Welcome back. So I'm really excited about a brand new lecture series on vector calculus and how we use vector calculus to derive and solve partial differential equations. So this is one of my absolute favorite fields of all of mathematics is dynamical systems and partial differential equations. And essentially these partial differential equations express conservation laws.
So all partial differential equations from physics essentially express some conservation laws like mass, momentum, energy. So let's say mass, momentum, momentum, and energy. Okay, so for example, momentum conservation in a fluid, Navier-Stokes equations. Mass conservation is the continuity equation, and so on and so forth.
So we're going to essentially learn how vector calculus is the language of partial differential equations. Okay, so vector calculus. And when I say vector calculus, I mean things like, you know, the gradient.
The divergence and the curl are essentially the language of how we describe these conservation laws. So this is the language. for PDEs. And usually these are taught kind of separately.
When I was learning, you know, vector calculus was an interesting thing that was independent. You know, you solved a bunch of surface integrals and volume integrals. They didn't really tell you why you were doing that. But it turns out that the reason you want to learn vector calculus is because that is how you translate the world, the physical laws of the world into differential equations that you can solve and analyze on pencil and paper, on a computer.
And this is where you build all of the intuition for how these things work. Okay. So I think, you know, this is a pretty interesting perspective. This is kind of a holistic view of not just PDEs.
but PDEs and vector calculus. And so right now I'm just going to give you a very high level view of some of the things we're going to talk about. So we're going to talk about the intuition for div, grad, and curl, these vector calculus operations, things like Stokes'theorem and Gauss'theorem that allow you to really do some of these. Manipulations you need to write these conservation laws down as partial differential equations. And then we're going to start deriving from scratch the heat equation, Laplace's equation, mass continuity, Navier-Stokes.
And we're going to make very clear what the assumptions along the way are. Are there shock waves? Is there some kind of laser, you know, pumping energy into my system?
All of this we can kind of express in the language of vector calculus, and I'm super excited to walk you through this. And so in general, we think about a kind of a vector field. So maybe I'll just start with a good example that I like, which is I always think about fluid flows.
So let's think about a fluid flow in the Gulf of Mexico. And this is a really terrible Gulf of Mexico. Maybe I'm from here. Cuba is somewhere not on this map. And we assume that there is some literally a vector field.
There is a fluid velocity field at every point in space and realistically, it's probably changing in time. Okay, so there is some vector field we're going to call this u of x and t and when I write it this way usually I'm going to put an underbar or an over arrow when I mean a vector. Okay, and so literally what I mean by this is that this is a two-dimensional vector. consisting of the, let's say I take one of these, it has, so this is vector u, it has a u component and a v component, a component in the x direction and a component in the y direction. So this is u of x and y comma t and little v of x and y comma t.
And this little u is different than this bold u. This bold u is a vector of the x component and the y component of the velocity. So this is my fluid velocity field, and it is the solution of a partial differential equation. There is some partial differential equation that represents mass, momentum, and energy conservation, namely the Navier-Stokes equations.
And I would have to solve that PDE to get this vector field. Okay, so this is a vector field. And I call it a vector field because literally it's a field of vectors. At every point x and y and at every instant in time, there is a vector at that location and it's like a field of vectors. Like if you look at a wheat field blowing in the wind, it shows you the vector field of the wind.
And that's what this is here, okay? And so this vector field is the solution of a partial differential equation. So in this case, maybe I have some PDE. which is partial u partial t. So this vector field u, and this is bold u, it's changing in time, and its rate of change in time at every point in space is gonna be given by some generically nonlinear function of that vector field, okay?
Good, and this might have other partial derivatives with respect to space, it might have nonlinear product terms and things like that. We'll derive all of these from scratch. in particular, you know, these equations here.
And importantly, if this is a real flow in the Gulf of Mexico, you know, the flow field changes in time also. So I'm going to say, and not just the flow field changes in time, the dynamics change in time because it's heated differently in the summer than in the winter. So there might also be some time dynamics.
This might be a function of the state and the time. And this is a very general framework. I can derive many different vector and scalar fields that are the solution to partial differential equations.
For example, the temperature in a metal plate, if I hit it with a blowtorch in the middle of that plate, maybe I'll draw that. So if I have some metal plate here and I hit it with a blowtorch, so I have some kind of, you know, hot pattern here. and then it gets cooler and cooler out here. And I want to solve for that temperature distribution. Let's call that big T as a function of x and y and time.
That is again the solution of a partial differential equation. In particular, this one is pretty easy to write. It's partial T with respect to time is equal to some positive constant alpha squared times the partial squared in x plus the partial squared in y, which I'm going to very compactly represent using my vector notation as the Laplacian of the temperature, which is partial squared t partial x squared plus partial squared t partial y squared. Okay, so again, I can derive, we're going to derive this heat equation from scratch to satisfy this physics here.
