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PROFESSOR STRANG: OK,
let's start with a

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review and preview.
 

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I put a P up there because
we're really looking into

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the Fourier part that just
started this morning.

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And there'll be some homework
from these early sections about

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the Fourier stuff, so we maybe
we should just do a few

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of those problems or
discuss here today.

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Just in advance.
 

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Can I say one thing about
MATLAB and the MATLAB

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homework first?
 

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And maybe open a
conversation about it?

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So there's really two different
problems that I'm personally

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quite interested in.
 

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Two model, I'll say model
problems because they're

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there for regular
polyons in a circle.

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And I'll draw an octagon
again, so M sides.

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And I'm interested in
if M goes to infinity.

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And I'm interested in
two different problems.

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So one of them is our
MATLAB problem, minus

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is Laplace's equation.
 

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What was it, four?
 

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With u=0 on the circle.
 

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OK, so that's our problem,
totally open for discussion.

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How many have started on that?
 

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Oh, good.
 

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OK.
 

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Well, then you all know
more about it than I.

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And that's great.
 

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I'd be happy to learn.
 

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So have I said
everything there?

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Yeah, we've got Poisson's
equation inside.

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We've got u=0 on the circle, so
the problem's well defined and

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the solution should be
one minus x squared

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minus y squared.
 

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So that's the correct solution.
 

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Maybe I can also tell you
about the second problem

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that I'm interested in.
 

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Because it hasn't come
up in class but it's

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very important, too.
 

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It would be the
eigenvalue problem.

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So this is problem number one,
the steady state problem

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when you've got a source and
you want to find out the

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temperature distribution.
 

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The problem number two
would be the eigenvalue

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problem, -u_xx-u_yy.
 

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I take those minuses so
that the eigenvalue

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will be positive.
 

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So that's what the eigenvalue
problem might look like.

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And again let me say, with
u=0 on the boundary.

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On the third one.
 

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OK, so a person would say
this is Laplace's eigenvalue

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problem because we have
Laplace's equation.

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We've got eigenvalue.
 

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As always, it's not linear
because we have two unknowns,

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lambda's multiplying u.
 

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And we have boundary
conditions, and this would

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describe the normal modes, for
example, of a circular drum.

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If I had a drum, or
a polygon drum.

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So maybe I connect to actually
build the drum, I might fold in

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the sides there and
have a polygon.

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And again, I hope that the
eigenvalues of the polygon,

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this equation in the
polygon, which are not

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known, by the way.
 

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To the best to my knowledge, we
know it only for M=3, which

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would be an equilateral
triangle, and M=4, which

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would be a square.
 

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And those eigenvalues, because
of Fourier or something

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are humanly doable.
 

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But I think five on up is, I
may be wrong about six I'm

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not sure about M=6,
a hexagon sometimes

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gives you enough help.
 

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But beyond that
you're on your own.

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With finite elements
to help you.

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So there's a whole sequence of
eigenfunctions, u, eigenvalues,

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lambda, just the way there
were in one dimension.

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And on the circle
they involve Bessel.

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That's where Bessel showed up.
 

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He figured out the functions
and they're not especially

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nice functions.
 

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But they're studied
for centuries.

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Bessel functions
come into that.

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But, here I have
the same question.

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I mean, let me just say, for me
this could be a UROP project if

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anybody was an undergraduate,
or it could be a project

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over January or something.
 

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I'd like to know something
about what happens as M goes

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to infinity as the polygon
approaches the circle.

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So I'm hoping maybe on the
homework that comes, if it's

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not too difficult, and maybe
it's not, to let m go up a bit.

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There is one thing.
 

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That the code we're working
with is linear elements, right?

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We're using linear
finite elements.

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So we're not getting
high accuracy.

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So I would really like to move
up to quadratic elements, at

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least, you remember quadratic
elements would be ones where -

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well, let me draw the one that
we've drawn in class before.

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We only have to look at one
triangle and then we cut it up

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into triangular elements by
taking some pieces here, taking

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the points above, which I hope
are now correct on the website.

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Connecting those edges, and
then connecting these.

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Is that right?
 

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Is that our mesh?
 

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So that mesh is
controlled by N.

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One, two, N points.
 

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Also, N is going to have to get
large too, to give me accuracy.

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And another way toward more
accuracy is, instead of linear

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elements, second degree.
 

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So do you remember I
wrote those down?

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Let me take that little
triangle out here as

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a bigger triangle.
 

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It would look something
like that, I guess.

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The second degree elements
have those six mesh points.

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You remember I drew those but
we didn't really have time

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to get further with them.
 

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The trial functions phi, which
are one at a typical mesh point

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and zero at all the others,
they are computable.

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We're up to second degree, so
it's a little, second degree

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things, then the first
derivatives, which come into

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the integrations, are linear.
 

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And not constant.
 

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So a little bit harder.
 

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But finite elements,
linear or quadratic.

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Or higher.
 

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Could be used for this problem.
 

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As we know, and
for this problem.

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What I wanted to add, that I've
not mentioned in class, and I

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think we may just not get a
chance to do it, is what does

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the finite element method look
like for an eigenvalue problem?

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Because eigenvalues
are highly important.

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That's the different
way to understand.

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There's the matrix
K and its entries.

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But then there eigenvalues.
 

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And you might think that, what
do you think is the discrete

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eigenvalue problem
copying this one?

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Here's my point.
 

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Your first guess would be,
well this is like K, right?

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This is like KU, right? (K2D)U,
I should call it, maybe.

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Well, I'll call it K, because
K2D I have specifically

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reserved for the Laplace
stiffness matrix on a square

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mesh, square mesh with
triangles, the K2D.

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That was one specific matrix
for one specific mesh, and here

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we have a different mesh.
 

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So I should just call it K.
 

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Ok, I think if anybody was
going to make a guess, they

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would say OK, KU=LAMBDA*U.
 

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Maybe I'll use capital LAMBDA,
because I'm using capital U.

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Is this the finite element
method eigenvalue problem.

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And if you answered yes, I
would have to say, well

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that's a reasonable answer.
 

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But it's wrong.
 

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The eigenvalue problem, when I
take the differential equation

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for the Laplace, Laplace's
equation, lambda u on the right

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side, and I go to do finite
elements, it produces K.

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Out of this stuff, out of the
weak form, all that stuff.

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But it produces another matrix
on the right-hand side from the

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constant term, and we have
not really mentioned it,

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it's the mass matrix.
 

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So this, instead of just
the identity here,

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there's a mass matrix.
 

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So that is the problem
that you could do.

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I could've made a
MATLAB project.

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I bet I'd do it
next fall. right?

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You guys did the
first one, this one.

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Or you are doing it now.
 

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And I'm going to pause in a
minute for questions about

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it, or discussion of it.
 

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But this one brings
in something called

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the mass matrix.
 

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So let me just say
what those are.

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If I write down the entries in
the mass matrix, you'll sort of

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get an idea of why they are.
 

