Today, you and I get into divergence and curl.
But if you'll indulge me, it's worth sharing a little about the writing trajectory that
led me here.
I originally started writing this video to be a follow up to the last one about alternate
ways to think about the derivative, namely it was going to be a view of what it means
for a complex function to have a derivative.
These functions are very interesting once you know them, but until then the words "complex
derivative" aren't exactly the best way to hold a new learner's interest, so I wanted
to center it around some more tangible motivating examples for where such functions pop up.
One that's pretty interesting, and fun to illustrate, is that a certain representation
of the simple function z + 1 / z can be seen as giving an idealized model for fluid flow
around a cylinder.
I'll explain what I mean by this fully later on, but in a nutshell, these warped grid lines
represent where the real or imaginary parts of the output stay constant.
The horizontal lines are showing streamlines for the flow, and the flow is fast in regions
where the vertical lines are close together, and slow in regions where the vertical lines
are farther apart.
Alright, so that's kind of interesting, but what's more fun is that if you shift and scale
this setup the right way, then apply the same simple function z + 1 / z to everything, you
now get a simplified model for flow around this airfoil-looking shape.
Intriguing, right?
And more still, that same original warped grid also has a completely different physical
interpretation.
Let's say you have a uniform electric field, pointing up; Meaning it would push positively
charged particles up and pull negative charges down.
If you put some copper wire in this field, one with a circular cross-section, then under
the assumption that the charges in this wire are free to move around, the negative charges
will accumulate in a certain way on the bottom, leaving the top generally positive charged,
resulting in some change to the electric field around the wire.
Those same lines which previously described streamlines of an idealized fluid flow around
a cylinder happen to be exactly the lines of equal electric potential for this new field;
that is, stepping from one line to an adjacent one corresponds to a constant voltage drop.
This raises lots of good questions.
What does idealized fluid flow have to do with electric potential?
And what do both of these have to do with complex numbers?
To really understand what's going on here, you need to be comfortable with two central
ideas from vector calculus, divergence and curl.
For me, scope creep eventually turned into a sort of scope meiosis as the section giving
this background grew out into its own video, and, well, here we are now.
Anyway, you might be wondering why I'm spending your precious minutes and my precious hours
to tell you all this, rather than just jumping straight into the actual topic.
Well, individual topics tend to be less enlightening than the connections between them.
Learning about divergence and curl runs the risk of feeling arbitrary if it's just some
other thing you do with derivatives.
But there's something kind of exhilarating if from the get-go, you have some awareness
for just how far-reaching the ideas will be.
To make sure we're all on the same page, let's first talk 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 a fluid at each point; maybe
they represent the force of gravity at many different points in space; or maybe magnetic
field strength
Quick note on drawing these: Often if you draw all the vector to scale, the long ones
clutter things up, so it's common to lie a little, 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 constant, it comes in gusts.
An electric field changes as the charged particles characterizing it move.
But here, we'll just be looking at static vector fields, which you might think of as
describing a steady state system] Also, while such vector fields could in principle be three-dimensional,
or even higher, we're just going to be looking in 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 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?
What can that tell us about the original gravitational force?
What if these vectors defining fluid flow were thought of as describing the downhill
direction of a certain hill?
Does such a hill exist?
If so, what does it tell us about the flow?
These sorts of questions can be surprisingly helpful.
For example, the ideas of divergence and curl can be particularly viscerally understood
when a vector field is thought of as representing fluid flow, even if the field you're looking
at is really meant to represent, say, an electric field.
Take a look at this vector field, and think of each vector as determining the velocity
of a fluid at that point.
Notice that this behaves in a very strange non-physical way: Around some points, like
these ones, the fluid seems to spontaneously spring into existence from nothingness, as
if there is some source there.
Some other points act more like sinks, in that the fluid seems to disappear into them.
The divergence of a vector field at a particular point on the plane tells you how much this
imagined fluid tends to flow out of or into a small region near a point.
For example, the divergence of our vector field evaluated at all those points that act
like sources will give a positive number.
And it doesn't just have to be that all the fluid is flowing away from that point; the
divergence would also be positive if it was just that the fluid coming at it from one
direction was slower than the flow away from it in another direction.
On the flip side, if in a small region around a point there seems to be more fluid flowing
_into_ that region than out of it, the divergence at that point would be a negative number.
