[jimktrains2]

Are you me?

Halfway through Calc III it was interesting to watch the visual students, who normally would do very well, have a rough time, while other students, who just treated it as an abstract system had more overhead/trouble/were slower when learning initially, but it paid off when getting to un-visualizable systems.

[Steuard]

In my experience, virtually all of Calc III is "visualizable" (even somewhat esoteric stuff like Lagrange multipliers[1]), because it's mostly about vectors (which have a natural geometric interpretation). My claim would be that to be really good at math, you need to be skilled at both visualization (and other intuitions) and abstract systems. They complement each other well.

[1] E.g. http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.ht...

[j2kun]

Lagrange multipliers are esoteric? Oh man, I was under the impression that they formed the basis for most many useful optimization techniques :)

[jimktrains2]

It's hard to visualize 4d and 6d systems. That's what I was referring to.

[magoghm]

Great explanation! Thank you.