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So what are the entries
in the stiffness matrix?

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K_ij, you remember, is
the integral of the d

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phi_i/dx, d phi_j/dx.
 

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Plus d phi_i/dy, d phi_j/dy,
dxdy, and that's what's

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you're computing.
 

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And that's what that
code is computing.

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And when phi is linear,
phi linear, then

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slopes are constant.
 

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So all you have to do, and what
that code in the book is doing,

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is figuring out what
are the slopes.

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These things are constant, so
we just need to know the area

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of the integration where
we're integrating.

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The area, triangle by triangle.
 

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Fine.
 

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That's what we're doing.
 

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That's what that code
is just set up to do.

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Now, I have to tell you what
is M_ij, the mass matrix.

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I just think you don't want to
have - we haven't done too

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badly with finite that
elements in here.

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We did it in 1-D, where we
got it kind of straight.

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And now we're seeing
what it looks like 2-D.

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But I had not really
mentioned a mass matrix.

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So here it comes.
 

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The mass matrix will
be the integral of

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phi_i i times phi_j.
 

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dxdy.
 

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It's the zero order, no
derivatives, just plain zero

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order, as you'd expect from the
fact that the term in the

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continuous part is zero order.
 

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So it's this mass
matrix that comes in.

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And maybe we could just look
to see which entries will

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be zero and which will not.
 

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How sparse is it?
 

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00:14:25 --> 00:14:28
What does the mass
matrix look like?

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And can we just, let
me do 1-D first.

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So there's a phi, right?
 

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There's another one.
 

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There's another one.
 

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So, what do you think about the
mass matrix, one phi multiplied

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by another phi and integrated?
 

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Is it diagonal?
 

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No, because each phi
overlaps its two neighbors.

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So tell me what kind of a
matrix m is going to be?

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In 1-D.
 

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Tri-diagonal.
 

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It'll be tri-diagonal.
 

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Now, so was K.
 

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So K and M actually have
non-zeroes in the same places.

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the same sparsity pattern.
 

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But, of course, not the
same numbers in there.

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K had minus ones and twos
and fours and minus ones.

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What can you tell me about
this tri-diagonal matrix?

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When I integrate that against
this, well, again I would do it

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element by element because this
against this, they

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only overlap here.
 

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Right?
 

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I'll just draw the one
place that they overlap.

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And what's the point?
 

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They're both positive.
 

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So the mass matrix is, its
rows don't add to zero.

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Its rows tend to add to one.
 

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But it's not diagonal,
that's the difference.

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OK, so I just felt I couldn't
feel as though I'd done a

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decent job in describing
finite elements if I

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00:16:21 --> 00:16:23
didn't describe this.
 

246
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Didn't mention
this mass matrix.

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And maybe I'd better say
where it comes from.

248
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Because eigenvalue problems, it
may come number two, but that's

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pretty high up the list.
 

250
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So let me tell you where does
this mass matrix come from.

251
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First, let me tell you
about eigenvalues of

252
00:16:52 --> 00:16:56
a, matrix eigenvalues.
 

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So the answer was, is
this the finite element

254
00:17:00 --> 00:17:01
eigenvalue problem?
 

255
00:17:01 --> 00:17:03
Only if there's an M there.
 

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And now I want to, OK, first
of all, what MATLAB command

257
00:17:11 --> 00:17:13
solves that problem?
 

258
00:17:13 --> 00:17:16
Let's just be a little
practical for a moment.

259
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What MATLAB command gives me
the matrix of eigenvectors, the

260
00:17:23 --> 00:17:32
matrix of eigenvalues would
come from eig of what?

261
00:17:32 --> 00:17:35
I'd call this the generalized
eigenvalue problem.

262
00:17:35 --> 00:17:38
Generalized because it's
got somebody over here.

263
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And it's just K,M.
 

264
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Or of course you get the same
answer, well you get the same

265
00:17:46 --> 00:17:50
eigenvalues, I guess the same
eigenvectors, yeah, if you

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00:17:50 --> 00:17:57
or, eig(M^-1,K), of course.
 

267
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If you want to do it with
just one matrix then bring

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M inverse over here.
 

269
00:18:01 --> 00:18:06
But and M inverse, the
inverse of this tridiagonal

270
00:18:06 --> 00:18:08
matrix, is full.
 

271
00:18:08 --> 00:18:11
No zeroes in the inverse.
 

272
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So everybody would much
prefer this tridiagonal

273
00:18:14 --> 00:18:16
tridiagonal one.
 

274
00:18:16 --> 00:18:19
So that's how MATLAB
would do it.

275
00:18:19 --> 00:18:27
And what I want to know is,
back in this problem, how close

276
00:18:27 --> 00:18:39
do the finite element guys
come, on polygons, come to the

277
00:18:39 --> 00:18:41
correct solution on circles.
 

278
00:18:41 --> 00:18:47
I'm hoping that for problem
one you can maybe keep M

279
00:18:47 --> 00:18:53
and N equal, or maybe four
times M or something.

280
00:18:53 --> 00:18:58
And let them grow and see.
 

281
00:18:58 --> 00:19:02
Well, for example, at the
center of the circle, or how

282
00:19:02 --> 00:19:05
quickly do you approach the
correct answer one at the

283
00:19:05 --> 00:19:07
center of the circle?
 

284
00:19:07 --> 00:19:09
I think it's going to
be a good problem.

285
00:19:09 --> 00:19:13
Let me open to, so I started
out just talking there.

286
00:19:13 --> 00:19:17
What about the MATLAB problem.
 

287
00:19:17 --> 00:19:24
You made a start on
it, is it going?

288
00:19:24 --> 00:19:30
Have you got a graph, maybe, or
what's reasonable to graph,

289
00:19:30 --> 00:19:35
to give Peter to look at?
 

290
00:19:35 --> 00:19:39
Who's done something on
that MATLAB problem?

291
00:19:39 --> 00:19:43
Yeah, go ahead tell
us all what to do.

292
00:19:43 --> 00:19:45
AUDIENCE: I made the
triangle pi section

293
00:19:45 --> 00:19:48
PROFESSOR STRANG: OK, right
 

294
00:19:48 --> 00:19:49
AUDIENCE: [INAUDIBLE]
 

295
00:19:49 --> 00:19:55
and I found that
the [INAUDIBLE]

296
00:19:55 --> 00:19:57
changes to M.
 

297
00:19:57 --> 00:19:59
PROFESSOR STRANG:
With M more, I see.

298
00:19:59 --> 00:20:07
So if you just fixed M like
eight, and let n get, it

299
00:20:07 --> 00:20:09
didn't change significantly.
 

300
00:20:09 --> 00:20:12
It wouldn't, of course,
converge to the right answer.

301
00:20:12 --> 00:20:15
It'll converge, if it
does, to some kind of an

302
00:20:15 --> 00:20:18
answer, for the polygon.
 

303
00:20:18 --> 00:20:19
Right.
 

