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TTU Math2450 Calculus3 Sec 10.1

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    I love this model.
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    Again, thank you, Casey.
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    I'm not going to take
    any credit for that.
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    So if you want to
    imagine the stool
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    I was talking about as
    a bamboo object, that
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    is about the same thing,
    at the same scale, compared
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    to the diameter and the height,
    scaled or dialated five times.
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    Uniform, no alterations.
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    And one can sit on it,
    [? and circle, ?] to sit on it.
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    Now, as you see this is
    a doubly ruled surface.
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    And you say, oh wait a minute.
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    You said rule surface, why all
    of a sudden, why doubly ruled
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    surface?
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    Because it is a surface
    that is ruled and generated
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    by two different one
    parameter families.
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    Each of them has a
    certain parameter
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    and that gives them continuity.
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    So you have two
    families of lines.
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    One family is in this direction.
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    Do you see it?
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    So these lines-- this
    line is in motion.
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    It moves to the right, to
    the right, to the right,
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    and it generated.
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    And the other family
    of lines is this one
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    in the other direction.
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    You have a continuity
    parameter for each of them.
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    So you have to imagine
    some real parameter going
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    along the entire
    [? infinite real ?] axis.
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    Or along a circle which would
    be about the same thing.
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    But in any case, you have
    a one parameter family
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    and another one
    parameter family.
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    Both of them are
    together generating
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    this beautiful
    one-sheeted hyperboloid.
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    It's incredible because you
    see where these sort of round,
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    but if you go towards the
    ends, it's topologically
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    a cylinder or a tube.
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    But if you look towards
    the end, the two ends
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    will look more straight.
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    And you will see the
    straight lines more clearly.
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    So imagine that you
    have a continuation
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    to infinity in this direction,
    and in the other direction.
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    And this actually should be an
    infinite surface in your model.
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    You're just cutting it
    between two z planes,
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    so you have a patch of a
    one-sheeted hyperboloid.
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    Yeah, the one-sheeted
    hyperboloid
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    that we wrote last time,
    do you guys remember
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    x squared over a
    squared plus y squared
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    over b squared minus z squared?
    z should be this [INAUDIBLE].
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    Minus z squared over
    c squared minus 1
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    equals 0 is an
    infinite surface area.
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    At both ends you keep going.
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    Very beautiful.
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    Thank you so much.
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    I appreciate.
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    And keep the brownies.
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    No, then I have to pay more.
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    Than I have to pay money.
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    STUDENT: It's made
    out of [INAUDIBLE].
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    PROFESSOR: When
    is your birthday?
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    [LAUGHTER]
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    Really?
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    When is it?
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    STUDENT: February 29.
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    PROFESSOR: Oh, it's coming.
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    [INTERPOSING VOICES]
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    STUDENT: It's coming
    in a year, too.
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    PROFESSOR: That was a smart one.
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    Anyway, I'll remember that.
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    I appreciate the gift very much.
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    And I will cherish
    it and I'll use it
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    with both my
    undergraduate students
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    and my graduate students who
    are just learning about-- some
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    of them don't know the
    one-sheeted hyperboloid model,
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    but they will learn about it.
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    Coming back to our lesson.
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    I announced Section 10.1.
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    Say goodbye to
    quadrant for a while.
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    I know you love them,
    but they will be there
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    for you in Chapter 11.
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    They will wait for you.
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    Now, let's go to Section
    10.1 of Chapter 10.
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    Chapter 10 is a
    beautiful chapter.
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    As you know very well,
    I announced last time,
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    it is about
    vector-valued functions.
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    And you say, oh
    my god, I've never
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    heard about vector-valued
    functions before.
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    You deal with them every day.
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    Every time you move,
    you are dealing
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    with a vector-valued
    function, which
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    is the displacement, which
    takes values in a subset in R3.
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    So let's try and see what
    you should understand
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    when you start Section 10.1.
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    Because the book is pretty
    good, not that I'm a co-author.
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    But it was meant to be really
    written for the students
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    and explain concepts
    really well.
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    How many of you took physics?
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    OK, quite a lot of
    you took physics.
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    Now, one of my students
    in a previous honors class
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    told me he enjoyed my
    class greatly in general.
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    The most [INAUDIBLE] thing
    he had from my class, he
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    learned from my class was the
    motion of the drunken bug.
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    And I said, did I say that?
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    Absolutely, you said that.
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    So apparently I had
    started one of my lessons
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    with imagine you have a fly
    who went into your coffee mug.
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    I think I did.
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    He reproduced the whole
    thing the way I said it.
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    It was quite spontaneous.
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    So imagine your coffee mug had
    some Baileys Irish Creme in it.
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    And the fly was really
    happy after she got up.
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    She managed to get up.
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    And the trajectory of the
    fly was something more
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    like a helix.
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    And this is how I actually
    introduced the helix
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    in my classroom.
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    And I thought, OK,
    is that unusual?
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    Very.
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    And I said, but that's
    an honors class.
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    Everything is supposed
    to be unusual, right?
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    So let's think about the
    position vector or some sort
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    of vector-valued function that
    you're familiar with already
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    from physics.
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    He is one of your best friends.
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    You have a function r of t.
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    And I will point out that r is
    practically the position vector
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    measure that time t, or
    observed at time t in R3.
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    So he takes values in R3.
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    How?
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    As the mathematician, because
    I like to write mathematically
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    all the notion I
    have, r is defined
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    on I was a sub-interval
    of R with values in R3.
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    And he asked me, my student
    said, what is this I?
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    Well, this I could
    be any interval,
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    but let's assume for
    the time being it's
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    just an open
    interval of the type
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    a, b, where a and b are
    real numbers, a less than b.
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    So this is practically the time
    for my bug from the moment,
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    let's say a equals 0 when
    she or he starts flying up,
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    until the moment she
    completely freaks
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    out or drops from the
    maximum point she reached.
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    And she eventually dies.
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    Or maybe she doesn't die.
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    Maybe she's just drunk and she
    will wake up after a while.
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    OK, so what do I mean by
    this displacement vector?
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    I mean, a function--
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    STUDENT: Is that Tc?
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    Do you have [INAUDIBLE]?
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    PROFESSOR: This is r, little r.
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    STUDENT: I know, but the Tc.
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    PROFESSOR: Tc?
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    STUDENT: Or is that an I?
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    PROFESSOR: No.
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    This is I interval,
    which is the same as a,
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    b open interval, like
    from 2 to 7, included.
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    This is inclusion
    [INAUDIBLE] included in R.
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    So I mean R is the real number
    set and a, b is my interval.
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    OK, so r of t is
    going to be what?
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    x of t, y of t, z of t.
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    The book tells you, hey, guys--
    it doesn't say hey, guys,
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    but it's quite informal-- if
    you live in Rn, if your image is
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    in Rn, instead of x
    of t, y of t, z of t,
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    you are going to get something
    like x1 of t, y1 of t.
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    x1 of t, x2 of t,
    x3 of t, et cetera.
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    What do we assume about R?
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    We have to assume
    something about it, right?
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    STUDENT: It's a
    function [INAUDIBLE].
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    PROFESSOR: It's a function
    that is differentiable
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    most of the times, right?
