6

I motivate these grad, curl, and div for myself as being the things which would make the respective version of Stokes' theorem true infinitesimally. You can read about this interpretation in the context of differential forms here: https://math.stackexchange.com/a/614473/34287 Given that interpretation, we can see The gradient at a point $p$ is the vector ...


6

This is a service course for students who are mostly engineering majors. Therefore any drastic change in notation like this is likely to be a bad idea. Leaving out $d\textbf{S}$ and $dV$ would be particularly unfortunate, since leaving out the $dx$ is such a common student mistake anyway in freshman calculus. Also, any notation that has the wrong units is a ...


5

For 2nd year engineering students, you should not be wasting time trying to prove the expansion, except perhaps giving a heuristic argument for the quadratic approximation in the case of two variables. Stick to operational algebraic formulations and computations, which is the only thing they'll ever need. My go-to book for an elementary treatment of this ...


5

I think you are being harsh in your criticism of the classical notation. Of course, at the mathematician's end of the spectrum, the notation you promote towards the end of your question has merit. But I have taught vector calculus for many years, and find the classical notations that provoke you do in fact help learners decode theorems and calculations. ...


4

If we study the tangent line to $y=f(x)$ at $(a,f(a))$ as given by: $$ y = L^a_f(x) = f(a)+f'(a)(x-a) $$ then you can verify that a little input interval $(a-h,a+h)$ is transferred to the output interval $(f(a)-f'(a)h,f(a)+f'(a)h)$ by the linearization $L^a_f$ of $f$ at $x=a$. What is the significance of the derivative in this viewpoint ? Observe the length ...


4

As the students are by now familiar with the cross product, it is a natural question of how it might generalize to higher dimensions. It is surprising that the cross product only exists in dimensions $3$ and $7$: The $7$-dimenstional cross-product is the only "bilinear product of two vectors that is vector-valued, orthogonal, and has the same magnitude as in ...


4

Don't underestimate how much your knowledge and skills can fade when unused, even after a couple of years, let alone twenty. I've experienced this personally when trying to study advanced topics for which I had the prerequisites on paper from many years ago, but which were no longer fresh in my memory. My general advice is first to check the immediate ...


3

Based on a suggestion I once found here on MESE, my class last term proved (via Green’s theorem) the existence of a linear planimeter, showing how it could mechanically measure the area of a shape. When we were done with the proof, I brought out the one I built at home from scrap wood: Figure 1: The red arm freely pivots on a block that is constrained to ...


2

I looked at some books I have. It's not clear to me how vital the multivariable Taylor topic even is. Swokowski does not cover it, for instance. Neither does Kreyszig, although he does cover it for complex analysis. So you could consider to stick with a cursory treatment or even cut it entirely. After all, time is limited and there are plenty of other &...


1

This may not move in the directions of your greatest interests, but it is one option. (Not my field so I cannot assess comparable books.) Eschrig, Helmut. Topology and Geometry in Physics. Vol. 822. Springer Science & Business Media, 2011. Springer link. Review by Jan Jerzy Sławianowski. Publisher summary: "Written as a set of largely self-...


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