304
00:20:19 --> 00:20:19
That's right.
 

305
00:20:19 --> 00:20:26
So you know, as I wrote the
problem I didn't know whether I

306
00:20:26 --> 00:20:31
dared say let M get increased
too, but of course that's

307
00:20:31 --> 00:20:32
the real question.
 

308
00:20:32 --> 00:20:34
And what happened then?
 

309
00:20:34 --> 00:20:37
Did error shrink?
 

310
00:20:37 --> 00:20:41
OK, and now maybe it's
possible to see how fast or

311
00:20:41 --> 00:20:42
something that's always--
 

312
00:20:42 --> 00:20:46
AUDIENCE: [INAUDIBLE]
 

313
00:20:46 --> 00:20:47
PROFESSOR STRANG: Ah.
 

314
00:20:47 --> 00:20:49
OK, at the center.
 

315
00:20:49 --> 00:20:53
OK, then I hope
for more comment.

316
00:20:53 --> 00:20:55
Let me say one more thing.
 

317
00:20:55 --> 00:21:01
My theory is that the error
at the center is quite a

318
00:21:01 --> 00:21:06
bit smaller than the error
closer to the boundary.

319
00:21:06 --> 00:21:13
I would be interested in an
error, is it fairly even?

320
00:21:13 --> 00:21:16
Oh, my theory's wrong.
 

321
00:21:16 --> 00:21:18
It wouldn't be the first time.
 

322
00:21:18 --> 00:21:21
And maybe because it's linear.
 

323
00:21:21 --> 00:21:28
Yeah, my theory is more for
better elements, like these.

324
00:21:28 --> 00:21:30
I'd be interested to know.
 

325
00:21:30 --> 00:21:37
Why do I think, why do I have
this theory, which you guys

326
00:21:37 --> 00:21:41
are going to prove wrong
anyway, but still.

327
00:21:41 --> 00:21:43
After you've proved it wrong,
you won't listen to me

328
00:21:43 --> 00:21:44
if I tell it to you.
 

329
00:21:44 --> 00:21:45
So now I'll tell it.
 

330
00:21:45 --> 00:21:54
My theory is that the error
around the boundary is, there's

331
00:21:54 --> 00:21:57
no error at these vertices, and
then there's sort of a going to

332
00:21:57 --> 00:22:01
be an error because the real
answer is not zero along here.

333
00:22:01 --> 00:22:04
It's sort of near
zero, but not quite.

334
00:22:04 --> 00:22:07
You know, there's an error.
 

335
00:22:07 --> 00:22:10
So there's errors around
here, from getting

336
00:22:10 --> 00:22:13
the boundary wrong.
 

337
00:22:13 --> 00:22:16
Squaring it off.
 

338
00:22:16 --> 00:22:20
But my theory is that errors,
the boundary stuff, drops off

339
00:22:20 --> 00:22:22
quickly as you go inside.
 

340
00:22:22 --> 00:22:26
That's why I think, from those,
you remember those - well,

341
00:22:26 --> 00:22:31
we'll see them again either
today or Friday, those

342
00:22:31 --> 00:22:36
r^n*cos(nx) type things?
 

343
00:22:36 --> 00:22:37
That cos(n*theta)?
 

344
00:22:39 --> 00:22:42
Yeah, you remember those
are the typical solutions

345
00:22:42 --> 00:22:44
to Laplace's equation.
 

346
00:22:44 --> 00:22:48
And then so that if, and
it has some coefficient,

347
00:22:48 --> 00:22:50
of course, a n.
 

348
00:22:50 --> 00:22:56
And I look at that, that
might be a piece of error.

349
00:22:56 --> 00:22:59
And it's way bigger
when r is one and way

350
00:22:59 --> 00:23:01
smaller when r is zero.
 

351
00:23:01 --> 00:23:05
So anyway, that's
sort of my theory.

352
00:23:05 --> 00:23:10
That if you have,
like physically.

353
00:23:10 --> 00:23:18
You have a circular plate
and you're maintaining the

354
00:23:18 --> 00:23:22
boundary temperature at
some sort of oscillation.

355
00:23:22 --> 00:23:27
Like, near one but up
and down from one.

356
00:23:27 --> 00:23:33
Then I think further
inside, it doesn't know.

357
00:23:33 --> 00:23:36
It hardly knows about
that oscillation.

358
00:23:36 --> 00:23:38
This is my theory.
 

359
00:23:38 --> 00:23:43
That toward the center of the
circle it only sees kind of an

360
00:23:43 --> 00:23:47
average boundary temperature
and not your little

361
00:23:47 --> 00:23:49
ups and downs.
 

362
00:23:49 --> 00:23:56
So when M is big, I expect that
part of the up and down part to

363
00:23:56 --> 00:23:59
be not so significant
in the center.

364
00:23:59 --> 00:24:01
Anyway, now that's my theory.
 

365
00:24:01 --> 00:24:04
AUDIENCE: [INAUDIBLE]
 

366
00:24:04 --> 00:24:11
PROFESSOR STRANG:
Ah, good question.

367
00:24:11 --> 00:24:15
So if we only looked
at the center, would

368
00:24:15 --> 00:24:17
it all be the same?
 

369
00:24:17 --> 00:24:20
I mean, if we're only looking
at that one point where it

370
00:24:20 --> 00:24:32
should be 1 at the center, but
along the thing, I don't know.

371
00:24:32 --> 00:24:38
If you look at both, and see a
significant difference in the

372
00:24:38 --> 00:24:40
behavior I'd be interested.
 

373
00:24:40 --> 00:24:41
Yeah, yeah.
 

374
00:24:41 --> 00:24:43
You know, all these problems
are things that there's

375
00:24:43 --> 00:24:47
no single solution to.
 

376
00:24:47 --> 00:24:53
AUDIENCE: [INAUDIBLE]
 

377
00:24:53 --> 00:24:55
PROFESSOR STRANG: The error
between one minus r squared

378
00:24:55 --> 00:25:04
AUDIENCE: [INAUDIBLE]
 

379
00:25:04 --> 00:25:06
PROFESSOR STRANG: Oh, right,
we've got slope error, too.

380
00:25:06 --> 00:25:10
That's a very
significant point.

381
00:25:10 --> 00:25:13
I see, right.
 

382
00:25:13 --> 00:25:14
So the slope error's in there.
 

383
00:25:14 --> 00:25:19
Everybody knows, then,
everybody in working the

384
00:25:19 --> 00:25:26
problem, I mentioned that the
boundary conditions in this

385
00:25:26 --> 00:25:32
piece of pie were zero along
here and normal derivative,

386
00:25:32 --> 00:25:36
somehow it got printed du/dh,
but that was an accident.

387
00:25:36 --> 00:25:42
It should've been
du/dn, dn is zero.

388
00:25:42 --> 00:25:46
So Neumann conditions on this
thing and then I was a little

389
00:25:46 --> 00:25:50
scared about that point,
but I think phooey on it.