Remember, this vector field is a really a function that takes in two-dimensional inputs
and spits out two-dimensional outputs.
The divergence of that vector field is a new function, which still takes only a single
2d point as its input, but its output depends on the behavior of the vector field in a small
neighborhood around that point.
That output is just a single number, measuring how much that point acts as a source or a
sink.
I'm purposefully delaying discussion of computations here; the understanding for what it represents
is more important.
Notice, this means that for an actual physical fluid, like water, rather than some imagined
one used to illustrate an arbitrary vector field, then if that fluid is incompressible,
the velocity vector field must have a divergence of zero everywhere.
That's an important constraint on what kinds of vector fields could solve a real fluid
flow problem.
For the curl at a given point, you also think about the fluid flow near each point, but
this time you ask how much that fluid tends to _rotate_ around that point.
As in, if you dropped a twig in that fluid at that point, somehow fixing its center in
place, would it tend to spin around.
Regions where that rotation is clockwise have a positive curl, regions where it's counter-clockwise
have a negative curl.
And it doesn't have to be that all the vectors around an input are pointing clockwise or
counterclockwise, a point in a region like _this_ would also have non-zero curl, since
the flow is slow on the bottom and quick up top, resulting in a net clockwise influence.
Really, proper curl is a three-dimensional idea, one where you associate each point in
3d space with a new _vector_ characterizing the rotation around that point according to
a certain right-hand rule.
I have plenty of content from my time at Khan Academy describing this in more detail, if
you want.
[But for our main purpose, which will be showing the connection between these vector calculus
ideas and complex analysis, I'll just be referring to the two-dimensional variant of curl which
associates each point in 2d space with a number.
As I said, even though these intuitions are in the context respect to fluid flow, both
these ideas are significant for many other sorts of vector fields.
For example, electricity and magnetism are described by four equations, known as Maxwell's
equations, written in the language of divergence and curl.
This top one, for example, is Gauss's law, stating that the divergence of the electric
field is proportional to the charge density at a point.
Unpacking the intuition for this, you might imagine positive charges as being _sources_
of some imaginary fluid, negative charges as the _sinks_ of that fluid, and in regions
without charge, that fluid would be flowing incompressibly, just like water.
Of course there's not some literal electric fluid, but it's a useful and pretty way to
read an equation like this.
Similarly, another important equation is that the divergence of the magnetic field is zero
everywhere.
You could understand by saying that if this field represented a fluid flow, that fluid
would be incompressible with no sources or sinks.
This also has the interpretation that magnetic monopoles, something that acts just like the
north or south end of a magnet in isolation, don't exist.
Likewise, the way one of these fields changes depends on the curl of the other field.
Really, this is a three-dimensional idea, and a little outside of our main focus here,
but the point is that divergence and curl arise in contexts unrelated to flow.
Side note, the last two equations are what gives rise to light waves.
And quite often these ideas are useful in contexts which at first don't even seem spatial
in nature.
To take a classic example that differential equations students study, let's say you want
to track the population sizes of two different species, where maybe one is a predator of
another.
The state of this system at a given time, meaning the two population sizes, could be
thought of a point in 2d space; what you would call the "phase space" of this system.
For a given pair of population sizes, those populations may be inclined to change based
on how reproductive these species are, and based on just how much one of them enjoys
eating the other.
These rates of change would typically be written as a set of differential equations.
It's okay if you don't understand these particular equations, I'm just throwing them up for those
who are curious (and because replacing variables with pictures makes me laugh).
The relevance here is that a nice way to visualize what such a set of equations is really saying
is to associate each point on the plane, each pair of population sizes, with a vector indicating
the rates of change for both variables.
For example, when there are lots of foxes, but few rabbits, the number of foxes might
tend to go down because of a constrained food supply, and the number of rabbits may also
tend to go down because they're getting eaten by all the foxes, potentially at a faster
rate than they can reproduce.
So each vector here is telling you how, and how quickly, a given pair of population sizes
tends to change.
Notice, this is a case where a vector field is not about physical space, but instead it's
a representation of how a dynamic system with two variables evolves over time.
This can also maybe give a sense for why mathematicians care about studying the geometry of higher
dimensions.
The flow associated with this field is called "phase flow" for our differential equations,
and it's a way to conceptualize at a glance how many possible starting states would evolve
over time.
Properties like divergence and curl can help inform you about the system.
