## TTU Math2450 Calculus3 Sec 11.4 and 11.5

• 0:00 - 0:02
PROFESSOR TODA: Any
questions so far?
• 0:02 - 0:06
I mean, conceptual,
theoretical questions first,
• 0:06 - 0:09
and then we will
do the second part
• 0:09 - 0:10
of [INAUDIBLE] applications.
• 0:10 - 0:14
for more questions.
• 0:14 - 0:16
No questions so far?
• 0:16 - 0:19
I have not finished 11-4.
• 0:19 - 0:26
I still owe you a long
• 0:26 - 0:28
Hopefully it's going to
make more sense today
• 0:28 - 0:31
• 0:31 - 0:34
I was just saying
that I'm doing 11-4.
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This is a lot of chapter.
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So second part of 11-4 today--
tangent plane and applications.
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• 0:51 - 0:54
Now, we don't say what
those applications are
• 0:54 - 0:59
from the start, but these are
some very important concepts
• 0:59 - 1:01
called the total differential.
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And the linear
approximation number
• 1:14 - 1:15
is going under the [INAUDIBLE].
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Thank you, sir.
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Linear approximation for
functions of the type z
• 1:24 - 1:29
equals f of xy, which means
graphs of two variables.
• 1:29 - 1:34
At the end of the chapter, I'll
take the notes copy from you.
• 1:34 - 1:37
So don't give me
anything until it's over.
• 1:37 - 1:39
When is that going to be over?
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We have four more
sections to go.
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So I guess right before
spring break you give me
• 1:47 - 1:50
the notes for chapter 11.
• 1:50 - 1:52
All right, and then
I'm thinking of making
• 1:52 - 1:55
copies of both chapters.
• 1:55 - 2:00
You get the-- I'm
distributing them to you.
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I haven't started
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Could anybody tell
me what the equation
• 2:10 - 2:14
that we used last time--
we proved it, actually.
• 2:14 - 2:16
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What is the equation
of the tangent plane
• 2:21 - 2:27
to a smooth surface or a patch
of a surface at the point
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m of coordinates x0, y0,
z0, where the graph is
• 2:34 - 2:37
given by z equals f of x and y.
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I'm going to label it on
the patch of a surface.
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OK, imagine it
labeled brown there.
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And can somebody tell me the
equation of the other plane?
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But because you
have better memory,
• 2:54 - 3:00
years younger than me or so.
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So could you-- could anybody
tell me what the tangent
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planes equation-- I'll start.
• 3:09 - 3:10
And it's going to come to you.
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z minus z0 equals.
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And now let's see.
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I'll pick a nice color.
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I'll wait.
• 3:19 - 3:22
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STUDENT: fx of x.
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PROFESSOR TODA: f sub x, the
partial derivative measured
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at f0 i0 times the
quantity x minus x0 plus--
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STUDENT: f sub y.
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PROFESSOR TODA: f
sub y, excellent.
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f sub y.
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STUDENT: x0, y0.
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PROFESSOR TODA: x0,
y0 times y minus y0.
• 3:45 - 3:49
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OK.
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All right.
• 3:52 - 3:59
Now thinking of what those
quantities mean, x minus x0, y
• 3:59 - 4:04
minus y0, z minus
z0, what are they?
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They are small
displacements, aren't they?
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I mean, what does it
mean small displacement?
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Imagine that you are near
the point on both surfaces.
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So what is a small
neighborhood--
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what's a typical small
neighborhood [INAUDIBLE]?
• 4:28 - 4:30
It's a disk, right?
• 4:30 - 4:33
There are many kinds of
neighborhoods, but one of them,
• 4:33 - 4:37
I'd say, would be
this open disk, OK?
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I'll draw that.
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Now, if I have a
red point-- I don't
• 4:45 - 4:53
know how to do that pink point--
somewhere nearby in planes--
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this is the plane.
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In plane, I have this
point that is close.
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And that point is xyz.
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And you think, OK, can
I visualize that better?
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Well, guys, it's hard to
visualize that better.
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But I'll draw a triangle
[? doing ?] a better job.
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That's the frame.
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This is a surface.
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Imagine it's a surface, OK?
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That's the point of x0, y0.
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[? It's ?] the 0 and that.
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Where is the point xyz again?
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The point xyz is not
on the pink stuff.
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This is a pink surface.
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It looks like Pepto
Bismol or something.
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No.
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That's not what I want.
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I want the close enough
point on the blue plane.
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It's actually in the blue plane
pie and this guy would be xyz.
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So now say, OK, how
far I x be from x0?
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Well, I don't know.
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We would have to check
the points, the set 0,
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check the blue point.
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This is x.
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So between x and x0, I
have this difference,
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which is delta x displacement,
displacement along the x-axis,
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away from the
point, fixed point.
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This is the fixed
point, this point.
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This point is p.
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OK.
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y minus y0, let's call
that delta y, which
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is the displacement
along the y-axis.
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And then the z minus z0 can be.
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Just because I'm a mathematician
and I don't like writing down
• 7:06 - 7:11
a lot, I would use
s batch as I can,
• 7:11 - 7:17
compact symbols, to
speed up my computation.
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So I can rewrite
this whole thing
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as a delta z equals f sub
x, x0 y0, which is a number.
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It's a slope.
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that last time.
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We even went skiing
last time, when
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we said that's like the slope
in-- what's the x direction?
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Slope in the x direction
and slope in the y direction
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on the graph that was the
white covered with snow hill.
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That was what we had last time.
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Delta x plus f sub
0, another slope
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in the y direction, delta y.
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And fortunately-- OK, the book
is a very good book, obviously,
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right?
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But I wish we could've done
certain things better in terms
• 8:16 - 8:22
of comparisons between
this notion in Calc III
• 8:22 - 8:27
and some corresponding
notion in Calc I.
• 8:27 - 8:30
So you're probably
thinking, what the heck
• 8:30 - 8:31
• 8:31 - 8:35
Well, I'm thinking
of something that you
• 8:35 - 8:40
may want to remember
from Calc I.
• 8:40 - 8:43
And that's going to come
into place beautifully
• 8:43 - 8:48
right now because you have the
Calc I, Calc III comparison.
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And that's why it would be
great-- the books don't even
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• 8:55 - 9:00
In Calc I, I reminded
• 9:00 - 9:01
He was a very nice guy.
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I have no idea, right?
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Never met him.
• 9:04 - 9:07
One of the fathers of calculus.
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And he introduced the
so-called Leibniz notation.
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And one of you in office
hours last Wednesday
• 9:16 - 9:19
told me, so the
Leibnitz notation
• 9:19 - 9:23
for a function g of
x-- I'm intentionally
• 9:23 - 9:26
changing notation-- is what?
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Well, this is just
the derivative
• 9:32 - 9:34
which is the limit of
the different quotients
• 9:34 - 9:38
delta x-- as done by some
• 9:38 - 9:43
blutches-- 0, right, which
would be the same as lim
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of g of x minus g of x0 over
x minus x0 as x approaches x0,
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right?
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Right.
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So we've done that in Calc I.
But it was a long time ago.
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My mission is to teach
you all Calc III,
• 10:01 - 10:04
but I feel that
my mission is also
• 10:04 - 10:09
to teach you what you may not
remember very well from Calc I,
• 10:09 - 10:12
because everything is related.
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So what was the way we
could have written this,
• 10:18 - 10:21
not delta g over delta
x equals g prime.
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No.
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But it's an approximation of
g prime around a very small
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[INAUDIBLE], very close to x0.
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So if you wanted to
rewrite this approximation,
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how would you have rewritten it?
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Delta g--
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STUDENT: g prime sub x.
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PROFESSOR TODA: g prime
of x0 times delta x.
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OK?
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Now, why this approximation?
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What if I had put equal?
• 11:12 - 11:14
If I had put equal, it
would be all nonsense.
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Why?
• 11:15 - 11:19
Well, say, Magdalena, if you
put equal, it's another object.
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What object?
• 11:20 - 11:20
OK.
• 11:20 - 11:22
Let's look at the objects.
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Let's draw a picture.
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• 11:26 - 11:27
This is g.
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This is x0.
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This is g of x.
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What's g prime?
• 11:32 - 11:39
g prime-- thank god-- is the
slope of g prime x0 over here.
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So if I want to write the
line, the line is exactly this.
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The red object is the line.
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So what is the red object again?
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It's y minus y over x
minus x0 equals m, which
• 11:58 - 12:00
is g prime number 0.
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m is the slope.
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That's the point slope
formula, thank you very much.
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So the red object is this.
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This is the line.
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Attention is not the same.
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The blue thing is my
curve, more precisely
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a tiny portion of my curve.
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This neighborhood around the
point is what I have here.
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What I'm actually-- what?
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• 12:26 - 12:30
I'm trying to
approximate my curve
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function with a little line.
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And I say, I would rather
approximate with a red line
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because this is the
best approximation
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to the blue arc of a curve
which is on the curve, right?
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So this is what it is
is just an approximation
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of a curve, approximation of
a curve of an arc of a curve.
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But Magdalena's lazy
today-- approximation
• 12:58 - 13:04
of an arc of a curve
with a segment of a line,
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with a segment of
the tangent line
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of the tangent [INAUDIBLE].
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How do we call
such a phenomenon?
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An approximation of
an arc of a circle
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with a little segment
of a tangent line
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is like a discretization, right?
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But we call it
linear approximation.
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It's called a linear
approximation.
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A-P-P, approx.
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Have you ever seen a
linear approximation
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before coming from Calc II?
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Well, in Calc II you've
seen the Taylor's formula.
• 13:50 - 13:52
What is the Taylor's formula?
• 13:52 - 13:55
It's a beautiful
thing that said what?
• 13:55 - 13:56
I don't know.
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Let's remember together.
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So relationship
with Calc II, I'm
• 14:00 - 14:05
going to go and make an arrow--
relationship with Calc II,
• 14:05 - 14:08
because everything
is actually related.
• 14:08 - 14:14
In Calc II-- how did we
introduce Taylor's formula?
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Well, instead of little a that
you're so used to in Calc II,
• 14:17 - 14:21
we are going to put x0
is the same thing, right?
• 14:21 - 14:24
So what was Taylor's
formula saying?
• 14:24 - 14:28
You have this kind of
smooth, beautiful curve.
• 14:28 - 14:31
But being smooth is not enough.
• 14:31 - 14:34
You have that real analytic.
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Real analytic means
that the function can be
• 14:36 - 14:41
expanded in Taylor's formula.
• 14:41 - 14:42
So what does it mean?
• 14:42 - 14:53
It means that we have f of x
prime is f of x0 equals-- or g.
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You want-- it doesn't matter.
• 14:55 - 15:01
f prime of x0 times
x minus x0 plus
• 15:01 - 15:06
dot, dot, dot, dot something
that I'm going to put.
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This is [? O. ?] It's a small
quantity that's maybe not
• 15:09 - 15:13
so small, but I declare
it to be negligible.
• 15:13 - 15:15
And so they're going
to be negligible.
• 15:15 - 15:19
I have to make a face,
a smiley face and eyes,
• 15:19 - 15:24
meaning that it's OK to
neglect the second order
• 15:24 - 15:25
term, the third order term.
• 15:25 - 15:28
So what happens, that
little h, when I square it,
• 15:28 - 15:29
say the heck with it.
• 15:29 - 15:31
It's going to be very small.
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Like if h is 0.1 and then
h squared will be 0.0001.
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And I have a certain range
of error that I allow,
• 15:40 - 15:42
a threshold.
• 15:42 - 15:43
I say that's negligible.
• 15:43 - 15:47
If h squared and h cubed and h
to the fourth are negligible,
• 15:47 - 15:50
then I'm fine.
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If I take all the
other spot, that's
• 15:53 - 15:56
the linear approximation.
• 15:56 - 16:00
And that's exactly
what I wrote here
• 16:00 - 16:02
with little g instead of f.
• 16:02 - 16:05
The only difference is this is
little f and this is little g.
• 16:05 - 16:09
But it's the same exact
formula, linear approximation.
• 16:09 - 16:15
Do you guys remember then next
terms of the Taylor's formula?
• 16:15 - 16:15
STUDENT: fw--
• 16:15 - 16:16
PROFESSOR TODA: fw--
• 16:16 - 16:20
STUDENT: w over--
• 16:20 - 16:23
PROFESSOR TODA: So
fw prime at x0 over--
• 16:23 - 16:24
STUDENT: 1 factorial.
• 16:24 - 16:26
PROFESSOR TODA: 2 factorial.
• 16:26 - 16:27
This was 1 factorial.
• 16:27 - 16:29
This was over 1 factorial.
• 16:29 - 16:31
But I don't write
it because it's one.
• 16:31 - 16:31
STUDENT: Right.
• 16:31 - 16:36
PROFESSOR TODA: Here I would
have f double prime of blah,
• 16:36 - 16:41
blah, blah over-- what did
you say-- 2 factorial times x
• 16:41 - 16:44
minus x0 squared plus, plus,
plus, the cubic [INAUDIBLE]
• 16:44 - 16:50
of the-- this is the quadratic
term that I neglect, right?
• 16:50 - 16:51
So that was Taylor's formula.
• 16:51 - 16:55
Do I mention anything
• 16:55 - 16:56
We should.
• 16:56 - 16:58
But practically, the
authors of the book
• 16:58 - 17:00
thought, well, everything
is in the book.
• 17:00 - 17:02
You can go back and forth.
• 17:02 - 17:05
It's not like that unless
• 17:05 - 17:10
For example, I didn't
see that when I was 21.
• 17:10 - 17:13
I couldn't make any connection
between these Calc I,
• 17:13 - 17:15
Calc II, Calc III notions.
• 17:15 - 17:18
Because nobody told me, hey,
• 17:18 - 17:20
and look at that in
perspective and make
• 17:20 - 17:25
a comparison between what you
learned in different chapters.
• 17:25 - 17:26
• 17:26 - 17:29
After 20 years, I
said, oh, I finally
• 17:29 - 17:34
see the picture of linearization
of a function of, let's say,
• 17:34 - 17:35
n variables.
• 17:35 - 17:38
So all these total
differentials will come in place
• 17:38 - 17:41
when time comes.
• 17:41 - 17:46
You have a so-called
differential in Calc I.
• 17:46 - 17:48
And that's not delta g.
• 17:48 - 17:50
Some people say, OK,
no, that's delta g.
• 17:50 - 17:52
No, no, no, no.
• 17:52 - 17:54
The delta x is a displacement.
• 17:54 - 17:57
The delta g is the
induced displacement.
• 17:57 - 18:00
If you want this to be
come a differential,
• 18:00 - 18:03
then you shrink
that displacement
• 18:03 - 18:06
to infinitesimally small.
• 18:06 - 18:06
OK?
• 18:06 - 18:10
So it's like going from
a molecule to an atom
• 18:10 - 18:14
to an electron to subatomic
particles but even more,
• 18:14 - 18:16
something infinitesimally small.
• 18:16 - 18:17
So what do we do?
• 18:17 - 18:23
We shrink delta x into dx
which is infinitesimally small.
• 18:23 - 18:26
• 18:26 - 18:29
It's like the notion of
God but microscopically
• 18:29 - 18:34
or like microbiology
compared to the universe, OK?
• 18:34 - 18:42
So dx is multiplied
by g prime of x0.
• 18:42 - 18:46
And instead of delta g, I'm
going to have a so-called dg,
• 18:46 - 18:49
and that's a form.
• 18:49 - 18:53
In mathematics, this is
called a form or a one form.
• 18:53 - 18:59
And it's a special
kind of object, OK?
• 18:59 - 19:02
So Mr. Leibniz was very smart.
• 19:02 - 19:10
He said, but I can rewrite this
form like dg dx equals g prime.
• 19:10 - 19:13
So if you ever forget
• 19:13 - 19:18
is called differential,
differential form,
• 19:18 - 19:21
you remember Mr.
Leibniz, he taught you
• 19:21 - 19:25
how to write the derivative in
two different ways, dg dx or g
• 19:25 - 19:27
prime.
• 19:27 - 19:30
What you do is just formally
multiply g prime by dx
• 19:30 - 19:32
and you get dg.
• 19:32 - 19:35
Say it again, Magdalena--
multiply g prime by dx
• 19:35 - 19:36
and you get dg.
• 19:36 - 19:39
And that's your
so-called differential.
• 19:39 - 19:42
Now, why do you say total
differential-- total
• 19:42 - 19:47
differential, my god, like
complete differentiation?
• 19:47 - 19:52
In 11.4, we deal with
functions of two variables.
• 19:52 - 19:55
So can we say differentials?
• 19:55 - 19:57
Mmm, it's a little bit
like a differential
• 19:57 - 20:00
with respect to what variable?
• 20:00 - 20:03
If you say with respect
to all the variables,
• 20:03 - 20:09
then you have to be thinking
to be smart and event,
• 20:09 - 20:12
create this new object.
