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PROFESSOR STRANG: So.
 

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Today's lecture is partly
cleaning up some pieces

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left from 1-D problems.
 

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3.1 was second order equations,
3.2 was fourth order

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equation for beams.
 

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And just a few more
comments to make.

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But then I do want to say
something about splines.

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And I included a homework
question that asked you to

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use the spline command
and get a result.

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I'm not planning to make that a
serious topic in the course.

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In other words, you're not like
going to be responsible on an

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exam for a discussion
of splines.

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But it's such a major event,
and major requirement in

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scientific computing -- if
you're given some values, put a

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curve through them -- that I
have to say something

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about that.
 

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So that's interpolation.
 

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Because you'll have
this all the time.

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If your experiment produces
some values at specific times

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or a finite number of values,
and you want to fit a curve

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through those points.
 

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How do you do it?
 

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So splines give a very,
very good answer.

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And then, the next and big
topic of the course we'll

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make a beginning on.
 

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Which is gradient, divergence,
leading to Laplace's equation.

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Problems in 2-D.
 

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So partial differential
equations are going to show up.

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OK.
 

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Some small points.
 

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First point, about the
MATLAB homework for Monday.

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A question came by email.
 

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I put the spike and also the
jump in c in the coefficient,

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not at mesh point.
 

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If you've started on that
problem, you probably noticed

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that, and maybe a little
swearing went on.

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[LAUGHTER]
 

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Because it makes
it not so easy.

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And then the question came, did
I really expect you to compute,

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use the exact position and
figure out, when you have some

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integrals to do, they'll change
at the place where

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c changes values.
 

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Where we have integrals of c,
if c has one of those jumps, on

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the quiz the jump came right at
a neat point, so it was clear.

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c was one on one side
and two on the other.

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Now the jump is going to come
at an in-between point.

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So the answer is yes.
 

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I had a nice email this morning
from somebody who has done

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that MATLAB homework.
 

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And he said, quote,
it's a good problem.

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I learned more from it
than I did from the other

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problems in the pset.
 

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So I'll blame it on him.
 

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I'll stay with that problem,
and ask you to do your best to

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deal with the calculation.
 

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It's second order and I
used linear elements.

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I didn't want to push you up to
these cubic elements, even

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though those would give
much better accuracy.

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One reason I didn't put the
spike and the jump right at the

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node is that, if I did, the
finite element solution

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would be exactly right.
 

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Because the correct solution is
going to be piecewise linear.

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And if I put the breaks between
pieces right at node points,

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well, then I have a function
that's in my finite element

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space, in my linear space.
 

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So it'll come out exactly.
 

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So I thought well, let's at
least have some chance to

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see the error between the
finite element solution

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and the exact one.
 

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You see what I'm saying?
 

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I'm just anticipating,
that the exact solution

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will -- let's see.
 

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Something happened at 1/3 and
something happened at 2/3.

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I'm afraid I don't remember
the boundary conditions.

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Does anybody remember what I
took for boundary conditions?

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Probably free fixed,
or something.

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So maybe, if it was free fixed,
probably the solution would

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be maybe something like that.
 

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I don't know.
 

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Whatever.
 

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Piecewise linear with breaks,
where there's a jump in c,

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or where the point
load is hitting.

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Yeah.
 

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I don't know if that
picture is accurate.

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But my point was, if I chose
those points to be also mesh

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points, then the finite element
solution would be exact, and we

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wouldn't learn anything
about accuracy.

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OK.
 

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So anyway, for that MATLAB
problem, I'm hoping you can do

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the integrals, which will mean
noticing which integral -- if

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that doesn't fall, so if the
mesh points are like this, in

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the finite element integral
over that mesh, over that

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interval, you're going to have
to split it into two pieces.

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That's it.
 

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So see if you can do that
split, do the split

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into two pieces.
 

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OK.
 

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So that's a comment
on that homework.

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OK.
 

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What else do I need to
comment on, when I'm sort

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of catching up here?
 

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Oh, about boundary conditions.
 

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I thought I would just repeat
clearly on the board the rules

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about and the difference
between fixed and free when

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we're doing this weak form
approach to the problem.

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And then I gave the other
names: a fixed condition, in

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this theory it's often called
an essential boundary

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condition, but many people just
name Dirichlet as the person's

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name that's associated
with such conditions.

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Like u(0)=0.
 

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Right, fixed.
 

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And then free conditions are
natural conditions in the

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finite element method, and
that means that we don't

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have to impose them.
 

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Right.
 

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I discussed this in
the earlier lecture.

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I just thought I'd bring it
back here, because it's easy

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to describe in one lie.
 

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Now somebody asked, on
Wednesday, a good question.

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Suppose u' is given,
but not given as zero.

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So suppose we have a condition
like u'=A, let's say.

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What to do?
 

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So I'll just put question mark.
 

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How do I deal with -- how do
I? -- how does the weak form

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approach and the Galerkin
approach deal with a

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non-zero, natural condition?
 

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A nonzero Neumann condition,
that'd be the right word.

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So this would be, in a second
order problem, this would

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be a Neumann condition.
 

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And I'm thinking, what do
you do if A isn't zero.

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Somehow that has to show
up in your finite element

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equations, right?
 

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So I guess the right way
is to go back to the

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way they came from.
 

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So you remember how
the weak form came?

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I took the differential
equation, I multiplied by

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any test function, and
I integrated by parts.

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That's where the
weak form started.

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So the integration by
parts gave me this nice

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symmetric integral.
 

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It's symmetric in u and v.
 

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It only requires one derivative
of u, so that I'm allowed

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to use hat functions.
 

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And I'm allowed to use hat
functions for v, because their

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derivatives have just a jump,
and a jump function I can

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integrate, no problem.
 

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Now what about this u'=A?
 

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And I guess we're
seeing it there.

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If u' was zero, if we had a
totally free end, u' was zero,

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nothing happening there, then
that term would drop out of

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the integration by parts.
 

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And you see why I don't have to
impose anything on v, because

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that term is already accounted
for by the u' going away.

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Now, what about if u' is
given, but not given zero?

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Suppose it's giving
the value A.

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All I want to say is, then
this term has to stay.

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I have to pay attention to it.
 

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What what happens when
Galerkin takes over?

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Galerkin will put in one
of his test functions.

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Maybe I just indicate that by
changing the v to a cap V.

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It'll be one of them.
 

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There'll be n of them.
 

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Each V_i, each test guy
gives me an equation.

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And of course u is now going
to -- the Galerkin idea is

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that instead of any u, I
only have capital U's.

