Looking at usage frequencies in Google:
"in polar form" - 162000 occurrences
"in the polar form" - 85100
"in a polar form" - 26200
The context I am interested is math and it seems that most occurrences I see in Google belong to it. Same for "Cartesian":
"in Cartesian form" - 33700 occurrences
"in the Cartesian form" - 26900
"in a Cartesian form" - 162
I am not asking which one is correct, since it seems that all of them are correct. I am asking which would be the subtle difference in meaning between the three.
I am familiar with the theory of articles in English and have improved my confidence considerably, but still cannot figure it out in this particular case.