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    What does it mean smooth?
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    I saw that your books
    before college level
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    never mention smooth.
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    A smooth function is a
    function that is differentiable
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    and whose first
    derivative is continuous.
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    Some mathematicians even assume
    that you have c infinity, which
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    means you have a function that's
    infinitely many differentiable.
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    So you have first derivative,
    second derivative, third
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    derivative, fifth derivative.
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    Somebody stop me.
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    All the derivatives exist
    and they are all continuous.
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    By smooth, I will
    assume c1 in this case.
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    I know it's not accurate,
    but let's assume c1.
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    What does it mean?
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    Differentiable function whose
    derivative is continuous.
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    And I will assume
    one more thing.
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    That is not enough for me.
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    I will also assume that
    r prime of t in this case
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    is different from 0 for
    every t in the interval I.
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    Could somebody tell me in
    everyday words what that means?
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    We call that regular function.
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    [INAUDIBLE]
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    You have a brownie [INAUDIBLE].
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    I have no brownies with me.
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    But if you answer, so what--
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    STUDENT: So that means you've
    got no relative mins or maxes,
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    and you never-- the
    object never stops moving.
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    PROFESSOR: Well, actually,
    you can have relative mins
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    and maxes in some way.
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    I'm talking about something
    like that, r prime.
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    This is r of t.
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    And r prime of t
    is the derivative.
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    It's never going to stop.
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    The velocity.
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    I'm talking about this
    piece of information.
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    Velocity [INAUDIBLE] 0 means
    that drunken bug between time
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    a and time b never stops.
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    He stops at the end, but the end
    is b, is outside [INAUDIBLE].
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    So he stops at b and he falls.
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    So I don't stop.
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    I move on from time a to time b.
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    I don't stop at all.
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    Yes, sir.
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    STUDENT: Wouldn't the derivative
    of that line at some point
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    equal 0 where it flattens out?
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    PROFESSOR: Let me
    draw very well.
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    So at time r of t, this
    is the position vector.
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    What is the derivative?
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    The derivative represents
    the velocity vector.
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    A beautiful thing about the
    velocity vector r prime of t
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    is that it has a
    beautiful property.
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    It's always tangent
    to the trajectory.
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    So at every point
    you're going to have
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    a velocity vector that is
    tangent to the trajectory.
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    [INAUDIBLE] in physics.
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    This r prime of t
    should never become 0.
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    So you will never have a
    point instead of a segment
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    when it comes to r prime.
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    So you don't stop.
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    You are going to
    say, wait a minute?
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    But are you always going to
    consider curves, regular curves
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    in space?
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    Regular curves in space.
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    And by space, I know you guys
    mean the Euclidean three space.
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    Actually, many times I will
    consider curves in plane.
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    And the plane is
    part of the space.
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    And you say, give us an example.
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    I will give you an
    example right now.
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    You're going to laugh
    how simple that is.
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    Now, I have another bug
    who is really happy,
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    but it's not drunk at all.
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    And this bug knows how to
    circle around a certain point
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    at the same speed.
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    So very organized bug.
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    Yes, sir.
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    STUDENT: Where did
    you get c prime?
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    PROFESSOR: What?
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    STUDENT: You have c prime is
    differentiable, is [INAUDIBLE].
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    PROFESSOR: c1.
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    STUDENT: c1.
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    PROFESSOR: OK. c1.
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    This is the notation for any
    function that is differentiable
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    and whose derivative
    is continuous.
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    So again, give an
    example of a c1 function.
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    STUDENT: x squared.
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    PROFESSOR: Yeah.
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    On some real interval.
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    How about absolute value
    of x over the real line?
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    What's the problem with that?
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    [INTERPOSING VOICES]
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    PROFESSOR: It's not
    differentiable at 0.
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    OK, so we'll talk a little
    bit later about smoothness.
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    It's a little bit
    delicate as a notion.
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    It's really beautiful
    on the other side.
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    Let's find the nice picture
    trajectory for the bug.
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    This is a ladybug.
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    I cannot draw her, anyway.
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    She is moving along this circle.
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    And I'll give you
    the law of motion.
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    And that reminds me of a
    student who told me, what
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    do I care about law of motion?
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    He never had me as a
    teacher, obviously.
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    But he was telling me,
    well, after I graduated,
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    I always thought, what do I
    care about the law of motion?
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    I mean, I took calculus.
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    Everything was about
    the law of motion.
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    I'm sorry, you should care
    about the law of motion.
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    Once you're not there anymore,
    absolutely you don't care.
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    But why do you want to
    [INAUDIBLE] doing calculus?
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    When you bring
    [INAUDIBLE] to calculus,
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    when you walk into
    calculus, it's law of motion
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    everywhere whether
    you like it or not.
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    So let's try cosine t
    sine t and z to b 1.
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    Let's make it 1 to
    make your life easier.
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    What kind of curve
    is this and why am I
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    claiming that the ladybug
    following this curve
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    is moving at a constant speed?
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    Oh my god.
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    Go ahead, Alexander.
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    STUDENT: That's a circle.
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    PROFESSOR: That's the circle.
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    It's more than a circle.
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    It's a parametrized circle.
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    It's a vector-valued function.
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    Now, like every mathematician
    I should specify the domain.
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    I am just winding
    around one time,
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    and I stop where I started.
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    So I better be smart and
    realize time is not infinity.
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    It could be.
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    I'm wrapping around the
    circle infinitely many times.
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    They do that in
    topology actually when
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    you're going to be--
    seniors takes topology.
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    But I'm not going around
    in circles only one time.
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    So my time will
    start at 0 when I
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    start my motion and
    end at 2 pi seconds
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    if the time is in seconds
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    So I say r is defined
    on the interval I which
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    is-- say it again, Magdalena.
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    You just said it.
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    STUDENT: 0.
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    PROFESSOR: 0 to pi.
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    If you want to take
    0 together, fine.
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    But for consistency, let's
    take it like before, 0 to 2 pi.
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    I'm actually
    excluding the origin.
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    And with values in R3.
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    Although, this is a [? plane ?]
    curve, z will be constant.
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    Do I care about that very much?
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    You will see the beauty of it.
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    I have the velocity vector
    being really pretty.
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    What is the velocity vector?
  • 17:28 - 17:30
    STUDENT: [INAUDIBLE].
  • 17:30 - 17:32
    PROFESSOR: Negative sign t.
  • 17:32 - 17:32
    Thank you.
  • 17:32 - 17:33
    STUDENT: [INAUDIBLE].
  • 17:33 - 17:36
    PROFESSOR: Cosine t.
  • 17:36 - 17:37
    And 0, finally.
  • 17:37 - 17:41
    Because as you saw
    very well in the book,
  • 17:41 - 17:44
    the way we compute
    the velocity vector
  • 17:44 - 17:47
    is by taking x of
    t, y of t, z of t
  • 17:47 - 17:50
    and differentiating
    them in terms of time.
  • 17:50 - 17:54
  • 17:54 - 17:55
    Good.
  • 17:55 - 17:58
    Is this a regular function?
  • 17:58 - 18:03
    As the bug moves between
    time 0 and time equals 2 pi,
  • 18:03 - 18:07
    is the bug ever going to
    stop between these times?