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Combinations of these phis.
 

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I'm not going to say anything
very big here, or very clear.

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All I want to say is that if u
prime is given at an end point,

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and given by A, then this whole
stuff is equalling something

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with an f, right? f*V_i*dx.
 

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I should remember the other
side of the equation.

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If u' is given, then c at that
end point, times the given

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value of A, times the V will be
-- whatever value of V that is

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-- will show up in equation i.
 

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That will be a term that goes
on the right hand side.

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That's all I'm saying, and
all you would expect.

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That any time I'm given some
data, I'm given some non-zero

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boundary conditions, or I'm
given some non-zero f in the

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inside, that stuff is going to
show up on the right hand side.

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It'll show up as part of the f.
 

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The vector of these.
 

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So up to now, f has just
come from little f(x),

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integrated against V.
 

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And all I'm saying is that if
we had one of these guys, then

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that would contribute
to big F, also.

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Yeah.
 

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You can see, it's
not beautiful.

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But we have to realize
what we would do.

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OK, that's not my
favorite topic.

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But I'll just say
quit on that one.

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I'm not expecting you to do
problems or anything that

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involved that possibility.
 

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Just to see that
it could happen.

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OK.
 

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Quit.
 

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Now, interpolation.
 

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All right.
 

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So I guess if I had to list
problems that people face all

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the time and need numerical
guidance on, this

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is one of them.
 

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And so let's say we're in 1-D
-- 1-D is certainly a lot

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easier -- so suppose
I have this.

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I'll call this x, just to have
a name for the variable.

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Suppose I have some values.
 

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Say six values.
 

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And I've measured them,
I've worked hard to

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find those six values.
 

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But now you may say, well
I'm thinking I have some

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function F(x) here.
 

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00:14:43 --> 00:14:45
But right now I don't
have a function.

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Right now I just have six
values of that function.

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You know, what is the
stretching constant,

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what is the c(x).
 

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00:14:55 --> 00:14:57
 
 

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If you had a physical
experiment with an actual

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spring, you might measure, you
might put on different forces

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and measure the stretching, and
have six values of that.

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How do I fit those
with a curve?

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In other words, how do I know
-- interpolation means, inter-

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means asking, what about
between the points?

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What value should
it have there?

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OK, well there's one simple
rule would be, just

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interpolate linearly between.
 

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OK.
 

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That's certainly pretty stable.
 

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It will not get out of control.
 

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But it's not very accurate.
 

236
00:16:00 --> 00:16:03
For the function that's
probably lying behind this,

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00:16:03 --> 00:16:07
this isn't that great
of a representation.

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00:16:07 --> 00:16:10
OK, you say, I want
something smoother.

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00:16:10 --> 00:16:19
Well another idea that
naturally comes up is, the

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other extreme would be -- so
that was completely local.

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00:16:24 --> 00:16:28
Just, every two values
determine the broken line.

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The second idea that's
completely natural would

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00:16:31 --> 00:16:34
be, fit a polynomial.
 

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Fit a polynomial through those
points, and then you have a

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00:16:36 --> 00:16:39
nice, totally simple function.
 

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Nice curve.
 

247
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And so what degree polynomial
would we be looking for here?

248
00:16:46 --> 00:16:51
If I have six points, I could
fit a polynomial of degree

249
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-- what do you think?
 

250
00:16:54 --> 00:16:59
I want to have six
coefficients, so I can get

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six equations to make the
polynomial go through

252
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those six points.
 

253
00:17:03 --> 00:17:09
So my polynomial P(x), which
actually goes through those

254
00:17:09 --> 00:17:11
points, would be some
polynomial of the form, it's

255
00:17:11 --> 00:17:15
got some constant term, and
it's got some linear term.

256
00:17:15 --> 00:17:19
And how far am I going to go?
 

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Fifth, right.
 

258
00:17:19 --> 00:17:22
That'll give me
six coefficients.

259
00:17:22 --> 00:17:22
OK.
 

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00:17:22 --> 00:17:26
So I could put a fifth degree
polynomial, that gives

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00:17:26 --> 00:17:27
me a_0, a_1, up to a_5.
 

262
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Six coefficients
through six points.

263
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Well, six isn't too bad.
 

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00:17:36 --> 00:17:44
But if six became
60, don't do it.

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That's the message.
 

266
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Don't do it.
 

267
00:17:47 --> 00:17:55
If I'm going on up to, say I
have an a 59x to the 59th,

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00:17:55 --> 00:17:58
And that fits my 60 points.
 

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And you say, well look, I've
got 60 values, that should

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00:18:01 --> 00:18:04
be way better than
having only six.

271
00:18:04 --> 00:18:09
But the polynomial that would
come out of that, even if these

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00:18:09 --> 00:18:16
values were pretty smooth, the
polynomial that fits -- well

273
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I'd have to put in a whole lot
more to get 60, and I won't try

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-- the polynomial
would go crazy.

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00:18:26 --> 00:18:28
Can I just draw
something crazy?

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00:18:28 --> 00:18:31
I mean, whatever.
 

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00:18:31 --> 00:18:34
It's unstable.
 

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00:18:34 --> 00:18:37
And there are classical
examples of that.

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00:18:37 --> 00:18:41
In fact, the famous example
is the function one over

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00:18:41 --> 00:18:44
one plus x squared.
 

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00:18:44 --> 00:18:48
And later, there's a figure in
the book, I think it's in

282
00:18:48 --> 00:18:56
section 5.4, which
shows the result.

283
00:18:56 --> 00:19:00
That's a terrific function.
 

284
00:19:00 --> 00:19:04
It's infinitely
differentialable, I would even

285
00:19:04 --> 00:19:06
call it an analytic function.
 

286
00:19:06 --> 00:19:07
It's great.
 

287
00:19:07 --> 00:19:13
But if I tried to fit it with
a high degree polynomial at

288
00:19:13 --> 00:19:18
points, that polynomial
gets out of hand.

289
00:19:18 --> 00:19:22
And a figure is--
a figure there.

290
00:19:22 --> 00:19:37
So, MATLAB or any computing
system would have a subroutine

291
00:19:37 --> 00:19:39
that does interpolation.
 

292
00:19:39 --> 00:19:43
And this is named
after Legrange.

293
00:19:43 --> 00:19:47
So this is called Legrange
interpolation, fitting a

294
00:19:47 --> 00:19:53
polynomial. and if the degree
is small, and maybe degree five

295
00:19:53 --> 00:19:58
would be okay, then that's an
important thing to

296
00:19:58 --> 00:20:02
be able to do.
 

