u/Background_Salad4659

I'm currently in Calc 3, and this is the first time I've actually run into something that I just am struggling to understand. I'm going to office hours tomorrow with a harder problem in mind, but every time I do change of variables, like xy into uv plane, or vice versa, It just seems to take me like 30-40 minutes every time, and I always mess up something and need help. Is there any advice or tips you can give to help?

Anything would be seriously appreciated! I just can never parametrize the boundaries correctly