  • 18:07 - 18:07
    STUDENT: No.
  • 18:07 - 18:08
    PROFESSOR: No.
  • 18:08 - 18:09
    How do you know?
  • 18:09 - 18:11
    You guys are faster
    than me, right?
  • 18:11 - 18:11
    What did you do?
  • 18:11 - 18:13
    You did the speed.
  • 18:13 - 18:14
    What's the relationship?
  • 18:14 - 18:17
    What's the difference
    between velocity and speed?
  • 18:17 - 18:19
    STUDENT: Speed is the
    absolute value [INAUDIBLE].
  • 18:19 - 18:19
    PROFESSOR: Wonderful.
  • 18:19 - 18:20
    This is very good.
  • 18:20 - 18:22
    You should tell everybody
    that because people
  • 18:22 - 18:24
    confuse that left and right.
  • 18:24 - 18:27
    So the velocity is
    a vector, like you
  • 18:27 - 18:28
    learned in engineering.
  • 18:28 - 18:30
    You learned in physics.
  • 18:30 - 18:31
    Velocity is a vector.
  • 18:31 - 18:32
    It changes direction.
  • 18:32 - 18:34
    I'm going to Amarillo this way.
  • 18:34 - 18:35
    I'm driving.
  • 18:35 - 18:37
    The velocity will be a
    vector pointing this way.
  • 18:37 - 18:41
    As I come back, will
    point the opposite way.
  • 18:41 - 18:44
    The speed will be a
    scalar, not a vector.
  • 18:44 - 18:46
    It's a magnitude of
    a velocity vector.
  • 18:46 - 18:48
    So say it again, Magdalena.
  • 18:48 - 18:49
    What is the speed?
  • 18:49 - 18:56
    The speed is the magnitude
    of the velocity vector.
  • 18:56 - 18:59
    It's a scalar.
  • 18:59 - 19:01
    Speed.
  • 19:01 - 19:03
    Speed.
  • 19:03 - 19:06
    I heard that before in
    cars, in the movie Cars.
  • 19:06 - 19:11
    Anyway, r prime of t magnitude.
  • 19:11 - 19:12
    In magnitude.
  • 19:12 - 19:17
    Remember, there is a big
    difference between the velocity
  • 19:17 - 19:19
    as the notion.
  • 19:19 - 19:23
    Velocity is a vector.
  • 19:23 - 19:26
    The speed is a
    magnitude, is a scalar.
  • 19:26 - 19:28
    I'm going to go
    ahead and erase that
  • 19:28 - 19:33
    and I'm going to ask
    you what the speed is
  • 19:33 - 19:36
    for my fellow over here.
  • 19:36 - 19:41
    What is the speed
    of a trajectory
  • 19:41 - 19:46
    of the bug who is sober and
    moves at the constant speed?
  • 19:46 - 19:47
    OK.
  • 19:47 - 19:49
    As I already told
    you, it's constant.
  • 19:49 - 19:50
    What is that constant?
  • 19:50 - 19:54
  • 19:54 - 19:57
    What's the constant speed
    I was talking about?
  • 19:57 - 19:59
    STUDENT: [INAUDIBLE].
  • 19:59 - 20:02
    PROFESSOR: I say the
    magnitude of that.
  • 20:02 - 20:04
    I'm too lazy to write it down.
  • 20:04 - 20:06
    It's a Tuesday, almost morning.
  • 20:06 - 20:10
    So I go square root
    of minus I squared
  • 20:10 - 20:12
    plus cosine squared plus 0.
  • 20:12 - 20:14
    I don't need to write that down.
  • 20:14 - 20:15
    You write it down.
  • 20:15 - 20:17
    And how much is that?
  • 20:17 - 20:17
    STUDENT: [INAUDIBLE].
  • 20:17 - 20:18
    PROFESSOR: 1.
  • 20:18 - 20:25
    So I love this curve because
    in mathematician slang,
  • 20:25 - 20:29
    especially in [? a geometer's ?]
    slang-- and my area
  • 20:29 - 20:30
    is differential geometry.
  • 20:30 - 20:35
    So in a way, I do calculus in
    R3 every day on a daily basis.
  • 20:35 - 20:37
    So I have what?
  • 20:37 - 20:43
    This is a special kind of curve.
  • 20:43 - 20:46
    It's a curve parameterized
    in arc length.
  • 20:46 - 20:58
    So definition, we say
    that a curve in R3,
  • 20:58 - 21:10
    or Rn, well anyway, is
    parameterized in arc length.
  • 21:10 - 21:13
  • 21:13 - 21:13
    When?
  • 21:13 - 21:14
    Say it again, Magdalena.
  • 21:14 - 21:32
    Whenever, if and only if,
    its speed is constantly 1.
  • 21:32 - 21:36
  • 21:36 - 21:41
    So this is an example
    where the speed is 1.
  • 21:41 - 21:46
    In such cases, we avoid
    the notation with t.
  • 21:46 - 21:46
    You say, oh my god.
  • 21:46 - 21:47
    Why?
  • 21:47 - 21:50
    When the curve is
    parameterized in arc length,
  • 21:50 - 21:55
    from now on the we
    will actually try
  • 21:55 - 21:59
    to use s whatever we
    know it's an arc length.
  • 21:59 - 22:01
    We use s instead of t.
  • 22:01 - 22:05
    So I'm sorry for the people
    who cannot change that,
  • 22:05 - 22:09
    but you should all be
    able t change that.
  • 22:09 - 22:13
    So everything will be
    in s because we just
  • 22:13 - 22:15
    discovered
    [? Discovery Channel, ?] we
  • 22:15 - 22:19
    just discovered that speed is 1.
  • 22:19 - 22:24
    So there is something
    special about this s.
  • 22:24 - 22:29
  • 22:29 - 22:33
    In this example-- oh, you
    can rewrite the whole example
  • 22:33 - 22:37
    if you want in s so you don't
    have to smudge the paper.
  • 22:37 - 22:39
    OK, it's beautiful.
  • 22:39 - 22:41
    So I am already arc length.
  • 22:41 - 22:44
    And in that case, I'm going
    to call my time parameter
  • 22:44 - 22:46
    little s. s comes from special.
  • 22:46 - 22:48
    No, s comes from
    speed [INAUDIBLE].
  • 22:48 - 22:52
    STUDENT: So you use s
    when it's [INAUDIBLE]?
  • 22:52 - 22:57
    PROFESSOR: We use s whenever the
    speed of that curve will be 1.
  • 22:57 - 22:58
    STUDENT: So [INAUDIBLE].
  • 22:58 - 23:00
    PROFESSOR: And we call that
    arc length parameterization.
  • 23:00 - 23:03
  • 23:03 - 23:06
    I'm moving into the duration
    of your final thoughts.
  • 23:06 - 23:08
    Yes, sir.
  • 23:08 - 23:10
    STUDENT: When we
    get the question, so
  • 23:10 - 23:11
    before solving [INAUDIBLE].
  • 23:11 - 23:13
  • 23:13 - 23:14
    PROFESSOR: We don't know.
