## TTU Math2450 Calculus3 Sec 11.4 part 1

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PROFESSOR: So let's
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and review what we learned
in 11.3, chapter 11.
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Chapter 11, again, was
functions of several variables.
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In our case, I'll say
functions of two variables.
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11.3 taught you, what?
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Taught you some
beautiful things.
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Practically, if you
understand this picture,
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you will remember everything.
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This picture is going to
try and [INAUDIBLE] a graph
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that's sitting
above here somewhere
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in Euclidean free space,
dimensional space.
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You have the origin.
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And you say I want markers.
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No, you don't say
I want markers.
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I say I want markers.
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We want to fix a point
x0, y0 on the surface,
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assuming the surface is smooth.
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That x0 of mine
should be projected.
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I'm going to try to draw
better than I did last time.
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X0, y0 corresponds
to a certain altitude
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z0 that is projected like that.
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And this is my
[INAUDIBLE] 0 here.
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But I don't care much
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fact that locally, I
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represent the function
as a graph-- z of f-- f
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of x and y defined
over a domain.
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I have a domain
that is an open set.
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And you connect to-- that's
more than you need to know.
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Could be anything.
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Could be a square, could be
a-- this could be something,
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a nice patch of them like.
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So the projection of my
point here is x0, y0.
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I'm going to draw these
parallels as well as I can.
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But I cannot draw very well.
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But I'm trying.
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X0 and y0-- and
remember from last time.
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What did we say?
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I'm going to draw a plane
of equation x equals x0.
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All right, I'll try.
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I'll try and do a good job--
x equals x0 is this plane.
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STUDENT: Don't you
have the x amount
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and the y amounts backward?
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Or [INAUDIBLE]
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PROFESSOR: No.
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STUDENT: [INAUDIBLE]
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PROFESSOR: x is this 1
coming towards you like that.
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that always, Ryan.
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Do I have them backward?
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This time, I was lucky.
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I didn't have them backward.
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So y goes this way.
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For y0, let me pick another
color, a more beautiful color.
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For y0, my video is not
going to see the y0.
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But hopefully, it's going to
see it, this beautiful line.
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Spring is coming.
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So this is going
to be the plane.
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Label it [INAUDIBLE]
y equals 0y.
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Now, the green plane cuts the
surface into a plane curve,
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of course, because
teasing the plane
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that I drew with the line.
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And in the plane that I drew
with red-- was it red or pink?
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It's Valentine's Day.
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It's pink.
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OK, so I have it like that.
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So what is the pink curve?
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The pink curve is the
intersection between z
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equals-- x equals x0
plane with my surface.
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My surface is black.
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I'm going to say s on surface.
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And then I have a pink curve.
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Let's call it c1.
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Because you cannot see
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You only can imagine that
it's not the same thing.
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C2 is y equals y0 plane
intersected with s.
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And what have we
learned last time?
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Last time, we learned that
we introduce some derivatives
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at the point at 0, y is
0, so that they represent
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those partial derivatives of
the function z with respect
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to x and y.
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So we have the partial z sub x
at x0, y0 and the partial and z
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sub y at x0, y0.
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Do we have a more
elegant definition?
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That's elegant enough for
me, thank you very much.
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But if I wanted to give the
original definition, what
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was that?
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That is d of bx at x0, y0, which
is a limit of the difference
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quotient.
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And this time, we're going to--
not going to do the x of y.
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I I'm different today.
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So I do h goes to 0.
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H is my smallest
displacement of [INAUDIBLE].
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Here, I have f of-- now,
who is the variable?
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X.
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So who is going to say fixed?
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Y. So I'm going to say I'm
displacing mister x0 with an h.
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And y0 will be fixed minus
f of x0, y0, all over h.
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of a delta x, I call the h.
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And the derivative
with respect to y
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will assume that
x0 is a constant.
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I saw how well
[INAUDIBLE] explained that
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and I'm ambitious.
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I want to do an even better
job than [INAUDIBLE].
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Hopefully, I might manage.
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D of ty equals [INAUDIBLE] h
going to 0 of that CF of-- now,
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who's telling me what we have?
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Of course, mister
x, y and y, yy.
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F of x0, y0 is their constant
waiting for his turn.
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And then you'll have, what?
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H0 is fixed, right?
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STUDENT: So h0 is--
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PROFESSOR: --fixed.
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Y is the variable.
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So I go into the direction
of y starting from y0.
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And I displace that with
a small quantity, right?
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So these are my
partial velocity--
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my partial derivatives, I'm
sorry, not partial velocities.
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Forget what I said.
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I said something that
you will learn later.
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What are those?
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Those are the slopes at x0, y0
of the tangents at the point
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here, OK?
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The tangents to the two curves,
the pink one-- the pink one
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and the green one, all right?
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For the pink one, for the pink
curve, what is the variable?
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The variable is the y, right?
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So this is c1 is a
curve that depends on y.