• 20:12 - 20:17
If one would write
Taylor's formula,
• 20:17 - 20:23
there is a Taylor's
formula that we don't give.
• 20:23 - 20:23
OK.
• 20:23 - 20:26
Now, you guys are looking
at me with excitement.
• 20:26 - 20:31
For one point extra
credit, on the internet,
• 20:31 - 20:35
find Taylor's formula for
n variables, functions
• 20:35 - 20:39
of n variables or at
least two variables,
• 20:39 - 20:44
which was going to look
like z minus z0 equals
• 20:44 - 20:49
f sub x at the point x0
at 0 times x minus x0 plus
• 20:49 - 21:00
f sub y at x0 y0 times x minus
x0 plus second order terms
• 21:00 - 21:04
plus third order terms
plus fourth order terms.
• 21:04 - 21:07
And the video cannot see me.
• 21:07 - 21:09
So what do we do?
• 21:09 - 21:14
We just truncate this
part of Taylor's I say,
• 21:14 - 21:18
polynomial of degree one.
• 21:18 - 21:21
everything else, the heck
• 21:21 - 21:23
with that.
• 21:23 - 21:25
And I call that a
linear approximation,
• 21:25 - 21:28
but it's actually Taylor's
formula being discussed.
• 21:28 - 21:31
We don't tell you in
the book because we
• 21:31 - 21:32
don't want to scare you.
• 21:32 - 21:35
I think we would better
tell you at some point,
• 21:35 - 21:38
so I decided to tell you now.
• 21:38 - 21:39
All right.
• 21:39 - 21:42
So this is Taylor's formula
for functions of two variables.
• 21:42 - 21:46
We have to create
not out of nothing
• 21:46 - 21:50
but out of this the
total differential.
• 21:50 - 21:51
Who tells me?
• 21:51 - 21:54
Shrink the
displacement, Magdalena.
• 21:54 - 21:58
The delta x shrunk to
an infinitesimally small
• 21:58 - 21:59
will be dx.
• 21:59 - 22:01
Delta y will become dy.
• 22:01 - 22:06
The line is a smiley from the
skies, just looking at us.
• 22:06 - 22:08
He loves our notations.
• 22:08 - 22:11
And this is dz.
• 22:11 - 22:19
So I'm going to write dz or df's
the same thing equals f sub x.
• 22:19 - 22:22
At the point, you
could be at any point
• 22:22 - 22:30
you are taking in particular,
dx plus f sub y xy dy.
• 22:30 - 22:34
So this is at any point
at the arbitrary point xy
• 22:34 - 22:39
in the domain where your
function e is at least c1.
• 22:39 - 22:41
What does it mean, c1?
• 22:41 - 22:43
It means the function
is differentiable
• 22:43 - 22:47
and the partial
derivatives are continuous.
• 22:47 - 22:51
I said several times, I
want even more than that.
• 22:51 - 22:57
I want it maybe second
order derivatives
• 22:57 - 23:03
to exist and be continuous
and so on and so forth.
• 23:03 - 23:08
And I will assume
that the function can
• 23:08 - 23:12
be expanded [INAUDIBLE] series.
• 23:12 - 23:14
• 23:14 - 23:17
All right, now example
of a final problem
• 23:17 - 23:22
that was my first problem
on the final many times
• 23:22 - 23:26
and also on the common
final departmental final.
• 23:26 - 23:28
And many students
screwed up, and I
• 23:28 - 23:32
don't want you to ever
make such a mistake.
• 23:32 - 23:37
So this is a mistake not
to make, OK, mistake not
• 23:37 - 23:44
to make because after 20
something years of teaching,
• 23:44 - 23:46
I'm quite familiar with
the mistakes students
• 23:46 - 23:49
make in general and I don't
want you to make them.
• 23:49 - 23:51
You are too good to do this.
• 23:51 - 23:52
So problem 1.
• 23:52 - 23:57
On the final, I said-- we
said-- the only difference was
• 23:57 - 24:01
on some departmental finals,
we gave a more sophisticated
• 24:01 - 24:02
function.
• 24:02 - 24:07
I'm going to give only
some simple function
• 24:07 - 24:08
for this polynomial.
• 24:08 - 24:10
That's beautiful.
• 24:10 - 24:19
And then I said we said
write the differential
• 24:19 - 24:28
of this function at an
arbitrary point x, y.
• 24:28 - 24:29
And done.
• 24:29 - 24:31
And [INAUDIBLE].
• 24:31 - 24:35
Well, let me tell you what
some of my students-- some
• 24:35 - 24:36
of my studentss-- don't do that.
• 24:36 - 24:38
I'm going to cross it with red.
• 24:38 - 24:42
And some of my students
wrote me very beautifully df
• 24:42 - 24:44
equals 2x plus 2y.
• 24:44 - 24:48
And that can send
me to the hospital.
• 24:48 - 24:53
If you want to go to the ER
soon, do this on the exam
• 24:53 - 24:56
because this is nonsense.
• 24:56 - 24:57
Why is this nonsense?
• 24:57 - 24:58
This is not--
• 24:58 - 25:00
STUDENT: [INAUDIBLE] dx or dy.
• 25:00 - 25:01
PROFESSOR TODA: Exactly.
• 25:01 - 25:07
So the most important thing
is that the df is like-- OK,
• 25:07 - 25:09
let me come back to driving.
• 25:09 - 25:14
I'm driving to Amarillo-- and I
give this example to my calc 1
• 25:14 - 25:18
students all the time because
it's a linear motion in terms
• 25:18 - 25:19
of time.
• 25:19 - 25:21
And let's say I'm on
cruise control or not.
• 25:21 - 25:23
It doesn't matter.
• 25:23 - 25:30
When we drive and I'm looking at
the speedometer and I see 60--
• 25:30 - 25:37
I didn't want to say more, but
let's say 80, 80 miles an hour.
• 25:37 - 25:39
That is a miles an hour.
• 25:39 - 25:43
That means the hour is a huge
chunk delta h or delta t.
• 25:43 - 25:45
Let's call it delta
t because it's time.
• 25:45 - 25:46
I'm silly.
• 25:46 - 25:48
Delta t is 1.
• 25:48 - 25:51
Delta s, the space,
the space, is going
• 25:51 - 25:55
to be the chunk of 60 miles.
• 25:55 - 26:00
But then that is the
• 26:00 - 26:02
So that's why I said 60.
• 26:02 - 26:05
That's the average
speed I had in my trip,
• 26:05 - 26:06
during my trip [INAUDIBLE].
• 26:06 - 26:11
There were moments when my
speed was 0 or close to 0.
• 26:11 - 26:12
Let's assume it was never 0.
• 26:12 - 26:15
But that means there were many
moments when my speed could've
• 26:15 - 26:19
been 100, and nobody knows
because they didn't catch me.
• 26:19 - 26:21
So I was just lucky.
• 26:21 - 26:26
So in average, if somebody is
asking you what is the average,
• 26:26 - 26:30
that doesn't tell them anything.
• 26:30 - 26:34
That reminds me of that
joke-- overall I'm good,
• 26:34 - 26:38
the statistician joke
who was, are you cold?
• 26:38 - 26:39
Are you warm?
• 26:39 - 26:44
And he was actually sitting
on with one half of him
• 26:44 - 26:47
on a block of ice and the
other half on the stove,
• 26:47 - 26:49
and he says, in
average, I'm fine.
• 26:49 - 26:52
But he was dying.
• 26:52 - 26:54
This is the same kind of thing.
• 26:54 - 26:58
My average was 60 miles
an hour, but I almost
• 26:58 - 27:02
got caught when I was
driving almost 100.
• 27:02 - 27:06
But nobody knows because I'm
not giving you that information.
• 27:06 - 27:12
That's the infinitesimally small
information that I have not
• 27:12 - 27:17
put correctly here
means that what is
• 27:17 - 27:19
what I see on the speedometer?
• 27:19 - 27:21
It's the instantaneous
rate of change
• 27:21 - 27:24
that I see that
fraction of second.
• 27:24 - 27:31
So that means maybe a few feet
per a fraction of a second.
• 27:31 - 27:34
It means how many
feet did I travel
• 27:34 - 27:36
in that fraction of a second?
• 27:36 - 27:41
And if that fraction of a second
is very tiny that I cannot even
• 27:41 - 27:44
express it properly, that's
what I'm going to have--
• 27:44 - 27:47
df equals f prime dx.
• 27:47 - 27:52
So df and dx have to be small
because their ratio will be
• 27:52 - 27:56
a good number, like 60, like
80, but [? them in ?] themselves
• 27:56 - 27:59
delta m delta [? srv, ?]
very tiny things.
• 27:59 - 28:03
It's the ratio that matters
in the end to be 60, or 80,
• 28:03 - 28:04
or whatever.
• 28:04 - 28:09
So I have 2x dx plus 2y dy.
• 28:09 - 28:11
Never say that the
differential, which
• 28:11 - 28:13
is something
infinitesimally small,
• 28:13 - 28:17
is equal to this scalar
function that it doesn't even
• 28:17 - 28:18
make any sense.
• 28:18 - 28:20
Don't do that because
you get 0 points
• 28:20 - 28:22
and then we argue,
and I don't want
• 28:22 - 28:25
you to get 0 points on
this problem, right.
• 28:25 - 28:27
So it's a very simple problem.
• 28:27 - 28:31
All I want to test you on
would be this definition.
• 28:31 - 28:36
Remember, you're going to
see that again on the midterm
• 28:36 - 28:39
and on the final, or
just on the final.
• 28:39 - 28:42
• 28:42 - 28:42
All right.
• 28:42 - 28:54
So I want to give you the
following homework out
• 28:54 - 29:01
of section 11.4 on
top of the web work.
• 29:01 - 29:07
• 29:07 - 29:17
examples of the section.
• 29:17 - 29:24
• 29:24 - 29:24
OK.
• 29:24 - 29:30
So for example,
somebody tells you
• 29:30 - 29:40
I have to apply this
knowing that I have
• 29:40 - 29:45
an error of measurement of
some sort in the s direction
• 29:45 - 29:48
and an error of measurement of
some sort in the y direction.
• 29:48 - 29:51
There are two or three
examples like that.
• 29:51 - 29:55
They will give you all this
data, including the error
• 29:55 - 29:56
measurement.
• 29:56 - 29:58
For delta, it should be 0.1.
• 29:58 - 30:04
Don't confuse the 0.1 with
dx. dx is not a quantity.
• 30:04 - 30:09
dx is something like
micro cosmic thing.
• 30:09 - 30:14
It's like infinitely
[? small ?].
• 30:14 - 30:15
Infinitesimally small.
• 30:15 - 30:20
So saying that dx should be
0.1 doesn't make any sense,
• 30:20 - 30:23
but delta x being
0.1 make sense.
• 30:23 - 30:26
Delta y being 0.3 makes sense.
• 30:26 - 30:30
plug it in and find
• 30:30 - 30:32
the general difference.
• 30:32 - 30:34
For example, where
could that happen?
• 30:34 - 30:36
And you see examples
in the book.
• 30:36 - 30:41
Somebody measures something--
an area of a rectangle
• 30:41 - 30:43
or a volume of a cube.
• 30:43 - 30:46
But when you measure,
you make mistakes.
• 30:46 - 30:48
You have measurement errors.
• 30:48 - 30:53
In the delta x, you have
an error of plus minus 0.1.
• 30:53 - 31:01
In the y direction, you have
displacement error 0.2 or 0.3,
• 31:01 - 31:02
something like that.
• 31:02 - 31:05
What is the overall
error you are
• 31:05 - 31:08
going to make when you measure
that function of two variables?
• 31:08 - 31:10
That's what you have.
• 31:10 - 31:12
So you plug in all
those displacements
• 31:12 - 31:15
and you come up with the
computational problem.
• 31:15 - 31:20
Several of you Wednesday we
• 31:20 - 31:25
solved those problems through
web work and came to me,
• 31:25 - 31:28
and I said, how did you know
to plug in those [? numbers ?]?
• 31:28 - 31:29
Well, it's not so hard.
• 31:29 - 31:30
It's sort of common sense.
• 31:30 - 31:33
Plus, I looked in the book
and that gave me the idea
• 31:33 - 31:35
to remind you to
look in the book
• 31:35 - 31:37
for those numerical examples.
• 31:37 - 31:40
You will have to
• 31:40 - 31:43
So you don't have it with
you, you generally, we
• 31:43 - 31:45
don't use in the classroom,
but it's very easy.
• 31:45 - 31:48
All you have to do is use the
calculator and [INAUDIBLE]
• 31:48 - 31:51
examples and see how it goes.
• 31:51 - 31:57
I wanted to show you
something more interesting
• 31:57 - 32:09
even, more beautiful
regarding something
• 32:09 - 32:13
we don't show in the
book until later on,
• 32:13 - 32:18
and I'm uncomfortable with the
idea of not showing this to you
• 32:18 - 32:20
now.
• 32:20 - 32:27
An alternate way, or
• 32:27 - 32:38
define the tangent plane--
• 32:38 - 32:49
the tangent plane-- to a
surface S at the point p.
• 32:49 - 32:52
And I'll draw again.
• 32:52 - 32:56
Half of my job is drawing
in this class, which I like.
• 32:56 - 33:00
I mean, I was having an argument
with one of my colleagues who
• 33:00 - 33:03
said, I hate when they are
giving me to teach calculus 3
• 33:03 - 33:08
because I cannot draw.
• 33:08 - 33:10
I think that the
most beautiful part
• 33:10 - 33:15
is that we can represent
things visually,
• 33:15 - 33:20
and this is just pi, the
tangent plane I'm after,
• 33:20 - 33:25
and p will be a
coordinate 0 by 0, z0.
• 33:25 - 33:27
And what was the label?
• 33:27 - 33:28
Oh, the label.
• 33:28 - 33:28
The label.
• 33:28 - 33:34
The label was internal
where z equals f of xy.
• 33:34 - 33:40
But more generally, I'll say
this time plus more generally,
• 33:40 - 33:59
what if you have f of xyz
equals c for that surface.
• 33:59 - 34:01
Let's call it [INAUDIBLE].
• 34:01 - 34:05
F of xy is [INAUDIBLE].
• 34:05 - 34:08
And somebody even said, can
you have a parametrization?
• 34:08 - 34:10
And this is where
I wanted to go.
• 34:10 - 34:14
• 34:14 - 34:16
Ryan was the first
• 34:16 - 34:19
but then there were
three more of you
• 34:19 - 34:21
who have restless
minds plus you--
• 34:21 - 34:26
because that's the essence
of being active here.
• 34:26 - 34:30
We don't lose our connections.
• 34:30 - 34:34
We lose neurons anyway, but
we don't lose our connections
• 34:34 - 34:38
if we think, and
anticipate things,
• 34:38 - 34:40
and try to relate concepts.
• 34:40 - 34:43
So if you don't want to
get Alzheimer's, just
• 34:43 - 34:46
• 34:46 - 34:50
So can I have a
parametrization for a surface?
• 34:50 - 34:52
All righty, what do you mean?
• 34:52 - 34:58
What if somebody says for a
curve, we have r of t, right,
• 34:58 - 34:59
which was what?
• 34:59 - 35:06
It was x of ti plus y of tj plus
z of tk, and we were so happy
• 35:06 - 35:10
and we were happy
because we were traveling
• 35:10 - 35:12
in time with respect
to the origin,
• 35:12 - 35:16
and this was r of t at time t.
• 35:16 - 35:18
[INAUDIBLE]
• 35:18 - 35:20
me, [INAUDIBLE],
• 35:20 - 35:27
can you have such a position
vector moving on a surface?
• 35:27 - 35:30
Like look, it's a rigid motion.
• 35:30 - 35:33
If you went to the
robotics science
• 35:33 - 35:36
fair, Texas Tech, or something
like that, you know about that.
• 35:36 - 35:37
Yeah, cities.
• 35:37 - 35:40
So how do we introduce
such a parametrization?
• 35:40 - 35:44
We have an origin of course.
• 35:44 - 35:46
An origin is always important.
• 35:46 - 35:48
Everybody has an origin.
• 35:48 - 35:53
• 35:53 - 35:58
And I take that position
vector, and where does it start?
• 35:58 - 36:02
It starts at the origin, and
the tip of it is on the surface,
• 36:02 - 36:05
And it's gliding on the
surface, the tip of it.
• 36:05 - 36:10
And that's going to be r, but
it's not going to be r of t.
• 36:10 - 36:13
It's going to be r of
longitude and latitude.
• 36:13 - 36:16
Like imagine, that would
• 36:16 - 36:18
from the center of the earth.
• 36:18 - 36:21
And it depends on
two parameters.
• 36:21 - 36:25
One of them would be latitude.
• 36:25 - 36:26
Am I drawing this right?
• 36:26 - 36:27
Latitude--
• 36:27 - 36:29
STUDENT: [INAUDIBLE] longitude.
• 36:29 - 36:31
PROFESSOR TODA:
--from a latitude 0.
• 36:31 - 36:32
I'm at the equator.