  • 23:14 - 23:18
    That's why it was our
    discovery that, hey, at the end
  • 23:18 - 23:22
    it is an arc length, so I better
    change [INAUDIBLE] t into s
  • 23:22 - 23:27
    because that will help me in
    the future remember to do that.
  • 23:27 - 23:30
    Every time I have arc length,
    that it means speed 1.
  • 23:30 - 23:33
    I will call it s instead of y.
  • 23:33 - 23:35
    There is a reason for that.
  • 23:35 - 23:37
    I'm going to erase
    the definition
  • 23:37 - 23:43
    and I'm going to give
    you the-- more or less,
  • 23:43 - 23:46
    the explanation that my
    physics professor gave me.
  • 23:46 - 23:50
    Because as a freshman,
    my mathematics professor
  • 23:50 - 23:54
    in that area, in geometry,
    was not very, very active.
  • 23:54 - 23:57
    But practically, what my physics
    professor told me is that,
  • 23:57 - 24:06
    hey, I would like to have
    some sort of a uniform tangent
  • 24:06 - 24:10
    vector, something that is
    standardized to be in speed 1.
  • 24:10 - 24:16
    So I would like that tangent
    vector to be important to us.
  • 24:16 - 24:20
    And if r is an
    arc length, then r
  • 24:20 - 24:24
    prime would be that unit
    vector that I'm talking about.
  • 24:24 - 24:31
    So he introduced for any r of
    t, which is x of t, y of t,
  • 24:31 - 24:32
    z of t.
  • 24:32 - 24:37
    My physics professor introduced
    the following terminology.
  • 24:37 - 24:43
    The tangent unit vector
    for a regular curve--
  • 24:43 - 24:47
    he was very well-organized
    I might add about him--
  • 24:47 - 24:53
    is by definition r
    prime of t as a vector
  • 24:53 - 24:54
    divided by the
    speed of the vector.
  • 24:54 - 24:56
    So what is he doing?
  • 24:56 - 24:59
    He is unitarizing the velocity.
  • 24:59 - 25:00
    Say it again, Magdalena.
  • 25:00 - 25:03
    He has unitarized
    the velocity in order
  • 25:03 - 25:08
    to make research more consistent
    from the viewpoint of Frenet
  • 25:08 - 25:10
    frame.
  • 25:10 - 25:13
    So in Frenet frame, you
    will see-- you probably
  • 25:13 - 25:14
    learned about the
    Frenet frame if you
  • 25:14 - 25:18
    are a mechanics major, or some
    solid mechanics or physics
  • 25:18 - 25:19
    major.
  • 25:19 - 25:23
    The Frenet frame is
    an orthogonal frame
  • 25:23 - 25:29
    moving along a line in time
    where the three components are
  • 25:29 - 25:33
    t, and the principal normal
    vector, and b the [INAUDIBLE].
  • 25:33 - 25:36
    We only know of the
    first of them, which
  • 25:36 - 25:39
    is T, which is a unit vector.
  • 25:39 - 25:40
    Say it again who it was.
  • 25:40 - 25:45
    It was the velocity vector
    divided by its magnitude.
  • 25:45 - 25:47
    So the velocity vector could
    be any wild, crazy vector
  • 25:47 - 25:55
    that's tangent to the trajectory
    at the point where you are.
  • 25:55 - 25:58
    His magnitude varies from
    one point to the other.
  • 25:58 - 26:00
    He's absolutely crazy.
  • 26:00 - 26:01
    He or she, the velocity vector.
  • 26:01 - 26:02
    Yes, sir.
  • 26:02 - 26:03
    STUDENT: [INAUDIBLE].
  • 26:03 - 26:07
  • 26:07 - 26:08
    PROFESSOR: Here?
  • 26:08 - 26:08
    Here?
  • 26:08 - 26:09
    STUDENT: Yeah, down there.
  • 26:09 - 26:11
    PROFESSOR: D-E-F, definition.
  • 26:11 - 26:13
    That's how a mathematician
    defines things.
  • 26:13 - 26:18
    So to define you write def
    on top of an equality sign
  • 26:18 - 26:21
    or double dot equal.
  • 26:21 - 26:24
    That's a formal way a
    mathematician introduces
  • 26:24 - 26:25
    a definition.
  • 26:25 - 26:28
    Well, he was a physicist,
    but he does math.
  • 26:28 - 26:29
    So what do we do?
  • 26:29 - 26:32
    We say all the blue
    guys that are not equal,
  • 26:32 - 26:35
    divide yourselves
    by your magnitude.
  • 26:35 - 26:40
    And I'm going to have
    the T here is next one,
  • 26:40 - 26:43
    the T here is next one,
    the T here is next one.
  • 26:43 - 26:44
    They are all equal.
  • 26:44 - 26:51
    So that T changes direction, but
    its magnitude will always be 1.
  • 26:51 - 26:52
    Right?
  • 26:52 - 26:55
    Know that the magnitude--
    that's what unit vector means,
  • 26:55 - 26:58
    the magnitude is 1.
  • 26:58 - 27:01
    Why am I so happy about that?
  • 27:01 - 27:04
    Well let me tell
    you that we can have
  • 27:04 - 27:07
    another parametrization
    and another parametrization
  • 27:07 - 27:11
    and another parametrization
    of the same curve.
  • 27:11 - 27:12
    Say what?
  • 27:12 - 27:15
    The parametrization of
    a curve is not unique?
  • 27:15 - 27:16
    No.
  • 27:16 - 27:19
    There are infinitely
    many parametrizations
  • 27:19 - 27:22
    for a physical curve.
  • 27:22 - 27:34
    There are infinitely
    many parametrizations
  • 27:34 - 27:40
    for an even physical curve.
  • 27:40 - 27:43
  • 27:43 - 27:45
    Like [INAUDIBLE]
    the regular one?
  • 27:45 - 27:47
    Well let me give you
    another example that
  • 27:47 - 27:51
    says that this is
    currently R of T
  • 27:51 - 27:58
    equals cosine 5T sine 5T and 1.
  • 27:58 - 27:59
    Why 1?
  • 27:59 - 28:03
    I still want to have
    the same physical curve.
  • 28:03 - 28:04
    What's different, guys?
  • 28:04 - 28:07
    Look at that and then
    say oh OK, is this
  • 28:07 - 28:12
    the same curve as
    a physical curve?
  • 28:12 - 28:13
    What's different in this case?
  • 28:13 - 28:15
    I'm still here.
  • 28:15 - 28:17
    It's still the
    [? red ?] physical curve
  • 28:17 - 28:18
    I'm moving along.
  • 28:18 - 28:19
    What is different?
  • 28:19 - 28:20
    STUDENT: The velocity.
  • 28:20 - 28:21
    PROFESSOR: The velocity.
  • 28:21 - 28:24
    The velocity and
    actually the speed.
  • 28:24 - 28:29
    I'm moving faster or slower, I
    don't know, we have to decide.
  • 28:29 - 28:34
    Now how do I realize
    how many times
  • 28:34 - 28:36
    I'm moving along this curve?
  • 28:36 - 28:40
    I can be smart and say
    hey, I'm not stupid.
  • 28:40 - 28:43
    I know how to move only one
    time and stop where I started.
