For me, Einstein notation had the major advantage in electrostatics of reeling in the complexity of really long integrals that show up in boundary-value problems in electrostatics as well as being able to rederive formulas like "curl of curl is div-grad minus laplacian" by hand in seconds (which I always forget -- I took Calc ).

I had only a relatively cursory introduction to general relativity last semester; I have at best a vague understanding of Christoffel symbols, to give you an idea. So if Einstein notation can at some level become a way to fling symbols around and forget you're doing physics, I guess I haven't gotten there yet. I would like to think, though, that were it introduced alongside vector calculus instead of several years later, people might connect the adscripts with their meaning more easily.

Usually, if I want to understand the geometry of a problem, though, I find the best tool is a diagram, if at all possible.

For historical reasons, many areas of physics come with heir own notation, eg introductory courses on mechanics and electrodynamics are often done using vector notation you know from school with some additional differential operators thrown in, thermodynamics uses differentials, analytical mechanics and general relativity use index notation and quantum mechanics uses bras and kets.

Specialization sometimes makes sense, but it's non-obvious (at least it wasn't to me) that when checking if a force field is conserved by computing it's rotation, you're doing the same thing as when computing the derivative of a differential to see if it belongs to a conserved thermodynamical potential, or that the difference between a bra and a ket is the same as between a covector (lower index in Einstein notation) and a vector (upper index) - things look so different that it's hard to see when they are the same.

Another example is the relation between Newtonian and Lagrangian mechanics. In the lectures I took, it was presented as if Lagrangian mechanics is somehow special because you have an invariant formulation using generalized coordinates, wheres Newtonian mechanics was only ever done in Euclidean or Minkowski space.

It turns out that Newtonian mechanics is as invariant and general as Lagrangian mechanics (however, it's possible to further generalize Lagrangian mechanics, whereas as far as I can tell, you're pretty stuck with second-order system when doing Newtonian mechanics):

The Euler-Lagrange-equations are Newtonian equations and the differential of the Lagrange function dL is just a funny way to write down a force field - ie the main difference between Newtonian and Lagrangian formulation is that you require your force to be derived from a generalized potential (more formally: every hyper-regular Lagrangian system is a Newtonian system, any Newtonian system where the force maps to a closed form under the isomorphism T* TM ~ TT* M is locally Lagrangian).

In my opinion, lectures on theoretical physics are somewhat broken, and that's a more serious problem than the non-issue of whether to use τ or 2π…

PS: Please don't get me started on Christoffel symbols if you're not prepared for another rant ;)