EDIT: Solved all of them.
I guess I don't know how to solve any of the following, because I keep getting the wrong answer all the time. It's mostly the boundaries that I'm having trouble with.
1) where
2) where
3) where is the triangle with corners in and .
So here's what I tried to do:
1)
So the limits become and
There's an error with the Jacobian. It's meant to be .
I can't see any other error. But I get .
Maybe someone else can see what I can't (if there's anything to see).
If I have time I'll take a look at the others. I would have thought that the second one is similar to the first in many ways ......
The triangular region D is bounded by the lines and .
Note that . This suggests making the transformation
u = x - y .... (1)
v = x + y .... (2)
(1) + (2): .
(2) - (1): .
The Jacobian of the transformation is therefore .
The triangular region D transforms to a simple triangular region D' bounded by the following lines:
.
.
.
So the integral becomes and it should be blue sky from here.
The one thing I didn't check - Spec's transformation from xy-coordinates to uv-coordinates. I used the original transformation (see the quote in post #2) without checking it
Looking for my mistake, I see that it should have been v = 3y, NOT v = y/3. Then I notice Spec has edited the question to reflect this - s/he obviously realised the same thing (*ahem* it's good form to include the reason for an edit, by the way .....)
My mistake for not checking right from the start I knew it would be something simple.
Moral of the lesson - don't assume that even the simplest things are done correctly.