  • 28:43 - 28:47
    So if I start with
    my T in the interval
  • 28:47 - 28:53
    zero-- I start at
    zero, where do I stop?
  • 28:53 - 28:54
    I can hear your brain buzzing.
  • 28:54 - 28:55
    STUDENT: [INAUDIBLE].
  • 28:55 - 28:58
    PROFESSOR: 2pi over 5.
  • 28:58 - 28:59
    Why is that?
  • 28:59 - 29:00
    Excellent answer.
  • 29:00 - 29:03
    STUDENT: Because when you
    plug it in, it's [INAUDIBLE].
  • 29:03 - 29:05
    PROFESSOR: 5 times 2pi over 5.
  • 29:05 - 29:06
    That's where I stop.
  • 29:06 - 29:08
    So this is not the same
    interval as before.
  • 29:08 - 29:10
    Are you guys with me?
  • 29:10 - 29:17
    This is a new guy, which
    is called J. Oh, all right.
  • 29:17 - 29:19
    So there is a
    relationship between the T
  • 29:19 - 29:23
    and the S. That's why I
    use different notations.
  • 29:23 - 29:27
    And I wish my teachers
    started it just
  • 29:27 - 29:30
    like that when I took math
    analysis as a freshman,
  • 29:30 - 29:31
    or calculus.
  • 29:31 - 29:32
    That's calculus.
  • 29:32 - 29:36
    Because what they started
    with was a diagram.
  • 29:36 - 29:37
    What kind of diagram?
  • 29:37 - 29:42
    Say OK, the
    parametrizations are both
  • 29:42 - 29:45
    starting from
    different intervals.
  • 29:45 - 29:48
    And first I have
    the parametrization
  • 29:48 - 29:50
    from I going to our 3.
  • 29:50 - 29:53
    And that's called-- how
    did we baptize that?
  • 29:53 - 29:58
    R. And the other
    one, from J to R3,
  • 29:58 - 30:02
    we call that big R.
    They're both vectors.
  • 30:02 - 30:05
    And hey guys, we
    should have some sort
  • 30:05 - 30:09
    of correspondence
    functions between I
  • 30:09 - 30:14
    and J that are both 1 to 1, and
    they are 1 being [INAUDIBLE]
  • 30:14 - 30:16
    the other.
  • 30:16 - 30:18
    I swear to God,
    when they started
  • 30:18 - 30:21
    with this theoretical
    model, I didn't understand
  • 30:21 - 30:23
    the motivation at all.
  • 30:23 - 30:25
    At all.
  • 30:25 - 30:28
    Now with an example,
    I can get you
  • 30:28 - 30:31
    closer to the motivation
    of such a diagram.
  • 30:31 - 30:35
    So where does our
    primary S live?
  • 30:35 - 30:39
    S lives in I, and
    T lives in J. So I
  • 30:39 - 30:43
    have to have a correspondence
    that takes S to T or T to S.
  • 30:43 - 30:46
    STUDENT: Wait I
    thought since R of T
  • 30:46 - 30:48
    is also pretty much
    [INAUDIBLE] that we should also
  • 30:48 - 30:50
    use S [INAUDIBLE].
  • 30:50 - 30:53
    PROFESSOR: It's very--
    actually it's very easy.
  • 30:53 - 30:56
    This is 5T.
  • 30:56 - 31:02
    And we cannot use S
    instead of this T,
  • 31:02 - 31:05
    because if we use S
    instead of this T,
  • 31:05 - 31:08
    and we compute the
    speed, we get 5.
  • 31:08 - 31:11
    So it cannot be called S.
    This is very important.
  • 31:11 - 31:15
    So T is not an arc
    length parameter.
  • 31:15 - 31:18
    I wonder what the speed
    will be for this guy.
  • 31:18 - 31:20
    So who wants to
    compute R prime of T?
  • 31:20 - 31:23
    Nobody, but I'll force you to.
  • 31:23 - 31:27
    And the magnitude of that
    will be god knows what.
  • 31:27 - 31:28
    I claim it's 5.
  • 31:28 - 31:30
    Maybe I'm wrong.
  • 31:30 - 31:31
    I did this in my head.
  • 31:31 - 31:33
    I have to do it on paper, right.
  • 31:33 - 31:35
    So I have what?
  • 31:35 - 31:39
    I have to differentiate
    component-wise.
  • 31:39 - 31:42
    And I have [INAUDIBLE] that,
    because I'm running out of gas.
  • 31:42 - 31:43
    STUDENT: Minus 5--
  • 31:43 - 31:46
    PROFESSOR: Minus 5, very good.
  • 31:46 - 31:48
    Sine of 5T.
  • 31:48 - 31:49
    What have we applied?
  • 31:49 - 31:52
    In case you don't
    know that, out.
  • 31:52 - 31:53
    That was Calc 1.
  • 31:53 - 31:54
    Chain rule.
  • 31:54 - 31:55
    Right?
  • 31:55 - 32:00
    So 5 times cosine 5T.
  • 32:00 - 32:04
    And finally, 1
    prime, which is 0.
  • 32:04 - 32:10
    Now let's be brave and
    write the whole thing down.
  • 32:10 - 32:13
    I know I'm lazy today, but I'm
    going to have to do something.
  • 32:13 - 32:14
    Right?
  • 32:14 - 32:18
    So I'll say minus 5
    sine 5T is all squared.
  • 32:18 - 32:21
    Let me take it and square it.
  • 32:21 - 32:24
    Because I see one
    face is confused.
  • 32:24 - 32:27
    And since one face
    is confused, it
  • 32:27 - 32:30
    doesn't matter that the
    others are not confused.
  • 32:30 - 32:31
    OK?
  • 32:31 - 32:36
    So I have square root of this
    plus square of [INAUDIBLE] plus
  • 32:36 - 32:39
    [INAUDIBLE] computing
    the magnitude.
  • 32:39 - 32:40
    What do I get out of here?
  • 32:40 - 32:40
    STUDENT: Five.
  • 32:40 - 32:41
    PROFESSOR: Five.
  • 32:41 - 32:42
    Excellent.
  • 32:42 - 32:45
    This is 5 sine squared
    plus 5 cosine squared.
  • 32:45 - 32:50
    Now yes, then I have 5 times 1.
  • 32:50 - 32:55
    So I have square root
    of 25 here will be 5.
  • 32:55 - 32:56
    What is 5?
  • 32:56 - 33:03
    5 is the speed of the [? bug ?]
    along the same physical curve
  • 33:03 - 33:05
    the other way around.
  • 33:05 - 33:07
    The second time around.
  • 33:07 - 33:10
    Now can you tell me the
    relationship between T and S?
  • 33:10 - 33:13
    They are related.
  • 33:13 - 33:19
    They are like if you're my
    uncle, then I'm your niece.
  • 33:19 - 33:21
    It's the same way.
  • 33:21 - 33:23
    It depends where you look at.
  • 33:23 - 33:26
    T is a function of S,
    and S is a function of T.
  • 33:26 - 33:32
    So it has to be a 1 to 1
    correspondence between the two.
  • 33:32 - 33:38
    Now any ideas of how I what
    to compute the-- how do I
  • 33:38 - 33:43
    want to write the
    relationship between them.
