# Resources for learning multivariable calculus

• Calculus

## Main Question or Discussion Point

I have recently started studying multivariable calculus and I cannot quite visualise the concepts.Problem solving is not a problem but I want a true understanding of the concepts.Which book or online resources are great at developing visualisation in this course?

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I have recently started studying multivariable calculus and I cannot quite visualise the concepts.Problem solving is not a problem but I want a true understanding of the concepts.Which book or online resources are great at developing visualisation in this course?
https://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/

This is where I first learnt it close to 10 years ago. The lectures are a tad old but I think they are excellent assuming you would have the patience to go through it all. It is quite worth it if you do. I have never had trouble with visualising stuff or drawing intuitions in multivariate calculus thanks to this guy. The visualisations are just good old chalk-on-board though, nothing too fancy.

Also, I think visualisation may also be tied with understanding the applications of the math. The best visualisation of flux through a surface is gotten through studying it in terms of electric flux, gravitational flux or fluid flux.

If you mention what concepts exactly you have trouble visualising, I might be able to help out more.

Of course there may be other excellent resources that use computer graphic animations and stuff elsewhere that maybe others could suggest.

https://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/

This is where I first learnt it close to 10 years ago. The lectures are a tad old but I think they are excellent assuming you would have the patience to go through it all. It is quite worth it if you do. I have never had trouble with visualising stuff or drawing intuitions in multivariate calculus thanks to this guy. The visualisations are just good old chalk-on-board though, nothing too fancy.

Also, I think visualisation may also be tied with understanding the applications of the math. The best visualisation of flux through a surface is gotten through studying it in terms of electric flux, gravitational flux or fluid flux.

If you mention what concepts exactly you have trouble visualising, I might be able to help out more.

Of course there may be other excellent resources that use computer graphic animations and stuff elsewhere that maybe others could suggest.
Thank you so much for helping me out.Can you suggest a good book too for problem solving?

mathwonk
Homework Helper
There are basically three fundamental theorems in several variable calculus, one about derivatives, one about integrals, and one relating the two concepts.

The first theorem is the inverse/implicit function theorem, which in its simplest form says that a smooth function from R^n to R^n whose derivative is invertible (as a linear map) at a point, is it self smoothly invertible on a neighborhood of that point. This is a higher dimensional version of the theorem in one variable that a smooth function whose tangent line is not horizontal at a point, miust be either increasing or decreasing (i.e. injective) on some interval around that point.

The second theorem tells you how to compute several variable integrals by reducing to the one variable case; this is called Fubini's theorem, i.e. a multiple integral can be computed as a repeated integral, integrating one variable at a time.

The third theorem is called Stokes' theorem, which generalizes theorems also called Green's, Gauss's, or the divergence theorem. It equates an integral over a region with boundary with another integral taken just over the boundary. This generalizes the fundamental theorem of calculus in one variable. In particular it is again true that the integrand over the bounded region is a certain derivative of the integrand over the boundary. Essentially this is proved by using Fubini's theorem to reduce to the one variable fundamental theorem.

There are also other important results such as the "change of variables" rule for integrals, and rules for interchanging the order of taking derivatives and integrals. It is also useful to know some criteria for existence, smoothness, and uniqueness of solutions to differential equations.

I see from the MIT course notes that are linked above that the text for that course is the one by my friends Henry Edwards and Dave Penney. That is a little more introductory course than the theoretical one I have sketched, for which a good but very terse treatment is in Spivak's Calculus on Manifolds. Edwards and Penney would be a better first encounter with this material.

• verty
Homework Helper
Can you suggest a good book too for problem solving?
For a book with lots of problems, try Marsden & Tromba, Vector Calculus. But be aware that the explanations and examples have been called unhelpful in reviews. (I found it to be okay though.)

mathwonk
Homework Helper
I taught from Marsden and Tromba to bright high school students in the 1980's and thought it was a very good book. It was produced to be used as I recall at Berkeley, so maybe not the right level for all students.

Thank you so much for helping me out.Can you suggest a good book too for problem solving?
Happy to help! I don't quite recall a textbook for this. I see a couple of suggestions have come in for that. Hopefully one of them will suit your need!

verty
Homework Helper
I taught from Marsden and Tromba to bright high school students in the 1980's and thought it was a very good book. It was produced to be used as I recall at Berkeley, so maybe not the right level for all students.
Yes, it's probably a book for the B student who wants good problems to solve. But I found I understood the material better after solving the problems.