So, a typical strategy for solving certain PDE's is separation of variables. I understand this only works on domains that can be represented by a cartesian product like: $[0,L] X [0,H]$. Therefore, we are pretty much limited to rectangular and circular domains for separation of variables to work.
I want to see a better explanation of WHY separation of variables fails on - lets say - a triangular domain, other than: "well, you can't express a triangular region as a cartesian product like $[0,L] X [0,H]$."
I'd also like to have an answer that ties in the idea of orthogonality - I think these are all related but currently unable to put all the pieces together in a nice way that I can understand.
Also, is there anyway we can make use of a transformation to the triangular region to turn it into a region that allows the variables to separate?
Please help me put all these ideas together! Thank you in advance.