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And c2 is a curve
that depends on x.
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So this comes with x0 fixed.
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I better write it like that.
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F of x is 0.
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Y, instead of c2 of x, I'll
say f of y-- f of x and yz.
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So, which slope is which?
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The d of dy at this point
is the slope to this one.
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Are you guys with me?
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The slope of that tangent.
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Considered in the
plane where it is.
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S of x will be the slope of this
line in the green plane, OK?
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That is considered as a
plane of axis of coordinates.
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Good, good-- so we
know what they are.
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A quick example to review--
I've given you some really ugly,
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nasty functions today.
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The last time, you
did a good job.
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So today, I'm not
challenging you anymore.
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I'm just going to give
you one simple example.
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does this guy look like
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and what will the meanings
of z sub x and z sub y be?
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What will they be at?
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Let's say I think I know what I
want to take at the point 0, 0.
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And maybe you're going to
tell me what else it will be.
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And eventually at
another point like z
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sub a, so coordinates, 1
over square root of 2 and 1
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over square root of 2.
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And v sub y is same-- 1
over square root of 2,
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1 over square root of 2.
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Can one draw them and have
a geometric explanation
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of what's going on?
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Well, I don't want you to
forget the definitions,
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but since you absorbed them
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and with your eyes, you're not
going to need them anymore.
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We should be able to draw
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I'm sure you love it.
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When it's-- what
does it look like?
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STUDENT: [INAUDIBLE]
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PROFESSOR: Wait a minute, you're
not awake or I'm not awake.
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So if you do x squared
plus y squared,
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It would be that.
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And what is this?
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STUDENT: That's a [INAUDIBLE].
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PROFESSOR: A circular
paraboloid-- you are correct.
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We've done that before.
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I'd say it looks
like an egg shell,
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but it's actually--
this is a parabola
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if it's going to infinity.
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And you said a bunch of circles.
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Yes, sir.
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STUDENT: So is it an
upside down graph?
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PROFESSOR: It's an
upside down paraboloid.
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STUDENT: [INAUDIBLE]
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PROFESSOR: So, very
good-- how do we do that?
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We make this guy
look in the mirror.
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This is the lake.
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The lake is xy plane.
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So this guy is
looking in the mirror.
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Take his image and
shift it just like he
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said-- shift it one unit up.
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This is one.
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You're going to have
another paraboloid.
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So from this construction,
I'm going to draw.
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And he's going to look
like you took a cup
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and you put it upside down.
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But it's more like
an eggshell, right?
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It's not a cup because
a cup is supposed
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to have a flat bottom, right?
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But this is like an eggshell.
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And I'll draw.
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And for this fellow, we
have a beautiful picture
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that looks like this hopefully
But I'm going to try and draw.
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STUDENT: Are you looking
from a top to bottom?
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PROFESSOR: We can look it
whatever you want to look.
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That's a very good thing.
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You're getting too close
to what I wanted to go.
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We'll discuss in one minute.
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So you can imagine this
is a hill full of snow.
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Although in two days,
we have Valentine's Day
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and there is no snow.
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But assume that we
go to New Mexico
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and we find a hill
that more or less looks
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like a perfect hill like that.
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And we start thinking
of skiing down the hill.
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Where am I at 0, 0?
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I am on top of the hill.
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I'm on top of the
hill and I decide
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to analyze the slope
of the tangents
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to the surface in the
direction of-- who is this?
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Like now, and you
make me nervous.
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So in the direction of
y, I have one slope.
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In the direction of x, I have
another slope in general.
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Only in this case, they
are the same slope.
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And what is that same slope if
I'm here on top of the hill?
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This is me-- well, I don't
know, one of you guys.
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That looks horrible.
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What's going to happen?
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We don't want to think about it.
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But it definitely is too steep.
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So this will be the slope
of the line in the direction
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with respect to y.
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So I'm going to think
f sub y and f sub
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x if I change my skis go
this direction and I go down.
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So I could go down this
way and break my neck.
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Or I could go down this way
and break my neck as well.
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OK, it has to go like-- right?
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Can you tell me what
these guys will be?
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I'm going to put them in pink
because they're beautiful.
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STUDENT: 0 [INAUDIBLE].
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PROFESSOR: Thank God,
they are beautiful.
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Larry, what does it mean?
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That means that the two
tangents, the tangents
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to the curves, are horizontal.
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And if I were to draw the plane
between those two tangents--
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one tangent is in pink
pen, our is in green.
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I'm in a good mood.
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And that's going to be the
so-called tangent plane--
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tangent plane to the surface
at x0, y0, which is the origin.
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That was a nice point.
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That is a nice point.
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Not all the points
will be [INAUDIBLE]
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and nice but beautiful.
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[INAUDIBLE] I take the
nice-- well, not so nice,
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I don't know.
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You'll have to figure it out.
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How do I get-- well, first
of all, where is this point?