• 36:32 - 36:34
Then latitude 90 degrees.
• 36:34 - 36:36
I'm at the North Pole.
• 36:36 - 36:38
In mathematics, we are funny.
• 36:38 - 36:41
We say latitude 0,
latitude 90 North Pole,
• 36:41 - 36:45
latitude negative 90,
which is South Pole.
• 36:45 - 36:49
And longitude from 0 to 2 pi.
• 36:49 - 36:54
Meridian 0 to all around.
• 36:54 - 36:58
So r will be not a function of
t but a function of u and b,
• 36:58 - 37:02
thank god, because u and b
are the latitude and longitude
• 37:02 - 37:03
sort of.
• 37:03 - 37:12
So we have x of uv i plus
y of uv j plus z of uv k.
• 37:12 - 37:21
• 37:21 - 37:23
You can do that.
• 37:23 - 37:26
And you say, but can you give
us an example, because this
• 37:26 - 37:28
looks so abstract for god sake.
• 37:28 - 37:32
If you give me the graph
the way you gave it to me
• 37:32 - 37:37
before z equals f of xy,
• 37:37 - 37:42
• 37:42 - 37:45
Parametrize it for
me because I'm lost.
• 37:45 - 37:46
You are not lost.
• 37:46 - 37:48
We can do this together.
• 37:48 - 37:51
Now what's the simplest
way to parametrize
• 37:51 - 37:57
a graph of the type
z equals f of xy?
• 37:57 - 38:02
Take the xy to be
u and v. Take x
• 38:02 - 38:05
and y to be your
independent variables
• 38:05 - 38:08
and take z to be the
dependent variable.
• 38:08 - 38:13
• 38:13 - 38:17
I'm again expressing these
things in terms of variables
• 38:17 - 38:18
like I did last time.
• 38:18 - 38:23
Then I say, let's take this kind
of parametrization. [INAUDIBLE]
• 38:23 - 38:24
vu, right.
• 38:24 - 38:33
y would be v. Then I'm
going to write r of x and y
• 38:33 - 38:37
just like that guy will
be [INAUDIBLE] of xn.
• 38:37 - 38:39
[? y ?] will say, wait a minute.
• 38:39 - 38:43
I will have to re-denote
everybody with capitals.
• 38:43 - 38:46
Then my life will become
better because you
• 38:46 - 38:47
don't have to erase.
• 38:47 - 38:51
You just make little
x big, little y bigs,
• 38:51 - 38:54
bigs, big, capitalized XYZ.
• 38:54 - 39:02
And then I'll say OK, XYZ
will be my setting here in 3D.
• 39:02 - 39:07
• 39:07 - 39:08
All right.
• 39:08 - 39:10
So how am I going
to re-parametrize
• 39:10 - 39:13
the whole surface?
• 39:13 - 39:22
Whole surface will be r of
xy equals in this case, well,
• 39:22 - 39:23
• 39:23 - 39:29
In this case, I'm
going to have xy.
• 39:29 - 39:31
And where's the little f?
• 39:31 - 39:33
I just erased it.
• 39:33 - 39:35
I was smart, right,
that I erased f of xy.
• 39:35 - 39:38
• 39:38 - 39:46
So I have x, y, and
z, which is f of xy.
• 39:46 - 39:53
• 39:53 - 40:01
And this is the generic point
p of coordinates xy f of xy.
• 40:01 - 40:05
• 40:05 - 40:08
So I say, OK, what does it mean?
• 40:08 - 40:10
I will project this point.
• 40:10 - 40:13
And this is the point
when big x becomes little
• 40:13 - 40:18
x, when big y becomes--
where is my y-axis?
• 40:18 - 40:20
Somebody ate my y axis.
• 40:20 - 40:22
[INAUDIBLE]
• 40:22 - 40:28
So when big Y becomes
little y, little y
• 40:28 - 40:34
is just an instance of big Y.
And big Z will take what value?
• 40:34 - 40:36
Well, I need to project that.
• 40:36 - 40:39
How do you project from
a point to the z-axis?
• 40:39 - 40:43
You have to take the
parallel from the point
• 40:43 - 40:48
to the horizontal
plane until you
• 40:48 - 40:53
hit the-- [INAUDIBLE] the whole
plane parallel to the floor
• 40:53 - 40:54
through the point p.
• 40:54 - 40:55
And what do I get here?
• 40:55 - 40:56
STUDENT: [INAUDIBLE].
• 40:56 - 40:59
PROFESSOR TODA: Not
z0, but it's little z
• 40:59 - 41:03
equals f of xy, which is an
instance of the variable xz.
• 41:03 - 41:06
For you programmers, you know
that big z will be a variable
• 41:06 - 41:12
and little z will be
[INAUDIBLE] a variable.
• 41:12 - 41:12
OK.
• 41:12 - 41:17
So I parametrized my graph
in a more general way,
• 41:17 - 41:19
general parametrization
for a graph.
• 41:19 - 41:26
• 41:26 - 41:33
And now, what are-- what's the
meaning of r sub x and r sub y?
• 41:33 - 41:34
What are they?
• 41:34 - 41:35
STUDENT: [INAUDIBLE].
• 41:35 - 41:38
• 41:38 - 41:42
PROFESSOR TODA: Now, we
don't say that in the book.
• 41:42 - 41:43
Shame on us.
• 41:43 - 41:44
Shame on us.
• 41:44 - 41:47
We should have because I was
browsing through the projects
• 41:47 - 41:50
about a year and a half ago.
• 41:50 - 41:53
The senior projects of
a few of my students
• 41:53 - 41:56
who are-- two of them were
in mechanical engineering.
• 41:56 - 42:01
One of them was in
petroleum engineering.
• 42:01 - 42:04
And he actually showed me
that they were doing this.
• 42:04 - 42:08
They were taking vectors
that depend on parameters--
• 42:08 - 42:11
this is a vector [INAUDIBLE]--
and differentiated them with
• 42:11 - 42:14
respect to those parameters.
• 42:14 - 42:17
And I was thinking OK, did we
do the partial derivatives r sub
• 42:17 - 42:18
x, r sub y?
• 42:18 - 42:19
Not so much.
• 42:19 - 42:22
But now I want to do it
because I think that prepares
• 42:22 - 42:25
you better as engineers.
• 42:25 - 42:29
So what is r sub x
and what is r sub y?
• 42:29 - 42:31
And you say, well,
OK. [INAUDIBLE],
• 42:31 - 42:35
I think I know how to do
that in my sleep, right.
• 42:35 - 42:37
If you want me to do
that theoretically
• 42:37 - 42:40
from this formula,
but on the picture,
• 42:40 - 42:42
I really don't know what it is.
• 42:42 - 42:46
I'm going to have in terms
• 42:46 - 42:47
of r sub x and r sub y.
• 42:47 - 42:49
They will be vectors.
• 42:49 - 42:52
This should be a
vector as well, right.
• 42:52 - 42:57
And for me, vector triple
means the identification
• 42:57 - 43:00
between the three coordinates
and the physical vector.
• 43:00 - 43:02
So this is the physical vector.
• 43:02 - 43:06
Go ahead and write x prime
with respect to x is 1.
• 43:06 - 43:09
• 43:09 - 43:14
y prime with respect to x is 0.
• 43:14 - 43:16
The third [INAUDIBLE]
prime with respect
• 43:16 - 43:20
to x is just whatever
this little f is,
• 43:20 - 43:22
it's not any of my business.
• 43:22 - 43:25
It's a [INAUDIBLE]
function f sub x.
• 43:25 - 43:28
• 43:28 - 43:31
Well, what is the second vector?
• 43:31 - 43:32
STUDENT: 0, 1, f sub y.
• 43:32 - 43:35
PROFESSOR TODA: 0, 1, f sub y.
• 43:35 - 43:37
Now, are they slopes?
• 43:37 - 43:37
No.
• 43:37 - 43:38
These are slopes.
• 43:38 - 43:41
That's a slope and
that's a slope.
• 43:41 - 43:45
And we learned
• 43:45 - 43:50
and we understood that those
are ski slopes, they were.
• 43:50 - 43:52
In the direction of x
and the direction of y,
• 43:52 - 44:00
the slopes of the tangents
to the coordinate lines.
• 44:00 - 44:05
But this looks like I have
a direction of a line,
• 44:05 - 44:09
and this would be the lope, and
that's the direction of a line,
• 44:09 - 44:10
and that would be the slope.
• 44:10 - 44:13
What are those lines?
• 44:13 - 44:16
STUDENT: [INAUDIBLE] to
the function [INAUDIBLE].
• 44:16 - 44:17
PROFESSOR TODA: Let me draw.
• 44:17 - 44:19
Then shall I erase
the whole thing?
• 44:19 - 44:20
No.
• 44:20 - 44:24
I'm just going to keep--
I'll erase the tangent.
• 44:24 - 44:27
Don't erase anything
• 44:27 - 44:29
So this is the point p.
• 44:29 - 44:30
It's still there.
• 44:30 - 44:31
This is the surface.
• 44:31 - 44:33
It's still there.
• 44:33 - 44:38
So my surface will be x,
slices of x, [? S ?] constant
• 44:38 - 44:40
are coming towards you.
• 44:40 - 44:46
They are these [? walls ?]
like that, like this, yes.
• 44:46 - 44:48
It's like the CT scan.
• 44:48 - 44:52
I think that when they
• 44:52 - 44:54
tch tch tch tch tch
tch, take pictures
• 44:54 - 44:58
of the slices of your body,
that's the same kind of thing.
• 44:58 - 45:00
So x0, x0, x0, x0.
• 45:00 - 45:05
I'm going to [INAUDIBLE]
planes and I had x equals x0.
• 45:05 - 45:12
And in the other direction, I
cut and I get, what do I get?
• 45:12 - 45:18
• 45:18 - 45:20
• 45:20 - 45:24
• 45:24 - 45:25
Great, Magdalena, this is--
• 45:25 - 45:27
What is this pink?
• 45:27 - 45:32
It's not Valentine's Day
anymore. y equals [INAUDIBLE].
• 45:32 - 45:35
And this is the point.
• 45:35 - 45:39
So, as Alex was
trying to tell you,
• 45:39 - 45:45
our sub x would represent the
vector, the physical vector
• 45:45 - 45:52
in 3D, that is originating
at p and tangent to which
• 45:52 - 45:56
of the two, to the purple
one or to the red one?
• 45:56 - 45:57
STUDENT: Red.
• 45:57 - 45:58
Uh, purple.
• 45:58 - 46:00
PROFESSOR TODA:
• 46:00 - 46:01
STUDENT: The purple one.
• 46:01 - 46:04
PROFESSOR TODA: [INAUDIBLE]
constant and [INAUDIBLE]
• 46:04 - 46:07
constant in the red
one, y equals y0, right?
• 46:07 - 46:09
So, this depends on x.
• 46:09 - 46:11
So this has r sub x.
• 46:11 - 46:15
• 46:15 - 46:19
This is the velocity with
respect to the variable x.
• 46:19 - 46:23
And the other one, the
blue one, x equals x0,
• 46:23 - 46:28
means x0 is held fixed
and y is the variable.
• 46:28 - 46:31
So I have to do r sub y,
and what am I gonna get?
• 46:31 - 46:33
I'm gonna get the blue vector.
• 46:33 - 46:35
What's the property
of the blue vector?
• 46:35 - 46:38
It's tangent to the purple line.
• 46:38 - 46:44
So r sub y has to be
tangent to the curve.
• 46:44 - 46:47
• 46:47 - 46:55
x0, y, f of x0 and
y is the curve.
• 46:55 - 47:00
And r sub x is tangent
to which curve?
• 47:00 - 47:02
Who is telling me which curve?
• 47:02 - 47:12
x, y0 sub constant,
f of x and y0.
• 47:12 - 47:14
So that's a curve that
depends only on y,
• 47:14 - 47:17
y is the time in this case.
• 47:17 - 47:19
And that's the curve
that depends only on x.
• 47:19 - 47:21
x is the time in this case.
• 47:21 - 47:25
r sub x and r sub y are
the tangent vectors.
• 47:25 - 47:27
• 47:27 - 47:31
If I shape this
triangle between them,
• 47:31 - 47:32
that will be the tangent plane.
• 47:32 - 47:36
• 47:36 - 47:39
And I make a smile because I
discovered the tangent plane
• 47:39 - 47:43
in a different way than
we did it last time.
• 47:43 - 47:51
So the tangent plane represents
the plane of the vector r sub
• 47:51 - 47:55
x and r sub y.
• 47:55 - 48:02
The tangent plane
represents the plane
• 48:02 - 48:13
given by vectors r sub x and
r sub y with what conditions?
• 48:13 - 48:14
It's a conditional.
• 48:14 - 48:17
• 48:17 - 48:21
r sub x and r sub
y shouldn't be 0.
• 48:21 - 48:25
r sub x different from 0,
r sub y different from 0,
• 48:25 - 48:27
and r sub x and r sub
y are not collinear.
• 48:27 - 48:32
• 48:32 - 48:35
What's gonna happen
if they are collinear?
• 48:35 - 48:37
Well, they're gonna
collapse; they are not
• 48:37 - 48:38
gonna determine a plane.
• 48:38 - 48:41
So there will be
no tangent planes.
• 48:41 - 48:44
So they have to be
linearly independent.
• 48:44 - 48:48
For the people who are taking
now linear algebra, I'm saying.
• 48:48 - 48:51
So we have no other
choice, we have
• 48:51 - 48:55
to assume that these vectors,
called partial velocities,
• 48:55 - 49:04
by the way, for the
motion across the surface.
• 49:04 - 49:05
OK?
• 49:05 - 49:07
These are the partial
velocities, or partial velocity
• 49:07 - 49:09
vectors.
• 49:09 - 49:13
Partial velocity vectors
have to determine a plane,
• 49:13 - 49:17
so I have to assume
that they are non-zero,
• 49:17 - 49:20
they never become 0, and
they are not collinear.
• 49:20 - 49:23
If they are collinear,
life is over for you.
• 49:23 - 49:24
OK?
• 49:24 - 49:29
So I have to assume that I
throw away all the points where
• 49:29 - 49:35
the velocities become 0, and
all the points where--those are
• 49:35 - 49:40
singularity points--where
my velocity vectors are 0.
• 49:40 - 49:44
• 49:44 - 49:46
Have you ever studied design?
• 49:46 - 49:47
Any kind of experimental design.
• 49:47 - 49:52
Like, how do you design a car,
the coordinate lines on a car?
• 49:52 - 49:53
I'm just dreaming.
• 49:53 - 50:00
You have a car, a beautiful
car, and then you have-- Well,
• 50:00 - 50:05
I cannot draw really
well, but anyway.
• 50:05 - 50:09
I have these coordinate
lines on this car.
• 50:09 - 50:12
It's a mesh what I have there.
• 50:12 - 50:16
Actually, we do that in
animation all the time.
• 50:16 - 50:21
We have meshes over the
models we have in animation.
• 50:21 - 50:23
Think Avatar.
• 50:23 - 50:27
Now, those are all
coordinate lines.
• 50:27 - 50:34
Those coordinate lines would be,
• 50:34 - 50:39
For example, if you take a body
in a mesh like that, in a net,
• 50:39 - 50:43
in, like, a fishnet, then
you pull from the fishnet,
• 50:43 - 50:53
all the coordinate lines
will come together,
• 50:53 - 50:55
and this would be a singularity.
• 50:55 - 50:58
We avoid this kind
of singularity.
• 50:58 - 51:00
So these are points where
• 51:00 - 51:05
Either the velocity
vectors become collinear.
• 51:05 - 51:07
You see what I'm talking about?
• 51:07 - 51:11
Or the velocity
vectors shrank to 0.
• 51:11 - 51:14
that's a singularity point.
• 51:14 - 51:17
They have this
problem when meshing.
• 51:17 - 51:21
So when they make
these models that
• 51:21 - 51:27
involve two-dimensional meshing
and three-dimensional ambient
• 51:27 - 51:31
space, like it is in
animation, the mesh
• 51:31 - 51:35
is called regular
if we don't have
• 51:35 - 51:40
this kind of singularity, where
the velocity vectors become 0,
• 51:40 - 51:42
or collinear.
• 51:42 - 51:46
It's very important for a
person who programs in animation
• 51:46 - 51:47
to know mathematics.
• 51:47 - 51:50
If they don't understand
these things, it's over.
• 51:50 - 51:56
Because you write the matrix,
and you will know the vectors
• 51:56 - 52:00
will become collinear when the
two vectors--let's say two rows
• 52:00 - 52:00
of a matrix--
• 52:00 - 52:01
STUDENT: Parallel.
• 52:01 - 52:02
PROFESSOR TODA:
Are proportional.
• 52:02 - 52:03
Or parallel.
• 52:03 - 52:04
Or proportional.
• 52:04 - 52:08
So, everything is numerical
in terms of those matrices,
• 52:08 - 52:13
but it's just a discretization
of a continuous phenomenon,
• 52:13 - 52:14
which is this one.