390
00:25:50 --> 00:25:56
It's just, don't
worry about it.

391
00:25:56 --> 00:25:59
But what I was going to say.
 

392
00:25:59 --> 00:26:04
How do you, what do you do to
take into account this du/dn=0?

393
00:26:04 --> 00:26:07
 
 

394
00:26:07 --> 00:26:13
This slope condition on
these long boundaries?

395
00:26:13 --> 00:26:15
What should you do in
finite elements to

396
00:26:15 --> 00:26:17
take account for that?
 

397
00:26:17 --> 00:26:21
And the answer is,
in one nice word?

398
00:26:21 --> 00:26:22
Nothing.
 

399
00:26:22 --> 00:26:24
Right, nothing.
 

400
00:26:24 --> 00:26:27
Your finite element method
should not, you don't

401
00:26:27 --> 00:26:30
impose any condition
along these boundaries.

402
00:26:30 --> 00:26:35
Just use the code as it is
with zeroes on this boundary.

403
00:26:35 --> 00:26:39
And it should work, yeah.
 

404
00:26:39 --> 00:26:39
It should work.
 

405
00:26:39 --> 00:26:43
Any comments on other people.
 

406
00:26:43 --> 00:26:48
Did you get reasonable
results, or?

407
00:26:48 --> 00:26:49
Tell me something.
 

408
00:26:49 --> 00:26:55
Because you guys looked at
those graphs and I have not.

409
00:26:55 --> 00:26:57
Any feedback yet?
 

410
00:26:57 --> 00:26:58
On those?
 

411
00:26:58 --> 00:27:01
I'm happy to get
email, too, about.

412
00:27:01 --> 00:27:04
So all the email, first of
all they've corrected the

413
00:27:04 --> 00:27:09
typos in the original
coordinate positions.

414
00:27:09 --> 00:27:14
And now they've pointed
out I'd better look at M

415
00:27:14 --> 00:27:19
is very, very welcome.
 

416
00:27:19 --> 00:27:22
It doesn't mean that everybody
has to do this, if you've

417
00:27:22 --> 00:27:25
completed that MATLAB
assignment, you never want to

418
00:27:25 --> 00:27:30
see it again, and you've
kept M=8, it's ok.

419
00:27:30 --> 00:27:36
But if you're interested to see
what happens if M goes to 16

420
00:27:36 --> 00:27:39
or 32, I'm interested also.
 

421
00:27:39 --> 00:27:41
Right, yeah.
 

422
00:27:41 --> 00:27:45
OK, so anyway that's the
problem we're really

423
00:27:45 --> 00:27:45
thinking about.
 

424
00:27:45 --> 00:27:51
And that's the problem that is
equally important, but it

425
00:27:51 --> 00:27:55
seemed reasonable just
to do one of the two.

426
00:27:55 --> 00:27:59
And we were set up to do, we
have the code for the stiffness

427
00:27:59 --> 00:28:07
matrix, we would need a new
code to do these integrals.

428
00:28:07 --> 00:28:13
Because this will be linear
times linear, right?

429
00:28:13 --> 00:28:19
I'll have to compute that
one times this one and I

430
00:28:19 --> 00:28:23
would need new formulas
that are not there.

431
00:28:23 --> 00:28:26
I'd need formulas for, this
will be linear times linear

432
00:28:26 --> 00:28:31
so I'll be integrating
x squared type stuff.

433
00:28:31 --> 00:28:36
And xy's, because I'm
2-D, and y squareds.

434
00:28:36 --> 00:28:43
So it would take a little
more code, but not much.

435
00:28:43 --> 00:28:47
I think the math, oh here's
a question for you.

436
00:28:47 --> 00:28:49
Here's a question for you.
 

437
00:28:49 --> 00:28:52
Suppose I have my trial
functions, phi_i(x).

438
00:28:52 --> 00:28:56
 
 

439
00:28:56 --> 00:29:00
What do they add up to?
 

440
00:29:00 --> 00:29:05
Let me again draw a mesh,
so I've got a mesh.

441
00:29:05 --> 00:29:10
These are you know, I'm
sorry, I want to put in

442
00:29:10 --> 00:29:14
some more triangles here.
 

443
00:29:14 --> 00:29:18
Lots of triangles, whatever.
 

444
00:29:18 --> 00:29:23
Let me get some more
vertices, too.

445
00:29:23 --> 00:29:25
I'm getting in trouble.
 

446
00:29:25 --> 00:29:28
OK, whatever.
 

447
00:29:28 --> 00:29:33
So phi_i, is the piecewise
linear guy that

448
00:29:33 --> 00:29:35
is one at node i.
 

449
00:29:35 --> 00:29:38
So I've got all these
different nodes.

450
00:29:38 --> 00:29:41
I need a node there, so I've
got one, two, three, there's

451
00:29:41 --> 00:29:44
a node, there's more nodes.
 

452
00:29:44 --> 00:29:48
If I add them all up,
this is just like in

453
00:29:48 --> 00:29:51
an insights question.
 

454
00:29:51 --> 00:29:57
I've got all these, you could
add up these hats in 1-D.

455
00:29:57 --> 00:30:00
What's the sum of the
hats in one dimension?

456
00:30:00 --> 00:30:01
One.
 

457
00:30:01 --> 00:30:03
Good.
 

458
00:30:03 --> 00:30:05
The sum is one.
 

459
00:30:05 --> 00:30:09
It's a nice fact that
these guys add up to one.

460
00:30:09 --> 00:30:14
And now why is it still true
here in 2-D, that these little

461
00:30:14 --> 00:30:18
pyramids will add to one?
 

462
00:30:18 --> 00:30:22
That's an inside question, but
it's worth thinking about.

463
00:30:22 --> 00:30:25
Why do those pyramids
add to one?

464
00:30:25 --> 00:30:29
Let me leave that question.
 

465
00:30:29 --> 00:30:32
I'm thinking about, we
haven't imposed any

466
00:30:32 --> 00:30:34
boundary conditions yet.
 

467
00:30:34 --> 00:30:39
We've got them all. and I claim
that if we add up all the

468
00:30:39 --> 00:30:44
pyramids including the boundary
chopped off pyramids from the

469
00:30:44 --> 00:30:48
boundary, that we'll get
one throughout the whole,

470
00:30:48 --> 00:30:49
now it'll be phi(x,y).
 

471
00:30:49 --> 00:30:52
 
 

472
00:30:52 --> 00:30:56
Because now I'm moving
to 2-D, with pyramids.

473
00:30:56 --> 00:31:00
I think we'll still have one.
 

474
00:31:00 --> 00:31:02
Let me give you a minute
to think about that one.

475
00:31:02 --> 00:31:08
And then we could turn to
Fourier questions if you

476
00:31:08 --> 00:31:15
would like, we could do some
problems from the text.

477
00:31:15 --> 00:31:17
Any thoughts about this guy?
 