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If I take x to be 1 over square
2 and y to be 1 over square 2
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and I plug them in,
what's z going to be?
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STUDENT: 0.
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PROFESSOR: 0, and I
did that on purpose.
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Because in that case, I'm
going to be on flat line again.
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This look like [INAUDIBLE].
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Except [INAUDIBLE]
is not z equal 0.
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What is [INAUDIBLE] like?
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z equals--
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STUDENT: [INAUDIBLE]
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PROFESSOR: Huh?
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STUDENT: I don't know.
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PROFESSOR: Do you want to
go in meters or in feet?
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STUDENT: [INAUDIBLE].
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PROFESSOR: Yes, I don't know.
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kilometer, 1,000 something
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meters.
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But somebody said it's more so.
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It's flat land, and I'd say
about a mile above the sea
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level.
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All right, now, I am going to
be in flat land right here,
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1 over a root 2, 1
over a root 2, and 0.
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What happened here?
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broke my neck, you know.
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Well, if I came
in this direction,
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I would need to draw a
prospective trajectory that
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was hopefully not mine.
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And the tangent would--
the tangent, the slope
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of the tangent, would be funny.
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Let's see what you need to do.
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You need to say, OK, prime
with respect to x, minus 2x.
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And then at the point x
equals 1 over [INAUDIBLE] 2 y
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equals 1 over 2,
you just plug in.
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And what do you have?
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STUDENT: Square
root of [INAUDIBLE].
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PROFESSOR: Negative
square root of 2-- my god
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that is really bad as a slope.
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It's a steep slope.
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And this one-- how
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Same idea, symmetric function.
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And it's going to be exactly
the same-- very steep slope.
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Why are they negative numbers?
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Because the slope is
going down, right?
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That's the kind of slope I
have in both directions--
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one and-- all right.
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If I were to draw
this thing continuing,
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how would I represent
those slopes?
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This circle-- this circle is
just making my life harder.
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But I would need to imagine
those slopes as being
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like I'm here, all right?
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Are you guys with me?
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And I will need to draw
x0-- well, what is that?
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1 over root 2 and 1 over root
2 And I would draw two planes.
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And I would have two curves.
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And when you slice
up, imagine this
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would be a piece of cheese.
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STUDENT: [INAUDIBLE]
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PROFESSOR: And you cut--
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STUDENT: [INAUDIBLE]
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PROFESSOR: Right?
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STUDENT: Yeah.
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PROFESSOR: And you cut
in this other side.
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Well, this is the one
that's facing you.
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You cut like that.
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And when you cut like
this, it's facing--
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STUDENT: [INAUDIBLE]
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PROFESSOR: Hm?
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But anyway, let's not
draw the other one.
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It's hard, right?
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STUDENT: [INAUDIBLE] angle
like this-- just the piece
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of the corner of the cheese.
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PROFESSOR: Right.
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STUDENT: The corner
is facing you.
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PROFESSOR: So yeah, so it's--
the corner is facing you.
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STUDENT: So basically,
you [INAUDIBLE] this.
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PROFESSOR: But-- exactly, but--
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STUDENT: Like this.
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PROFESSOR: Yeah, well--
yeah, it's hard to draw.
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So practically,
this is what you're
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looking at it is slope that's
negative in both directions.
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So you're going to go
this way or this way.
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And it's much steeper than
you imagine [INAUDIBLE].
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OK, they are equal.
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I'm trying to draw them equal.
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I don't know how
equal they can be.
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One belongs to one plane
just like you said.
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This belongs to this plane.
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And the green one belongs to
the plane that's facing you.
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So the slope goes this way.
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But the two slopes are equal.
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You have to have a little
bit of imagination.
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We would need some cheese
to make a mountain of cheese
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and cut them and slice them.
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We'll eat everything
after, yeah.
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All right, let's move on to
something more challenging
• 20:29 - 20:32
now that we got to
the tangent plane.
• 20:32 - 20:34
So if somebody would
say, wait a minute,
• 20:34 - 20:38
you said this is the tangent
plane to the surface.
• 20:38 - 20:40
You just introduced
a new notion.
• 20:40 - 20:41
You were fooling us.
• 20:41 - 20:44
I'm fooling you guys.
• 20:44 - 20:49
It's not April 1, but this
kind of a not a neat thing.
• 20:49 - 20:59
I just tried to introduce
you into the section 11.4.
• 20:59 - 21:03
So if you have a piece
of a curve that's smooth
• 21:03 - 21:08
and you have a point
x0, y0, can you
• 21:08 - 21:13
find out the equation
of the tangent plane?
• 21:13 - 21:17
Pi, and this is s form surface.
• 21:17 - 21:21
How can I find the equation
of the tangent plane?
• 21:21 - 21:27
• 21:27 - 21:33
That x0, y0-- 12 is
going to be also z0.
