I Advice toward Mastering Challenging Vector Calc Problems

[Rippling Hysteresis]

TL;DR Summary

I'd like greatly improve proficiency in MV/vector calc and am seeking advice on the best path forward. I'll know I have reached my goal when I can pick up any random professor's problem set and be able to construct a plan and recognize fundamental concepts underlying the problem.

I've taken multivariable/vector calc and can do most of the basic operations and have an OK understanding of the fundamental concepts, but certainly can't "see it" like I can calc I and II. In those subjects, I often feel competent to take on any problem I come across because the concepts are strong. If relevant, I have an undergrad background in physics from years past, so have some familiarity with some of the basic applications.

What would people recommend to build up my conceptual grasp and widen the scope of problems I can jump into? For example, if I work through a particular textbook, even semi-rigorous ones like Marsden and Tromba, I can handle the majority of the problems. But when faced with a problem set I come across online from one professor or another, it's hard to jump in and attack them. I'm sure if I had sat in any individual professor's class and seen the way they've set up problems and the such I'd be able to do it, but I don't have that general competency yet without a model of an instructor's approach.

Is there a good a good set of problems that have a wide scope and a variety of situations or ways that the standard computations can be set-up, modeled, applied, etc. (and include solutions) that anyone would recommend? The goal is to have general and fluid skills so that I could develop the ability to see the path forward for any vector calc type problems, so I'm not limiting myself to one modality. I will say I'm not as interested in the real analysis/proof-based approach-- I hesitate to take that route because I don't see myself following through in a serious way for such an undertaking.

Very much excited to hear what people would recommend!

 

[fresh_42]

This sounds to me as if you should search the internet for lecture notes and exams, with or without solutions. The natural next step after vector calculus would probably be differential geometry, which is as important in physics.

 

[Rippling Hysteresis]

This sounds to me as if you should search the internet for lecture notes and exams, with or without solutions. The natural next step after vector calculus would probably be differential geometry, which is as important in physics.

Thanks for your reply. In past searches I have found a lot of basic problems (calculate the line integral of this function, use the divergence theorem to solve some integral, take the cross product of these vectors). Rarer finds seem to be the sets that have less obvious paths forward. I did find a decent one by Marsden, but it seemed to be pretty aligned with the approach his book takes, which I've mostly worked through.

Recently I came across a few problems from a UT course (no solutions) that were all very tough. There were often layers of complexity I'd never seen before (and thus really would question how to scale the problem and from angle to attack it) or they were very open-ended and gave little direction (you'd really have to think about which theorems to apply, such as Stokes' or something like that, and which techniques to employ).

It made me realize how superficial my skills were and that I'd like to sharpen them. I'll dig around some more, but it's tough to find a really rich set of problems like that with solutions that allow me to really push my boundaries and gain insight.

 

[fresh_42]

You could have a look at the five insight articles about differentiation:
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
to see where you want to go. You could as well have a look at our monthly challenges.
The problems there are not sorted and of all kind, but include some vector calculus problems, too:
https://www.physicsforums.com/forums/math-proof-training-camp-and-practice.296/
or jump to some solutions of older ones:
https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/

I don't think that either of them fits exactly your needs, but I have not really an idea what you are looking for. The keyword 'vector calculus' produces so many Google hits, that I have difficulties to imagine neither of them would match your demands.

 

[Rippling Hysteresis]

You could have a look at the five insight articles about differentiation:
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
to see where you want to go. You could as well have a look at our monthly challenges.
The problems there are not sorted and of all kind, but include some vector calculus problems, too:
https://www.physicsforums.com/forums/math-proof-training-camp-and-practice.296/
or jump to some solutions of older ones:
https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/

I don't think that either of them fits exactly your needs, but I have not really an idea what you are looking for. The keyword 'vector calculus' produces so many Google hits, that I have difficulties to imagine neither of them would match your demands.

Thanks! I will check those links out. Might be cool to tackle some.