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My two main ones are Electrostatic force field $\mathbf{E}\left(\mathbf{r}\right)=\frac{Q}{4\pi\epsilon_0 \left|\left|\mathbf{r}\right|\right|^3}\mathbf{r}$ and Gravitational force field, $\mathbf{F}\left(\mathbf{r}\right)=-\frac{GMm}{\left|\left|\mathbf{r}\right|\right|^3}\mathbf{r}$ but I'd like to have some others that could be used throughout an entire course. With those two, I can talk about grad, div, and curl, and could also explain the mathematical results through what we'd expect through physics. For a student, they could possibly predict the answer through intuition, and then prove themselves correct through the math. And, for as a teacher, I like the idea of bringing up repeat examples since it helps link together different topics and make the course seem less like random exercises in computation.

What are some fields that are well-known and could link together various vector calculus topics?

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    $\begingroup$ Velocity fields (of fluids),e.g. earth.nullschool.net $\endgroup$
    – Raciquel
    Oct 12, 2020 at 5:37
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    $\begingroup$ Does anyone have economic examples? I could imagine something like a covector field for marginal utility of consuming a pair of goods (hotdogs and hamburgers?). $\endgroup$
    – Steven Gubkin
    Oct 12, 2020 at 15:22
  • $\begingroup$ I don't know, I think the divergence of either one of these is rather curious from a math stand point... they're zero everywhere they're defined, yet, they produce nontrivial flux. It would seem violate the divergence theorem... except... that pesky singular point is where all the action is... I hear you though, more examples would be great. $\endgroup$
    – James S. Cook
    Oct 13, 2020 at 6:31

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Air speed/direction on a weather map) is a very intuitive one. There's also other fluid velocity (and flux) vector fields in various chemE, mechE, and nukeE applications.

I personally think the air speed is most intuitive as something where you really need speed and direction (i.e. a vector, not a scalar) and it's something people encounter in daily life. Electrostatics is a little mysterious as I always find electrical things more "hidden" then mechanics or fluids. But the nice thing is you can actually do demonstrations of magnetic fields using iron filings on the overhead projector (showing the hidden field).

My concern a little about the two force fields, you listed is that many familiar two point calculations can be done, rather more conveniently, using scalars to solve for forces, potential energy change etc. I.e. it's not clear why we need to invoke vector calculus. Granted, it is possible to complicate the problems. But, I just would think to list some situations like air flow, where it's very clear the situation is more complex.

[Edit: for Steve. It's an interesting question, but I would be pretty hesitant about showing such an example to beginning general calc 3 students. After all, most econ undergrads don't even have a calc 3 requirement, sometimes taking "business calculus" rather than even a normal calc 1/2 sequence. Note: it's very important to differentiate between the requirements of research level econ and typical undergrad work. This in contrast to say engineering or physics, which are fairly mathematical even at the BS level.]

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    $\begingroup$ +1; It is really fun to ask students what div F and curl F are when F is the wind-on-a-weather-map vector field. Is the wind a conservative vector field? Is it the gradient of the barometric pressure function? $\endgroup$
    – Chris Cunningham
    Oct 12, 2020 at 14:42
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The two examples you give both have zero curl, which limits their usefulness. Examples that do have a curl would be:

  1. an electromagnetic wave

  2. the magnetic field of a wire, inside the wire

  3. the magnetic field of a slab of current, inside the slab

  4. the field of a point charge that is moving inertially.

The external magnetic field of a wire is also an interesting example, because it looks curly, but actually has a curl of zero.

I would suggest not referring to your $1/r^2$ examples by names like "the electrostatic field." Many, many students have an unshakeable belief that the $1/r^2$ equation applies to virtually any static field, including, say, the field of a dipole or of a charged plane. A $1/r^2$ electric field is the field of a point charge at rest (or the external field of a spherically symmetric charge distribution).

You use the word "force field," but that isn't really right. Physicists don't say "force field." And your examples of E and F aren't analogous. The gravitational field is $g$, not $F$, and it doesn't have units of force. The things physicists call fields are properties of empty space. They're not interactions between objects. I would present these as:

The electric field of a point charge at rest:

$\mathbf{E}\left(\mathbf{r}\right)=\frac{kQ}{\left|\left|\mathbf{r}\right|\right|^3}\mathbf{r}$

The gravitational field of a point mass:

$\mathbf{g}\left(\mathbf{r}\right)=\frac{GM}{\left|\left|\mathbf{r}\right|\right|^3}\mathbf{r}$

(In newtonian gravity, which is what you're doing here, it doesn't matter if the mass is at rest.)

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  1. The oscillation of species' populations due to predation.

Vector field formed by population dynamics of two species

  1. The convergence of machine learning models using gradient descent - this is a special vector field that's tuned by uniformly multiplying the field with a scalar (bonus points for involving a segue into machine learning theory and having them tune the vector field for most rapid descent)

A couple examples: https://blog.paperspace.com/intro-to-optimization-momentum-rmsprop-adam/ https://datascience-enthusiast.com/DL/Optimization_methods.html

It might help to know the demographics of your target audience to tune the examples toward things they're most likely interested in.

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The Hairy Ball Theorem:

      

wikiwand link: "there is no nonvanishing continuous tangent vector field on even-dimensional $n$-spheres."

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    $\begingroup$ I have always been bothered by the fact that the hairy vector field on most human heads visibly vanishes on half of it. A head better serves as an example of a hairy hemisphere theorem, with suitable hypotheses. +1 anyway. $\endgroup$
    – Raciquel
    Oct 18, 2020 at 14:10
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As I wrote here, in machine learning applications we are often trying to minimize some cost function $J(\theta)$ and the vector field induced by the gradient $-\nabla_{J(\theta)}$ is important in numerical approximations to minimizing error.

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