• 52:14 - 52:18
• 52:18 - 52:20
Do you remember Toy Story?
• 52:20 - 52:21
OK.
• 52:21 - 52:24
The Toy Story people,
the renderers,
• 52:24 - 52:27
the ones who did the rendering
techniques for Toy Story,
• 52:27 - 52:30
both have their
master's in mathematics.
• 52:30 - 52:34
And you realize why
now to do that you
• 52:34 - 52:39
have to know calc I, calc
II, calc III, linear algebra,
• 52:39 - 52:41
be able to deal with matrices.
• 52:41 - 52:46
Have a programming course
or two; that's essential.
• 52:46 - 52:50
because some people
• 52:50 - 52:55
don't cover thi-- I was about to
skip it right now in calc III.
• 52:55 - 53:00
But they teach that in
• 53:00 - 53:03
far as you can get,
• 53:03 - 53:06
and differential equation's
also very important.
• 53:06 - 53:10
So, if you master those and
you go into something else,
• 53:10 - 53:12
like programming,
electrical engineering,
• 53:12 - 53:14
• 53:14 - 53:17
[INAUDIBLE] If you went
I want to be a rendering
• 53:17 - 53:20
guy for the next movie,
then they'll say no,
• 53:20 - 53:22
we won't take you.
• 53:22 - 53:24
I have a friend who
works for Disney.
• 53:24 - 53:27
She wanted to get a PhD.
• 53:27 - 53:29
At some point, she
changed her mind
• 53:29 - 53:32
and ended up just with a
master's in mathematics
• 53:32 - 53:34
while I was in Kansas,
University of Kansas,
• 53:34 - 53:37
and she said, "You know what?
• 53:37 - 53:42
Disney's just giving me
\$65,000 as an intern."
• 53:42 - 53:46
And I was like OK and probably
• 53:46 - 53:47
a postdoc.
• 53:47 - 53:48
And she said,
"Good luck to you."
• 53:48 - 53:49
Good luck to you, too.
• 53:49 - 53:53
But we stayed in touch,
and right now she's
• 53:53 - 53:57
making twice as much as
I'm making, for Disney.
• 53:57 - 53:59
Is she happy?
• 53:59 - 54:00
Yeah.
• 54:00 - 54:00
Would I be happy?
• 54:00 - 54:01
No.
• 54:01 - 54:06
Because she works
for 11 hours a day.
• 54:06 - 54:08
11 hours a day, on a chair.
• 54:08 - 54:09
That would kill me.
• 54:09 - 54:15
I mean, I spend about six hours
sitting on a chair every day
• 54:15 - 54:19
of the week, but
it's still too much.
• 54:19 - 54:21
She's a hard worker, though.
• 54:21 - 54:23
She loves what she's doing.
• 54:23 - 54:24
• 54:24 - 54:27
After a while, your
• 54:27 - 54:34
So, what is the normal for
the plane in this case?
• 54:34 - 54:37
I'll try my best
ability to draw normal.
• 54:37 - 54:39
The normal has to
be perpendicular
• 54:39 - 54:42
to the tangent space, right?
• 54:42 - 54:44
Tangent plane.
• 54:44 - 54:46
So, n has to be
perpendicular to our sub
• 54:46 - 54:50
x and has to be
perpendicular to our sub y.
• 54:50 - 54:53
• 54:53 - 54:56
So, can you have any
guess how in the world
• 54:56 - 54:59
I'm gonna get n vector?
• 54:59 - 55:01
STUDENT: [INAUDIBLE]
• 55:01 - 55:03
PROFESSOR TODA:
[INAUDIBLE] That's
• 55:03 - 55:05
why you need to
know linear algebra
• 55:05 - 55:09
sort of at the same time, but
you guys are making it fine.
• 55:09 - 55:10
It's not a big deal.
• 55:10 - 55:16
You have a matrix, i, j, k
in the front row vectors,
• 55:16 - 55:22
and then you have r sub x that
you gave me, and I erased it.
• 55:22 - 55:24
1, 0, f sub x.
• 55:24 - 55:27
• 55:27 - 55:29
0, 1, f sub y.
• 55:29 - 55:41
And you have exactly 18
seconds to compute this vector.
• 55:41 - 55:48
• 55:48 - 55:48
STUDENT: [INAUDIBLE]
• 55:48 - 55:53
• 55:53 - 55:56
PROFESSOR TODA: You want k, but
I want to leave k at the end
• 55:56 - 55:59
because I always
order my vectors.
• 55:59 - 56:02
Something i plus something
j plus something k.
• 56:02 - 56:03
[INTERPOSING VOICES]
• 56:03 - 56:05
• 56:05 - 56:06
PROFESSOR TODA: Am I right?
• 56:06 - 56:07
Minus f sub x--
• 56:07 - 56:10
STUDENT: Minus f of x plus k.
• 56:10 - 56:12
PROFESSOR TODA: --times i.
• 56:12 - 56:14
For j, do I have to change sign?
• 56:14 - 56:18
Yeah, because 1 plus 2 is odd.
• 56:18 - 56:21
So I go minus 1.
• 56:21 - 56:23
And do it slowly.
• 56:23 - 56:26
You're not gonna make fun of
me; I gotta make fun of you, OK?
• 56:26 - 56:28
And minus 1 times--
• 56:28 - 56:29
STUDENT: Did you forget f y?
• 56:29 - 56:37
PROFESSOR TODA: --f sub y--I go
like that--sub y times j plus
• 56:37 - 56:39
k.
• 56:39 - 56:42
As you said very well
in the most elegant way
• 56:42 - 56:46
without being like yours,
but I say it like this.
• 56:46 - 56:50
So you have minus f
sub x, minus f sub y,
• 56:50 - 56:55
and 1 as a triple with angular
brackets--You love that.
• 56:55 - 57:00
I don't; I like it parentheses
[INAUDIBLE]--equals n.
• 57:00 - 57:03
But n is non-unitary,
but I don't care.
• 57:03 - 57:05
Why don't I care?
• 57:05 - 57:08
I can write the
tangent plane very well
• 57:08 - 57:13
without that n being
unitary, right?
• 57:13 - 57:15
It doesn't matter in the end.
• 57:15 - 57:18
These would be my a, b, c.
• 57:18 - 57:19
Now I know my ABC.
• 57:19 - 57:20
I know my ABC.
• 57:20 - 57:26
So, the tangent plane
• 57:26 - 57:30
The tangent plane would
be perpendicular to n.
• 57:30 - 57:32
So this is n.
• 57:32 - 57:36
The tangent plane passes
through the point p
• 57:36 - 57:37
and is perpendicular to n.
• 57:37 - 57:43
So, what is the equation
of the tangent plane?
• 57:43 - 57:45
STUDENT: Do you want
scalar equations?
• 57:45 - 57:49
PROFESSOR TODA: A by x minus 0.
• 57:49 - 57:50
Very good.
• 57:50 - 57:56
That's exactly what I
wanted you to write.
• 57:56 - 58:01
All right, so, does
it look familiar?
• 58:01 - 58:02
Not yet.
• 58:02 - 58:02
[STUDENT SNEEZES]
• 58:02 - 58:03
STUDENT: Bless you.
• 58:03 - 58:04
STUDENT: Bless you.
• 58:04 - 58:05
PROFESSOR TODA: Bless you.
• 58:05 - 58:06
Who sneezed?
• 58:06 - 58:09
OK.
• 58:09 - 58:10
Am I almost done?
• 58:10 - 58:12
Well, I am almost done.
• 58:12 - 58:15
I have to go backwards,
and whatever I get
• 58:15 - 58:18
I'll put it big here in
a big formula on top.
• 58:18 - 58:22
I'm gonna say oh, my God.
• 58:22 - 58:24
No, that's not
what I'm gonna say.
• 58:24 - 58:33
I'm gonna say minus f sub x at
my point p--that is a, right?
• 58:33 - 58:37
Times x minus x0.
• 58:37 - 58:46
Plus minus f sub y at
the point p; that's b.
• 58:46 - 58:55
y minus y0 plus--c is 1, right?
• 58:55 - 58:55
c is 1.
• 58:55 - 58:58
I'm not gonna write
it because if I write
• 58:58 - 59:04
it you'll want to make fun
of me. z minus z0 equals 0.
• 59:04 - 59:09
Now it starts looking like
something familiar, finally.
• 59:09 - 59:15
Now we discovered
that the tangent plane
• 59:15 - 59:21
can be written as z minus z0.
• 59:21 - 59:25
I'm keeping the guys z minus
z0 on the left-hand side.
• 59:25 - 59:29
And these guys are gonna
move to the right-hand side.
• 59:29 - 59:34
So, I'm gonna have
again, my friend,
• 59:34 - 59:45
the equation of the tangent
plane for the graph z equals f
• 59:45 - 59:46
of x,y.
• 59:46 - 59:52
• 59:52 - 59:55
But you will say
OK, I think by now
• 59:55 - 59:57
we've learned these
by heart, we know
• 59:57 - 60:00
the equation of the tangent
plane, and now we're asleep.
• 60:00 - 60:06
would be implicit the way
• 60:06 - 60:09
you gave it to us at first.
• 60:09 - 60:12
Maybe you remember the sphere
that was an implicit equation,
• 60:12 - 60:15
x squared plus x squared
plus x squared equals--
• 60:15 - 60:16
What do you want it to be?
• 60:16 - 60:17
STUDENT: 16.
• 60:17 - 60:18
PROFESSOR TODA: Huh?
• 60:18 - 60:19
STUDENT: 16.
• 60:19 - 60:21
PROFESSOR TODA: 16.
• 60:21 - 60:22
• 60:22 - 60:27
• 60:27 - 60:31
And in such a case, the equation
is of the type f of x, y, z
• 60:31 - 60:33
equals constant.
• 60:33 - 60:36
Can we write again the
equation [INAUDIBLE]?
• 60:36 - 60:40
• 60:40 - 60:42
Well, you say well,
you just taught
• 60:42 - 60:51
us some theory that says I have
to think of u and v, but not x
• 60:51 - 60:52
and y.
• 60:52 - 60:55
Because if I think of x
and y, what would they be?
• 60:55 - 60:58
I think the sphere
as being an apple.
• 60:58 - 61:02
Not an apple, something
you can cut easily.
• 61:02 - 61:05
Well, an apple, an
orange, something.
• 61:05 - 61:07
A round piece of soft cheese.
• 61:07 - 61:10
I started being hungry,
and I'm dreaming.
• 61:10 - 61:14
So, this is a huge something
you're gonna slice up.
• 61:14 - 61:19
If you are gonna
do it with x and y,
• 61:19 - 61:22
the slices would be like this.
• 61:22 - 61:25
Like that and like this, right?
• 61:25 - 61:27
And in that case,
• 61:27 - 61:31
are sort of weird.
• 61:31 - 61:34
If you want to do it in
different coordinates,
• 61:34 - 61:35
so we want to
change coordinates,
• 61:35 - 61:40
and those coordinates should
be plotted to the longitude,
• 61:40 - 61:44
then we cannot use x and y.
• 61:44 - 61:45
Am I right?
• 61:45 - 61:47
We cannot use x and y.
• 61:47 - 61:51
So those u and v will be
different coordinates,
• 61:51 - 61:55
and then we can do it
like that, latitude.
• 61:55 - 61:58
• 61:58 - 62:00
[INAUDIBLE] minus [INAUDIBLE].
• 62:00 - 62:01
And longitude.
• 62:01 - 62:03
spherical coordinates
• 62:03 - 62:05
later, not today.
• 62:05 - 62:06
Latitude and longitude.
• 62:06 - 62:10
• 62:10 - 62:13
1 point extra credit,
because eventually we
• 62:13 - 62:17
are gonna get
there, chapter 12.7.
• 62:17 - 62:21
12.7 comes way
after spring break.
• 62:21 - 62:27
But before we get there, who
is in mechanical engineering
• 62:27 - 62:29
again?
• 62:29 - 62:33
angles, and stuff like that.
• 62:33 - 62:34
OK.
• 62:34 - 62:40
Can you write me
the equations of x
• 62:40 - 62:48
and y and z of the sphere
with respect to u and v,
• 62:48 - 62:51
u being latitude and
v being longitude?
• 62:51 - 62:54
• 62:54 - 62:59
These have to be
trigonometric functions.
• 62:59 - 63:04
• 63:04 - 63:11
In terms of u and v, when u is
latitude and v is longitude.
• 63:11 - 63:15
1 point extra credit
until a week from today.
• 63:15 - 63:16
• 63:16 - 63:21
• 63:21 - 63:24
U and v are latitude
and longitude.
• 63:24 - 63:34
And express the xyz point in
the ambient space on the sphere.
• 63:34 - 63:36
x squared plus x squared
plus x squared would be 16.
• 63:36 - 63:40
So you'll have lots of
cosines and sines [INAUDIBLE]
• 63:40 - 63:46
of those angles, the latitude
angle and the longitude angle.
• 63:46 - 63:50
And I would suggest to you that
you take--for the extra credit
• 63:50 - 63:55
thing--you take the longitude
angle to be from 0 to 2pi,
• 63:55 - 64:00
from the Greenwich 0 meridian
going back to himself,
• 64:00 - 64:08
and--well, there are two ways
we do this in mathematics
• 64:08 - 64:10
because mathematicians
are so diverse.
• 64:10 - 64:15
Some of us, say, for me,
I measure the latitude
• 64:15 - 64:17
starting from the North Pole.
• 64:17 - 64:20
I think that's because we all
believe in Santa or something.
• 64:20 - 64:23
So, we start measuring
always from the North Pole
• 64:23 - 64:27
because that's the most
important place on Earth.
• 64:27 - 64:36
They go 0, pi over 2, and then--
what is our lat--shame on me.
• 64:36 - 64:36
STUDENT: It's 33.
• 64:36 - 64:37
PROFESSOR TODA: 33?
• 64:37 - 64:39
OK.
• 64:39 - 64:44
Then pi would be the
equator, and then pi
• 64:44 - 64:46
would be the South Pole.
• 64:46 - 64:51
But some other mathematicians,
especially biologists
• 64:51 - 64:55
and differential geometry
people, I'm one of them,
• 64:55 - 64:56
we go like that.
• 64:56 - 65:02
Minus pi over 2, South Pole
0, pi over 2 North Pole.
• 65:02 - 65:07
So we shift that
kind of interval.
• 65:07 - 65:10
Then for us, the trigonometric
functions of these angles
• 65:10 - 65:12
would be a little
bit different when we
• 65:12 - 65:14
do the spherical coordinates.
• 65:14 - 65:16
OK, that's just extra credit.
• 65:16 - 65:19
It has nothing to do with
what I'm gonna do right now.
• 65:19 - 65:23
What I'm gonna do right now
is to pick a point on Earth.
• 65:23 - 65:26
We have to find Lubbock.
• 65:26 - 65:27
STUDENT: It's on the left.
• 65:27 - 65:29
PROFESSOR TODA: Here?
• 65:29 - 65:30
Is that a good point?
• 65:30 - 65:32
• 65:32 - 65:34
This is LBB.
• 65:34 - 65:38
That's Lubbock
International Airport.
• 65:38 - 65:48
So, for Lubbock--let's call it
p as well--draw the r sub u,
• 65:48 - 65:53
r sub v. So, u was latitude.
• 65:53 - 65:56
So if I fix the latitude,
that means I fix
• 65:56 - 65:59
the 33 point whatever you said.
• 65:59 - 66:00
u equals u0.
• 66:00 - 66:10
It is fixed, so I have u
fixed, and v equals v0 is that.
• 66:10 - 66:14
I fixed the meridian
where we are.
• 66:14 - 66:16
What is this tangent vector?
• 66:16 - 66:21
• 66:21 - 66:23
To the pink parallel,
the tangent vector
• 66:23 - 66:26
would be r sub what?
• 66:26 - 66:26
STUDENT: v.
• 66:26 - 66:28
PROFESSOR TODA: r
sub v. You are right.
• 66:28 - 66:29
You've got the idea.
• 66:29 - 66:33
And the blue vector would
be the partial velocity.
• 66:33 - 66:39
That's the tangent vector
to the blue meridian,
• 66:39 - 66:44
which is r sub u.
• 66:44 - 66:49
And what is n gonna be? n's
gonna be r sub u [INAUDIBLE].
• 66:49 - 66:53
But is there any other way
to do it in a simpler way
• 66:53 - 66:56
without you guys going oh, man.
• 66:56 - 66:58
Suppose some of you don't
wanna do the extra credit
• 66:58 - 67:00
and then say the
heck with it; I don't
• 67:00 - 67:04
credit until chapter 12,
• 67:04 - 67:08
when I have to study the
spherical coordinates,
• 67:08 - 67:11
and is there another
way to get n.
• 67:11 - 67:13
I told you another way to get n.
• 67:13 - 67:15
Well, we are getting there.
• 67:15 - 67:22
n was the gradient of f
over the length of that.