• 21:33 - 21:45
But what I mean that x0,
y0 is in on the floor
• 21:45 - 21:47
as a projection.
• 21:47 - 21:50
So I'm always
looking at the graph.
• 21:50 - 21:52
And that's why.
• 21:52 - 21:54
The moment I stop
looking at the graph,
• 21:54 - 21:55
things will be different.
• 21:55 - 22:00
But I'm looking at the graph
of independent variables x, y.
• 22:00 - 22:03
And that's why those guys
are always on the floor.
• 22:03 - 22:08
A and z would be a function
to keep in the variable.
• 22:08 - 22:09
Now, does anybody know?
• 22:09 - 22:13
Because I know you guys
• 22:13 - 22:16
and you have better
teachers than me.
• 22:16 - 22:17
You have the internet.
• 22:17 - 22:18
• 22:18 - 22:19
• 22:19 - 22:21
• 22:21 - 22:25
I know from a bunch of you
• 22:25 - 22:27
over half of the chapter 11.
• 22:27 - 22:31
I just hope that now you
can compare what you learned
• 22:31 - 22:34
with what I'm teaching
you, And I'm not
• 22:34 - 22:37
expecting you to go in
• 22:37 - 22:39
• 22:39 - 22:44
office hours on yesterday.
• 22:44 - 22:46
Because Tuesday, I
didn't have office hours.
• 22:46 - 22:50
• 22:50 - 22:56
So what equation corresponds
to the tangent plate?
• 22:56 - 22:57
STUDENT: [INAUDIBLE]
• 22:57 - 23:00
• 23:00 - 23:02
PROFESSOR: Several
of you know it.
• 23:02 - 23:04
You know what I hated?
• 23:04 - 23:05
It's fine that you know it.
• 23:05 - 23:08
I'm proud of you guys
and I'll write it.
• 23:08 - 23:12
But when I was a freshman--
or what the heck was I?
• 23:12 - 23:17
A sophomore I think-- no, I was
a freshman when they fed me.
• 23:17 - 23:19
They spoon-fed me this equation.
• 23:19 - 23:22
And I didn't understand
anything at the time.
• 23:22 - 23:26
I hated the fact that
the Professor painted it
• 23:26 - 23:31
on the board just like
that out of the blue.
• 23:31 - 23:34
I want to see a proof.
• 23:34 - 23:40
And he was able to-- I think
he could have done a good job.
• 23:40 - 23:43
But he didn't.
• 23:43 - 23:47
He showed us a bunch
of justifications
• 23:47 - 23:53
like if you generally have
a surface in implicit form,
• 23:53 - 23:58
I told you that
• 23:58 - 24:02
represents the normal
connection, right?
• 24:02 - 24:07
And he prepared us pretty
good for what could
• 24:07 - 24:09
have been the proof of that.
• 24:09 - 24:10
He said, OK, guys.
• 24:10 - 24:12
You know the duration
of the normal
• 24:12 - 24:16
as even as the gradient over
• 24:16 - 24:18
if you want unit normal.
• 24:18 - 24:19
How did he do that?
• 24:19 - 24:22
bunch of examples.
• 24:22 - 24:23
• 24:23 - 24:26
He showed us that
for the sphere,
• 24:26 - 24:30
you have the normal,
which is the continuation
• 24:30 - 24:31
of the position vector.
• 24:31 - 24:34
Then he said, OK, you
can have approximations
• 24:34 - 24:39
of a surface that is smooth and
round with oscillating spheres
• 24:39 - 24:45
just the way you have for a
curve, a resonating circle,
• 24:45 - 24:50
a resonating circle-- that's
called oscillating circle.
• 24:50 - 24:53
Resonating circle-- in that
case, what will the normal be?
• 24:53 - 24:56
Well, the normal
will have to depend
• 24:56 - 24:57
on the radius of the circle.
• 24:57 - 25:02
So you have a principal normal
or a normal if it's a plane
• 25:02 - 25:03
curve.
• 25:03 - 25:06
And it's easy to
understand that's the same
• 25:06 - 25:07
• 25:07 - 25:13
So we have enough
justification for the direction
• 25:13 - 25:16
a function is always
• 25:16 - 25:20
normal-- normal to the
surface, normal to all
• 25:20 - 25:23
the curves on the surface.
• 25:23 - 25:27
If we want to find
that without swallowing
• 25:27 - 25:32
this like I had to when I
was a student, it's not hard.
• 25:32 - 25:35
And let me show
you how we do it.
• 25:35 - 25:37
We start from the graph, right?
• 25:37 - 25:41
Z equals f of x and y.
• 25:41 - 25:44
And we say, well, Magdalena,
but this is a graph.
• 25:44 - 25:48
It's not an implicit equation.
• 25:48 - 25:50
And I'll say, yes it is.
• 25:50 - 25:54
Let me show you how I make
it an implicit equation.
• 25:54 - 25:56
I move z to the other side.
