I'd assume you take multivariate, but skip on programming languages, digital design, and operating systems.

Going from 1 variable to 2 or 20 never seemed like a big jump to me. I can see taking 1-2 weeks to go over it quickly, but you don't need a lot of problem set's to get it.

It's like learning a new computer language, it takes about as long to do that on it's own as to have a class in something else that happens to use your that new language.

Line / surface / volume integrals are certainly a jump from one-variable integrals.

Not a big jump, if you conceptually stick to the 'area under curve' definition.

Analytically computing surface/volume integrals, yes, that is a big jump. But I don't understand why students need to memorize N algorithms and M substitution rules for that.

What is the straightforward "area under the curve" definition of a line integral?

Wikipedia has a nice animation for this.

https://en.wikipedia.org/wiki/Line_integral#/media/File:Line...

Okay. I am used to this being called a path integral, and the phrase line integral being reserved for the integral of a vector field over a curve. This does not, as far as I know, admit a straightforward definition as an area under a curve.

If you replace area under curve with volume under surface, it seems to me that the same intuition holds.

Sure, and AI isn't a big jump from statistics.

I would say AI:Statistics :: Engineering:Physics .

Sometimes it is a big jump, sometimes it is all common sense.

I don't get this focus on 'maths via arcane problem solving' either.

They taught gradient descent in my lower division math courses in college.

Multivariable calculus is a lower-division math class.

yep, softmore level math.