478
00:31:17 --> 00:31:24
Why should all those
individual pyramids add

479
00:31:24 --> 00:31:30
up to a flat group?
 

480
00:31:30 --> 00:31:31
Why did it work here?
 

481
00:31:31 --> 00:31:41
Well, it worked because you
could see it, right, somehow?

482
00:31:41 --> 00:31:47
Does it still work if the
nodes are not equally spaced?

483
00:31:47 --> 00:31:51
So we've got a hat function for
that guy, and a hat function

484
00:31:51 --> 00:31:54
for this guy, and a hat
function for this guy.

485
00:31:54 --> 00:31:57
And these guys are
in there, too.

486
00:31:57 --> 00:32:00
We haven't imposed anything.
 

487
00:32:00 --> 00:32:08
So those one, two, three, four,
five functions, five phis,

488
00:32:08 --> 00:32:14
they add up to one and y.
 

489
00:32:14 --> 00:32:18
Well, you're going to say it's
obvious, but that's what

490
00:32:18 --> 00:32:20
professors are allowed to say.
 

491
00:32:20 --> 00:32:24
Things are obvious, you
have to actually say why.

492
00:32:24 --> 00:32:26
Which is not as easy.
 

493
00:32:26 --> 00:32:36
So, why do they add to one?
 

494
00:32:36 --> 00:32:41
Let me look inside one element.
 

495
00:32:41 --> 00:32:47
Why does the sum of these
two guys add to a flat

496
00:32:47 --> 00:32:51
top inside that interval?
 

497
00:32:51 --> 00:32:57
AUDIENCE: [INAUDIBLE]
 

498
00:32:57 --> 00:33:00
PROFESSOR STRANG: At the
end points, you've got it.

499
00:33:00 --> 00:33:04
Because what's happening
at the end points?

500
00:33:04 --> 00:33:10
This guy, one of the guys,
the right guy is one.

501
00:33:10 --> 00:33:14
And all other guys
are zero, right.

502
00:33:14 --> 00:33:17
And this guy is also at one.
 

503
00:33:17 --> 00:33:20
Because it's the right guy.
 

504
00:33:20 --> 00:33:23
It has height one and
all others zero.

505
00:33:23 --> 00:33:27
So at the nodes we are at
one, because of one person,

506
00:33:27 --> 00:33:29
really, one element.
 

507
00:33:29 --> 00:33:30
And then?
 

508
00:33:30 --> 00:33:32
AUDIENCE: [INAUDIBLE]
 

509
00:33:32 --> 00:33:38
PROFESSOR STRANG: Right.
 

510
00:33:38 --> 00:33:42
But the sum of them is, why
is the sum of them always

511
00:33:42 --> 00:33:43
one, why is slope zero?
 

512
00:33:44 --> 00:33:48
Yeah.
 

513
00:33:48 --> 00:33:51
The slopes cancel, right.
 

514
00:33:51 --> 00:33:54
We know that in between it
will be a linear function.

515
00:33:54 --> 00:33:56
That would be one
way to look at it.

516
00:33:56 --> 00:33:59
If I add up a linear function
and a linear function the

517
00:33:59 --> 00:34:01
sum is a linear function.
 

518
00:34:01 --> 00:34:04
So I'm getting a linear
function, which is one at

519
00:34:04 --> 00:34:08
those points, so what
is that function?

520
00:34:08 --> 00:34:09
One.
 

521
00:34:09 --> 00:34:11
Right, you know that's
the straight line.

522
00:34:11 --> 00:34:16
So, that idea will
work here too.

523
00:34:16 --> 00:34:20
Look inside some
little triangle here.

524
00:34:20 --> 00:34:25
OK, that's got one, two,
three corners, OK.

525
00:34:25 --> 00:34:31
And if I look at this sum,
what is it at this point?

526
00:34:31 --> 00:34:36
If I look at that sum at this
corner, one guy is one,

527
00:34:36 --> 00:34:37
the one for that pyramid.
 

528
00:34:37 --> 00:34:40
And all others are?
 

529
00:34:40 --> 00:34:41
Zero.
 

530
00:34:41 --> 00:34:44
So the sum is one there, the
sum is one there, the sum is

531
00:34:44 --> 00:34:50
one there, so that blowing up
this little triangle, this is

532
00:34:50 --> 00:34:53
at height one, this is at
height one, this is at height

533
00:34:53 --> 00:34:56
one, so what's the roof?
 

534
00:34:56 --> 00:34:59
Flat.
 

535
00:34:59 --> 00:35:04
It's just a nice way to see the
nice property of these phis.

536
00:35:04 --> 00:35:14
That there's a phi for every
node, and they add to one.

537
00:35:14 --> 00:35:17
To that's it.
 

538
00:35:17 --> 00:35:24
OK, well I was going to say one
more thing and I am, about this

539
00:35:24 --> 00:35:27
eigenvalue problem, just
because I'll never

540
00:35:27 --> 00:35:29
have a chance again.
 

541
00:35:29 --> 00:35:33
So this is the moment
to say something about

542
00:35:33 --> 00:35:34
the eigenvalues.
 

543
00:35:34 --> 00:35:35
Lambda.
 

544
00:35:35 --> 00:35:41
Eigenvalue.
 

545
00:35:41 --> 00:35:44
I'm answering the question
where does K come from,

546
00:35:44 --> 00:35:45
where does M come from?
 

547
00:35:45 --> 00:35:56
Well, eigenvalue is, boy we
really got dramatic music here.

548
00:35:56 --> 00:36:00
That's a great Gates of
Kiev, I think might be.

549
00:36:00 --> 00:36:01
Mussorgski.
 

550
00:36:01 --> 00:36:05
If you like drums and big
noise, it's not music

551
00:36:05 --> 00:36:11
actually, but you got a
lot of noise out of it.

552
00:36:11 --> 00:36:16
Well, of course, he'd know
more than we do, but still.

553
00:36:16 --> 00:36:26
OK, so the eigenvalues in the
matrix case for Kx=lambda*M*x,

554
00:36:26 --> 00:36:31
the eigenvalue problem, lambda,
the lowest eigenvalue, lambda

555
00:36:31 --> 00:36:34
lowest, has a nice property.
 

556
00:36:34 --> 00:36:46
It's the minimum of sort of our
energy over our other energy.

557
00:36:46 --> 00:36:51
I just think, well this is
something you should see.

558
00:36:51 --> 00:36:54
This is a quotient here.
 

559
00:36:54 --> 00:36:56
It's got a name called
the Rayleigh quotient.

560
00:36:56 --> 00:36:59
And it would appear
in the book.

561
00:36:59 --> 00:37:02
So really, I guess what
I'm doing is calling your

562
00:37:02 --> 00:37:06
attention to something
that's in the book.

563
00:37:06 --> 00:37:10
That this a ratio of x
transpose K x to x transpose M

564
00:37:10 --> 00:37:19
x, if I look over all vectors
x, the lowest one is

565
00:37:19 --> 00:37:20
the eigenvector.
 

