This is very useful information, but there are a few things I feel I must point out.

It should read:

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You accidentally put

Will the transformation work any time I do this? And more than that, will the transformation be easy to work with any time I set up these inequalities? In your second example, you did a non-linear transformation, but you did not show how the new tranformation could be used to derive the domain.Example 1:...

Look at the below, note we can think of this region as the following system of inequalities . Substitute our new defined variables into this system to get . Thus, .

Can you finish this example just so I can varify what I believe the answer would be?Example 2:Let be described as and , i.e. a square. Then the transformation and yield .

Another typo. You wrote: but clearly meant to write:Integrating Over an Ellipse....

Integrating over a circle centered at the origin is ideal at times if converted to polar form. But what can we do if the region of integration is an ellipse ?

I only learned about rotations when tutoring (I litterally learned it while tutoring and reading the student's text book). Of course I referred the student to another tutor for that particular help, but I read up on it for my own benefit and was later able to help other students in that class. But even still, I am somewhat limited in my understanding of rotations.3)Sometimes it might be convinent to rotate the region of integration. The reader probably knows that the rotation formula by angle is given by:

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Compute the Jacobian.

Anyways, nice tutorial. It was helpful and informative. (I had forgotten some of what you mentioned from when I learned it in Calculus 3).