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 , and . Find a transformation
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 - would this work? It gives u=1 and v=1 along the 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 , where D is the region bounded by
and the planes , .'
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!