Exterior calculus and invariant formulations are important and lead to many breakthroughs and great insights in physics and mathematics. But for daily vector calculus tasks, I still struggle to apply the theory efficiently.

Question: Is it possible to practice exterior calculus hard enough such that it becomes as fast as classical vector calculus? Or is there just an unavoidable trade-off we need to accept?

Example: I want to validate a simple equation like $$\mathbf B = \operatorname{rot} \mathbf A$$ for a potential $\mathbf A$ defined by $\mathbf{A} := \frac 1 2( \mathbf{r} \times \mathbf{B}).$ ($\mathbf B$ is a given constant magnetic field.)

Using classical vector calculus (with indices), this computation is easy. But using rules from differential geometry instead, there are always the same difficulties for me:

  • Which kind of tensors do we have: differential forms, vector fields or tensor-fields? (Knowing this is in general interesting, but sometime I just don't have the time to get into these questions...)
  • If the tensors don't match, I need to use general formulas which involve many isomorphisms (musicals, hodge-star), that's complicated!

For this example, I did the computations by applying all isomorphisms one after each other, i.e. $$\mathbf B = [*(\mathrm d ( * (\mathbf r^\flat \wedge \mathbf B^\flat)))]^\sharp .$$ It took me more than 10 minutes to figure it out. And additionally, it does not use the idea that $\mathbf B$ is more naturally seen as a $2$-form.

Therefore, I really hope that I am just a bit stupid and there is a smarter way to this. But this is not the first example where I could not find an efficient way to calculate.

Books:

I mainly use John M. Lee's 'Introduction to smooth manifolds' and Marsdens books 'Introduction to Mechanics and Symmetry' and 'Mathematical foundations of elasticity' to learn differential geometry. (To be honest, I was scared off but the old prints of Spivak's books.)

  • 1
    Some years ago, I took a course in fluid mechanics, given by a famous expert in the field. He used to laugh at me, always trying to do computations coordinate-free. He also added that being able to compute in coordinates is important, because it frees from the necessity of knowing lots of theory. I think that he was right. – Giuseppe Negro Sep 14 at 13:07
  • Sometimes abstractions simplify, sometimes abstractions make things a lot more complicated. That's not only the case in mathematics, but also in programming, according to my experience. – md2perpe Sep 14 at 15:14
  • Seems like my question was too open. Anyway, thanks for your comments! I guess it is part of becoming a mathematician to learn which level of abstraction is useful for which task! – Steffen Plunder 12 hours ago

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