant1.
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