  • 33:43 - 33:46
    Well, S is a
    function of T, right?
  • 33:46 - 33:51
    I just don't know what
    function of T that is.
  • 33:51 - 33:52
    And I wish my professor
    had started like that,
  • 33:52 - 33:55
    but he started
    with this diagram.
  • 33:55 - 33:59
    So simply here you
    have S equals S of T,
  • 33:59 - 34:01
    and here you have
    T equals T of S,
  • 34:01 - 34:03
    the inverse of that function.
  • 34:03 - 34:06
    And when you-- when
    somebody starts that
  • 34:06 - 34:10
    without an example as a
    general diagram philosophy,
  • 34:10 - 34:12
    then it's really, really tough.
  • 34:12 - 34:13
    All right?
  • 34:13 - 34:16
    So I'd like to know
    who S of T-- how
  • 34:16 - 34:20
    in the world do I want
    to define that S of T.
  • 34:20 - 34:26
    He spoonfed us S of T. I don't
    want to spoonfeed you anything.
  • 34:26 - 34:28
    Because this is
    honors class, and you
  • 34:28 - 34:31
    should be able to figure
    this out yourselves.
  • 34:31 - 34:36
    So who is big R of T?
  • 34:36 - 34:42
    Big R of T should
    be, what, should
  • 34:42 - 34:45
    be the same thing in
    the end as R of S.
  • 34:45 - 34:57
    But I should say maybe it's
    R of function T of S, right?
  • 34:57 - 35:00
    Which is the same
    thing as R of S. So
  • 35:00 - 35:06
    what should be the
    relationship between T and S?
  • 35:06 - 35:11
    We have to call them-- one of
    them should be T equals T of S.
  • 35:11 - 35:13
    How about this function?
  • 35:13 - 35:16
    Give it a Greek name,
    what do you want.
  • 35:16 - 35:16
    Alpha?
  • 35:16 - 35:17
    Beta?
  • 35:17 - 35:17
    What?
  • 35:17 - 35:18
    STUDENT: [INAUDIBLE].
  • 35:18 - 35:19
    PROFESSOR: Alpha?
  • 35:19 - 35:20
    Beta?
  • 35:20 - 35:20
    Alpha?
  • 35:20 - 35:22
    I don't know.
  • 35:22 - 35:26
    So S going to T, alpha.
  • 35:26 - 35:27
    And this is going
    to be alpha inverse.
  • 35:27 - 35:31
  • 35:31 - 35:32
    Right?
  • 35:32 - 35:37
    So T equals alpha of S.
    It's more elegant to call it
  • 35:37 - 35:45
    like that than T of S. T
    equals alpha of S. Alpha of S.
  • 35:45 - 35:49
    So from this thing,
    I realize that I
  • 35:49 - 35:54
    get that R composed with
    alpha equals R. Say what?
  • 35:54 - 35:55
    Magdalena?
  • 35:55 - 35:57
    Yeah, yeah, that
    was pre-calculus.
  • 35:57 - 36:01
    R composed with alpha
    equals little r.
  • 36:01 - 36:09
    So how do I get a little r
    by composing R with alpha?
  • 36:09 - 36:12
    How do we say that?
  • 36:12 - 36:17
    Alpha followed by R.
    R composed with alpha.
  • 36:17 - 36:22
    R of alpha of S equals
    R of S. Say it again.
  • 36:22 - 36:31
    R of alpha of S, which is T--
    this T is alpha of S-- equals
  • 36:31 - 36:31
    R.
  • 36:31 - 36:39
    This is the composition
    that we learned in pre-calc.
  • 36:39 - 36:41
    Who can find me the
    definition of S?
  • 36:41 - 36:44
    Because this may be
    a little bit hard.
  • 36:44 - 36:47
    This may be a little bit hard.
  • 36:47 - 36:49
    STUDENT: S [INAUDIBLE].
  • 36:49 - 36:52
    PROFESSOR: Eh, yeah,
    let me write it down.
  • 36:52 - 36:57
    I want to find out
    what S of T is.
  • 36:57 - 37:00
  • 37:00 - 37:11
    Equals what in terms of the
    function R of T. The one
  • 37:11 - 37:14
    that's given here.
  • 37:14 - 37:15
    Why is that?
  • 37:15 - 37:23
  • 37:23 - 37:26
    Let's try some sort
    of chain rule, right?
  • 37:26 - 37:29
    So what do I know I have?
  • 37:29 - 37:30
    I have that.
  • 37:30 - 37:33
    Look at that.
  • 37:33 - 37:39
    R prime of S, which
    is the velocity of-- I
  • 37:39 - 37:44
    erased it-- the velocity of R
    with respect to the arc length
  • 37:44 - 37:47
    parameter is going to be what?
  • 37:47 - 37:52
    R of alpha of S prime
    with respect to S, right?
  • 37:52 - 37:54
    So I should put DDS.
  • 37:54 - 37:55
    Well I'm a little bit lazy.
  • 37:55 - 37:58
    Let's do it again.
  • 37:58 - 38:06
    DDS, R of alpha of S.
  • 38:06 - 38:08
    OK.
  • 38:08 - 38:11
    And what do I have in this case?
  • 38:11 - 38:19
    Well, I have R prime of-- who is
    alpha of S. T, [INAUDIBLE] of T
  • 38:19 - 38:27
    and alpha of S times
    R prime of alpha
  • 38:27 - 38:30
    of S times the prime outside.
  • 38:30 - 38:32
    How do we prime
    in the chain rule?
  • 38:32 - 38:35
    From the outside to the
    inside, one at a time.
  • 38:35 - 38:39
    So I differentiated the
    outer shell, R prime,
  • 38:39 - 38:40
    and then times what?
  • 38:40 - 38:41
    Chain rule, guys.
  • 38:41 - 38:45
    Alpha prime of S. Very good.
  • 38:45 - 38:50
    Alpha prime of S.
  • 38:50 - 38:51
    All right.
  • 38:51 - 38:55
    So I would like
    to understand how
  • 38:55 - 39:03
    I want to compute-- how I want
    to define S of T. If I take
  • 39:03 - 39:07
    this in absolute value, R
    prime of S in absolute value
  • 39:07 - 39:12
    equals R prime of T in absolute
    value times alpha prime of S
  • 39:12 - 39:15
    in absolute value.
  • 39:15 - 39:15
    What do I get?
  • 39:15 - 39:21
  • 39:21 - 39:22
    Who is R prime of S?
  • 39:22 - 39:26
    This is my original
    function in arc length,
  • 39:26 - 39:29
    and that's the
    speed in arc length.
  • 39:29 - 39:31
    What was the speed
    in arc length?
  • 39:31 - 39:32
    STUDENT: One.
  • 39:32 - 39:34
    PROFESSOR: One.
  • 39:34 - 39:37
    And what is the speed
    in not in arc length?
  • 39:37 - 39:38
    STUDENT: Five.
  • 39:38 - 39:42
    PROFESSOR: In that case,
    this is going to be five.
  • 39:42 - 39:46
    And so what is this
    alpha prime of S guy?