297
00:20:02 --> 00:20:05
In other words, you're finding
these six coefficients

298
00:20:05 --> 00:20:06
from these six heights.
 

299
00:20:06 --> 00:20:10
You've got some six by six
matrix that connects the six

300
00:20:10 --> 00:20:13
heights to the six a's.
 

301
00:20:13 --> 00:20:18
But when you get up to 60, that
matrix stops behaving well.

302
00:20:18 --> 00:20:22
It becomes very ill
conditioned, and your

303
00:20:22 --> 00:20:25
coefficients go all
over the place.

304
00:20:25 --> 00:20:29
OK, so my question
is, what do you do?

305
00:20:29 --> 00:20:36
And one answer, one good answer
is fit those points, not with

306
00:20:36 --> 00:20:40
straight lines -- that's too
crude -- not with a single

307
00:20:40 --> 00:20:44
polynomial -- that's too
unstable -- but fit

308
00:20:44 --> 00:20:45
it with splines.
 

309
00:20:45 --> 00:20:50
So interpolation by splines.
 

310
00:20:50 --> 00:20:51
OK.
 

311
00:20:51 --> 00:21:04
So it's just sort of appearing
in this section 3.2 -- I mean

312
00:21:04 --> 00:21:09
by that, and people often do
mean, when they say spline,

313
00:21:09 --> 00:21:13
they mean cubic spline.
 

314
00:21:13 --> 00:21:18
So you could have splines
of other degrees.

315
00:21:18 --> 00:21:22
But often, sort of the
natural choice is the cubic.

316
00:21:22 --> 00:21:27
So can I just briefly
describe what that does.

317
00:21:27 --> 00:21:32
What the cubic spline
would be like.

318
00:21:32 --> 00:21:35
So let me draw in again
these six points that

319
00:21:35 --> 00:21:38
I'm going to fit.
 

320
00:21:38 --> 00:21:46
And just say, for these few
minutes, what's a cubic spline.

321
00:21:46 --> 00:21:49
I touched on that Wednesday,
and now I just want to

322
00:21:49 --> 00:21:51
say a little bit more.
 

323
00:21:51 --> 00:22:00
And then the homework asked you
to actually do it with values

324
00:22:00 --> 00:22:02
of a particular function.
 

325
00:22:02 --> 00:22:05
And see how close does
the cubic spline come

326
00:22:05 --> 00:22:09
to the given function.
 

327
00:22:09 --> 00:22:12
OK, so what's a cubic spline?
 

328
00:22:12 --> 00:22:14
What does that word,
spline, what should

329
00:22:14 --> 00:22:16
that mean in your mind?
 

330
00:22:16 --> 00:22:22
So a cubic spline is a
cubic in each piece.

331
00:22:22 --> 00:22:24
A cubic in each piece.
 

332
00:22:24 --> 00:22:29
Now we've met cubics in each
piece as finite elements.

333
00:22:29 --> 00:22:34
And part of this short
discussion is to keep those

334
00:22:34 --> 00:22:39
two slightly different
ideas separate.

335
00:22:39 --> 00:22:47
The finite element idea
was also piecewise cubic.

336
00:22:47 --> 00:22:50
But it used values and slopes.
 

337
00:22:50 --> 00:22:52
So it was completely local.
 

338
00:22:52 --> 00:22:57
If I had a value, as I have
here, six values, and if I also

339
00:22:57 --> 00:23:04
had six slopes, if I knew the
slopes, then I would use

340
00:23:04 --> 00:23:06
those cubic elements.
 

341
00:23:06 --> 00:23:13
And I would fit A -- if I knew
the height and slope there, and

342
00:23:13 --> 00:23:15
I knew the height and slope
there, there would be

343
00:23:15 --> 00:23:16
exactly one cubic.
 

344
00:23:16 --> 00:23:20
Because I would have four
conditions, two conditions

345
00:23:20 --> 00:23:22
there, two conditions
there, would be four.

346
00:23:22 --> 00:23:25
I could fit a cubic between,
another cubic there,

347
00:23:25 --> 00:23:27
another cubic there.
 

348
00:23:27 --> 00:23:32
And because I'm using the same
slope from the left and from

349
00:23:32 --> 00:23:35
the right, the slope
would be good.

350
00:23:35 --> 00:23:37
It would be continuous.
 

351
00:23:37 --> 00:23:42
The second derivative, the
curvature, would not be

352
00:23:42 --> 00:23:46
continuous with those
cubic elements.

353
00:23:46 --> 00:23:47
And that's the difference.
 

354
00:23:47 --> 00:23:54
So the difference is, splines
have continuous, no jump --

355
00:23:54 --> 00:23:56
let me just put it this way.
 

356
00:23:56 --> 00:24:03
No jump in the function.
 

357
00:24:03 --> 00:24:05
I'll use S, maybe, for spline.
 

358
00:24:05 --> 00:24:09
No jump in its slope.
 

359
00:24:09 --> 00:24:12
And no jump in the
second derivative.

360
00:24:12 --> 00:24:14
So that's the difference.
 

361
00:24:14 --> 00:24:21
That the spline functions,
the only jumps you see

362
00:24:21 --> 00:24:26
for those are jumps in
the third derivative.

363
00:24:26 --> 00:24:29
So that makes this
extremely smooth.

364
00:24:29 --> 00:24:33
So if I just try to draw now
what a spline would do, it'll

365
00:24:33 --> 00:24:36
go through those points,
coming out of the spline

366
00:24:36 --> 00:24:39
fit MATLAB command.
 

367
00:24:39 --> 00:24:45
And it'll be as
smooth as I drew it.

368
00:24:45 --> 00:24:53
There is a change in the third
derivative at these points.

369
00:24:53 --> 00:24:58
Actually, have you ever seen
this word, "spline," before?

370
00:24:58 --> 00:25:05
It came out of
naval engineering.

371
00:25:05 --> 00:25:15
When people were designing
the shape of the ship.

372
00:25:15 --> 00:25:22
A naval architect is fitting
the proposed shape of a ship

373
00:25:22 --> 00:25:30
-- this was before MATLAB,
before life started.

374
00:25:30 --> 00:25:32
[LAUGHTER]
 

375
00:25:32 --> 00:25:42
They had little physical,
slightly bendable --

376
00:25:42 --> 00:25:43
I'll call them splines.
 

377
00:25:43 --> 00:25:44
I guess that's what they were.
 

