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TTU Math2450 Calculus3 Sec 11.4 part 1

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    PROFESSOR: So let's
    forget about this example
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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
    about that right now.
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    I care about the
    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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    And I also think about
    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
    pink on your notes.
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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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    So again, instead of-- instead
    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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    H is your parameter.
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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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    How about the other one?
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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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    And I'm asked you, what
    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
    with your mind hopefully
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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
    this quadric that you love.
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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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    don't write it down please.
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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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    Today, I'm all about colors.
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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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    It's about a mile.
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    PROFESSOR: Yes, I don't know.
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    I thought it's about one
    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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    Here, I just already
    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
    about this one?
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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.
  • 18:34 - 18:35
    STUDENT: [INAUDIBLE]
  • 18:35 - 18:36
    PROFESSOR: And you cut--
  • 18:36 - 18:37
    STUDENT: [INAUDIBLE]
  • 18:37 - 18:42
  • 18:42 - 18:42
    PROFESSOR: Right?
  • 18:42 - 18:43
    STUDENT: Yeah.
  • 18:43 - 18:45
    PROFESSOR: And you cut
    in this other side.
  • 18:45 - 18:49
    Well, this is the one
    that's facing you.
  • 18:49 - 18:50
    You cut like that.
  • 18:50 - 18:51
    And when you cut like
    this, it's facing--
  • 18:51 - 18:52
    STUDENT: [INAUDIBLE]
  • 18:52 - 18:55
  • 18:55 - 18:57
    PROFESSOR: Hm?
  • 18:57 - 18:59
    But anyway, let's not
    draw the other one.
  • 18:59 - 19:01
    It's hard, right?
  • 19:01 - 19:04
    STUDENT: [INAUDIBLE] angle
    like this-- just the piece
  • 19:04 - 19:06
    of the corner of the cheese.
  • 19:06 - 19:06
    PROFESSOR: Right.
  • 19:06 - 19:08
    STUDENT: The corner
    is facing you.
  • 19:08 - 19:13
    PROFESSOR: So yeah, so it's--
    the corner is facing you.
  • 19:13 - 19:17
    STUDENT: So basically,
    you [INAUDIBLE] this.
  • 19:17 - 19:19
    PROFESSOR: But-- exactly, but--
  • 19:19 - 19:20
    STUDENT: Like this.
  • 19:20 - 19:23
    PROFESSOR: Yeah, well--
    yeah, it's hard to draw.
  • 19:23 - 19:26
    So practically,
    this is what you're
  • 19:26 - 19:29
    looking at it is slope that's
    negative in both directions.
  • 19:29 - 19:35
    So you're going to go
    this way or this way.
  • 19:35 - 19:39
    And it's much steeper than
    you imagine [INAUDIBLE].
  • 19:39 - 19:43
    OK, they are equal.
  • 19:43 - 19:45
    I'm trying to draw them equal.
  • 19:45 - 19:48
    I don't know how
    equal they can be.
  • 19:48 - 19:56
    One belongs to one plane
    just like you said.
  • 19:56 - 19:58
    This belongs to this plane.
  • 19:58 - 20:07
    And the green one belongs to
    the plane that's facing you.
  • 20:07 - 20:09
    So the slope goes this way.
  • 20:09 - 20:11
    But the two slopes are equal.
  • 20:11 - 20:13
    You have to have a little
    bit of imagination.
  • 20:13 - 20:17
    We would need some cheese
    to make a mountain of cheese
  • 20:17 - 20:18
    and cut them and slice them.
  • 20:18 - 20:22
    We'll eat everything
    after, yeah.
  • 20:22 - 20:29
    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
    are reading in advance
  • 22:13 - 22:16
    and you have better
    teachers than me.
  • 22:16 - 22:17
    You have the internet.
  • 22:17 - 22:18
    You have the links.
  • 22:18 - 22:19
    You have YouTube.
  • 22:19 - 22:21
    You have Khan Academy.
  • 22:21 - 22:25
    I know from a bunch of you
    that you have already gone
  • 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
    advance, but several of you
  • 22:37 - 22:39
    already know this formula.
  • 22:39 - 22:44
    We talked about it in
    office hours on yesterday.
  • 22:44 - 22:46
    Because Tuesday, I
    didn't have office hours.
  • 22:46 - 22:50
    I had a coordinator meeting.
  • 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
    the gradient of F
  • 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
    the next of the gradient,
  • 24:16 - 24:18
    if you want unit normal.
  • 24:18 - 24:19
    How did he do that?
  • 24:19 - 24:22
    Well, he had a
    bunch of examples.
  • 24:22 - 24:23
    He had the sphere.
  • 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
    as the gradient.
  • 25:07 - 25:13
    So we have enough
    justification for the direction
  • 25:13 - 25:16
    of the gradient of such
    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
    Gradient of f will
    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
    And what is the gradient?
  • 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
    So the gradient of a
    function is the vector
  • 27:14 - 27:17
    whose coordinates are
    the partial velocity--
  • 27:17 - 27:20
    your friends form last time.
  • 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
    So I know the gradient.
  • 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
    How about I pick my
    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
    abroad that I'm sponsoring.
  • 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
    read it backwards-- then
  • 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
    I wasn't ready to
    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
    you neglect the quadratic
    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
    F of 0, 0 was this 0.
  • 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
    express the final answer
  • 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
    What's the answer?
  • 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

English subtitles

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