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Thread: 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 $\displaystyle y=x$, $\displaystyle y=ax (0 \leq a \leq 1)$ and $\displaystyle xy^2=1 (x,y \geq 0)$. Find a transformation
    $\displaystyle u=f(x,y) v=g(x,y)$
    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 $\displaystyle u=\frac{xy^2(y-ax)}{\frac{1}{\sqrt{x}}-ax}, v=\frac{xy^2(y-x)}{\frac{1}{\sqrt{x}}-x}$ - would this work? It gives u=1 and v=1 along the $\displaystyle xy^2=1$ 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 $\displaystyle \int_D \frac{y^2z^2}{x} \, dx \, dy \, dz$, where D is the region bounded by

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

    and the planes $\displaystyle z=0$, $\displaystyle z=1$.'

    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 11:48 AM.
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  2. #2
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    For (i), I would write the inequalities as $\displaystyle a\leqslant\frac yx\leqslant1$ and $\displaystyle 0\leqslant xy^2\leqslant1$. So if $\displaystyle u(x,y) = \frac xy$ and $\displaystyle v(x,y) = xy^2$ then S corresponds to the rectangle $\displaystyle a\leqslant u\leqslant1,\ 0\leqslant v\leqslant1$ in the (u,v)-plane.

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