378
00:25:44 --> 00:25:46
That's where the
word came from.

379
00:25:46 --> 00:25:50
Some physical thing which was
like a curve that you use -- I

380
00:25:50 --> 00:25:54
don't know if you guys ever
did mechanical drawing.

381
00:25:54 --> 00:25:58
That was a freshman subject
when I came to MIT.

382
00:25:58 --> 00:25:59
Mechanical drawing.
 

383
00:25:59 --> 00:26:04
I was terrible, terrible,
at mechanical drawing.

384
00:26:04 --> 00:26:06
I don't know.
 

385
00:26:06 --> 00:26:13
I had a friend who helped.
l probably helped him

386
00:26:13 --> 00:26:14
in some other course.
 

387
00:26:14 --> 00:26:16
Anyway, whatever.
 

388
00:26:16 --> 00:26:20
So there were physical things,
curves you used to use.

389
00:26:20 --> 00:26:23
And these spline curves were
used, and maybe still are

390
00:26:23 --> 00:26:29
used, by naval architects
in creating a drawing.

391
00:26:29 --> 00:26:33
But I'm assuming that
those things are now all

392
00:26:33 --> 00:26:37
computerized, and the
spline command is used.

393
00:26:37 --> 00:26:39
Anyway, the result is that.
 

394
00:26:39 --> 00:26:42
Now, I have to draw one
picture of a spline.

395
00:26:42 --> 00:26:44
Of the most important spline.
 

396
00:26:44 --> 00:26:48
And then I'm done with splines.
 

397
00:26:48 --> 00:26:56
So, what I'm going to
draw is now a B-spline.

398
00:26:56 --> 00:27:02
And that's a basic spline.
 

399
00:27:02 --> 00:27:05
It could be one of our
functions, it could be one

400
00:27:05 --> 00:27:07
of our functions phi(x).
 

401
00:27:09 --> 00:27:12
One of our trial functions.
 

402
00:27:12 --> 00:27:15
It could be, and let
me comment on that.

403
00:27:15 --> 00:27:22
So let me remind myself, good
or bad, with a question.

404
00:27:22 --> 00:27:24
And let me try to answer that.
 

405
00:27:24 --> 00:27:27
But let me first
draw the B-spline.

406
00:27:27 --> 00:27:32
OK, so it's like
a hat function.

407
00:27:32 --> 00:27:38
Actually a hat function
is a linear spline.

408
00:27:38 --> 00:27:43
A hat function is that
low level spline that's

409
00:27:43 --> 00:27:44
linear between pieces.
 

410
00:27:44 --> 00:27:53
Now the B-spline is going to
have the value one there.

411
00:27:53 --> 00:27:57
It's going to have no
jump in the function.

412
00:27:57 --> 00:27:59
So the function will
go through there.

413
00:27:59 --> 00:28:01
No jump in the slope.
 

414
00:28:01 --> 00:28:04
No jump in the
second derivative.

415
00:28:04 --> 00:28:07
And I want to get down to zero.
 

416
00:28:07 --> 00:28:13
I want to get down to zero.
 

417
00:28:13 --> 00:28:16
I want it to be as local
as I can make it.

418
00:28:16 --> 00:28:22
So I want it to get to zero.
 

419
00:28:22 --> 00:28:27
Here's the point.
 

420
00:28:27 --> 00:28:31
With those cubic finite
elements, I got

421
00:28:31 --> 00:28:33
them down to zero.
 

422
00:28:33 --> 00:28:37
They came in here with zero
slope, and then they continued

423
00:28:37 --> 00:28:42
as zero, no problem.
 

424
00:28:42 --> 00:28:51
If I'm wanting the second
derivative also to be zero, to

425
00:28:51 --> 00:28:55
be continuous, I won't be able
to do it in one interval.

426
00:28:55 --> 00:28:59
It's going to take me a
total of four intervals.

427
00:28:59 --> 00:29:04
This one can come down to
some point, here, where

428
00:29:04 --> 00:29:07
another cubic starts.
 

429
00:29:07 --> 00:29:10
Maybe I should do the one here.
 

430
00:29:10 --> 00:29:15
Going up, you remember the
allowed cubic spline would be

431
00:29:15 --> 00:29:21
an x cubed over six, some
multiple of x cubed.

432
00:29:21 --> 00:29:23
I don't know what multiple
it'll take, maybe x

433
00:29:23 --> 00:29:25
cubed over six is right.
 

434
00:29:25 --> 00:29:28
It'll come up to
some point here.

435
00:29:28 --> 00:29:36
Now I've got to get it
beginning to curve downwards.

436
00:29:36 --> 00:29:40
So I'm going to have to
change the third derivative.

437
00:29:40 --> 00:29:46
Change to another cubic there,
change to another cubic there,

438
00:29:46 --> 00:29:49
and a final cubic here.
 

439
00:29:49 --> 00:29:52
So there's a picture
of a B-spline.

440
00:29:52 --> 00:29:59
And we could figure out, by
requiring all these continuity

441
00:29:59 --> 00:30:03
conditions, we could figure out
the formula for the cubic

442
00:30:03 --> 00:30:05
in these four pieces.
 

443
00:30:05 --> 00:30:07
And it would be symmetric,
of course, across

444
00:30:07 --> 00:30:09
that center point.
 

445
00:30:09 --> 00:30:13
But I think I won't
try to do it.

446
00:30:13 --> 00:30:18
I'll just leave
that idea, then.

447
00:30:18 --> 00:30:20
There's a figure in the
book showing a picture

448
00:30:20 --> 00:30:24
of the B-spline.
 

449
00:30:24 --> 00:30:29
So those are functions,
extremely valuable in this

450
00:30:29 --> 00:30:32
interpolation problem.
 

451
00:30:32 --> 00:30:35
Because all splines are
combinations of B-splines.

452
00:30:35 --> 00:30:38
Yeah, now let me pull
this topic together.

453
00:30:38 --> 00:30:48
Every spline, every spline
function, is combination

454
00:30:48 --> 00:30:52
of these B-splines.
 

455
00:30:52 --> 00:30:58
Let B_i(x) be the one
centered on node i.

456
00:30:58 --> 00:31:01
And then there's a neighbor
centered on node i+1.

457
00:31:03 --> 00:31:06
And a neighbor to the left
centered on node i-1.

458
00:31:07 --> 00:31:09
What's the point?
 

459
00:31:09 --> 00:31:20
The point is that, at a typical
node, i+1, three of these

460
00:31:20 --> 00:31:23
functions will be non-zero.
 

