Is exterior calculus efficient for simple vector calculus problems?

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.)

• 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