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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

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 ...

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 ...

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

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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