[PaulHoule]

e.g. "tensors" are like "vectors" in they transform in a specific way when the coordinate system changes; what felt so magic about vectors as an undergrad was that they embody "the shape of space" and thus simplify calculations.

If you didn't have vectors, Maxwell's equations would spill all over the place. Tensors on the other hand are used in places like continuum mechanics and general relativity where something more than vectors are called for but you're living in the same space(/time) with the same symmetries.

[Joker_vD]

> If you didn't have vectors, Maxwell's equations would spill all over the place.

What do you mean, "would": they did! :) The original equations had 20 separate equations, although Maxwell himself tried to reformulate them in quaternions. But if you look e.g. at works of Lorentz, or Einstein's famous 1905 paper, you'll see the fully-expanded version of them. The vector form really didn't fully catch until about the middle of the XX century.

[lioeters]

This sounds like an interesting thread to follow. From a cursory search, it seems vector calculus was being used by the early 1900's to reformulate Maxwell's equations, then later with notations like differential, integral, and matrix forms. I'll read more and see if I can understand the gist of each major step of the process over the years, how the notation affected the way mathematicians thought about the equations, and "made them easier to work with".

And how Fortran has unique properties that make converting math equations into code "more natural". Intriguing, I'll to dig deeper for intellectual curiosity.

[foxglacier]

It goes even further. Einstein also simplified the writing of tensor equations involving sums (big sigma sum) with Einstein notation by basically dropping the sigma because it's redundant so undefined indices automatically get summed over all their applicable values. It works with nested sums too to make them deceptively simple looking. Add to that the comma subscript for differentiation and you get formulas with just a couple of terms but huge piles of subscripts. I've seen equations in text books that have both subscripts and superscripts on both the left and right of a variable.

[foxglacier]

Sure, but not all arrays are tensors. In continuum mechanics, you often represent tensors as matrices or vectors to exploit symmetry and then it's really confusing to keep calling them tensors because they don't behave like their tensor equivalent.