  • 39:46 - 39:47
    STUDENT: [INAUDIBLE].
  • 39:47 - 39:51
    PROFESSOR: It's going to be 1/5.
  • 39:51 - 39:52
    OK.
  • 39:52 - 39:53
    All right.
  • 39:53 - 39:56
    Actually alpha of S,
    who is that going to be?
  • 39:56 - 40:04
    Alpha of S.
  • 40:04 - 40:07
    Do you notice the
    correspondence?
  • 40:07 - 40:12
    We simply have to re-define
    this as S. That's how it goes.
  • 40:12 - 40:15
    That five times
    is nothing but S.
  • 40:15 - 40:17
    STUDENT: How did you
    get the [INAUDIBLE]?
  • 40:17 - 40:21
    PROFESSOR: Because 1
    equals 5 times what?
  • 40:21 - 40:26
    1, which is arc length
    speed, equals 5 times what?
  • 40:26 - 40:27
    1/5.
  • 40:27 - 40:28
    STUDENT: Yeah, but then
    where'd you get the 1?
  • 40:28 - 40:29
    PROFESSOR: That's
    one way to do it.
  • 40:29 - 40:32
    Oh, this is by definition,
    because little r means
  • 40:32 - 40:36
    curve in arc length, and little
    s is the arc length parameter.
  • 40:36 - 40:39
    By definition, that
    means you get speed 1.
  • 40:39 - 40:41
    This was our assumption.
  • 40:41 - 40:44
    So we could've gotten
    that much faster saying
  • 40:44 - 40:46
    oh, well, forget
    about this diagram
  • 40:46 - 40:49
    that you introduced-- and
    it's also in the book.
  • 40:49 - 40:53
    Simply take 5T to BS, 5T to BS.
  • 40:53 - 40:56
    Then I get my old
    friend, the curve.
  • 40:56 - 40:59
    The arc length
    parameter is the curve.
  • 40:59 - 41:05
    So this is the same as cosine
    of S, sine of S, and 1.
  • 41:05 - 41:08
    So what is the correspondence
    between S and T?
  • 41:08 - 41:11
  • 41:11 - 41:15
    Since S is 5T in
    this example, I'll
  • 41:15 - 41:16
    put it-- where shall I put it.
  • 41:16 - 41:20
    I'll put it here.
  • 41:20 - 41:23
    S is 5T.
  • 41:23 - 41:25
    I'll say S of T is 5T.
  • 41:25 - 41:28
  • 41:28 - 41:32
    and T of S, what
    is T in terms of S?
  • 41:32 - 41:37
    T in terms of S is S over 5.
  • 41:37 - 41:40
    So instead of T of
    S, we call this alpha
  • 41:40 - 41:48
    of S. So the correspondence
    between S and T, what is T?
  • 41:48 - 41:52
    T is exactly S over
    5 in this example.
  • 41:52 - 41:53
    Say it again.
  • 41:53 - 41:55
    T is exactly S over 5.
  • 41:55 - 41:58
    So alpha of S would be S over 5.
  • 41:58 - 42:02
    In this case, alpha prime of
    S would simply be 1 over 5.
  • 42:02 - 42:04
    Oh, so that's how I got it.
  • 42:04 - 42:06
    That's another way to get it.
  • 42:06 - 42:08
    Much faster.
  • 42:08 - 42:09
    Much simpler.
  • 42:09 - 42:14
    So just think of replacing
    5T by the S knowing
  • 42:14 - 42:19
    that you put S here, the whole
    thing will have speed of 1.
  • 42:19 - 42:20
    All right.
  • 42:20 - 42:22
    So what do I do?
  • 42:22 - 42:25
    I say OK, alpha prime
    of S is 1 over 5.
  • 42:25 - 42:28
    The whole chain rule also
    spit out alpha prime of S
  • 42:28 - 42:30
    to B1 over 5.
  • 42:30 - 42:33
    Now I understand the
    relationship between S and T.
  • 42:33 - 42:34
    It's very simple.
  • 42:34 - 42:40
    S is 5T in this example,
    or T equals S over 5.
  • 42:40 - 42:40
    OK?
  • 42:40 - 42:46
    So if somebody gives you a curve
    that looks like cosine 5T, sine
  • 42:46 - 42:52
    5T, 1, and that is in speed
    5, as we were able to find,
  • 42:52 - 42:57
    how do you re-parametrize
    that in arc length?
  • 42:57 - 43:01
    You just change
    something inside so
  • 43:01 - 43:08
    that you make this curve be
    representative-- representable
  • 43:08 - 43:12
    as little r of S.
    This is in arc length.
  • 43:12 - 43:14
    In arc length.
  • 43:14 - 43:18
  • 43:18 - 43:18
    OK.
  • 43:18 - 43:20
    Finally, this is
    just an example.
  • 43:20 - 43:24
    Can you tell me how that
    arc length parameter
  • 43:24 - 43:26
    is introduced in general?
  • 43:26 - 43:30
    What is S of T by definition?
  • 43:30 - 43:34
    What if I have
    something really wild?
  • 43:34 - 43:36
    How do I get to that
    S of T by definition?
  • 43:36 - 43:39
  • 43:39 - 43:41
    What is S of T in terms
    of the function R?
  • 43:41 - 43:45
    STUDENT: [INAUDIBLE] velocity
    [? of the ?] [INAUDIBLE]?
  • 43:45 - 43:48
    PROFESSOR: S prime of T will
    be one of the [INAUDIBLE].
  • 43:48 - 43:49
    STUDENT: Yes.
  • 43:49 - 43:49
    PROFESSOR: OK.
  • 43:49 - 43:59
    So let's see what we
    have if we define S of T
  • 43:59 - 44:12
    as being integral from 0 to
    T of the speed R prime of T.
  • 44:12 - 44:14
    And instead of T, we put tau.
  • 44:14 - 44:15
    Right?
  • 44:15 - 44:16
    P tau.
  • 44:16 - 44:18
    STUDENT: What is that?
  • 44:18 - 44:20
    PROFESSOR: We cannot
    put T, T, and T.
  • 44:20 - 44:21
    STUDENT: Oh.
  • 44:21 - 44:22
    PROFESSOR: OK?
  • 44:22 - 44:26
    So tau is the Greek T
    that runs between zero
  • 44:26 - 44:29
    and T. This is the
    definition of S
  • 44:29 - 44:44
    of T. General definition
    of the arc length parameter
  • 44:44 - 44:50
    that is according to the chain
    rule, given by the chain rule.
  • 44:50 - 44:57
  • 44:57 - 45:00
    Can we verify really
    quickly in our case,
  • 45:00 - 45:02
    is it easy to see that
    in our case it's correct?
  • 45:02 - 45:03
    STUDENT: Yeah.
  • 45:03 - 45:06
    PROFESSOR: Oh yeah,
    S of T will be,
  • 45:06 - 45:08
    in our case,
    integral from 0 to T.
  • 45:08 - 45:14
    We are lucky our prime of tau
    is a constant, which is 5.
  • 45:14 - 45:16
    So I'm going to
    have integral from 0
  • 45:16 - 45:21
    to T absolute value of
    5 [INAUDIBLE] d tau.
