Changing Variables in Multiple Integration

Question

On a test I had a question where the unit square $[0,1]^2$ in $u,v$-space was being transformed by something like $T(u,v)=(uv,v)$ into a new region in $x,y$-space (for a double integral).

I had to find the image of the transformation, and intuitively I thought: well, clearly $$0\le uv\le 1$$ $$0\le v\le 1$$ so the image must be the unit square.

Luckily I caught my error, and ended up realizing that the image was actually a triangle in the $x,y$-plane.

My question is: how can one systematically go about finding the image of a transformation from $R^n\to R^n$? It's not that I don't like actually "thinking," but I feel like there must be some more methodical way of going about these problems.

Alternatively, is there a way to find the bounds of integration without bothering with the region itself?

score 1 · Answer 1 · answered Apr 10 '14 at 20:34

Express the old variables as functions of the new ones and deduce the domain where the new ones "live".

In your case, $(x,y)=(uv,v)$ hence $$(u,v)=(x/y,y),$$ and the $(u,v)$-domain is defined by $$0\leqslant u\leqslant1,\qquad 0\leqslant v\leqslant 1,$$ hence the $(x,y)$-domain is defined by $$0\leqslant x/y\leqslant1,\quad 0\leqslant y\leqslant 1,$$ or, more simply, $$ 0\leqslant x\leqslant y\leqslant 1. $$

score 0 · Accepted Answer · answered Apr 10 '14 at 20:33

0

Parametrize the border and compose with the transformation.

answered Apr 10 '14 at 20:33

Martín-Blas Pérez Pinilla

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Can you always focus exclusively on the border? I've had a little topology and I think I remember that borders get sent to borders under continuous transformations (maybe there are some more conditions)... is that right? – Apr 10 '14 at 21:13
The transformation is a diffeomorphism, so is bijective and the inverse is continuous. – Martín-Blas Pérez Pinilla Apr 10 '14 at 21:23
Why is focussing on the border simpler (if I understand your point correctly) than the standard approach? – Did Apr 11 '14 at 13:42
@Did Because you just have to find the image of the border curves, without worrying about the interior of the region. – Apr 11 '14 at 14:22
But this is exactly as simple as the full job, no? – Did Apr 11 '14 at 21:53
My studendts find that only equalities is easier. – Martín-Blas Pérez Pinilla Apr 12 '14 at 13:43

Changing Variables in Multiple Integration

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