Okay, namely, you know, kind of conservation of heat energy in this uh in this plate good so we're going to show you how to derive partial differential equations by encoding these conservation laws using the language of vector calculus and we'll be able to describe tons of physics phenomena from heat distributions to how an airplane wing works and generates lift to how a flow field evolves in the Gulf of Mexico or in some other fluid flow environment. Okay, so really, really exciting stuff here. And I want to point out also, if I have a vector field, some solution of a partial differential equation u, I can also think about the induced ordinary differential equation of if I dropped a particle in this flow field. So imagine that I'm lost at c. So this is me at time 0. I'm x naught and that means x at time 0, y at time 0. And what we're going to do is we're going to be able to solve for the future trajectory of that point by integrating that particle through this vector field.
So this vector field, this flow field, is the solution of my partial differential equation, my Navier-Stokes equations. but it also induces a new dynamical system so if i drop a particle that particle sees this dynamical system so i could also say let me see how i want to write this i want to say ddt of x and again this is my bold x is equal to U of X and T. So this U is a dynamical system for how a particle X will float through the Gulf of Mexico.
Maybe this is a little oil spill and I want to track that oil spill and see where it goes. There's lots of tools for, you know, propagating a little Gaussian through this dynamical system. That's something I might want to do on a computer to predict where the oil spill is going to go.
Or maybe someone got lost at sea off the coastline and swept away, and I've got four hours to rescue them. Where should I search to find those people? These are real problems that engineers are working on today based on the numerical solution of these problems, which is extremely cool.
I mean, I just love how many applications, and it's tied to all of my favorite things in dynamical systems at the end of the day. These PDEs set up dynamical systems for the fluid flow evolution. And the fluid flow itself sets up a dynamical system for how particles move through the fluid flow. Good.
And we could do things like integrate that particle through and find x at time t. Okay, and this might be, you know, little x at time t and little y at time t. And I could integrate numerically the position of this particle through that vector field. Okay.
What else do I want to show you? Maybe I'll give you just a very, very quick example of what grad and div and curl are in these examples here. So if I have a local patch of my flow field where things are locally on average moving away, kind of diverging, this is a location that has a positive divergence. So the div of u is gonna be greater than zero at this point. And we would say like, Div is positive.
This has a positive divergence. And so the divergence, I'll walk you through all of these in great detail and show you exactly how this works. But in some sense, the divergence measures locally how much stuff is kind of flowing away from that point or sucking into or converging into that point. So like your sink is literally a place where the divergence is less than zero.
And when you like hit the faucet and it hits the metal, and it's causing the flow to go out, that's a place where the divergence is greater than zero. Similarly, maybe it's hurricane season, and I've got a region over here where my flow is going in a big cyclone, and it's coming in to the gulf. This would be an area where I would have a positive curl.
Okay, and so the curl would be greater than zero, and that's given by this kind of del cross. and I'll show you how these dots and crosses work. This grad operator is literally a vector of partial derivatives, and I can take its dot product with u and v, or its cross product with this uv velocity field.
Good. And of course, I hope it's obvious that in three dimensions, this u would have a uv and a w component for the x, y, and z components of the velocity field. And this can generalize to ten dimensions or a million dimensions.
You can have... the heat equation in 5d there's nothing wrong with that it's just this equation here okay the last one i haven't shown you is the gradient and i really like the gradient because the gradient takes a scalar field like a temperature distribution this is just a scalar at every point and it returns a vector field so if i compute the gradient of my temperature it will essentially if i look at the gradient of my temperature at this point here It will tell me in what direction is the temperature locally increasing the fastest. So it'll essentially establish a vector field of where the temperature is increasing the fastest.
So grad of t is a vector at every point in x and y that says, you know, climb this way if you want to get hotter faster. If you want to go to the hot spot as fast as possible, follow the gradient direction. This is the grad. And this is a huge deal in modern computer optimizations and machine learning.
We do gradient descent or gradient searches to find the maximum value of a cost function or the minimum value of a loss function. And we use stochastic gradient descent to train all of our machine learning algorithms. So, not all, many of our machine learning algorithms. So again, these gradients extend to million-dimensional spaces where we train models for machine learning.
And it all comes back to this kind of physical intuition of, you know, partial differential equations and conservation laws, vector calculus. I'm super excited to walk you through this. I'm going to start simple.
We're going to start with kind of what these vector calculus terms mean. We're going to start learning things like Gauss's theorem and Stokes'theorem, like Gauss's divergence theorem. We're going to use those to encode these physical ideas of conservation of mass, momentum, and energy. Okay. and derive our partial differential equations for real physical systems, things like Navier-Stokes or the heat equation or mass continuity.
This is all coming up. Really excited to walk you through this. This is how I wish I had learned vector calculus and PDEs.
So please stay tuned and keep watching for more. Thank you.