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7

One idea that comes to mind is the fact that a connected graph has a Eulerian circuit if and only if every vertex has even degree. Another idea that comes to mind is just the Fundamental Theorem of Calculus. If you have a rather boring video of a car's speedometer as the car travels from A to B, then you can figure out the total distance the car has ...

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

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. ...

5

Here are two similar ideas. (1) The shortest geodesic between two points on the surface of a polyhedron (generally) bends as it crosses edges, when viewed globally in $\mathbb{R}^3$, but from the local point of view of an ant walking along the path, it is straight, i.e., straight when each crossed edge is unfolded flat. This is commonly illustrated on a ...

2

Local-to-global is a big thing in algebra, starting with (Hasse-)Minkowski's result that if you have a solution to a quadratic form modulo all prime powers and in the reals, you also have one over the integers. (See Theorem 2.4 here for the Minkowskian statement without $p$-adic numbers, which is otherwise a bit difficult to find in a quick web search.) ...

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