• 67:22 - 67:26
And if we want it unitary,
the length of f was what?
• 67:26 - 67:32
f sub x, f sub y, f
sub z vector, where
• 67:32 - 67:37
the implicit equation of
the surface was f of x, y, z
• 67:37 - 67:38
equals c.
• 67:38 - 67:40
So now we've done this before.
• 67:40 - 67:42
You say Magdalena, you're
repeating yourself.
• 67:42 - 67:47
I know I'm repeating myself, but
I want you to learn this twice
• 67:47 - 67:49
so you can remember it.
• 67:49 - 67:52
What is f of x, y, z?
• 67:52 - 67:57
In my case, it's x squared
plus y squared plus z squared
• 67:57 - 68:00
minus 16, or even nothing.
• 68:00 - 68:02
Because the constant
doesn't matter anyway
• 68:02 - 68:04
• 68:04 - 68:06
You guys are doing homework.
• 68:06 - 68:08
You saw how the gradient goes.
• 68:08 - 68:14
be 2x times-- and that's
• 68:14 - 68:19
the partial derivative times i
plus 2y times j plus 2z times
• 68:19 - 68:23
k-- that's very important.
• 68:23 - 68:28
[? Lovett ?] has some
coordinates we plug in.
• 68:28 - 68:34
Now, can we write-- two things.
• 68:34 - 68:36
I want two things from you.
• 68:36 - 68:41
Write me a total
differential b tangent plane
• 68:41 - 68:46
at the point-- so, a, write
the total differential.
• 68:46 - 68:51
• 68:51 - 68:54
I'm not going to ask you you
to do a linear approximation.
• 68:54 - 68:56
I could.
• 68:56 - 69:24
B, write the tangent plane
to the sphere at the point
• 69:24 - 69:25
that-- I don't know.
• 69:25 - 69:27
I don't want one that's trivial.
• 69:27 - 69:30
• 69:30 - 69:38
Let's take this 0, square root
of 8, and square root of 8.
• 69:38 - 69:40
I just have to make
sure that I don't
• 69:40 - 69:42
come with some
nonsensical point that's
• 69:42 - 69:43
not going to be on the sphere.
• 69:43 - 69:46
This will be because I
plugged it in in my mind.
• 69:46 - 69:50
I get 8 plus 8 is 16 last
time I checked, right?
• 69:50 - 69:55
So after we do this
we take a break.
• 69:55 - 69:58
Suppose that this is a
• 69:58 - 70:01
• 70:01 - 70:04
or on somebody [? YouTubed it ?]
for a lot of money,
• 70:04 - 70:10
you asked them, \$25 an hour
for me to work that problem.
• 70:10 - 70:11
That's good.
• 70:11 - 70:17
I mean-- it's-- it's a
class that you're taking
• 70:17 - 70:20
• 70:20 - 70:22
to take calc 3.
• 70:22 - 70:26
But it gives you-- and
I know from experience,
• 70:26 - 70:28
some of my students came
back to me and said,
• 70:28 - 70:30
after I took calc
3, I understood it
• 70:30 - 70:33
so well that I was able to
tutor calc 1, calc 2, calc 3,
• 70:33 - 70:36
so I got a double job.
• 70:36 - 70:38
Several hours a week,
the tutoring center,
• 70:38 - 70:40
math department,
and several hours
• 70:40 - 70:41
at the [INAUDIBLE] center.
• 70:41 - 70:43
You know what I'm talking about?
• 70:43 - 70:46
So I've had students who did
well and ended up liking this,
• 70:46 - 70:49
and said I can tutor
this in my sleep.
• 70:49 - 70:54
So-- and also private tutoring
is always a possibility.
• 70:54 - 70:55
OK.
• 70:55 - 70:59
Write total differential.
• 70:59 - 71:04
df equals, and now
I'll say at any point.
• 71:04 - 71:07
So I don't care what
the value will be.
• 71:07 - 71:09
I didn't say at what point.
• 71:09 - 71:10
It means in general.
• 71:10 - 71:12
Why is that?
• 71:12 - 71:15
You tell me, you
know that by now.
• 71:15 - 71:18
2x times what?
• 71:18 - 71:20
Now, you learned
• 71:20 - 71:22
never gonna make mistakes.
• 71:22 - 71:25
2y plus 2z dz.
• 71:25 - 71:26
That is very good.
• 71:26 - 71:28
That's the total differential.
• 71:28 - 71:34
Now, what is the equation
of the tangent plane?
• 71:34 - 71:37
It's not gonna be that.
• 71:37 - 71:41
Because I'm not
considering a graph.
• 71:41 - 71:45
I'm considering an
implicitly given surface
• 71:45 - 71:53
by this implicit equation f of
x, y, z, equals c, your friend.
• 71:53 - 71:58
So what was, in that case,
the equation of the plane
• 71:58 - 72:00
written as?
• 72:00 - 72:02
STUDENT: [INAUDIBLE]
• 72:02 - 72:05
PROFESSOR TODA: I'm--
yeah, you guys are smart.
• 72:05 - 72:06
I mean, you are fast.
• 72:06 - 72:08
Let's do it in general.
• 72:08 - 72:12
F sub x-- we did that last
time, [INAUDIBLE] times--
• 72:12 - 72:14
do you guys remember?
• 72:14 - 72:16
x minus x0.
• 72:16 - 72:21
And this is at the point plus
big F sub y at the point times
• 72:21 - 72:26
y minus y0 plus big F sub
z at the point z minus z0.
• 72:26 - 72:27
This is just review.
• 72:27 - 72:28
Equals 0.
• 72:28 - 72:28
Stop.
• 72:28 - 72:31
Where do these guys come from?
• 72:31 - 72:33
• 72:33 - 72:35
• 72:35 - 72:40
Which are the a,b,c, now I
know my ABCs, from the normal.
• 72:40 - 72:42
My ABCs from the normal.
• 72:42 - 72:47
So in this case-- I
don't want to erase
• 72:47 - 72:49
this beautiful picture.
• 72:49 - 72:55
The last thing I have to do
before the break is-- you
• 72:55 - 72:57
said 0.
• 72:57 - 72:59
I'm a lazy person by definition.
• 72:59 - 73:03
Can you tell me why
you said 0 times?
• 73:03 - 73:05
STUDENT: Because the
x value is [INAUDIBLE]
• 73:05 - 73:07
PROFESSOR TODA: You said
2x, plug in and x equals 0
• 73:07 - 73:10
Magdalena, so you don't
• 73:10 - 73:12
have to write down everything.
• 73:12 - 73:20
But I'm gonna write down 0
times x minus 0 plus-- what's
• 73:20 - 73:21
next for me?
• 73:21 - 73:22
STUDENT: 2 square root 8.
• 73:22 - 73:24
PROFESSOR TODA: 2y, 2 root 8.
• 73:24 - 73:26
Is root 8 beautiful?
• 73:26 - 73:28
It looks like heck.
• 73:28 - 73:33
At the end I'm gonna
brush it up a little bit.
• 73:33 - 73:39
This is the partial-- f sub y of
t times y minus-- who is y, z?
• 73:39 - 73:41
Root 8.
• 73:41 - 73:42
Do I like it?
• 73:42 - 73:44
I hate it, but it
doesn't matter.
• 73:44 - 73:46
Because I'm gonna simplify.
• 73:46 - 73:52
Plus again, 2 root 8, thank you.
• 73:52 - 73:57
This is my c guy.
• 73:57 - 74:02
Times z minus root 8 equals 0.
• 74:02 - 74:05
I picked another example
from the one from the book,
• 74:05 - 74:09
because you are gonna
• 74:09 - 74:12
I'm gonna erase that.
• 74:12 - 74:15
And I'm gonna brush
this up because it
• 74:15 - 74:17
looks horrible to me.
• 74:17 - 74:20
Thank God this goes away.
• 74:20 - 74:22
So the plane will
simply be a combination
• 74:22 - 74:24
of my y and z in a constant.
• 74:24 - 74:28
And if I want to
make my life easier,
• 74:28 - 74:30
I'm gonna divide by what?
• 74:30 - 74:32
By this.
• 74:32 - 74:34
So in the end, it
doesn't matter.
• 74:34 - 74:36
Come on.
• 74:36 - 74:42
I'll get y minus root 8 plus
c minus root 8 equals 0.
• 74:42 - 74:44
Do I like it?
• 74:44 - 74:45
I hate it.
• 74:45 - 74:47
No, you know, I don't like it.
• 74:47 - 74:49
Why don't I like it?
• 74:49 - 74:50
It's not simplified.
• 74:50 - 74:56
So in any case, if this
were multiple choice,
• 74:56 - 74:59
it would not be written
like that, right?
• 74:59 - 75:04
So what would be the
simplified claim in this case?
• 75:04 - 75:09
The way I would write
it-- a y plus a z minus--
• 75:09 - 75:11
think, what is root 8?
• 75:11 - 75:13
STUDENT: 2 root 2.
• 75:13 - 75:14
PROFESSOR TODA: And 2 root 2.
• 75:14 - 75:21
And 2 root 2, how
much-- minus 4 root 2.
• 75:21 - 75:29
And this is how you are expected
• 75:29 - 75:38
This is that tangent
plane at the point.
• 75:38 - 75:41
• 75:41 - 75:43
To the sphere.
• 75:43 - 75:46
• 75:46 - 75:49
There are programs--
one time I was teaching
• 75:49 - 75:54
thing I gave my students to do,
• 75:54 - 75:59
which was a lot of fun--
using a parametrization,
• 75:59 - 76:03
plot the entire
sphere with MathLab.
• 76:03 - 76:04
We did it with MathLab.
• 76:04 - 76:07
Some people said they know
[INAUDIBLE] I didn't care.
• 76:07 - 76:09
So MathLab for me
was easier, so we
• 76:09 - 76:12
plotted the sphere in MathLab.
• 76:12 - 76:15
We picked a point,
and we drew-- well,
• 76:15 - 76:22
we drew-- with MathLab we
drew the tangent plane that
• 76:22 - 76:26
was tangent to the
sphere at that point.
• 76:26 - 76:27
And they liked it.
• 76:27 - 76:30
It was-- you know
what this class is,
• 76:30 - 76:32
is-- if you're math
majors you take it.
• 76:32 - 76:34
• 76:34 - 76:36
Mainly it's theoretical.
• 76:36 - 76:39
It teaches you Euclidian
axioms and stuff,
• 76:39 - 76:42
and then some
non-Euclidian geometries.
• 76:42 - 76:46
But I thought that I would
do it into an honors class.
• 76:46 - 76:49
And I put one third of that
last class visualization
• 76:49 - 76:51
with MathLab of geometry.
• 76:51 - 76:54
And I think that was what
they liked the most, not so
• 76:54 - 76:56
much the axiomatic
part and the proofs,
• 76:56 - 77:03
but the hands-on computation
and visualization in the lab.
• 77:03 - 77:05
We have this lab, 113.
• 77:05 - 77:07
We used to have two labs,
but now we are poor,
• 77:07 - 77:09
we only have one.
• 77:09 - 77:11
No, we lost the lab.
• 77:11 - 77:14
lab-- 009, next to you,
• 77:14 - 77:19
is lost because-- I used
to each calc 3 there.
• 77:19 - 77:22
Not because-- that's
not why we lost it.
• 77:22 - 77:25
We lost it because we-- we
• 77:25 - 77:25
there.
• 77:25 - 77:27
We have no space.
• 77:27 - 77:31
students in mathematics.
• 77:31 - 77:32
Where do you put them?
• 77:32 - 77:34
We just cram them into cubicles.
• 77:34 - 77:38
here, and they put some,
• 77:38 - 77:40
so we lost the lab.
• 77:40 - 77:42
• 77:42 - 77:43
All right.
• 77:43 - 77:45
So that's it for now.
• 77:45 - 77:48
We are gonna take a
short break, and we
• 77:48 - 77:52
will continue for one more hour,
which is mostly application.
• 77:52 - 77:55
I'm sort of done with 11.4.
• 77:55 - 77:58
I'll jump into 11.5 next.
• 77:58 - 78:01
Take a short break.
• 78:01 - 78:03
Thanks for the attendance.
• 78:03 - 78:05
Oh, and you did the calculus.
• 78:05 - 78:06
Very good.
• 78:06 - 79:52
• 79:52 - 79:55
Did this homework give you
• 79:55 - 79:56
or anything, or not?
• 79:56 - 79:58
Not too much?
• 79:58 - 79:59
It's a long homework.
• 79:59 - 80:01
49 problems-- 42 problems.
• 80:01 - 80:06
• 80:06 - 80:07
• 80:07 - 82:39
• 82:39 - 82:46
OK, questions from the-- what
was it, the first part-- mainly
• 82:46 - 82:48
the first part of chapter 11.
• 82:48 - 82:50
This is where we are.
• 82:50 - 82:57
Right now we hit the
half point because 11.8
• 82:57 - 82:59
is the last section.
• 82:59 - 83:03
And we will do that, that's
Lagrange multipliers.
• 83:03 - 83:07
So, let's do a little
bit of a review.
• 83:07 - 83:09
• 83:09 - 83:11
Do you have them?
• 83:11 - 83:14
Imagine this would
be office hour.
• 83:14 - 83:15
• 83:15 - 83:18
• 83:18 - 83:20
STUDENT: I know it's
a stupid question,
• 83:20 - 83:22
but my visualization [INAUDIBLE]
coming along, and question
• 83:22 - 83:27
the plane and passing the line.
• 83:27 - 83:32
So you have a 3, 5,
and 4 x, y, and z,
• 83:32 - 83:34
and you have a radius of 5.
• 83:34 - 83:36
Is it passing the x, y plane?
• 83:36 - 83:41
Is it passing [INAUDIBLE]
x plane and [INAUDIBLE]
• 83:41 - 83:42
passing the other plane.
• 83:42 - 83:44
PROFESSOR TODA: So-- say again.
• 83:44 - 83:46
So you have 3 and 4 and 5--
• 83:46 - 83:48
STUDENT: x minus-- yes.
• 83:48 - 83:49
PROFESSOR TODA: What
are the coordinates?
• 83:49 - 83:51
STUDENT: 3, 4, and 5.
• 83:51 - 83:53
PROFESSOR TODA: 3, 4, and
5, just as you said them.
• 83:53 - 83:54
You can--
• 83:54 - 83:56
STUDENT: And the radius is 5.
• 83:56 - 83:57
• 83:57 - 83:57
STUDENT: 5.
• 83:57 - 84:00
• 84:00 - 84:01
[INAUDIBLE]
• 84:01 - 84:02
PROFESSOR TODA: Yeah, well, OK.
• 84:02 - 84:08
So assume you have a
• 84:08 - 84:09
means you have 25.
• 84:09 - 84:15
If you do the 3 squared plus
4 squared plus 5 squared,
• 84:15 - 84:16
what is that?
• 84:16 - 84:17
For this point.
• 84:17 - 84:19
You have two separate points.
• 84:19 - 84:23
For this point you
have 25 plus 25.
• 84:23 - 84:25
Are you guys with me?
• 84:25 - 84:30
So you have the
specific x0, y0, z0.
• 84:30 - 84:39
You do the sum of the
squares, and you get 50.
• 84:39 - 84:44
My question is, is this point
outside, inside the sphere
• 84:44 - 84:45
or on the sphere?
• 84:45 - 84:47
On the sphere,
obviously, it's not,
• 84:47 - 84:54
because it does not verify the
equation of the sphere, right?
• 84:54 - 84:59
STUDENT: [INAUDIBLE] those the
location of the center point.
• 84:59 - 85:01
STUDENT: Where's the
center of the sphere?
• 85:01 - 85:02
STUDENT: [INAUDIBLE]
• 85:02 - 85:06
• 85:06 - 85:09
PROFESSOR TODA: The center
of the sphere would be at 0.
• 85:09 - 85:12
STUDENT: [INAUDIBLE]
• 85:12 - 85:14
PROFESSOR TODA: We are
making up a question.
• 85:14 - 85:15
So, right?
• 85:15 - 85:17
So practically, I am
making up a question.
• 85:17 - 85:17
STUDENT: Oh, OK.
• 85:17 - 85:23
PROFESSOR TODA: So I'm saying if
you have a sphere of radius 5,
• 85:23 - 85:27
and somebody gives you this
point of coordinates 3, 4,
• 85:27 - 85:29
and 5, where is the point?
• 85:29 - 85:35
Is it inside the sphere, outside
the sphere or on the sphere?
• 85:35 - 85:37
On the sphere it cannot be
because it doesn't verify
• 85:37 - 85:40
the sphere.
• 85:40 - 85:45
Ah, it looks like a Mr. Egg.
• 85:45 - 85:47
I don't like it.
• 85:47 - 85:51
I'm sorry, it's a sphere.
• 85:51 - 85:55
So a point on a sphere that
will have-- that's a hint.
• 85:55 - 85:58
A point on a sphere that
will have coordinates 3 and 4
• 85:58 - 86:02
would be exactly 3, 4, and 0.
