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 ...


5

As far as I know, vector calculus is applied by financial analysts in exotic derivatives pricing. The Black-Scholes Model is actually a special form of Schrödinger equation. Thus, if you want to establish high precision models to price exotic derivatives, you will have the chance to apply vector calculus. However, these kinds of applications presumes ...


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 ...


3

The following is based on my experience at as a TA and adjunct at a UC, a Cal State, and the University of Nevada, as well as observations of siblings and friends at other institutions. The two texts which seem to be used most commonly are Stewart's Calculus and Thomas' Calculus (note that both Stewart and Thomas have been dead for a while, and neither has ...


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.) ...


2

Working in a factory packing boxes: fill a pallet with a pile of boxes, where for each box you need to fill it with packets of biscuits.


1

I believe Community Calculus is exactly what you're looking for. It is under a CC BY-NC-SA 3.0 license.


1

I like stressing the connection between rates of change and integrals, and I carry this into multiple integrals as well. As an example, you could talk about a row of apple trees producing (on average) $f(t,x)$ apples per day per meter at time $t$ and location $x$ along the row. Probably $f(t,x)$ is roughly periodic in $t$ with a period of $365$, and has ...


1

Eating all the chocolates in a multi-layered chocolate box. Each layer has a tray containing several rows of chocolates, and each one you eat is just a tiny increment that only makes an infinitesimal contribution to your calorie intake . . .


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