• 25:56 - 26:01
I put 0 equals f of xy minus z.
• 26:01 - 26:04
Now it is an implicit equation.
• 26:04 - 26:06
So you say you cheated.
• 26:06 - 26:08
Yes, I did.
• 26:08 - 26:08
I have cheated.
• 26:08 - 26:11
• 26:11 - 26:15
It's funny that whenever
somebody gives you a graph,
• 26:15 - 26:17
you can rewrite that
graph immediately
• 26:17 - 26:19
as an implicit equation.
• 26:19 - 26:24
So that implicit equation
is of the form big F of xyz
• 26:24 - 26:27
now equals a
constant, which is 0.
• 26:27 - 26:32
F of xy is your old
friend and minus z.
• 26:32 - 26:36
Now, can you tell me what is
the normal to this surface?
• 26:36 - 26:40
Yeah, give me a splash
in a minute like that.
• 26:40 - 26:43
So what is the gradient of f?
• 26:43 - 26:45
be the normal.
• 26:45 - 26:47
I don't care if
it's unit or not.
• 26:47 - 26:49
To heck with the unit or normal.
• 26:49 - 26:54
I'm going to say I wanted
prime with respect to x, y,
• 26:54 - 26:58
and z respectively.
• 26:58 - 27:00
• 27:00 - 27:02
Is the vector.
• 27:02 - 27:07
Big F sub x comma big F
sub y comma big F sub z.
• 27:07 - 27:09
We see that last time.
• 27:09 - 27:14
function is the vector
• 27:14 - 27:17
whose coordinates are
the partial velocity--
• 27:17 - 27:20
• 27:20 - 27:22
Can we represent this again?
• 27:22 - 27:23
I don't know.
• 27:23 - 27:24
You need to help me.
• 27:24 - 27:29
Who is big F prime
with respect to x?
• 27:29 - 27:30
There is no x here.
• 27:30 - 27:33
Thank God that's
like a constant.
• 27:33 - 27:36
I just have to take this
little one, f, and prime it
• 27:36 - 27:37
with respect to x.
• 27:37 - 27:41
And that's exactly what that's
going to be-- little f sub x.
• 27:41 - 27:45
What is big F with respect to y?
• 27:45 - 27:46
STUDENT: [INAUDIBLE]
• 27:46 - 27:49
PROFESSOR: Little f sub
y prime with respect
• 27:49 - 27:52
to y-- differentiated
with respect to y.
• 27:52 - 27:56
And finally, if I differentiated
with respect to z,
• 27:56 - 27:58
there is no z here, right?
• 27:58 - 27:58
There is no z.
• 27:58 - 28:00
So that's like a constant.
• 28:00 - 28:05
Prime [INAUDIBLE] 0 and minus 1.
• 28:05 - 28:06
• 28:06 - 28:07
I know the normal.
• 28:07 - 28:09
This is the normal.
• 28:09 - 28:14
Now, if somebody gives you
the normal, there you are.
• 28:14 - 28:20
You have the normal to the
surface-- normal to surface.
• 28:20 - 28:21
What does it mean?
• 28:21 - 28:26
Equals normal to the tangent
plane to the surface.
• 28:26 - 28:30
Normal or perpendicular
to the tangent plane-
• 28:30 - 28:38
to the plane-- of the surface.
• 28:38 - 28:41
At that point--
point is the point p.
• 28:41 - 28:44
• 28:44 - 28:53
All right, so if you were to
study a surface that's-- do you
• 28:53 - 28:54
have a [INAUDIBLE]?
• 28:54 - 28:55
STUDENT: Uh, no.
• 28:55 - 28:56
Do you?
• 28:56 - 28:56
PROFESSOR: OK.
• 28:56 - 28:59
• 28:59 - 29:04
OK, I want to study
the tangent plane
• 29:04 - 29:05
at this point to the surface.
• 29:05 - 29:07
Well, that's flat, Magdalena.
• 29:07 - 29:08
You have no imagination.
• 29:08 - 29:14
The tangent plane is this plane,
is the same as the surface.
• 29:14 - 29:16
So, no fun-- no fun.
• 29:16 - 29:20
favorite plane here
• 29:20 - 29:25
and I take-- what is-- OK.
• 29:25 - 29:27
I have-- this is
Children Internationals.
• 29:27 - 29:30
I have a little girl
• 29:30 - 29:34
So you have a point
here and a plane
• 29:34 - 29:39
that passes through that point.
• 29:39 - 29:41
This is the tangent plane.
• 29:41 - 29:44
And my finger is the normal.
• 29:44 - 29:47
And the normal, we call
that normal to the surface
• 29:47 - 29:49
when it's normal to
the tangent plane.
• 29:49 - 29:53
At every point, this
is what the normal is.
• 29:53 - 29:55
All right, can we write
that based on chapter nine?
• 29:55 - 29:59
Now I will see what you remember
from chapter nine if anything
• 29:59 - 30:00
at all.