461
00:31:23 --> 00:31:28
With these very local hat
functions at that node, the

462
00:31:28 --> 00:31:31
only one of the phi functions
that wasn't zero is

463
00:31:31 --> 00:31:32
the one I've drawn.
 

464
00:31:32 --> 00:31:35
All the others were
zero there, right?

465
00:31:35 --> 00:31:38
All the other hats were zero.
 

466
00:31:38 --> 00:31:40
There was just the one.
 

467
00:31:40 --> 00:31:44
But now there'll be a B-spline
starting from here, going up to

468
00:31:44 --> 00:31:47
here, coming down from here.
 

469
00:31:47 --> 00:31:51
There'll be the B-spline
starting from here, going up

470
00:31:51 --> 00:31:54
to here, up, down, so on.
 

471
00:31:54 --> 00:31:59
So at a typical node, I'm
getting the one centered at

472
00:31:59 --> 00:32:04
that node, and also the one to
the left, which is on its way

473
00:32:04 --> 00:32:08
down, and also the one to the
right, which is on its way up.

474
00:32:08 --> 00:32:13
In other words, it's
not as local as the

475
00:32:13 --> 00:32:17
other construction.
 

476
00:32:17 --> 00:32:26
So I would say good, because
it's smooth, but bad because

477
00:32:26 --> 00:32:30
it's not fully local.
 

478
00:32:30 --> 00:32:32
Not completely local.
 

479
00:32:32 --> 00:32:37
It's reasonably local, in that
there are only three functions

480
00:32:37 --> 00:32:39
that are affecting
these points.

481
00:32:39 --> 00:32:44
If I use those polynomials,
they weren't local at all.

482
00:32:44 --> 00:32:51
All the points -- it was a
typical x to the fifth is not

483
00:32:51 --> 00:32:52
zero at any of the points.
 

484
00:32:52 --> 00:32:57
These B-splines are zero at a
lot of points, but three of

485
00:32:57 --> 00:33:01
them, one, two, and three,
will be non-zero at a

486
00:33:01 --> 00:33:03
typical mesh point.
 

487
00:33:03 --> 00:33:13
OK, so they're not popular, as
a result, for finite elements.

488
00:33:13 --> 00:33:16
I'm just wanting to be sure
you make the distinction.

489
00:33:16 --> 00:33:20
They're very popular for
the interpolation job.

490
00:33:20 --> 00:33:24
They're very popular for
the interpolation job.

491
00:33:24 --> 00:33:30
I've got some combination
of these at particular

492
00:33:30 --> 00:33:31
mesh points.
 

493
00:33:31 --> 00:33:36
It's supposed to agree with
F at those mesh points.

494
00:33:36 --> 00:33:38
This is my system to solve.
 

495
00:33:38 --> 00:33:41
I create these functions,
these B-splines.

496
00:33:41 --> 00:33:45
I'm given F at typical points.
 

497
00:33:45 --> 00:33:49
And I choose a combination
which matches F

498
00:33:49 --> 00:33:51
at those points.
 

499
00:33:51 --> 00:33:54
Yeah, OK.
 

500
00:33:54 --> 00:34:00
Is there a question or
discussion on this?

501
00:34:00 --> 00:34:04
I don't like to not say
anything about such an

502
00:34:04 --> 00:34:09
important problem as putting
curves through points.

503
00:34:09 --> 00:34:16
But I don't want to make
it a course on splines.

504
00:34:16 --> 00:34:16
Yes?
 

505
00:34:16 --> 00:34:18
AUDIENCE: [UNINTELLIGIBLE]
 

506
00:34:18 --> 00:34:24
PROFESSOR STRANG: I don't know
that I'll -- oh, OK, yes.

507
00:34:24 --> 00:34:25
All right.
 

508
00:34:25 --> 00:34:30
Two or three comments, and that
suggests a good question,

509
00:34:30 --> 00:34:32
a very good question.
 

510
00:34:32 --> 00:34:36
Can I make a separate
comment that if I go into

511
00:34:36 --> 00:34:40
two dimensions, this
gets much tougher.

512
00:34:40 --> 00:34:44
What happens if I had a
function of x and y?

513
00:34:44 --> 00:34:47
So that I've got points on
a surface, and I'm trying

514
00:34:47 --> 00:34:49
to fit a surface to it.
 

515
00:34:49 --> 00:34:55
Just one message first:
that's not as easy.

516
00:34:55 --> 00:34:59
It's pretty easy if
those points are on

517
00:34:59 --> 00:35:00
a rectangular grid.
 

518
00:35:00 --> 00:35:04
So this is like typical.
 

519
00:35:04 --> 00:35:07
And then I'll come to your
Nyquist question, which

520
00:35:07 --> 00:35:09
is a very good one.
 

521
00:35:09 --> 00:35:14
Can I just -- because we're
coming into 2-D now.

522
00:35:14 --> 00:35:17
Suppose, there's
two dimensions.

523
00:35:17 --> 00:35:18
I have a grid.
 

524
00:35:18 --> 00:35:20
Let's suppose it's a nice grid.
 

525
00:35:20 --> 00:35:27
And at every point, at every
grid point, I have a height.

526
00:35:27 --> 00:35:28
And I'm fitting a surface.
 

527
00:35:28 --> 00:35:30
Right?
 

528
00:35:30 --> 00:35:37
My function is now a function
of x and y. x is here, y is

529
00:35:37 --> 00:35:43
here, F is the surface coming
out of board, in the

530
00:35:43 --> 00:35:44
third dimension.
 

531
00:35:44 --> 00:35:48
And fitting those points, if
they're regularly spaced like

532
00:35:48 --> 00:35:50
that, my life would be OK.
 

533
00:35:50 --> 00:35:53
I could use sort of products
of splines in 1-D.

534
00:35:56 --> 00:36:06
I could use products of basis,
of splines in the x direction

535
00:36:06 --> 00:36:09
times splines in
the y direction.

536
00:36:09 --> 00:36:13
And it would be
pretty successful.

537
00:36:13 --> 00:36:18
It would be, not quite
as nice, but almost OK.

538
00:36:18 --> 00:36:25
But if I had irregularly
spaced points from a general

539
00:36:25 --> 00:36:30
grid, it's not as easy.
 

540
00:36:30 --> 00:36:33
And I won't -- people have
obviously had to figure out how

541
00:36:33 --> 00:36:38
to do it, and to repeat again,
that's what the CAD/CAM world

542
00:36:38 --> 00:36:42
is having to do all the
time, is fit a curve, fit a

543
00:36:42 --> 00:36:44
surface through points.
 