• 86:02 - 86:06
So it would be where?
• 86:06 - 86:08
STUDENT: 16, 4.
• 86:08 - 86:11
PROFESSOR TODA: 3 squared plus
4 squared is 5 squared, right?
• 86:11 - 86:13
So those are
Pythagorean numbers.
• 86:13 - 86:15
That's the beauty of them.
• 86:15 - 86:23
• 86:23 - 86:28
I'm trying to draw well.
• 86:28 - 86:28
Right.
• 86:28 - 86:30
This is the point a.
• 86:30 - 86:33
• 86:33 - 86:37
You go up how many?
• 86:37 - 86:39
You shift by 5.
• 86:39 - 86:41
So are you inside or outside?
• 86:41 - 86:42
STUDENT: Outside.
• 86:42 - 86:43
PROFESSOR TODA: Yeah.
• 86:43 - 86:50
• 86:50 - 86:55
STUDENT: Are you outside
or are you exactly on-- oh.
• 86:55 - 86:56
Sorry, I thought--
• 86:56 - 86:56
PROFESSOR TODA: You go--
• 86:56 - 86:58
STUDENT: I thought you
were saying point a.
• 86:58 - 87:00
Point a is like
exactly-- [INAUDIBLE]
• 87:00 - 87:01
PROFESSOR TODA: You
are on the equator,
• 87:01 - 87:02
and from the Equator
of the Earth,
• 87:02 - 87:06
you're going parallel to the
z-axis, then you stay outside.
• 87:06 - 87:09
But the question is
more subtle than that.
• 87:09 - 87:12
This is pretty--
you figured it out.
• 87:12 - 87:15
1 point-- 0.5 extra credit.
• 87:15 - 87:19
That we don't have--
• 87:19 - 87:20
we'll find some time.
• 87:20 - 87:23
When I-- when we rewrite the
book, maybe we should do that.
• 87:23 - 87:39
So express the points outside
the sphere, inside the sphere,
• 87:39 - 87:50
and on the sphere
using exclusively
• 87:50 - 87:52
equalities and inequalities.
• 87:52 - 87:58
• 87:58 - 87:59
And that's extra credit.
• 87:59 - 88:01
So, of course, the
[INAUDIBLE] is obvious.
• 88:01 - 88:07
The sphere is the set of
the triples x, y, z in R3.
• 88:07 - 88:10
• 88:10 - 88:13
OK, I'm teaching you a little
bit of mathematical language.
• 88:13 - 88:20
x, y, z belongs to R3,
R3 being the free space,
• 88:20 - 88:24
with the property that x squared
plus y squared plus z squared
• 88:24 - 88:27
equals given a squared.
• 88:27 - 88:30
What if you have less than,
what if you have greater than?
• 88:30 - 88:32
Ah, shut up, Magdalena.
• 88:32 - 88:33
This is all up to you.
• 88:33 - 88:36
You will figure
out how the points
• 88:36 - 88:41
on the outside and the points
on the inside are characterized.
• 88:41 - 88:47
And unfortunately we don't
emphasize that in the textbook.
• 88:47 - 88:50
I'll erase.
• 88:50 - 88:52
You figured it out.
• 88:52 - 88:53
And now I want to
move on to something
• 88:53 - 88:57
a little bit challenging,
but not very challenging.
• 88:57 - 89:11
• 89:11 - 89:12
STUDENT: Professor, [INAUDIBLE]
• 89:12 - 89:20
• 89:20 - 89:21
PROFESSOR TODA: The
last requirement
• 89:21 - 89:23
on the extra credit?
• 89:23 - 89:27
So I said the sphere
represents the set of all
• 89:27 - 89:30
triples x, y, z in
R3 with the property
• 89:30 - 89:32
that x squared plus y squared
plus y squared plus z squared
• 89:32 - 89:34
equals a squared.
• 89:34 - 89:37
With the equality sign.
• 89:37 - 89:40
Represent the points on
the inside of the sphere
• 89:40 - 89:45
and the outside of the sphere
using just inequalities.
• 89:45 - 89:45
Mathematics.
• 89:45 - 89:49
No writing, no words,
just mathematics.
• 89:49 - 89:50
In set theory symbols.
• 89:50 - 89:55
Like, the set of points
with braces like that.
• 89:55 - 89:58
OK.
• 89:58 - 90:03
bit of stuff from the chain
• 90:03 - 90:12
rule in-- in chapter--
I don't know, guys,
• 90:12 - 90:15
it was a long time ago.
• 90:15 - 90:16
Shame on me.
• 90:16 - 90:19
Chapter 3, calc 1.
• 90:19 - 90:38
Versus chain rule rules in
calc in-- chapter 5 calc 3.
• 90:38 - 90:41
This is a little
bit of a warmup.
• 90:41 - 90:42
I don't want to
[INAUDIBLE] again
• 90:42 - 90:44
next time when we
meet on Thursday.
• 90:44 - 90:46
Bless you.
• 90:46 - 90:49
The bless you was
out of the context.
• 90:49 - 90:52
What was the chain rule?
• 90:52 - 90:54
We did compositions
of functions,
• 90:54 - 91:01
and we had a diagram that we
don't show you, but we should.
• 91:01 - 91:05
There is practically a function
that comes from a set A
• 91:05 - 91:08
to a set B to a set
C. These are the sets.
• 91:08 - 91:13
And we have g and an f.
• 91:13 - 91:17
And we have g of f of t.
• 91:17 - 91:22
t is your favorite letter here.
• 91:22 - 91:27
How do you do the
derivative with respect
• 91:27 - 91:29
to g composed with f?
• 91:29 - 91:33
• 91:33 - 91:37
I asked the same question to
my Calc 1 and Calc 2 students,
• 91:37 - 91:42
and they really had a hard
time expressing themselves,
• 91:42 - 91:45
expressing the chain rule.
• 91:45 - 91:47
And when I gave them
an example, they
• 91:47 - 91:50
said, oh, I know how to
do it on the example.
• 91:50 - 91:55
I just don't know how to do it
on the-- I like the numbers,
• 91:55 - 91:58
but I don't like them letters.
• 91:58 - 92:02
So how do we do
it in an example?
• 92:02 - 92:05
• 92:05 - 92:09
I chose natural log,
which you find everywhere.
• 92:09 - 92:14
So how do you do d
dt of this animal?
• 92:14 - 92:16
It's an animal.
• 92:16 - 92:18
STUDENT: [INAUDIBLE]
• 92:18 - 92:21
PROFESSOR TODA: So the idea
is you go from the outside
• 92:21 - 92:23
to the inside, one at a time.
• 92:23 - 92:25
My students know that.
• 92:25 - 92:27
You prime the function,
the outer function,
• 92:27 - 92:31
the last one you applied,
to the function inside.
• 92:31 - 92:34
And you prime that with
respect to the argument.
• 92:34 - 92:37
This is called the
argument in that case.
• 92:37 - 92:41
Derivative of natural
log is 1 over what?
• 92:41 - 92:44
The argument.
• 92:44 - 92:46
And you cover up natural
• 92:46 - 92:47
and you keep going.
• 92:47 - 92:52
And you say, next I go,
times the derivative
• 92:52 - 92:56
of this square, plus 1,
prime with respect to t.
• 92:56 - 92:58
So I go times 2t.
• 92:58 - 93:01
And that's what we have.
• 93:01 - 93:05
And they say, when you explain
it like that, they said to me,
• 93:05 - 93:06
I can understand it.
• 93:06 - 93:09
But I'm having a
problem understanding it
• 93:09 - 93:13
when you express this diagram--
that it throws me off.
• 93:13 - 93:19
So in order to avoid that kind
of theoretical misconception,
• 93:19 - 93:25
I'm saying, let us see
what the heck this is.
• 93:25 - 93:33
d dt of g of f of t, because
this is what you're doing,
• 93:33 - 93:35
has to have some understanding.
• 93:35 - 93:39
The problem is that Mister
f of t, that lives here,
• 93:39 - 93:40
has a different argument.
• 93:40 - 93:45
The letter in B should
be, let's say, u.
• 93:45 - 93:49
• 93:49 - 93:52
That doesn't say
anything practically.
• 93:52 - 93:54
How do you differentiate
with respect to what?
• 93:54 - 93:56
You cannot say d dt here.
• 93:56 - 94:01
So you have to call f
of t something generic.
• 94:01 - 94:05
You have to have a
generic variable for that.
• 94:05 - 94:14
So you have then dg du, at
what specific value of u?
• 94:14 - 94:18
At the specific value of
u that we have as f of t.
• 94:18 - 94:22
Do you understand the
specificity of this?
• 94:22 - 94:27
Times-- that's the chain
rule, the product coming
• 94:27 - 94:32
from the chain rule-- df pt.
• 94:32 - 94:34
You take du dt or d of dt.
• 94:34 - 94:35
It is the same thing.
• 94:35 - 94:37
Say it again, df dt.
• 94:37 - 94:41
• 94:41 - 94:44
what if I put du dt?
• 94:44 - 94:45
Would it be wrong?
• 94:45 - 94:50
No, as long as you understand
that u is a-something,
• 94:50 - 94:55
as the image of this t.
• 94:55 - 94:56
Do you know what he liked?
• 94:56 - 94:59
• 94:59 - 95:02
He said, do you know
• 95:02 - 95:07
I like that I can imagine
that these are two cowboys-- I
• 95:07 - 95:09
told the same thing to my son.
• 95:09 - 95:13
He was so excited,
• 95:13 - 95:15
• 95:15 - 95:17
Of course, he is 10.
• 95:17 - 95:18
These are the cowboys.
• 95:18 - 95:20
They are across.
• 95:20 - 95:23
One is on top of
the building there,
• 95:23 - 95:25
shooting at this
guy, who is here
• 95:25 - 95:28
across the street on the bottom.
• 95:28 - 95:31
So they are
annihilating each other.
• 95:31 - 95:33
They shoot and they die.
• 95:33 - 95:37
And they die, and
you're left with 1/3.
• 95:37 - 95:42
The same idea is that, actually,
these guys do not simplify.
• 95:42 - 95:46
du and-- [? du, ?] they're not
cowboys who shoot at each other
• 95:46 - 95:49
at the same time and both
die at the same time.
• 95:49 - 95:53
It is not so romantic.
• 95:53 - 96:00
But the idea of remembering
this formula is the same.
• 96:00 - 96:04
Because practically, if you want
to annihilate the two cowboys
• 96:04 - 96:06
and put your hands over them
so you don't see them anymore,
• 96:06 - 96:11
du dt, you would
have to remember, oh,
• 96:11 - 96:12
so that was the
derivative with respect
• 96:12 - 96:16
to t that I initially
have of the guy on top,
• 96:16 - 96:19
which was g of f of
the composed function.
• 96:19 - 96:23
So if you view g of f of t
as the composed function,
• 96:23 - 96:24
who is that?
• 96:24 - 96:29
The composition g
composed with f of t
• 96:29 - 96:32
is the function g of f of t.
• 96:32 - 96:35
This is the function that
you want to differentiate
• 96:35 - 96:37
with respect to time, t.
• 96:37 - 96:41
This is this, prime
with respect to t.
• 96:41 - 96:46
It's like they would be killing
each other, and you would die.
• 96:46 - 96:48
And I liked this
idea, and I said,
• 96:48 - 96:50
I should tell that to my
students and to my son.
• 96:50 - 96:53
And, of course, my son
started jumping around
• 96:53 - 96:56
and said that he understands
multiplication of fractions
• 96:56 - 96:58
better now.
• 96:58 - 97:01
simplifications-- I don't
• 97:01 - 97:03
know how they teach these kids.
• 97:03 - 97:06
• 97:06 - 97:08
It became so complicated.
• 97:08 - 97:11
It's as if mathematics--
mathematics is the same.
• 97:11 - 97:12
It hasn't changed.
• 97:12 - 97:14
It's the people
who make the rules
• 97:14 - 97:17
on how to teach it that change.
• 97:17 - 97:22
So he simply doesn't see
that this simplifies.
• 97:22 - 97:25
And when I tell him simplify,
he's like, what is simplify?
• 97:25 - 97:26
What is this word simplify?
• 97:26 - 97:27
My teacher doesn't use it.
• 97:27 - 97:32
So I feel like sometimes
I want to shoot myself.
• 97:32 - 97:35
But he went over that and
• 97:35 - 97:37
of simplification.
• 97:37 - 97:39
[? He ?] composing
something on top
• 97:39 - 97:43
and the bottom finding the
common factors up and down,
• 97:43 - 97:45
crossing them out, and so on.
• 97:45 - 97:47
And so now he knows
what it means.
• 97:47 - 97:51
But imagine going to
college without having
• 97:51 - 97:51
this early knowledge.
• 97:51 - 97:55
You come to college,
you were good in school,
• 97:55 - 97:57
and you've never learned
enough simplification.
• 97:57 - 98:00
And then somebody like me,
and tells you simplification.
• 98:00 - 98:03
You say, she is a foreigner.
• 98:03 - 98:08
She has a language barrier
that is [INAUDIBLE] she has
• 98:08 - 98:10
that I've never heard before.
• 98:10 - 98:15
So I wish the people who
really re-conceive, re-write
• 98:15 - 98:19
the curriculum for K12
would be a little bit
• 98:19 - 98:22
more respectful of the history.
• 98:22 - 98:26
Imagine that I
would teach calculus
• 98:26 - 98:29
without ever telling you
• 98:29 - 98:31
was Leibniz, he doesn't exist.
• 98:31 - 98:34
Or Euler, or one
of these fathers.
• 98:34 - 98:38
They are the ones who
created these notations.
• 98:38 - 98:43
And if we never tell you
• 98:43 - 98:47
wherever they are, it is an
injustice that we are doing.
• 98:47 - 98:48
All right.
• 98:48 - 98:54
Chain rule in
Chapter 5 of Calc 3.
• 98:54 - 98:56
This is a little bit
more complicated,
• 98:56 - 99:00
but I'm going to teach it
to you because I like it.
• 99:00 - 99:06
Imagine that you have z equals
x squared plus y squared.
• 99:06 - 99:07
What is that?
• 99:07 - 99:08
It's an example of a graph.
• 99:08 - 99:11
And I just taught
you what a graph is.
• 99:11 - 99:13
• 99:13 - 99:23
But imagine that
• 99:23 - 99:26
• 99:26 - 99:28
[INAUDIBLE] with
respect to time.
• 99:28 - 99:38
• 99:38 - 99:41
And you will say, Magdalena,
can you draw that?
• 99:41 - 99:46
What in the world do you mean
that x and y follow a curve?
• 99:46 - 99:47
I'll try to draw it.
• 99:47 - 99:49
First of all, you are on a walk.
• 99:49 - 99:50
You are in a beautiful valley.
• 99:50 - 99:51
It's not a vase.
• 99:51 - 99:57
It's a circular
paraboloid, as an example.
• 99:57 - 100:01
• 100:01 - 100:02
It's like an egg shell.
• 100:02 - 100:05
• 100:05 - 100:07
You have a curve on that.
• 100:07 - 100:08
You draw that.
• 100:08 - 100:10
You have nothing better
to do than decorating eggs
• 100:10 - 100:11
for Easter.
• 100:11 - 100:12
Hey, wait.
• 100:12 - 100:15
Easter is far, far away.
• 100:15 - 100:17
But let's say you want to
decorate eggs for Easter.
• 100:17 - 100:23
You take some color of paint
and put paint on the egg.
• 100:23 - 100:28
You are actually describing
an arc of a curve.
• 100:28 - 100:38
And x and y, their
projection on the floor
• 100:38 - 100:40
will be x of t, y of t.
• 100:40 - 100:43
• 100:43 - 100:45
Because you paint in time.
• 100:45 - 100:46
You paint in time.
• 100:46 - 100:48
You describe this in time.
• 100:48 - 100:54
Now, if x of ty of t is
being projected on the floor.
• 100:54 - 100:59
Of course, you have a curve
here as well, which is what?
• 100:59 - 101:06
Which it will be x
of t, y of t, z of t.
• 101:06 - 101:07
Oh, my god.
• 101:07 - 101:12
Yes, because the altitude also
depends on the motion in time.
• 101:12 - 101:14
All right.
• 101:14 - 101:16
So what's missing here?
• 101:16 - 101:19
It's missing the third
coordinate, duh, that's
• 101:19 - 101:21
0 because I'm on the floor.
• 101:21 - 101:26
I'm on the xy plane, which
is the floor z equals z.
• 101:26 - 101:29
But now let's
suppose that I want
• 101:29 - 101:37
to say this is f of x and y,
and I want to differentiate
• 101:37 - 101:39
f with respect to t.
• 101:39 - 101:41
And you go, say what?
• 101:41 - 101:41
Oh, my god.
• 101:41 - 101:42
What is that?
• 101:42 - 101:46
I differentiate f
with respect to t.