• 30:00 - 30:04
• 30:04 - 30:09
All right, how do we
write the tangent plane
• 30:09 - 30:12
if we know the normal?
• 30:12 - 30:22
OK, review-- if the normal
vector is ai plus bj plus ck,
• 30:22 - 30:28
that means the plane that
is perpendicular to it
• 30:28 - 30:31
is of what form?
• 30:31 - 30:38
Ax plus by plus cz
plus d equals 0, right?
• 30:38 - 30:40
You've learned that
in chapter nine.
• 30:40 - 30:43
Most of you learned that
last semester in Calculus 2
• 30:43 - 30:45
at the end.
• 30:45 - 30:51
Now, if my normal is f sub
x, f sub y, and minus 1,
• 30:51 - 30:53
those are ABC for God's sake.
• 30:53 - 30:54
Well, good.
• 30:54 - 30:59
Big A, big B, big C
at the given point.
• 30:59 - 31:10
So I'm going to have f sub
x at the given point d times
• 31:10 - 31:18
x plus f sub y at any
given point d times y.
• 31:18 - 31:19
Who is c?
• 31:19 - 31:20
C is minus 1.
• 31:20 - 31:24
Minus 1 times z is--
say you're being silly.
• 31:24 - 31:27
Magdalena, why do
you write minus 1?
• 31:27 - 31:29
Just because I'm having fun.
• 31:29 - 31:33
And plus, d equals 0.
• 31:33 - 31:35
And you say, well,
wait, wait, wait.
• 31:35 - 31:40
This starts looking like that
but it's not the same thing.
• 31:40 - 31:43
All right, what?
• 31:43 - 31:44
How do you get to d?
• 31:44 - 31:48
• 31:48 - 31:51
Now, actually, the
plane perpendicular
• 31:51 - 31:55
to n that passes
through a given point
• 31:55 - 31:59
can be written
much faster, right?
• 31:59 - 32:06
So if a plane is perpendicular
to a certain line,
• 32:06 - 32:10
how do we write if
we know a point?
• 32:10 - 32:15
If we know a point
in the normal ABC--
• 32:15 - 32:19
I have to go backwards to
• 32:19 - 32:23
the plane is going
to be x minus x0
• 32:23 - 32:30
plus b times y times y0 plus
c times z minus c0 equals 0.
• 32:30 - 32:33
• 32:33 - 32:35
So who is the d?
• 32:35 - 32:39
The d is all the constant
that gets out of here.
• 32:39 - 32:44
So the point x0, y0, z0
has to verify the plane.
• 32:44 - 32:47
And that's why when
you plug in x0, y0, z0,
• 32:47 - 32:50
you get 0 plus 0
plus 0 equals 0.
• 32:50 - 32:53
That's what it means for a
point to verify the plane.
• 32:53 - 32:59
When you take the x0, y0, z0 and
you plug it into the equation,
• 32:59 - 33:03
you have to have an
identity 0 equals 0.
• 33:03 - 33:07
So this can be rewritten
zx plus by plus cz
• 33:07 - 33:11
just like we did there plus a d.
• 33:11 - 33:13
And who in the world is the d?
• 33:13 - 33:19
The d will be exactly minus
ax0 minus by0 minus cz0.
• 33:19 - 33:23
If that makes you uncomfortable,
this is in chapter nine.
• 33:23 - 33:29
Look at the equation of a
plane and the normal to it.
• 33:29 - 33:33
Now I know that I can do
better than that if I'm smart.
• 33:33 - 33:36
So again, I collect the ABC.
• 33:36 - 33:37
Now I know my ABC.
• 33:37 - 33:40
• 33:40 - 33:42
I put them in here.
• 33:42 - 33:46
So I have f sub x at
the point in time.
• 33:46 - 33:51
Oh, OK, x minus x0
plus, who is my b?
• 33:51 - 33:57
F sub y computed at the
point p times y minus y0.
• 33:57 - 33:59
And, what?
• 33:59 - 34:00
Minus, right?
• 34:00 - 34:04
Minus-- minus 1.
• 34:04 - 34:05
I'm not going to write minus 1.
• 34:05 - 34:07
You're going to make fun of me.
• 34:07 - 34:10
Minus z minus cz.
• 34:10 - 34:12
And my proof is done.
• 34:12 - 34:17
QED-- what does it mean, QED?
• 34:17 - 34:20
In Latin.
• 34:20 - 34:23
QED means I proved
what I wanted to prove.
• 34:23 - 34:24
Do you know what it stands for?
• 34:24 - 34:27
Did you take Latin, any of you?
• 34:27 - 34:29
You took Latin?
• 34:29 - 34:33
Quod erat demonstrandum.
• 34:33 - 34:37
• 34:37 - 34:39
So this was to be proved.
• 34:39 - 34:42
That's exactly what
it was to be proved.