544
00:36:44 --> 00:36:45
It's significant.
 

545
00:36:45 --> 00:36:49
Now, you asked about Nyquist.
 

546
00:36:49 --> 00:36:51
So that's a good question.
 

547
00:36:51 --> 00:37:00
Can I just say, I have to say
-- so what's a function called?

548
00:37:00 --> 00:37:04
Band-limited.
 

549
00:37:04 --> 00:37:07
How many have heard
Nyquist's name?

550
00:37:07 --> 00:37:10
Okay, some of you may know a
lot more than I about it.

551
00:37:10 --> 00:37:16
But let me just get some
context for band-limited

552
00:37:16 --> 00:37:18
functions.
 

553
00:37:18 --> 00:37:21
Okay.
 

554
00:37:21 --> 00:37:25
This is actually a topic that
will belong in the third part

555
00:37:25 --> 00:37:27
of our course, in
the Fourier part.

556
00:37:27 --> 00:37:35
So this is a Fourier idea.
 

557
00:37:35 --> 00:37:38
So I'll come back
to it, actually.

558
00:37:38 --> 00:37:48
So right here I'm just going to
say, very briefly, how it might

559
00:37:48 --> 00:37:51
connect us to what
I've said today.

560
00:37:51 --> 00:38:00
But then let's make a plan
to, when we've got the idea

561
00:38:00 --> 00:38:04
of Fourier coefficients.
 

562
00:38:04 --> 00:38:11
What is a band-limited
function?

563
00:38:11 --> 00:38:17
You know the Fourier idea is
to take F(x), and write it

564
00:38:17 --> 00:38:23
as some combination of pure
frequencies. e^(i*K*x),

565
00:38:23 --> 00:38:25
let me say. e^(i*K*x).
 

566
00:38:27 --> 00:38:30
So this is Fourier
that's coming.

567
00:38:30 --> 00:38:33
I take the function
F(x), and I write it.

568
00:38:33 --> 00:38:38
I could think of it as
a combination of pure

569
00:38:38 --> 00:38:41
exponentials, pure frequencies.
 

570
00:38:41 --> 00:38:44
That would be a Fourier series.
 

571
00:38:44 --> 00:38:48
So that's a Fourier series
because I'm using --

572
00:38:48 --> 00:38:50
K has integer values.
 

573
00:38:50 --> 00:38:55
The frequencies are zero,
one, two, three, whatever.

574
00:38:55 --> 00:39:02
Now that we'll use, so that
Fourier series comes for

575
00:39:02 --> 00:39:05
functions that are periodic.
 

576
00:39:05 --> 00:39:06
They repeat every 2pi.
 

577
00:39:06 --> 00:39:11
Because those functions, if I
increase by 2pi, don't change.

578
00:39:11 --> 00:39:13
So I'm repeating every 2pi.
 

579
00:39:13 --> 00:39:21
Now I have to say a word about
the other possibility, which

580
00:39:21 --> 00:39:26
would be to have
all frequencies.

581
00:39:26 --> 00:39:27
I'll integrate, now, dK.
 

582
00:39:27 --> 00:39:32
Instead of summing on K,
I'll integrate on K.

583
00:39:32 --> 00:39:35
What does band-limited mean?
 

584
00:39:35 --> 00:39:41
Band-limited means that only
frequencies in a certain

585
00:39:41 --> 00:39:44
range, say a range around
zero, are included.

586
00:39:44 --> 00:39:48
So a band-limited function
would be a function whose

587
00:39:48 --> 00:39:55
frequencies go from some value,
say minus omega to omega,

588
00:39:55 --> 00:39:58
instead of going from minus
infinity to infinity.

589
00:39:58 --> 00:40:06
This would be band-limited
frequencies between

590
00:40:06 --> 00:40:12
minus omega and omega.
 

591
00:40:12 --> 00:40:18
So that's another kind
of smooth function.

592
00:40:18 --> 00:40:25
That's another way -- functions
that have only low frequencies

593
00:40:25 --> 00:40:27
are associated with smoothness.
 

594
00:40:27 --> 00:40:31
High frequencies are associated
with fast oscillations.

595
00:40:31 --> 00:40:37
So the class of band-limited
functions gives me another, a

596
00:40:37 --> 00:40:41
Fourier way, to talk
about smoothness.

597
00:40:41 --> 00:40:45
So for us, smoothness was
something about how many

598
00:40:45 --> 00:40:47
derivatives were continuous.
 

599
00:40:47 --> 00:40:50
That's the sort of
smoothness in the x domain.

600
00:40:50 --> 00:40:53
How many derivatives.
 

601
00:40:53 --> 00:40:57
Smoothness in the frequency
domain is, how fast do

602
00:40:57 --> 00:40:59
the frequencies drop off.
 

603
00:40:59 --> 00:41:03
And here, band-limited means
they drop like a shot.

604
00:41:03 --> 00:41:09
Band-limited means that the
frequencies in the function,

605
00:41:09 --> 00:41:13
that these e^(i*K*x)'s are not
there for high frequencies.

606
00:41:13 --> 00:41:18
High frequencies are out for
a band-limited function.

607
00:41:18 --> 00:41:25
And then Shannon has a
formula for the natural

608
00:41:25 --> 00:41:32
way to fit -- so our same
interpolation problem.

609
00:41:32 --> 00:41:36
So now, completing the
answer to your question.

610
00:41:36 --> 00:41:38
So I have these points.
 

611
00:41:38 --> 00:41:44
So Shannon could fit a
function, and it would look

612
00:41:44 --> 00:41:47
smooth, through those points.
 

613
00:41:47 --> 00:41:53
And his function would
be band-limited.

614
00:41:53 --> 00:41:55
So it would be smooth again.
 

615
00:41:55 --> 00:42:02
It would be -- yeah,
it would be smooth.

616
00:42:02 --> 00:42:07
In some way, this Shannon
band-limited stuff is the limit

617
00:42:07 --> 00:42:12
of splines as the spline
degree goes way up.

618
00:42:12 --> 00:42:16
So we did hat functions,
degree one splines.

619
00:42:16 --> 00:42:23
Cubic splines I recommended as
a pretty reliable construction.

620
00:42:23 --> 00:42:26
But you could do fifth degrees
splines, seventh degree

621
00:42:26 --> 00:42:29
splines, you could keep going.
 

622
00:42:29 --> 00:42:32
And in the limit,
you would get this.

