Divergence and curl: The language of Maxwell's equations, fluid flow, and more

[Translated by Grant Sanderson. Submit corrections at criblate.com] Today, you and I are going to get into divergence and curl.

To make sure we're all on the same page, let's begin by talking about vector fields.

Essentially a vector field is what you get if you associate each

point in space with a vector, some magnitude and direction.

Maybe those vectors represent the velocities of particles of fluid at

each point in space, or maybe they represent the force of gravity at

many different points in space, or maybe a magnetic field strength.

Quick note on drawing these, often if you were to draw the vectors to scale,

the longer ones end up just cluttering up the whole thing,

so it's common to basically lie a little and artificially shorten ones

that are too long, maybe using color to give some vague sense of length.

In principle, vector fields in physics might change over time.

In almost all real-world fluid flow, the velocities of particles in a given

region of space will change over time in response to the surrounding context.

Wind is not a constant, it comes in gusts.

An electric field changes as the charged particles characterizing it move around.

But here we'll just be looking at static vector fields,

which maybe you think of as describing a steady-state system.

Also, while such vectors could in principle be three-dimensional,

or even higher, we're just going to be looking at two dimensions.

An important idea which regularly goes unsaid is that you can often

understand a vector field which represents one physical phenomenon

better by imagining what if it represented a different physical phenomenon.

What if these vectors describing gravitational force instead defined a fluid flow?

What would that flow look like?

And what can the properties of that flow tell us about the original gravitational force?

And what if the vectors defining a fluid flow were thought

of as describing the downhill direction of a certain hill?

Does such a hill even exist?

And if so, what does it tell us about the original flow?

These sorts of questions can be surprisingly helpful.

For example, the ideas of divergence and curl are particularly viscerally understood

when the vector field is thought of as representing fluid flow,

even if the field you're looking at is really meant to describe something else,

like an electric field.

Here, take a look at this vector field, and think of each

vector as describing the velocity of a fluid at that point.

Notice that when you do this, that fluid behaves in a very strange, non-physical way.

Around some points, like these ones, the fluid seems to just spring into

existence from nothingness, as if there's some kind of source there.

Some other points act more like sinks, where the

fluid seems to disappear into nothingness.

The divergence of a vector field at a particular point of the plane tells you

how much this imagined fluid tends to flow out of or into small regions near it.

For example, the divergence of our vector field evaluated at all

of those points that act like sources will give a positive number.

And it doesn't just have to be that all of the fluid is flowing away from that point.

The divergence would also be positive if it was just that the fluid coming into