Well for what it is worth the university I am getting my bachelors at does not even have a differential geometry class and it's the best math school in the state. Granted that state is Nevada...
mordechai9-
To me, and again this is my own opinion, the true practicality of PDE's (and ODE's for that matter) comes from the fact that once we are capable of solving these equations we are able to more accurately model a cornucopia of real life situations. For example in basic mechanics when...
As far as my experience goes, you are unable to separate PDE's from practicality, as many of the PDE's that we work with (particularly second order) arise from physical problems. But in the end I suppose it is just my opinion. It will largely depend on the instructor how theoretical or...
I say PDE's, mostly because it will be most immediately useful to you as a physicist. That is not to say that Differential Geometry is not useful, but you will use PDE's before you use DG, at least that is my experience.
Also to consider is you own research focus in physics. If you are going...
Let a be the adjacent, let b be the opposite, and let c be the hypotenuse and A the area. Then
\[
c = \sqrt{\frac{2A}{\cos \theta \sin \theta}}
\]
\[
a= \sqrt{2A \cot \theta}
\]
\[
b = \sqrt{2A \tan \theta}
\]