623
00:42:32 --> 00:42:37
So maybe that's some partial
answer to the connection

624
00:42:37 --> 00:42:42
between splines and
this Fourier world.

625
00:42:42 --> 00:42:43
OK.
 

626
00:42:43 --> 00:42:45
Thanks.
 

627
00:42:45 --> 00:42:47
So these are topics now.
 

628
00:42:47 --> 00:42:51
Wow, today's lecture is kind
of -- can I do one really

629
00:42:51 --> 00:42:54
important thing now
in today's lecture?

630
00:42:54 --> 00:42:56
For you to remember?
 

631
00:42:56 --> 00:42:58
Gradient and divergence?
 

632
00:42:58 --> 00:43:02
I don't want you to spend the
weekend without thinking about

633
00:43:02 --> 00:43:04
gradient and divergence.
 

634
00:43:04 --> 00:43:07
[LAUGHTER]
 

635
00:43:07 --> 00:43:10
OK.
 

636
00:43:10 --> 00:43:14
Here's the idea.
 

637
00:43:14 --> 00:43:19
For lots and lots of
applications, for a region,

638
00:43:19 --> 00:43:30
let's say, in the plane, --
what physical example

639
00:43:30 --> 00:43:31
shall I pick now?
 

640
00:43:31 --> 00:43:35
Maybe I'll let u be
the temperature.

641
00:43:35 --> 00:43:37
So instead of being
displacement, let me

642
00:43:37 --> 00:43:41
make it temperature. u.
 

643
00:43:41 --> 00:43:42
OK.
 

644
00:43:42 --> 00:43:46
Then I have a
temperature gradient.

645
00:43:46 --> 00:43:47
A slope.
 

646
00:43:47 --> 00:43:51
But now, the whole point is,
that's a function of x and y.

647
00:43:51 --> 00:43:55
Then I have a
temperature gradient.

648
00:43:55 --> 00:43:59
And if I'm consistent with the
notation, that'll be e(x,y).

649
00:43:59 --> 00:44:01
 
 

650
00:44:01 --> 00:44:06
And then you'll expect
that there's some c(x,y).

651
00:44:08 --> 00:44:13
Some operator, c, that tells
me how much heat flows.

652
00:44:13 --> 00:44:17
This will tell me something
about the the, the

653
00:44:17 --> 00:44:18
thermal conductivity.
 

654
00:44:18 --> 00:44:20
Right?
 

655
00:44:20 --> 00:44:28
Really, as I speak about
this framework, I'm just

656
00:44:28 --> 00:44:31
uttering the correct words.
 

657
00:44:31 --> 00:44:34
Having started with
temperature, this thing should

658
00:44:34 --> 00:44:36
be a temperature gradient.
 

659
00:44:36 --> 00:44:39
Then I should have some
physical thermal conductivity,

660
00:44:39 --> 00:44:43
different for different metals
or different materials.

661
00:44:43 --> 00:44:46
And I'll have a
heat flow, w(x,y).

662
00:44:48 --> 00:44:53
And everybody knows that
w will be c times e.

663
00:44:53 --> 00:44:57
And then there will
be some A transpose.

664
00:44:57 --> 00:45:01
Of course, there will be
some A transpose here,

665
00:45:01 --> 00:45:03
and some A here.
 

666
00:45:03 --> 00:45:07
And up here I'll have
a balance equation.

667
00:45:07 --> 00:45:13
OK.
 

668
00:45:13 --> 00:45:20
I just want to think,
what's the operator A?

669
00:45:20 --> 00:45:25
If we can focus on that
question, then that's what's

670
00:45:25 --> 00:45:28
going to occupy us for,
certainly the whole

671
00:45:28 --> 00:45:31
of next week.
 

672
00:45:31 --> 00:45:35
So I've actually used
the word gradient.

673
00:45:35 --> 00:45:39
We have functions
of two variables.

674
00:45:39 --> 00:45:43
We're looking for the change,
the rate of change, the

675
00:45:43 --> 00:45:44
steepness of those functions.
 

676
00:45:44 --> 00:45:52
So this A, Au, is going to
give me two derivatives.

677
00:45:52 --> 00:45:56
I've got two variables, there
are two first derivatives.

678
00:45:56 --> 00:46:01
Both of them are important.
 

679
00:46:01 --> 00:46:03
That's what the A is.
 

680
00:46:03 --> 00:46:09
For the next big
example in the course.

681
00:46:09 --> 00:46:14
The final major example of the
course is, when a acts on a

682
00:46:14 --> 00:46:18
function of two variables,
because I'm in a region in the

683
00:46:18 --> 00:46:23
plane, to find its
rate of change.

684
00:46:23 --> 00:46:26
And this is called
the gradient.

685
00:46:26 --> 00:46:34
The shorthand is
the gradient of u.

686
00:46:34 --> 00:46:37
So we have to understand that.
 

687
00:46:37 --> 00:46:41
We have to understand
what the gradient is.

688
00:46:41 --> 00:46:45
And, of course, we want
to know its transpose.

689
00:46:45 --> 00:46:49
So can I just think, what
should be the transpose

690
00:46:49 --> 00:46:54
of the gradient?
 

691
00:46:54 --> 00:46:55
OK.
 

692
00:46:55 --> 00:46:57
I'll take that picture out.
 

693
00:46:57 --> 00:46:58
OK.
 

694
00:46:58 --> 00:47:03
Thinking now about the
transpose of the gradient.

695
00:47:03 --> 00:47:06
OK.
 

696
00:47:06 --> 00:47:13
So a itself is, you could
say, is d/dx and d/dy.

697
00:47:14 --> 00:47:18
You notice how I'm
separating out A from Au.

698
00:47:20 --> 00:47:24
When this acts on a function,
this is the gradient operator

699
00:47:24 --> 00:47:27
that acts on a function, u,
to produce du/dx and du/dy.

700
00:47:29 --> 00:47:30
OK.
 

701
00:47:30 --> 00:47:33
Now what's the
transpose of this?

702
00:47:33 --> 00:47:35
OK.
 

703
00:47:35 --> 00:47:38
You can guess what it should
be, and then we'll see that

704
00:47:38 --> 00:47:40
yes, that guess is correct.
 

705
00:47:40 --> 00:47:44
So what should a
transpose look like?

706
00:47:44 --> 00:47:53
If there's any justice, a
transpose should be --

707
00:47:53 --> 00:47:58
OK, this is two by one.
 

708
00:47:58 --> 00:48:06
The transpose you would expect
to be a row, a row vector.