• 34:42 - 34:45
That, what, that
c minus z0, which
• 34:45 - 34:50
was my fellow over here pretty
in pink, is going to be f sub x
• 34:50 - 34:55
times x minus x0 plus yf
sub y times y minus y0.
• 34:55 - 35:01
So now you know why the equation
of the tangent plane is that.
• 35:01 - 35:05
I proved it more or less,
making some assumptions,
• 35:05 - 35:07
some axioms as assumption.
• 35:07 - 35:10
But you don't know
how to use it.
• 35:10 - 35:11
So let's use it.
• 35:11 - 35:14
So for the same valley--
not valley, hill--
• 35:14 - 35:16
it was full of snow.
• 35:16 - 35:19
Z equals 1 minus x
squared-- what was you
• 35:19 - 35:21
guys have forgotten?
• 35:21 - 35:25
OK, 1 minus x squared
minus y squared.
• 35:25 - 35:33
Find the tangent plane
at the following points.
• 35:33 - 35:37
Ah, x0, y0 to be origin.
• 35:37 - 35:39
And you say, did you
say that that's trivial?
• 35:39 - 35:40
Yes, it is trivial.
• 35:40 - 35:43
But I'm going to do
it one more time.
• 35:43 - 35:47
And what was my
[INAUDIBLE] point before?
• 35:47 - 35:49
STUDENT: [INAUDIBLE]
• 35:49 - 35:53
PROFESSOR: 1 over
2 and 1 over 2.
• 35:53 - 35:58
OK, and what will be the
corresponding point in 3D?
• 35:58 - 36:02
1 over 2, 1 over 2, I plug in.
• 36:02 - 36:04
Ah, yes.
• 36:04 - 36:07
And with this, I hope
to finish the day so we
• 36:07 - 36:10
can go to our other businesses.
• 36:10 - 36:12
Is this hard?
• 36:12 - 36:16
Now, I was not able-- I
have to be honest with you.
• 36:16 - 36:21
I was not able to memorize the
equation of a tangent plane
• 36:21 - 36:27
when I was-- when I was young,
like a freshman and sophomore.
• 36:27 - 36:30
understand that this
• 36:30 - 36:33
is a linear approximation
of a curved something.
• 36:33 - 36:36
This practically like
the Taylor equation
• 36:36 - 36:39
for functions of
two variables when
• 36:39 - 36:43
third term and so on.
• 36:43 - 36:46
You just take the--
I'll teach you
• 36:46 - 36:52
next time when this is, a first
order linear approximation.
• 36:52 - 36:54
All right, can we do
this really quickly?
• 36:54 - 36:56
It's going to be
a piece of cake.
• 36:56 - 36:57
Let's see.
• 36:57 - 36:58
Again, how do we do that?
• 36:58 - 37:00
This is f of x and y.
• 37:00 - 37:02
We computed that again.
• 37:02 - 37:04
• 37:04 - 37:08
Guys, if I say something
silly, will you stop me?
• 37:08 - 37:13
F of f sub x-- f
of y at 0, 0 is 0.
• 37:13 - 37:14
So I have two slopes.
• 37:14 - 37:16
Those are my hands.
• 37:16 - 37:19
The slopes of my hands are 0.
• 37:19 - 37:27
So the tangent plane will
be z minus z0 equals 0.
• 37:27 - 37:29
What is the 0?
• 37:29 - 37:30
STUDENT: 1
• 37:30 - 37:31
PROFESSOR: 1, excellent.
• 37:31 - 37:32
STUDENT: [INAUDIBLE]
• 37:32 - 37:33
PROFESSOR: Why is that 1?
• 37:33 - 37:36
0 and 0 give me 1.
• 37:36 - 37:40
So that was the picture
that I had z equals 1
• 37:40 - 37:43
as the tangent plane at
the point corresponding
• 37:43 - 37:45
to the origin.
• 37:45 - 37:48
That look like the
north pole, 0, 0, 1.
• 37:48 - 37:50
OK, no.
• 37:50 - 37:53
It's the top of a hill.
• 37:53 - 37:56
And finally, one last
thing [INAUDIBLE].
• 37:56 - 37:58
Maybe you can do
this by yourselves,
• 37:58 - 38:01
but I will shut up if I can.
• 38:01 - 38:03
I can't in general,
but I'll shut up.
• 38:03 - 38:09
Let's see-- f sub x at 1
over root 2, 1 over root 2.
• 38:09 - 38:10
Why was that?
• 38:10 - 38:12
What is f sub x?
• 38:12 - 38:14
STUDENT: The square root of--
negative square root of 2.
• 38:14 - 38:17
PROFESSOR: Right,
we've done that before.
• 38:17 - 38:20
And you got exactly what
you said-- [INAUDIBLE]
• 38:20 - 38:25
2 f sub y at the same point.
• 38:25 - 38:30
I am too lazy to write it
down again-- minus root 2.