• 101:46 - 101:49
By differentiating
f with respect to t,
• 101:49 - 101:55
I mean that I have f of
x and y differentiated
• 101:55 - 101:56
with respect to t.
• 101:56 - 101:58
And you say, wait, Magdalena.
• 101:58 - 102:00
This doesn't make any sense.
• 102:00 - 102:04
And you would be right to say
it doesn't make any sense.
• 102:04 - 102:07
Can somebody tell me why
it doesn't make any sense?
• 102:07 - 102:14
It's not clear where in the
world the variable t is inside.
• 102:14 - 102:17
So I'm going to say, OK,
x are themselves functions
• 102:17 - 102:20
of t, functions of that.
• 102:20 - 102:21
x of t, y of t.
• 102:21 - 102:24
If I don't do that,
it's not clear.
• 102:24 - 102:28
So this is a composed
function just like this one.
• 102:28 - 102:29
Look at the similarity.
• 102:29 - 102:31
It's really beautiful.
• 102:31 - 102:36
This is a function of
a function, g of f.
• 102:36 - 102:39
This is a function
of two functions.
• 102:39 - 102:43
Say it again, f is a function
of two functions, x and y.
• 102:43 - 102:45
This was a function
of a function of t.
• 102:45 - 102:48
This was a function
of two functions of t.
• 102:48 - 102:49
Oh, my God.
• 102:49 - 102:52
• 102:52 - 102:55
How do we compute this?
• 102:55 - 102:57
There is a rule.
• 102:57 - 102:58
It can be proved.
• 102:58 - 103:02
We will look a little bit into
the theoretical justification
• 103:02 - 103:03
of this proof later.
• 103:03 - 103:06
But practically what
you do, you say,
• 103:06 - 103:08
I have to have some
order in my life.
• 103:08 - 103:09
OK.?
• 103:09 - 103:13
So the way we do that,
we differentiate first
• 103:13 - 103:17
with respect to the first
location, which is x.
• 103:17 - 103:22
I go there, but I cannot write
df dx because f is a mother
• 103:22 - 103:23
of two babies.
• 103:23 - 103:27
f is a function of two
variables, x and y.
• 103:27 - 103:29
She has to be a mother
to both of them;
• 103:29 - 103:32
otherwise, they get
jealous of one another.
• 103:32 - 103:38
So I have to say, partial
of f with respect to x,
• 103:38 - 103:39
I cannot use d.
• 103:39 - 103:44
Like Leibniz, I have
to use del, d of dx.
• 103:44 - 103:49
At the point x of dy of t,
this is the location I have.
• 103:49 - 103:51
Times what?
• 103:51 - 103:52
I keep derivation.
• 103:52 - 103:56
I keep derivating, like
don't drink and derive.
• 103:56 - 103:57
What is that?
• 103:57 - 103:59
The chain rule.
• 103:59 - 104:05
Prime again, this guy x
with respect to t, dx dt.
• 104:05 - 104:09
And then you go,
plus because she has
• 104:09 - 104:12
to be a mother to both kids.
• 104:12 - 104:15
The same thing for
the second child.
• 104:15 - 104:18
So you go, the derivative
of f with respect
• 104:18 - 104:27
to y, add x of ty
of t times dy dt.
• 104:27 - 104:30
• 104:30 - 104:35
So you see on the surface, x and
y are moving according to time.
• 104:35 - 104:39
And somehow we want to
measure the derivative
• 104:39 - 104:43
of the resulting function,
or composition function,
• 104:43 - 104:45
with respect to time.
• 104:45 - 104:46
This is a very
important chain rule
• 104:46 - 104:50
that I would like
you to memorize.
• 104:50 - 104:53
A chain rule.
• 104:53 - 104:54
Chain Rule No.
• 104:54 - 104:55
1.
• 104:55 - 104:59
• 104:59 - 105:00
Is it hard?
• 105:00 - 105:01
No, but for me it was.
• 105:01 - 105:05
When I was 21 and I saw
that-- and, of course,
• 105:05 - 105:06
my teacher was good.
• 105:06 - 105:10
And he told me, Magdalena,
imagine that instead of del you
• 105:10 - 105:14
would have d's.
• 105:14 - 105:17
So you have d and d and d and d.
• 105:17 - 105:21
The dx dx here, dy dy here,
they should be in your mind.
• 105:21 - 105:23
They are facing each other.
• 105:23 - 105:26
They are across on a diagonal.
• 105:26 - 105:29
And then, of course, I didn't
tell my teacher my idea
• 105:29 - 105:32
with the cowboys,
but it was funny.
• 105:32 - 105:39
So this is the chain rule
that re-makes, or generalizes
• 105:39 - 105:43
this idea to two variables.
• 105:43 - 105:48
Let's finish the example
because we didn't do it.
• 105:48 - 105:53
What is the derivative
of f in our case?
• 105:53 - 106:02
df dt will be-- oh, my god--
at any point p, how arbitary,
• 106:02 - 106:04
would be what?
• 106:04 - 106:08
First, you write
with respect to x.
• 106:08 - 106:11
2x, right?
• 106:11 - 106:11
2x.
• 106:11 - 106:17
But then you have to compute
this dx, add the pair you give.
• 106:17 - 106:20
And the pair they
gave you has a t.
• 106:20 - 106:23
So 2x is add x of
ty-- if you're going
• 106:23 - 106:25
to write it first
like that, you're
• 106:25 - 106:30
going to find it weird-- times,
I'm done with the first guy.
• 106:30 - 106:33
Then I'm going to take
the second guy in red,
• 106:33 - 106:35
and I'll put it here.
• 106:35 - 106:39
dx dt, but dx dt
everybody knows.
• 106:39 - 106:45
[INAUDIBLE] Let me
write it like this.
• 106:45 - 106:52
Plus [INAUDIBLE] that
guy again with green-- dy
• 106:52 - 106:59
computed at the pair x
of dy of [? t ?] times,
• 106:59 - 107:02
again, in red, dy dt.
• 107:02 - 107:07
• 107:07 - 107:09
So how do we write
the whole thing?
• 107:09 - 107:11
Could I have written it
from the beginning better?
• 107:11 - 107:11
Yeah.
• 107:11 - 107:21
2x of t, dx dt plus 2y of t dy.
• 107:21 - 107:22
Is it hard?
• 107:22 - 107:25
No, this is the idea.
• 107:25 - 107:28
Let's have something
more specific.
• 107:28 - 107:30
I'm going to erase
the whole thing.
• 107:30 - 107:36
• 107:36 - 107:40
I'll give you a problem
that we gave on the final
• 107:40 - 107:42
a few years ago.
• 107:42 - 107:45
And I'll show you how my
students cheated on that.
• 107:45 - 107:53
And I let them cheat, in
a way, because in the end
• 107:53 - 107:54
they were smart.
• 107:54 - 107:59
It didn't matter how they did
the problem, as long as they
• 107:59 - 108:02
• 108:02 - 108:03
So the problem was like that.
• 108:03 - 108:10
And my colleague did that many
years ago, several years ago,
• 108:10 - 108:12
did that several times.
• 108:12 - 108:20
So he said, let's do f of
t, dt squared and g of t.
• 108:20 - 108:27
I'll I'll do this
one, dq plus 1.
• 108:27 - 108:43
And then let's
[INAUDIBLE] the w of u
• 108:43 - 108:54
and B, exactly the same thing I
gave you before, [INAUDIBLE] I
• 108:54 - 108:56
remember that.
• 108:56 - 109:06
And he said, compute the
derivative of w of f of t,
• 109:06 - 109:10
and g of t with respect to t.
• 109:10 - 109:12
wait a minute here.
• 109:12 - 109:15
Why do you put d and not del?
• 109:15 - 109:18
Because this is a composed
function that in the end
• 109:18 - 109:21
is a function of t only.
• 109:21 - 109:23
So if you do it as
a composed function,
• 109:23 - 109:26
because this goes like this.
• 109:26 - 109:32
t goes to two
functions, f of t and u.
• 109:32 - 109:34
• 109:34 - 109:41
And there is a function w
that takes both of them, that
• 109:41 - 109:43
is a function of both of them.
• 109:43 - 109:47
In the end, this composition
that's straight from here
• 109:47 - 109:51
to here, is a function
of one variable only.
• 109:51 - 109:55
• 109:55 - 109:58
So my students then-- it was in
the beginning of the examine,
• 109:58 - 109:59
I remember.
• 109:59 - 110:02
And they said, well,
I forgot, they said.
• 110:02 - 110:04
I stayed up almost all night.
• 110:04 - 110:05
Don't do that.
• 110:05 - 110:06
Don't do what they did.
• 110:06 - 110:08
Many of my students
stay up all night
• 110:08 - 110:11
before the final because
I think I scare people,
• 110:11 - 110:13
and that's not what I mean.
• 110:13 - 110:15
I just want you to study.
• 110:15 - 110:19
But they stay up before
the final and the next day,
• 110:19 - 110:19
I'm a vegetable.
• 110:19 - 110:21
I don't even remember
the chain rule.
• 110:21 - 110:23
So they did not
remember the chain rule
• 110:23 - 110:25
that I've just wrote.
• 110:25 - 110:28
And they said, oh, but I
think I know how to do it.
• 110:28 - 110:30
And I said, shh.
• 110:30 - 110:32
Just don't say anything.
• 110:32 - 110:35
Let me show you how the
course coordinator wanted
• 110:35 - 110:37
that done several years ago.
• 110:37 - 110:40
So he wanted it done
by the chain rule.
• 110:40 - 110:42
He didn't say how you do it.
• 110:42 - 110:42
OK?
• 110:42 - 110:44
He said just get to
• 110:44 - 110:46
It doesn't matter.
• 110:46 - 110:47
He wanted it done like that.
• 110:47 - 110:56
He said, dw of f of tg
of p with respect to t,
• 110:56 - 111:07
of u you have f of t.
• 111:07 - 111:17
f of tg of t times df
dt plus dw with respect
• 111:17 - 111:19
to the second variable.
• 111:19 - 111:25
So this would be u, and
this would be v with respect
• 111:25 - 111:27
to the variable v,
the second variable
• 111:27 - 111:31
where [? measure ?]
that f of dg of t.
• 111:31 - 111:39
Evaluate it there times dg dt.
• 111:39 - 111:46
So it's like dv dt, which is dg
dt. [INAUDIBLE] So he did that,
• 111:46 - 111:48
and he expected
people to do what?
• 111:48 - 111:51
He expected people to take
a u squared the same 2 times
• 111:51 - 111:54
u, just like you
did before, 2 times.
• 111:54 - 111:58
And instead of u, since u is
f of t to [INAUDIBLE] puts
• 111:58 - 112:13
2f of t, this is the first
squiggly thing, times v of dt.
• 112:13 - 112:20
2t is this smiley face.
• 112:20 - 112:31
This is 2t plus--
what is the f dv?
• 112:31 - 112:38
Dw with respect to dv is
going to be 2v 2 time gf t.
• 112:38 - 112:47
t, this funny fellow
• 112:47 - 112:58
with this funny fellow, times qg
• 112:58 - 113:01
I'm going to erase
and write 3p squared.
• 113:01 - 113:04
• 113:04 - 113:07
And the last row he expected
my students to write
• 113:07 - 113:22
was 2t squared times 2t plus
2pq plus 1, times 3t squared.
• 113:22 - 113:28
• 113:28 - 113:32
Are you guys with me?
• 113:32 - 113:43
So [INAUDIBLE] 2t 2x
2t squared, correct.
• 113:43 - 113:50
I forgot to identify
this as that.
• 113:50 - 113:50
All right.
• 113:50 - 113:53
So in the end, the answer
• 113:53 - 113:54
Can you tell me what it is?
• 113:54 - 113:55
I'm too lazy to write it down.
• 113:55 - 113:57
You compute it.
• 113:57 - 113:59
How much is it simplified?
• 113:59 - 114:00
Find it as a polynomial.
• 114:00 - 114:01
STUDENT: [INAUDIBLE].
• 114:01 - 114:04
• 114:04 - 114:09
PROFESSOR TODA:
So you have 6, 6--
• 114:09 - 114:10
STUDENT: 16 cubed plus 3--
• 114:10 - 114:15
PROFESSOR TODA: T
to the 5th plus--
• 114:15 - 114:17
STUDENT: [INAUDIBLE].
• 114:17 - 114:19
PROFESSOR TODA: In
order, in order.
• 114:19 - 114:20
What's the next guy?
• 114:20 - 114:22
STUDENT: [INAUDIBLE].
• 114:22 - 114:23
PROFESSOR TODA: 4t cubed.
• 114:23 - 114:24
And the last guy--
• 114:24 - 114:25
STUDENT: 6t squared.
• 114:25 - 114:26
PROFESSOR TODA: 6t squared.
• 114:26 - 114:31
• 114:31 - 114:32
Yes?
• 114:32 - 114:33
Did you get the same thing?
• 114:33 - 114:34
OK.
• 114:34 - 114:37
Now, how did my students do it?
• 114:37 - 114:38
[INAUDIBLE]
• 114:38 - 114:40
• 114:40 - 114:41
Did they apply the chain rule?
• 114:41 - 114:42
No.
• 114:42 - 114:44
They said OK, this
is how it goes.
• 114:44 - 114:47
• 114:47 - 114:58
W of U of T and V of T is U is
F. So this guy is T squared,
• 114:58 - 115:02
T squared squared,
plus this guy is T
• 115:02 - 115:09
cubed plus 1 taken and
shaken and squared.
• 115:09 - 115:14
And then when I do the
whole thing, derivative
• 115:14 - 115:23
of this with respect
to T, I get--
• 115:23 - 115:28
I'm too lazy-- T to the
4 prime is 40 cubed.
• 115:28 - 115:29
I'm not going to do on the map.
• 115:29 - 115:37
2 out T cubed plus 1 times
chain rule, 3t squared.
• 115:37 - 115:50
40 cubed plus 16 to the 5 plus--
[INAUDIBLE] 2 and 6t squared.
• 115:50 - 115:56
So you realize that I
have to give them 100%.
• 115:56 - 115:59
Although they were very
honest and said, we blanked.
• 115:59 - 116:01
We don't remember
the chain rule.
• 116:01 - 116:03
We don't remember the formula.
• 116:03 - 116:04
So that's fine.
• 116:04 - 116:05
Do whatever you can.
• 116:05 - 116:07
So I gave them 100% for that.
• 116:07 - 116:11
But realize that the
author of the problem
• 116:11 - 116:14
was a little bit naive.
• 116:14 - 116:17
Because you could have
done this differently.
• 116:17 - 116:22
I mean if you wanted to
actually test the whole thing,
• 116:22 - 116:26
you wouldn't have given-- let's
say you wouldn't have given
• 116:26 - 116:32
the actual-- yeah, you wouldn't
have given the actual functions
• 116:32 - 116:38
and say write the chain
formula symbolically
• 116:38 - 116:45
for this function applied
for F of T and G of T.
• 116:45 - 116:49
So it was-- they
were just lucky.
• 116:49 - 116:52
Remember that you need
to know this chain rule.
• 116:52 - 116:54
It's going to be
one of the problems
• 116:54 - 116:57
to be emphasized in the exams.
• 116:57 - 117:02
Maybe one of the top 15 or
16 most important topics.
• 117:02 - 117:07
• 117:07 - 117:08
Is that OK?
• 117:08 - 117:09
Can I erase the whole thing?
• 117:09 - 117:10
OK.
• 117:10 - 117:11
Let me erase the whole thing.
• 117:11 - 117:44
• 117:44 - 117:45
OK.
• 117:45 - 117:46
Any other questions?
• 117:46 - 118:02
• 118:02 - 118:04
No?
• 118:04 - 118:05
I'm not going to let
you go right away,
• 118:05 - 118:08
we're going to work one
more problem or two more
• 118:08 - 118:09
simple problems.
• 118:09 - 118:11
And then we are going to go.
• 118:11 - 118:11
OK?
• 118:11 - 118:23
• 118:23 - 118:26
So question.
• 118:26 - 118:28
A question.
• 118:28 - 118:33
• 118:33 - 118:40
What do you think the
• 118:40 - 118:49
• 118:49 - 118:51
Two reasons, right.
• 118:51 - 118:54
Review number one.
• 118:54 - 118:59
If you have an increasingly
defined function,
• 118:59 - 119:03
then the gradient of F was what?
• 119:03 - 119:22
Equals direction of the
normal to the surface S--
• 119:22 - 119:26
let's say S is given
increasingly at the point
• 119:26 - 119:27
with [INAUDIBLE].
• 119:27 - 119:32
• 119:32 - 119:33
But any other reason?
• 119:33 - 120:00
• 120:00 - 120:02
Let's take that again.
• 120:02 - 120:06
Z equals x squared
plus y squared.
• 120:06 - 120:08
Let's compute a few
partial derivatives.
• 120:08 - 120:09
• 120:09 - 120:21
The gradient is Fs of x, Fs
of y, where this is F of xy
• 120:21 - 120:25
or Fs of xi plus Fs of yj.