709
00:48:06 --> 00:48:08
I should have the
transpose of that there.

710
00:48:08 --> 00:48:11
And what is the
transpose of that?

711
00:48:11 --> 00:48:13
Just tell me.
 

712
00:48:13 --> 00:48:15
Because we have an idea.
 

713
00:48:15 --> 00:48:17
What should be the
transpose of d/dx?

714
00:48:19 --> 00:48:20
Negative d/dx.
 

715
00:48:20 --> 00:48:24
 
 

716
00:48:24 --> 00:48:25
And what should be the
transpose of d/dy?

717
00:48:26 --> 00:48:27
It should be negative d/dy.
 

718
00:48:28 --> 00:48:30
Those are two different pieces.
 

719
00:48:30 --> 00:48:32
This is not run together.
 

720
00:48:32 --> 00:48:35
There's a big space in there.
 

721
00:48:35 --> 00:48:37
That's two pieces.
 

722
00:48:37 --> 00:48:44
So what is A transpose
applied to -- heat flow w.

723
00:48:44 --> 00:48:50
I want to say, what is A
transpose applied to w?

724
00:48:50 --> 00:48:55
You're going to see this
again, but we'll just take

725
00:48:55 --> 00:48:58
these minutes to show
it for the first time.

726
00:48:58 --> 00:49:03
So A transpose --
wait a minute.

727
00:49:03 --> 00:49:08
Is w a function, or
is it a vector?

728
00:49:08 --> 00:49:11
Yeah we've got to get that
straight before we start.

729
00:49:11 --> 00:49:14
Here's an ordinary function,
a scalar function.

730
00:49:14 --> 00:49:18
Just whatever, x squared
plus y squared.

731
00:49:18 --> 00:49:21
What is e?
 

732
00:49:21 --> 00:49:25
Suppose this is x
squared plus y squared.

733
00:49:25 --> 00:49:27
Let's have a specific example.
 

734
00:49:27 --> 00:49:31
What would e then be?
 

735
00:49:31 --> 00:49:33
It's got two components, right?
 

736
00:49:33 --> 00:49:36
It's got an x derivative
and a y derivative. e

737
00:49:36 --> 00:49:38
is a vector, .
 

738
00:49:38 --> 00:49:41
 
 

739
00:49:41 --> 00:49:44
Then I multiply by c.
 

740
00:49:44 --> 00:49:46
So this w has two components.
 

741
00:49:46 --> 00:49:50
This has got a
w_1(x,y), w_2(x,y).

742
00:49:50 --> 00:49:53
 
 

743
00:49:53 --> 00:49:56
So just keep things straight.
 

744
00:49:56 --> 00:50:00
So it's that that I want
to apply A transpose to.

745
00:50:00 --> 00:50:07
So A transpose w is
minus d/dx, minus d/dy.

746
00:50:08 --> 00:50:11
And everything's coming out
right because it's applied

747
00:50:11 --> 00:50:13
to a function w_1 and w_2.
 

748
00:50:14 --> 00:50:17
Two is a vector field.
 

749
00:50:17 --> 00:50:20
It's not a scalar field,
it's a vector field.

750
00:50:20 --> 00:50:24
And the result is just
what it should be: minus

751
00:50:24 --> 00:50:28
dw_1/dx, minus dw_2/dy.
 

752
00:50:31 --> 00:50:37
OK, good.
 

753
00:50:37 --> 00:50:40
This has got to come out of
integration by parts, right?

754
00:50:40 --> 00:50:43
So we'll have to think: what
does integration by parts

755
00:50:43 --> 00:50:45
mean in two variables?
 

756
00:50:45 --> 00:50:50
And it's a famous formula
named after Gauss and Green.

757
00:50:50 --> 00:50:52
The Green's formula, often.
 

758
00:50:52 --> 00:50:56
But do you recognize what
I'm looking at here?

759
00:50:56 --> 00:51:01
This is so important
it has a name.

760
00:51:01 --> 00:51:02
And what's the name?
 

761
00:51:02 --> 00:51:04
And then we're ready to go.
 

762
00:51:04 --> 00:51:08
What's the name of, if I take a
vector field, like .

763
00:51:09 --> 00:51:17
Let me take ,
as a specific example.

764
00:51:17 --> 00:51:24
Suppose w is , and
was multiplied by, let

765
00:51:24 --> 00:51:26
me take c to be one.
 

766
00:51:26 --> 00:51:33
Then what is A transpose w?
 

767
00:51:33 --> 00:51:34
Specifically.
 

768
00:51:34 --> 00:51:39
What am I getting out of it?
 

769
00:51:39 --> 00:51:43
What do I get from here?
 

770
00:51:43 --> 00:51:48
That's minus the x derivative
of the first guy, and the x

771
00:51:48 --> 00:51:52
derivative of that guy is two,
so I'm getting a minus two.

772
00:51:52 --> 00:51:58
And this is minus the y
derivative of the second guy,

773
00:51:58 --> 00:52:00
so that's another minus two.
 

774
00:52:00 --> 00:52:02
So I'm getting a number.
 

775
00:52:02 --> 00:52:03
It happens to be a number here.
 

776
00:52:03 --> 00:52:09
I chose such a simple function
it came out to be a number.

777
00:52:09 --> 00:52:11
And what's the name?
 

778
00:52:11 --> 00:52:15
So this is minus the what of w?
 

779
00:52:15 --> 00:52:20
Just tell me, what's the
name everybody uses

780
00:52:20 --> 00:52:22
for that operation?
 

781
00:52:22 --> 00:52:24
The divergence.
 

782
00:52:24 --> 00:52:28
Minus the divergence of w.
 

783
00:52:28 --> 00:52:38
So I what I'm saying here is
that this -- I'm saying it

784
00:52:38 --> 00:52:43
because somehow I remember
studying vector calculus.

785
00:52:43 --> 00:52:48
And in that process, I learned
about the gradient, and I

786
00:52:48 --> 00:52:51
learned about the divergence.
 

787
00:52:51 --> 00:52:56
But I never learned that one
was the transpose of the other.

788
00:52:56 --> 00:53:03
I think, looking back,
that was criminal.

789
00:53:03 --> 00:53:07
To describe those -- with
a minus sign, of course.

790
00:53:07 --> 00:53:16
I learned Green's formula, but
now we'll see what it means.

791
00:53:16 --> 00:53:17
OK, that's next week's job.
 

792
00:53:17 --> 00:53:20
Have a great weekend
and see you Monday.

793
00:53:20 --> 00:53:21