Do the population sizes tend to converge towards a particular pair of numbers?
Are there any values that they diverge away from?
Are there any cyclic patterns?
Are these cycles stable or unstable?
To be honest, for something like this you'd often want to bring in related tools beyond
divergence and curl to get the full story, but the frame of mind that practice with these
two ideas brings you carries over well to studying setups like this with similar pieces
of mathematical machinery.
As to computing divergence and curl, I'll leave some links to where you can learn about
this and practice if you want.]
Again, I did some videos, articles and worked examples for Khan Academy on the topic during
my time there, so too much detail here would start to feel redundant for me.
There is one thing worth bringing up, though, regarding the notation associated with these
computations.
Commonly the divergence is written as a dot product between this upside-down triangle
thing and your vector field function, and the curl is a similar cross product.
Sometimes students are told that this is just a notational trick; each computation involves
a sum of certain derivatives, and treating this upside-down triangle like a vector of
derivative operators can be a helpful way to keep everything straight.
But it is more than just a mnemonic device, there is a real connection between divergence
and the dot product, and between curl and the cross product.
Even though we won't be doing practice computations here, I would like to give you some sense
for how these four ideas are connected.
Imagine taking some small step from one point on your vector field to another.
The vector at this new point will be a little different from the one at first point, there
will be some *change* to the function after that step, which you might see by subtracting
your original vector from the new one.
This kind of difference to your function over small steps is what differential calculus
is all about.
The dot product of your step vector with that difference vector
tends to be positive in regions where the divergence is positive, and vice versa.
In some sense, the divergence is sort of average value for this dot product of a step with
the change to the output caused by that step over all possible step directions, assuming
things are rescaled appropriately.
If a step in some direction causes a change to the vector _in that same direction_, this
corresponds to a tendency for outward flow; for positive divergence.
Similarly, the cross product of your step vector with the difference vector tends to
be positive in regions where the curl is positive, and vice versa.
You might think of the curl as a sort of average of this step-vector-difference-vector cross
product.
If a step in some direction corresponds to a change _perpendicular_ to that step, it
corresponds to a tendency for flow rotation.
Our next step will be understanding how functions of complex numbers give us an elegant way
to produce vector fields where the curl and divergence are zero everywhere.
Thought of in terms of flow, this describes fluids which are both incompressible and irrotational.
Thought of in terms of electromagnetism, this gives steady-state fields in a vacuum, where
there are no charges or current.
This is what I'll be talking about in the next video, where you and I will return back
to those models for flow around a cylinder, and an airfoil, and importantly we'll talk
about where these models fall short, and why.
So... typically this is the part where there might be some kind of sponsor message.
But one thing I want to do with the channel moving ahead is to stop doing sponsored content,
and instead make things just about the direct relationship with the audience.
I mean that not only in the sense of the funding model, with direct support through Patreon,
but also in the sense that I think these videos can better accomplish their goal if each one
feels like it's just about you and me sharing in a love of math, with no other motive, especially
in the cases where viewers are students.
That's not the only reason, and I wrote up some of my full thoughts on this over on the
Patreon, which you certainly don't have to be a supporter to read.
But since I've already opened the doors for weird self-reflection on the process with
the intro of this video, maybe I'll just run with that and give some of the high level
here:
I think advertising on the internet occupies a wide spectrum from truly degenerate to genuinely
well-aligned win-win-win partnerships.
I've always taken care to only do promotions for companies I would genuinely recommend.
To take one example, you may have noticed I did a number of promos for Brilliant, and
it's really hard to imagine better alignment than that: I try to inspire people to be interested
in math, but am also a firm believer that videos aren't enough, you need you to actively
solve problems, and then here's a platform that provides practice.
And same goes for any others, I always feel good alignment.
But even still, even if you seek out the best possible partnerships, whenever advertising
is in the equation the incentives will be to try reaching as many people as possible.
When the model is more exclusively about a direct relationship with the audience, though,
the incentives are pointed towards maximizing how _valuable_ people find the experiences
their given.
These two goals are correlated, but not always perfectly.
I like to think I'll try to maximize the value of the experience no matter what, but for
that matter I also like to think I can consistently wake up early and resist eating too much sugar.
What matters more than wanting something is to align incentives.
Anyway, if you want to hear more of my thoughts, I'll link to the Patreon post.
And thank you to existing supporters for making this possible.
See you in the next video!
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