566
00:37:20 --> 00:37:23
The best x is the eigenvector
and the ratio is

567
00:37:23 --> 00:37:26
the eigenvalue.
 

568
00:37:26 --> 00:37:29
This is like my point
that I wanted to mention

569
00:37:29 --> 00:37:31
the Rayleigh quotient.
 

570
00:37:31 --> 00:37:34
Here it is in the matrix case,
and there would be similar

571
00:37:34 --> 00:37:38
Rayleigh quotient in
the continuous case.

572
00:37:38 --> 00:37:41
I'll just leave it at that.
 

573
00:37:41 --> 00:37:45
That in describing eigenvalues,
we can talk about

574
00:37:45 --> 00:37:49
Kx=lambda*M*x, like this.
 

575
00:37:49 --> 00:37:51
Or we can get energy into it.
 

576
00:37:51 --> 00:37:54
And you remember the whole
point about finite elements

577
00:37:54 --> 00:37:57
is, look at the energy.
 

578
00:37:57 --> 00:37:59
Look at that the quadratics.
 

579
00:37:59 --> 00:38:04
Multiply things by things.
 

580
00:38:04 --> 00:38:07
It came from the weak
form, it didn't come

581
00:38:07 --> 00:38:10
from the strong form.
 

582
00:38:10 --> 00:38:13
In the differential equation
here, we just have

583
00:38:13 --> 00:38:15
single terms.
 

584
00:38:15 --> 00:38:20
We got to these things through
that process of multiplying

585
00:38:20 --> 00:38:23
by u's and integrating.
 

586
00:38:23 --> 00:38:25
That's what gave us these
products and it works

587
00:38:25 --> 00:38:29
also in the matrix case.
 

588
00:38:29 --> 00:38:36
OK, that was a lot of
speechmaking about topics

589
00:38:36 --> 00:38:41
that we simply didn't
have time for in class.

590
00:38:41 --> 00:38:45
I'm ready for any question, or
I'm ready to maybe do a Fourier

591
00:38:45 --> 00:38:48
example, would you like that?
 

592
00:38:48 --> 00:38:51
Because this is where
we really are.

593
00:38:51 --> 00:38:55
I'll even take one that
will be on the homework.

594
00:38:55 --> 00:39:01
Just so you'll have a start.
 

595
00:39:01 --> 00:39:09
OK, let me take a square
pulse, yeah this is

596
00:39:09 --> 00:39:16
a good one, I think.
 

597
00:39:16 --> 00:39:19
In Section 4.1, there's a
question for the Fourier

598
00:39:19 --> 00:39:21
series of a square pulse.
 

599
00:39:21 --> 00:39:24
OK, so what does the
square pulse look like?

600
00:39:24 --> 00:39:29
Here's minus pi to pi.
 

601
00:39:29 --> 00:39:30
Here's zero.
 

602
00:39:30 --> 00:39:35
The square pulse goes along
here, up square pulse and down.

603
00:39:35 --> 00:39:48
Actually, let me go to L/2,
oh I'll just call it h.

604
00:39:48 --> 00:39:56
Let me find the Fourier
series for this function.

605
00:39:56 --> 00:40:02
It goes along at 0, it jumps up
to 1 over a interval of length

606
00:40:02 --> 00:40:06
2 h, going from minus h to h,
and then back down to

607
00:40:06 --> 00:40:08
0 and then repeat.
 

608
00:40:08 --> 00:40:11
So bip bip bip, square pulse.
 

609
00:40:11 --> 00:40:14
So that's my function.
 

610
00:40:14 --> 00:40:18
Is that function odd, or
even, or neither one?

611
00:40:18 --> 00:40:21
It's even, so I can
call that C(x).

612
00:40:22 --> 00:40:25
And figure that I'm going
to use cosines for

613
00:40:25 --> 00:40:27
that one, right?
 

614
00:40:27 --> 00:40:31
So tell me a formula for the
coefficients, what's the

615
00:40:31 --> 00:40:33
integral that I have to do?
 

616
00:40:33 --> 00:40:40
So my C(x) o is going to be
some a_0, we have to think

617
00:40:40 --> 00:40:46
what's a_0, then a_1*cos(x),
a_2*cos, and so on.

618
00:40:46 --> 00:40:48
So on. a_k*cos(kx).
 

619
00:40:48 --> 00:40:52
 
 

620
00:40:52 --> 00:41:01
OK, what's the formula for a_k?
 

621
00:41:01 --> 00:41:03
Before I plug in that
function I would like

622
00:41:03 --> 00:41:04
to get the formula.
 

623
00:41:04 --> 00:41:07
So I'm looking for the formula.
 

624
00:41:07 --> 00:41:10
It's a formula to remember.
 

625
00:41:10 --> 00:41:12
So I'm not wasting your time.
 

626
00:41:12 --> 00:41:14
Because you're going to see it
on the board and it'll just

627
00:41:14 --> 00:41:16
take a mental photograph of it.
 

628
00:41:16 --> 00:41:18
What do you think
it's going to be?

629
00:41:18 --> 00:41:20
How am I going to get it?
 

630
00:41:20 --> 00:41:26
I'll multiply both sides of the
equation by cos(kx), right?

631
00:41:26 --> 00:41:28
And I'll integrate.
 

632
00:41:28 --> 00:41:32
So and then when I integrate,
the cosines are orthogonal.

633
00:41:32 --> 00:41:34
Just like the sines
this morning.

634
00:41:34 --> 00:41:38
All those terms will go,
except for this term.

635
00:41:38 --> 00:41:40
When I multiply this
by cos(kx), I'll have

636
00:41:40 --> 00:41:42
cos(kx) squared.
 

637
00:41:42 --> 00:41:46
Here I'll have a cos(kx), and
here I'll have a whole lot of

638
00:41:46 --> 00:41:51
cos(kx)'s but when I
integrate, all this stuff is

639
00:41:51 --> 00:41:55
going to disappear.
 

640
00:41:55 --> 00:41:57
And this will all disappear.
 

641
00:41:57 --> 00:41:58
This is it.
 

642
00:41:58 --> 00:42:04
So a_k is going to be the
integral of my function,

643
00:42:04 --> 00:42:04
times cos(kx)dx.
 

644
00:42:04 --> 00:42:07
 
 

645
00:42:07 --> 00:42:09
Divided by what?
 

646
00:42:09 --> 00:42:14
Divided by the integral
of cos(kx) squared.

647
00:42:14 --> 00:42:18
Because I haven't
normalized things.

648
00:42:18 --> 00:42:21
So I don't know that that's
one, and in fact it isn't one.

649
00:42:21 --> 00:42:25
So I have to remember
to put that number in.

650
00:42:25 --> 00:42:28
OK, so that's the formula
and that number turns

651
00:42:28 --> 00:42:32
out to be pi, again.
 