• 120:25 - 120:29
• 120:29 - 120:31
[INAUDIBLE]
• 120:31 - 120:34
And we drew it.
• 120:34 - 120:42
I drew this case, and we also
drew another related example,
• 120:42 - 120:46
where we took Z equals 1 minus
x squared minus y squared.
• 120:46 - 120:47
And we went skiing.
• 120:47 - 120:52
And we were so happy last week
to go skiing, because we still
• 120:52 - 120:58
Mexico, and we-- and we
• 120:58 - 121:03
said now we computed the
Z to be minus 2x minus 2y.
• 121:03 - 121:06
• 121:06 - 121:10
And we said, I'm
looking at the slopes.
• 121:10 - 121:13
This is the x duration
and the y duration.
• 121:13 - 121:19
And I'm looking at the slopes of
the lines of these two curves.
• 121:19 - 121:24
So one that goes
down, like that.
• 121:24 - 121:25
So this was for what?
• 121:25 - 121:28
For y equals 0.
• 121:28 - 121:32
And this was for x equals 0.
• 121:32 - 121:37
• 121:37 - 121:40
Curve, x equals
0 curve in plane.
• 121:40 - 121:40
Right?
• 121:40 - 121:43
We just cross-section
our surface,
• 121:43 - 121:44
and we have this [INAUDIBLE].
• 121:44 - 121:52
And then we have the two
tangents, two slopes.
• 121:52 - 121:54
And we computed them everywhere.
• 121:54 - 122:00
• 122:00 - 122:02
At every point.
• 122:02 - 122:07
• 122:07 - 122:11
But realize that to go
up or down these hills,
• 122:11 - 122:15
I can go on a curve
like that, or I
• 122:15 - 122:18
can go-- remember the
train of Mickey Mouse going
• 122:18 - 122:20
on the hilly point on the hill?
• 122:20 - 122:22
We try to take different paths.
• 122:22 - 122:24
We are going hiking.
• 122:24 - 122:29
We are going hiking, and we'll
take hiking through the pass.
• 122:29 - 122:39
• 122:39 - 122:41
OK.
• 122:41 - 123:01
How do we get the maximum
rate of change of the function
• 123:01 - 123:04
Z equals F of x1?
• 123:04 - 123:06
So now I'm
anticipating something.
• 123:06 - 123:11
I'd like to see your intuition,
• 123:11 - 123:12
know what's going to happen.
• 123:12 - 123:14
And you know what
that from Mister--
• 123:14 - 123:15
STUDENT: Heinrich.
• 123:15 - 123:18
PROFESSOR TODA: [? Heinrich ?]
from high school.
• 123:18 - 123:21
rephrase the question
• 123:21 - 123:23
like a non-mathematician.
• 123:23 - 123:24
Let's go hiking.
• 123:24 - 123:30
This is [INAUDIBLE] we
go to the lighthouse.
• 123:30 - 123:34
Which path shall I take
on my mountain, my hill,
• 123:34 - 123:38
my god knows what
geography, in order
• 123:38 - 123:40
to obtain the maximum
rate of change?
• 123:40 - 123:44
That means the
highest derivative.
• 123:44 - 123:46
In what direction do I get
the highest derivative?
• 123:46 - 123:49
STUDENT: In what direction you
get the highest derivative--
• 123:49 - 123:51
PROFESSOR TODA: So
in which direction--
• 123:51 - 123:53
in which direction
on this hill do
• 123:53 - 123:55
I get the highest derivative?
• 123:55 - 123:57
The highest rate of change.
• 123:57 - 124:04
Rate of change means I want to
get the fastest possible way
• 124:04 - 124:05
somewhere.
• 124:05 - 124:08
STUDENT: The shortest slope?
• 124:08 - 124:10
Along just the straight line up.
• 124:10 - 124:11
PROFESSOR TODA: Along--
• 124:11 - 124:12
STUDENT: You don't want
to take any [INAUDIBLE].
• 124:12 - 124:13
PROFESSOR TODA: Right.
• 124:13 - 124:14
STUDENT: [INAUDIBLE].
• 124:14 - 124:15
It could be along any axis.
• 124:15 - 124:18
PROFESSOR TODA: So could
you see which direction
• 124:18 - 124:19
those are-- very good.
• 124:19 - 124:21
Actually you were getting
to the same direction.
• 124:21 - 124:24
So [INAUDIBLE] says
Magdalena, don't be silly.
• 124:24 - 124:28
The actual maximum rate of
change for the function Z
• 124:28 - 124:31
is obviously, because
it is common sense,
• 124:31 - 124:37
it's obviously happening if
you take the so-called-- what
• 124:37 - 124:38
are these guys?
• 124:38 - 124:41
[INAUDIBLE], not meridians.
• 124:41 - 124:42
STUDENT: Longtitudes?
• 124:42 - 124:43
PROFESSOR TODA: OK.
• 124:43 - 124:45
That is-- OK.
• 124:45 - 124:48
Suppose that we don't hike,
because it's too tiring.
• 124:48 - 124:51
We go down from the
top of the hill.
• 124:51 - 124:53
Ah, there's also very good idea.
• 124:53 - 124:59
So when you let yourself
go down on a sleigh,
• 124:59 - 125:03
don't think bobsled or
anything-- just a sleigh,
• 125:03 - 125:04
think of a child's sleigh.
• 125:04 - 125:08
No, take a plastic bag
and put your butt in it
• 125:08 - 125:11
and let yourself go.
• 125:11 - 125:14
What is their
direction actually?
• 125:14 - 125:20
fastest way to get down.
• 125:20 - 125:23
The fastest way to get
down will happen exactly
• 125:23 - 125:28
in the same
directions going down
• 125:28 - 125:30
in the directions
of these meridians.
• 125:30 - 125:34
• 125:34 - 125:36
OK?
• 125:36 - 125:37
And now, [INAUDIBLE].
• 125:37 - 125:46
• 125:46 - 125:59
The maximum rate of
change will always
• 125:59 - 126:07
happen in the direction
• 126:07 - 126:15
• 126:15 - 126:19
You can get a little
• 126:19 - 126:22
by just-- I would like this
• 126:22 - 126:24
until we get to that section.
• 126:24 - 126:27
In one section we will be there.
• 126:27 - 126:40
We also-- it's also reformulated
as the highest, the steepest,
• 126:40 - 126:42
ascent or descent.
• 126:42 - 126:45
The steepest.
• 126:45 - 126:59
The steepest ascent or
the steepest descent
• 126:59 - 127:10
always happens in the
• 127:10 - 127:15
• 127:15 - 127:17
Ascent is when you hike
to the top of the hill.
• 127:17 - 127:21
Descent is when you let yourself
go in the plastic [INAUDIBLE]
• 127:21 - 127:25
bag in the snow.
• 127:25 - 127:26
Right?
• 127:26 - 127:30
Can you verify this happens
just on this example?
• 127:30 - 127:33
It's true in general,
for any smooth function.
• 127:33 - 127:36
Our smooth function is
a really nice function.
• 127:36 - 127:40
• 127:40 - 127:43
Well again, it was 2x 2y, right?
• 127:43 - 127:46
• 127:46 - 127:51
And that means at a certain
point, x0 y0, whenever you are,
• 127:51 - 127:52
guys you don't
necessarily have to start
• 127:52 - 127:55
from the top of the hill.
• 127:55 - 127:59
You can be-- OK,
• 127:59 - 128:02
And here you are with
friends, or with mom and dad,
• 128:02 - 128:05
or whoever, on the hill.
• 128:05 - 128:09
You get out, you take the
sleigh, and you go down.
• 128:09 - 128:14
So no matter where
you are, there you go.
• 128:14 - 128:23
You have 2x0 times
i plus 2y0 times j.
• 128:23 - 128:32
And the direction of the
• 128:32 - 128:35
Do you like this one?
• 128:35 - 128:39
Well in this case,
if you were-- suppose
• 128:39 - 128:42
you were at the
point [INAUDIBLE].
• 128:42 - 128:49
• 128:49 - 128:54
You are at the point
of coordinates--
• 128:54 - 128:55
do you want to be here?
• 128:55 - 128:57
You want to be here, right?
• 128:57 - 128:59
So we've done that before.
• 128:59 - 129:03
I'll take it as 1 over
[? square root of ?]
• 129:03 - 129:10
2-- I'm trying to be creative
today-- [INAUDIBLE] y equals 0,
• 129:10 - 129:15
and Z equals-- what's left?
• 129:15 - 129:16
1/2, right?
• 129:16 - 129:18
Where am I?
• 129:18 - 129:20
Guys, do you realize where I am?
• 129:20 - 129:22
I'll [? take a ?] [INAUDIBLE].
• 129:22 - 129:24
• 129:24 - 129:25
y0.
• 129:25 - 129:29
So I need to be on this
meridian on the red thingy.
• 129:29 - 129:34
• 129:34 - 129:37
And somewhere here.
• 129:37 - 129:40
• 129:40 - 129:43
What's the duration
• 129:43 - 129:46
Delta z at this p.
• 129:46 - 129:57
• 129:57 - 129:59
Then you say ah,
well, I don't get it.
• 129:59 - 130:04
I have-- the second guy will
become 0, because y0 is 0.
• 130:04 - 130:07
The first guy will become
1 over square root of 2.
• 130:07 - 130:15
So I have 2 times 1 over square
root of 2 times i plus 0j.
• 130:15 - 130:29
It means in the direction of i--
in the direction of i-- from p,
• 130:29 - 130:39
I have the fastest-- fastest,
Magdalena, fastest-- descent
• 130:39 - 130:40
possible.
• 130:40 - 130:43
• 130:43 - 130:47
But we don't say in
the direction of i
• 130:47 - 130:50
in our everyday life, right?
• 130:50 - 130:53
Let's say geographic points.
• 130:53 - 130:59
We are-- I'm a bug,
and this is north.
• 130:59 - 131:00
This is south.
• 131:00 - 131:05
• 131:05 - 131:06
This is east.
• 131:06 - 131:09
• 131:09 - 131:11
And this is west.
• 131:11 - 131:18
So if I go east, going east
means going in the direction i.
• 131:18 - 131:23
• 131:23 - 131:26
Now suppose-- I'm going
to finish with this one.
• 131:26 - 131:29
Suppose that my house
is not on the prairie
• 131:29 - 131:32
but my house is here.
• 131:32 - 131:34
House, h.
• 131:34 - 131:38
Find me a wood
point to be there.
• 131:38 - 131:40
STUDENT: Northeast.
• 131:40 - 131:41
Or to get further down.
• 131:41 - 131:45
PROFESSOR TODA: Anything, what
would look like why I'm here?
• 131:45 - 131:48
x0, y0, z0.
• 131:48 - 131:50
Hm.
• 131:50 - 131:58
1/2, 1/2, and I
need the minimum.
• 131:58 - 132:03
So I want to be on the
bisecting plane between the two.
• 132:03 - 132:03
You understand?
• 132:03 - 132:04
This is my quarter.
• 132:04 - 132:07
And I want to be in
this bisecting plane.
• 132:07 - 132:10
So I'll take 1/2, 1/2, and
what results from here?
• 132:10 - 132:12
I have to do math.
• 132:12 - 132:16
1 minus 1/4 minus 1/4 is 1/2.
• 132:16 - 132:18
Right?
• 132:18 - 132:20
1/2, 1/2, 1/2.
• 132:20 - 132:22
This is where my house
is [? and so on. ?]
• 132:22 - 132:24
And this is full of smoke.
• 132:24 - 132:30
And what is the
maximum rate of change?
• 132:30 - 132:34
What is the steepest
descent is the trajectory
• 132:34 - 132:38
that my body will take
when I let myself go down
• 132:38 - 132:39
on the sleigh.
• 132:39 - 132:41
How do I compute that?
• 132:41 - 132:44
I will just do the same thing.
• 132:44 - 132:50
Delta z at the point x0
equals 1/2, y0 equals 1/2,
• 132:50 - 132:52
z0 equals 1/2.
• 132:52 - 132:54
Well what do I get as direction?
• 132:54 - 132:57
That will be the
• 132:57 - 133:03
2 times 1/2-- you
guys with me still?
• 133:03 - 133:09
i plus 2 times 1/2 with j.
• 133:09 - 133:14
And there is no Mr.
z0 In the picture.
• 133:14 - 133:15
Why?
• 133:15 - 133:17
Because that will
give me the direction
• 133:17 - 133:22
like on-- in a geographic way.
• 133:22 - 133:24
North, west, east, south.
• 133:24 - 133:26
These are the
direction in plane.
• 133:26 - 133:28
I'm not talking
directions on the hill,
• 133:28 - 133:31
I'm talking
directions on the map.
• 133:31 - 133:34
These are directions on the map.
• 133:34 - 133:36
So what is the direction
i plus j on the map?
• 133:36 - 133:40
If you show this to a
geography major and say,
• 133:40 - 133:43
I'm going in the direction
i plus j on the map,
• 133:43 - 133:46
he will say you are crazy.
• 133:46 - 133:48
He doesn't understand the thing.
• 133:48 - 133:50
But you know what you mean.
• 133:50 - 133:54
East for you is the
direction of i in the x-axis.
• 133:54 - 133:56
[INAUDIBLE]
• 133:56 - 133:58
And this is north.
• 133:58 - 134:00
Are you guys with me?
• 134:00 - 134:02
The y direction is north.
• 134:02 - 134:06
So I'm going perfectly
northeast at a 45-degree angle.
• 134:06 - 134:08
If I tell the
geography major I'm
• 134:08 - 134:11
going northeast perfectly in
the middle, he will say I know.
• 134:11 - 134:14
But you will know that
for you, that is i plus j.
• 134:14 - 134:16
Because you are
the mathematician.
• 134:16 - 134:17
Right?
• 134:17 - 134:19
So you go down.
• 134:19 - 134:21
And this is where you are.
• 134:21 - 134:22
And you're on the meridian.
• 134:22 - 134:25
This is the direction i plus j.
• 134:25 - 134:30
So if I want to project my
trajectory-- I went down
• 134:30 - 134:33
with the sleigh, all the way
down-- project the trajectory,
• 134:33 - 134:37
my trajectory is a
body on the snow.
• 134:37 - 134:39
Projecting it on the
ground is this one.
• 134:39 - 134:44
So it is exactly the
direction i plus j.
• 134:44 - 134:44
Right, guys?
• 134:44 - 134:48
So exactly northeast
perfectly at 45-degree angles.
• 134:48 - 134:51
Now one caveat.
• 134:51 - 134:53
One caveat, because
when we get there,
• 134:53 - 134:59
• 134:59 - 135:03
When we will say direction,
we are also crazy people.
• 135:03 - 135:05
I told you, mathematicians
are not normal.
• 135:05 - 135:07
You have to be a
little bit crazy
• 135:07 - 135:11
to want to do all the stuff
• 135:11 - 135:16
i plus j for us is not a
direction most of the time.
• 135:16 - 135:20
When we say direction, we mean
we normalize that direction.
• 135:20 - 135:23
We take the unit
vector, which is unique,
• 135:23 - 135:26
for responding to i plus j.
• 135:26 - 135:29
So what is that
unique unit vector?
• 135:29 - 135:33
You learned in Chapter 9
everything is connected.
• 135:33 - 135:34
It's a big circle.
• 135:34 - 135:35
i plus j, very good.
• 135:35 - 135:40
So direction is a unit vector
for most mathematicians,
• 135:40 - 135:45
which means you will be i
plus j over square root of 2.
• 135:45 - 135:52
remember, unlike Chapter 9,
• 135:52 - 135:56
direction is a unit vector.
• 135:56 - 136:00
In Chapter 9, Chapter 10,
it said direction lmn,
• 136:00 - 136:01
direction god knows what.
• 136:01 - 136:06
But in Chapter 11, direction
is a vector in plane,
• 136:06 - 136:08
like this one, i
plus [INAUDIBLE]
• 136:08 - 136:12
has to be a unique
normal-- a unique vector.
• 136:12 - 136:13
OK?
• 136:13 - 136:14
And we-- keep that in mind.
• 136:14 - 136:16
Next time, when we
meet on Thursday,
• 136:16 - 136:20
you will understand why
we need to normalize it.
• 136:20 - 136:23
Now can we say goodbye to
the snow and everything?
• 136:23 - 136:26
It's not going to
show up much anymore.
• 136:26 - 136:28
Remember this example.
• 136:28 - 136:31
flowers next time.
• 136:31 - 136:31
OK.
• 136:31 - 136:33
Have a nice day.
• 136:33 - 136:34
Yes, sir?
• 136:34 - 136:36
Let me stop the video.
• 136:36 - 136:37
Title:
TTU Math2450 Calculus3 Sec 11.4 and 11.5
Description:

Tangent Plane, Differentiability and Approximations and intro to Chain Rule

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Video Language:
English
 jackie.luft edited English subtitles for TTU Math2450 Calculus3 Sec 11.4 and 11.5