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
uh momentum conservation and a fluid navy or stokes equations mass conservation is
the continuity equation and so on and so forth so uh 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 uh 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's theorem and gauss's 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 uh mass continuity navy or 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 uh we can kind of express in the language
of vector calculus and i'm super excited to walk you through this uh and so in general
we think about a kind of a vector field so um maybe i'll just start with a good example
that i like which is uh 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 under bar 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 okay 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 going to 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 non-linear
product terms and things like that will derive all of these from scratch in particular you know these
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 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 i can derive many different vector and scalar fields that are
the solution the partial differential equations for example the temperature in a metal plate if
i hit it with a blow torch 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 blow torch
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 um 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 n 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.
So 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 sea... so this is me at time 0... I'm "x_0" 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 by Navier-Stokes equations
but it also induces a new dynamical system... so if I drop a particle, then that particle sees this dynamical
system so I could also say -- let me see how I want to write this -- I want to say dx/dt, and again
this is my bold x is equal to u(x,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 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 um 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 okay 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 going to be greater than
0 at this point and we would say like div is positive this has a 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 okay 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 you know big cyclone 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 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 uh good and of course i hope it's obvious that in
three dimensions this u would have a u v and a w component for the x y and z components of
the velocity field and this can generalize to 10 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: 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... not all, but 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's theorem like
Gauss's divergence theorem... we're going to use those to encode these physical
ideas of conservation of mass, momentum, and energy 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