652
00:42:32 --> 00:42:37
If I'm integrating from minus
pi to pi, then the average

653
00:42:37 --> 00:42:41
value of the cosine squared is
a 1/2, it's sort of as much

654
00:42:41 --> 00:42:47
above 1/2 as it is below 1/2,
and so the average of the half,

655
00:42:47 --> 00:42:51
the interval is 2pi, so pi.
 

656
00:42:51 --> 00:42:54
OK, that's the formula.
 

657
00:42:54 --> 00:42:59
Please just take a
mental photograph.

658
00:42:59 --> 00:43:00
Catch that one.
 

659
00:43:00 --> 00:43:07
Alright, now I've got my
particular C(x), my square

660
00:43:07 --> 00:43:10
wave, square pulse.
 

661
00:43:10 --> 00:43:11
Very, very important.
 

662
00:43:11 --> 00:43:16
Very important
Fourier series here.

663
00:43:16 --> 00:43:18
Famous one.
 

664
00:43:18 --> 00:43:20
OK, so what do I have?
 

665
00:43:20 --> 00:43:24
From minus pi to pi, so
what's my integral?

666
00:43:24 --> 00:43:27
Well, my integral really
doesn't go from minus

667
00:43:27 --> 00:43:31
pi to pi because my
function is mostly zero.

668
00:43:31 --> 00:43:34
Where does my integral go?
 

669
00:43:34 --> 00:43:36
Negative h to h, right?
 

670
00:43:36 --> 00:43:39
And in that region,
what is C(x)?

671
00:43:40 --> 00:43:44
One.
 

672
00:43:44 --> 00:43:47
So you see it's
going to be nice.

673
00:43:47 --> 00:43:53
From negative h to h, where
this is one, I just have to

674
00:43:53 --> 00:44:00
integrate cos(kx), so what do I
get? sin(kx), over k, and the

675
00:44:00 --> 00:44:05
pi so you see again that that k
is showing up in the

676
00:44:05 --> 00:44:09
denominator, and that's going
to give me the typical

677
00:44:09 --> 00:44:20
decay rate of 1/k for
functions with steps.

678
00:44:20 --> 00:44:21
For step functions.
 

679
00:44:21 --> 00:44:28
And now I have to evaluate
this between minus pi and pi.

680
00:44:28 --> 00:44:30
And no, h.
 

681
00:44:30 --> 00:44:32
Better be h.
 

682
00:44:32 --> 00:44:35
I mean, minus h and h.
 

683
00:44:35 --> 00:44:37
So what do I get for that?
 

684
00:44:37 --> 00:44:42
I get sine(kh), right?
 

685
00:44:42 --> 00:44:48
At the top, and what
do I get at minus?

686
00:44:48 --> 00:44:51
So I now I want to subtract,
what is the sin(-kh)?

687
00:44:51 --> 00:44:54
 
 

688
00:44:54 --> 00:44:57
It's a negative, right?
 

689
00:44:57 --> 00:45:04
So as I expect with an even
function like cosine, am

690
00:45:04 --> 00:45:08
I just getting twice?
 

691
00:45:08 --> 00:45:13
I could take it from 0 to h,
and it would give me one

692
00:45:13 --> 00:45:14
of them and the other one.
 

693
00:45:14 --> 00:45:21
Yep, I think so, and
divide by k pi.

694
00:45:21 --> 00:45:24
So those are the
Fourier coefficients.

695
00:45:24 --> 00:45:25
Except for a_0.
 

696
00:45:27 --> 00:45:33
a_0 has a slightly different
formula, because for a_0,

697
00:45:33 --> 00:45:34
why is a_0 different?
 

698
00:45:34 --> 00:45:39
How do you come up with a_0,
and what's its meaning? a_0

699
00:45:39 --> 00:45:45
has a nice meeting, so this
is worth having come

700
00:45:45 --> 00:45:46
this afternoon for.
 

701
00:45:46 --> 00:45:50
a_0 will be what?
 

702
00:45:50 --> 00:45:52
Well, I could get
it the same way.

703
00:45:52 --> 00:45:57
What will I multiply
both sides by?

704
00:45:57 --> 00:45:58
If I want to pick off a_0?
 

705
00:46:00 --> 00:46:01
Just one.
 

706
00:46:01 --> 00:46:06
It's not a cosine, it's the
cos(0x), it's the one.

707
00:46:06 --> 00:46:08
And then I integrate.
 

708
00:46:08 --> 00:46:13
I'm just going to get the
integral from minus pi to pi

709
00:46:13 --> 00:46:19
of C(x) times one, divided
by the integral from minus

710
00:46:19 --> 00:46:30
pi of one times one.
 

711
00:46:30 --> 00:46:37
Same method. multiply both
sides by one, which was

712
00:46:37 --> 00:46:41
the very first of my
orthogonal functions.

713
00:46:41 --> 00:46:45
Integrate it, all the other
integrals went away, right?

714
00:46:45 --> 00:46:48
The integral of cosine
over a whole interval.

715
00:46:48 --> 00:46:51
Its periodic.
 

716
00:46:51 --> 00:46:53
You get the same at
the two ends, so the

717
00:46:53 --> 00:46:56
difference is zero.
 

718
00:46:56 --> 00:47:00
So we just, the only term
left was a constant.

719
00:47:00 --> 00:47:03
And now what is the integral,
row what's the denominator now?

720
00:47:03 --> 00:47:06
That was the little,
slight twist.

721
00:47:06 --> 00:47:06
2pi.
 

722
00:47:07 --> 00:47:08
The denominator's 2pi.
 

723
00:47:09 --> 00:47:10
Yeah.
 

724
00:47:10 --> 00:47:15
That's that's why it's not,
yeah, it's slightly irregular,

725
00:47:15 --> 00:47:16
I have to divide by 2pi.
 

726
00:47:18 --> 00:47:22
And now, what word would you
use to describe, if I have a

727
00:47:22 --> 00:47:25
function, and integrate
it, and I divide by the

728
00:47:25 --> 00:47:30
length, what am I getting?
 

729
00:47:30 --> 00:47:35
There's an English word that
would describe what this is.

730
00:47:35 --> 00:47:37
Average.
 

731
00:47:37 --> 00:47:44
This is the average.
 

732
00:47:44 --> 00:47:46
And it has to be.
 

733
00:47:46 --> 00:47:49
This constant term is
always the average.

734
00:47:49 --> 00:47:52
And what will it be for this?
 

735
00:47:52 --> 00:47:59
So this was a_k, and
what is a_0, then?

736
00:47:59 --> 00:48:03
So you can now tell me, so
everybody's remembering this

737
00:48:03 --> 00:48:06
formula, you integrate the
function and divide by the 2pi.

738
00:48:07 --> 00:48:10
Now we've got a particular
function, so what is the

739
00:48:10 --> 00:48:12
integral of that function?
 

740
00:48:12 --> 00:48:16
So what does this equal?
 

741
00:48:16 --> 00:48:17
For this particular C(x)?
 