• 38:30 - 38:33
And how do we actually
• 38:33 - 38:37
so we can go home and
whatever-- to the next class?
• 38:37 - 38:39
Is it hard?
• 38:39 - 38:40
No.
• 38:40 - 38:41
• 38:41 - 38:44
Z minus-- now, attention.
• 38:44 - 38:46
What is z0?
• 38:46 - 38:47
STUDENT: 0.
• 38:47 - 38:49
PROFESSOR: 0, right.
• 38:49 - 38:49
Why is that?
• 38:49 - 38:54
Because when I plug 1 over
a 2, 1 over a 2, I got 0.
• 38:54 - 38:57
0-- do I have to write it down?
• 38:57 - 38:59
No, not unless I
want to be silly.
• 38:59 - 39:02
But if you do write
down everything
• 39:02 - 39:05
and you don't simplify
the equation of the plane,
• 39:05 - 39:09
we don't penalize you in
any way in the final, OK?
• 39:09 - 39:14
So if you show your work like
that, you're going to be fine.
• 39:14 - 39:17
What is that 1 over 2?
• 39:17 - 39:25
Plus minus root 2 times
y minus 1 over root 2.
• 39:25 - 39:28
Is it elegant?
• 39:28 - 39:31
No, it's not elegant at all.
• 39:31 - 39:36
So as the last row for
today, one final line.
• 39:36 - 39:40
Can we make it
look more elegant?
• 39:40 - 39:44
Do we care to make
it more elegant?
• 39:44 - 39:48
Definitely some of you care.
• 39:48 - 39:52
Z will be minus root 2x.
• 39:52 - 39:57
I want to be consistent and
keep the same style in y.
• 39:57 - 39:59
And yet the constant
goes wherever
• 39:59 - 40:00
it wants to go at the end.
• 40:00 - 40:02
What's that constant?
• 40:02 - 40:03
STUDENT: 2 [INAUDIBLE].
• 40:03 - 40:05
PROFESSOR: So you
see what you have.
• 40:05 - 40:06
You have this times that.
• 40:06 - 40:08
It's a 1, this then that is a 1.
• 40:08 - 40:10
1 plus 1 is 2.
• 40:10 - 40:12
All right, are you
happy with this?
• 40:12 - 40:14
I'm not.
• 40:14 - 40:16
I'm happy.
• 40:16 - 40:18
You-- if this were
a multiple choice,
• 40:18 - 40:22
you would be able to
recognize it right away.
• 40:22 - 40:25
What's the standardized general
equation of a plane, though?
• 40:25 - 40:29
Something x plus something y
plus something z plus something
• 40:29 - 40:31
equals 0.
• 40:31 - 40:34
So if you wanted to
make me very happy,
• 40:34 - 40:39
you would still move everybody
to the left hand side.
• 40:39 - 40:41
• 40:41 - 40:43
Do you want equal to or minus 3?
• 40:43 - 40:45
Yes, it does.
• 40:45 - 40:46
STUDENT: [INAUDIBLE]
• 40:46 - 40:47
PROFESSOR: Huh?
• 40:47 - 40:50
Negative 2-- is that OK?
• 40:50 - 40:51
Is that fine?
• 40:51 - 40:52
Are you guys done?
• 40:52 - 40:52
Is this hard?
• 40:52 - 40:54
Mm-mm.
• 40:54 - 40:55
It's hard?
• 40:55 - 40:56
No.
• 40:56 - 40:59
Who said it's hard?
• 40:59 - 41:05
So-- so I would work more
tangent planes next time.
• 41:05 - 41:08
But I think it's something
that we can practice on.
• 41:08 - 41:13
And do expect one exercise
like that from one
• 41:13 - 41:16
of those, God knows,
15, 16 on the final.
• 41:16 - 41:18
I'm not sure about the midterm.
• 41:18 - 41:20
I like this type of problem.
• 41:20 - 41:23
So you might even see
something with tangent planes
• 41:23 - 41:27
on the midterm-- normal to
a surface tangent plane.
• 41:27 - 41:28
It's a good topic.
• 41:28 - 41:29
It's really pretty.
• 41:29 - 41:33
For people who like to draw,
it's also nice to draw them.
• 41:33 - 41:35
But do you have to?
• 41:35 - 41:36
No.
• 41:36 - 41:39
Some of you don't like to.
• 41:39 - 41:43
OK, so now I say thank
you for the attendance
• 41:43 - 41:48
and I'll see you next time
on Thursday-- on Tuesday.
• 41:48 - 41:51
Happy Valentine's Day.
• 41:51 - 41:52
Title:
TTU Math2450 Calculus3 Sec 11.4 part 1
Description:

Tangent plains, Aproximations, and Differentiability

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Video Language:
English
Duration:
41:53
 jackie.luft edited English subtitles for TTU Math2450 Calculus3 Sec 11.4 part 1