742
00:48:18 --> 00:48:20
What's the area under
that function C(x)?

743
00:48:22 --> 00:48:24
2h.
 

744
00:48:24 --> 00:48:26
Right?
 

745
00:48:26 --> 00:48:29
So 2h/2pi cancel twos.
 

746
00:48:29 --> 00:48:40
So there's a constant term, a_0
is h/pi and the and the cosine

747
00:48:40 --> 00:48:43
terms are, yeah, actually we're
going to get something nice.

748
00:48:43 --> 00:48:50
A really nice way to complete
this will be if I put this

749
00:48:50 --> 00:48:55
together, put this
series together.

750
00:48:55 --> 00:49:05
So now I'm saying that this
square pulse is that constant

751
00:49:05 --> 00:49:13
term h/pi plus the next
term a_1, you can see all

752
00:49:13 --> 00:49:14
these terms have 2/pi's.
 

753
00:49:14 --> 00:49:17
 
 

754
00:49:17 --> 00:49:21
I'm a little surprised that h
over, yeah, I guess it's right.

755
00:49:21 --> 00:49:22
2/pi.
 

756
00:49:22 --> 00:49:29
 
 

757
00:49:29 --> 00:49:33
So I've got sin(h), I think.
 

758
00:49:33 --> 00:49:35
And now I'm just copying this.
 

759
00:49:35 --> 00:49:47
2/pi*sin(h), sin(h),
is that what I want?

760
00:49:47 --> 00:49:51
Over one.
 

761
00:49:51 --> 00:49:54
That's the coefficient
version of sine, of cos(x).

762
00:49:54 --> 00:49:58
 
 

763
00:49:58 --> 00:50:00
a_1 was the coefficient
of cos(1x).

764
00:50:01 --> 00:50:04
And then a_2 is the
coefficient of cos(2x).

765
00:50:06 --> 00:50:07
So that will be sin(2h).
 

766
00:50:07 --> 00:50:09
 
 

767
00:50:09 --> 00:50:13
k is two, so I have a
two down here, cos(2x).

768
00:50:14 --> 00:50:20
And so on.
 

769
00:50:20 --> 00:50:23
Yeah, I think that's
the Fourier series.

770
00:50:23 --> 00:50:33
That would be the Fourier
series for the square pulse.

771
00:50:33 --> 00:50:34
Yeah.
 

772
00:50:34 --> 00:50:38
That would be the Fourier
series for the square pulse.

773
00:50:38 --> 00:50:40
Could I test any
interesting cases?

774
00:50:40 --> 00:50:45
Suppose h is all
the way out to pi.

775
00:50:45 --> 00:50:46
Suppose I take that case.
 

776
00:50:46 --> 00:50:55
Let h go all the way out to
pi, then what's my function?

777
00:50:55 --> 00:51:01
If h=pi, then what have
I got a graph of?

778
00:51:01 --> 00:51:02
Just one.
 

779
00:51:02 --> 00:51:03
It's just a one.
 

780
00:51:03 --> 00:51:07
If h is pi, what happens?
 

781
00:51:07 --> 00:51:11
That becomes a one, and what
about these other things?

782
00:51:11 --> 00:51:15
What is this thing
when h is pi?

783
00:51:15 --> 00:51:15
Zero.
 

784
00:51:15 --> 00:51:18
All the other terms go away.
 

785
00:51:18 --> 00:51:22
It's just a sin(2pi)
that would go away.

786
00:51:22 --> 00:51:27
Yeah, so if h is pi, if I go
out to the place where I don't

787
00:51:27 --> 00:51:33
have any jumps at all because
it's now all the way out there,

788
00:51:33 --> 00:51:37
then these terms all disappear
and I just have this.

789
00:51:37 --> 00:51:40
And I would like to ask you
and it's going to come up on

790
00:51:40 --> 00:51:47
Friday, too, what happens
if h goes to zero?

791
00:51:47 --> 00:51:50
Well, let me just take
h going to zero.

792
00:51:50 --> 00:51:52
What happens to
this whole thing?

793
00:51:52 --> 00:51:55
What happens to my function
if h goes to zero?

794
00:51:55 --> 00:51:58
Goes to zero, right, then
squeezed it to nothing.

795
00:51:58 --> 00:52:05
And if h is zero then sin(h) is
zero, I get 0=0, that's not

796
00:52:05 --> 00:52:10
interesting enough to mention
on Friday But there is one

797
00:52:10 --> 00:52:13
case that is important.
 

798
00:52:13 --> 00:52:16
Suppose I make the
height, yeah.

799
00:52:16 --> 00:52:17
Make a guess.
 

800
00:52:17 --> 00:52:24
Suppose I make the
height higher as I

801
00:52:24 --> 00:52:27
make the base smaller.
 

802
00:52:27 --> 00:52:31
I'm going to keep the area as
one, so if this has a base of

803
00:52:31 --> 00:52:34
2h, I'm going to have
a height of 1/2h.

804
00:52:35 --> 00:52:41
So if I keep the area at one,
so the height now is 1/2h,

805
00:52:41 --> 00:52:45
so now my square pulse
I've divided it by 2h.

806
00:52:46 --> 00:52:50
I have a 1/2h
multiplying everything.

807
00:52:50 --> 00:52:56
And now if I let h go to
zero, something more

808
00:52:56 --> 00:52:58
interesting will happen.
 

809
00:52:58 --> 00:52:59
And what?
 

810
00:52:59 --> 00:53:05
Just tell me first, what
would you expect to happen?

811
00:53:05 --> 00:53:05
Delta.
 

812
00:53:05 --> 00:53:08
Right, delta.
 

813
00:53:08 --> 00:53:12
So what I'll see show up
will be the Fourier series

814
00:53:12 --> 00:53:15
for the delta function.
 

815
00:53:15 --> 00:53:23
When I divide by 2h, so I have
sin(h)'s over h's, and of

816
00:53:23 --> 00:53:25
course what's the great
fact about sin(h)/h?

817
00:53:27 --> 00:53:34
As h goes to zero, it
goes to, everybody know,

818
00:53:34 --> 00:53:36
that's the big deal.
 

819
00:53:36 --> 00:53:36
Yeah.
 

820
00:53:36 --> 00:53:42
One. sin(h) is the same
size as h for a very small

821
00:53:42 --> 00:53:44
h, and approaches one.
 

822
00:53:44 --> 00:53:49
Yeah so we'll see the
delta function Friday.

823
00:53:49 --> 00:53:53
OK, so you've got a sort of
mini-lecture instead of a real

824
00:53:53 --> 00:53:56
chance to ask about homework.
 

825
00:53:56 --> 00:53:59
Next Wednesday should be
different because there will be

826
00:53:59 --> 00:54:04
Fourier series homework, and
I'll be ready to answer

827
00:54:04 --> 00:54:05
questions about it.
 

828
00:54:05 --> 00:54:07
OK, thanks.