  • 45:21 - 45:23
    And what in the world
    is absolute value of 5?
  • 45:23 - 45:28
    It's 5 integral from 0
    to T [? of the ?] tau.
  • 45:28 - 45:31
    What is integral from
    0 to T of the tau?
  • 45:31 - 45:34
    T. 5T.
  • 45:34 - 45:37
    So S is 5T.
  • 45:37 - 45:40
    And that's what I
    said before, right?
  • 45:40 - 45:42
    S is 5T.
  • 45:42 - 45:47
    S equals 5T, and
    T equals S over 5.
  • 45:47 - 45:51
    So this thing, in general,
    is told to us by who?
  • 45:51 - 45:53
    It has to match the chain rule.
  • 45:53 - 45:55
    It matches the chain rule.
  • 45:55 - 46:20
  • 46:20 - 46:20
    OK.
  • 46:20 - 46:25
    So again, why does that
    match the chain rule?
  • 46:25 - 46:31
    We have that-- we
    have R-- or how
  • 46:31 - 46:35
    should I start, the little f,
    the little r, little r of S,
  • 46:35 - 46:36
    right?
  • 46:36 - 46:41
    Little r of S is
    little r of S of T.
  • 46:41 - 46:45
    How do I differentiate
    that with respect to T?
  • 46:45 - 46:53
    Well DDT of R will be R
    primed with respect to S.
  • 46:53 - 47:02
    So I'll say DRDS of
    S of T times DSDT.
  • 47:02 - 47:05
  • 47:05 - 47:06
    Now what is DSDT?
  • 47:06 - 47:09
    DSDT was the derivative of that.
  • 47:09 - 47:16
    It's exactly the speed
    absolute value of R prime of T.
  • 47:16 - 47:18
    So when you prime
    here, S prime of T
  • 47:18 - 47:23
    will be exactly that,
    with T replacing tau.
  • 47:23 - 47:24
    We learned that in Calc 1.
  • 47:24 - 47:27
    I know it's been a long time.
  • 47:27 - 47:29
    I can feel you're
    a little bit rusty.
  • 47:29 - 47:30
    But it doesn't matter.
  • 47:30 - 47:33
    So S prime of T,
    DSDT will simply
  • 47:33 - 47:36
    be absolute value
    of R prime of T.
  • 47:36 - 47:41
    That's the speed of
    the original curve.
  • 47:41 - 47:44
    This one.
  • 47:44 - 47:46
    OK?
  • 47:46 - 47:47
    All right.
  • 47:47 - 47:59
    So here, when I look at
    DRDS, this is going to be 1.
  • 47:59 - 48:02
  • 48:02 - 48:06
    And if you think of
    this as a function of T,
  • 48:06 - 48:12
    you have DR of S of
    T. Who is R of S of T?
  • 48:12 - 48:15
    This is R-- big
    R-- of T. So this
  • 48:15 - 48:22
    is the DRDT Which is exactly
    the same as R prime of T
  • 48:22 - 48:25
    when you put the absolute
    values [INAUDIBLE].
  • 48:25 - 48:26
    It has to fit.
  • 48:26 - 48:33
    So indeed, you have R prime
    of T, R prime of T, and 1.
  • 48:33 - 48:35
    It's an identity.
  • 48:35 - 48:39
    If I didn't put DSDT to
    [? P, ?] our prime of T
  • 48:39 - 48:42
    in absolute value,
    it wouldn't work out.
  • 48:42 - 48:48
    DSDT has to be R prime
    of T in absolute value.
  • 48:48 - 48:51
    And this is how we
    got, again-- are
  • 48:51 - 48:54
    you going to remember
    this without having
  • 48:54 - 48:56
    to re-do the whole thing?
  • 48:56 - 49:11
    Integral from 0 to T of R
    prime of T or tau d tau.
  • 49:11 - 49:14
    When you prime this
    guy with respect to T
  • 49:14 - 49:18
    as soon as it's positive--
    when it is positive-- assume--
  • 49:18 - 49:20
    why is this positive, S of T?
  • 49:20 - 49:24
    Because you integrate from
    time 0 to another time
  • 49:24 - 49:25
    a positive number.
  • 49:25 - 49:29
    So it has to be
    positive derivative.
  • 49:29 - 49:30
    It's an increasing function.
  • 49:30 - 49:34
    This function is increasing.
  • 49:34 - 49:37
    So DSDT again will be the speed.
  • 49:37 - 49:39
    Say it again, Magdalena?
  • 49:39 - 49:44
    DSDT will be the speed
    of the original line.
  • 49:44 - 49:47
    DSDT in our case was 5.
  • 49:47 - 49:48
    Right?
  • 49:48 - 49:50
    DSDT was 5.
  • 49:50 - 49:55
    S was 5 times T.
    S was 5 times T.
  • 49:55 - 49:55
    All right.
  • 49:55 - 49:58
    That was a simple
    example, sort of, kind of.
  • 49:58 - 50:00
    What do we want to remember?
  • 50:00 - 50:04
    We remember the formula
    of the arc length.
  • 50:04 - 50:06
    Formula of arc length.
  • 50:06 - 50:09
  • 50:09 - 50:11
    So the formula of
    arc length exists
  • 50:11 - 50:15
    in this form because of
    the chain rule [INAUDIBLE]
  • 50:15 - 50:19
    from this diagram.
  • 50:19 - 50:25
    So always remember, we have
    a composition of functions.
  • 50:25 - 50:28
    We use that composition of
    function for the chain rule
  • 50:28 - 50:29
    to re-parametrize it.
  • 50:29 - 50:31
    And finally, the drunken bug.
  • 50:31 - 50:34
  • 50:34 - 50:35
    what did I take [INAUDIBLE] 14?
  • 50:35 - 50:37
    R of t.
  • 50:37 - 50:44
    Let's say this is 2
    cosine t, 2 sine t.
  • 50:44 - 50:46
    Let me make it more beautiful.
  • 50:46 - 50:54
    Let me put 4-- 4, 4, and 3t.
  • 50:54 - 50:57
    Can anybody tell
    me why I did that?
  • 50:57 - 51:00
    Maybe you can guess my mind.
  • 51:00 - 51:04
    Find the following things.
  • 51:04 - 51:11
    The unit vector T, by
    definition R prime over R prime
  • 51:11 - 51:16
    of t in absolute value.
  • 51:16 - 51:22
    Find the speed of
    this motion R of t.
  • 51:22 - 51:25
    This is a law of motion.
  • 51:25 - 51:32
    And reparametrize in arclength--
    this curve in arclength.
  • 51:32 - 51:37
  • 51:37 - 51:40
    And you go, oh my God, I
    have a problem with a, b,c.
  • 51:40 - 51:43
    The is a typical problem for
    the final exam, by the way.
  • 51:43 - 51:46
    This problem popped up on
    many, many final exams.
  • 51:46 - 51:47
    Is it hard?
  • 51:47 - 51:49
    Is it easy?
  • 51:49 - 51:53
    First of all, how did I
    know what it looked like?
  • 51:53 - 51:57
    I should give at
    least an explanation.
  • 51:57 - 52:01
    If instead of 3t I
    would have 3, then I
  • 52:01 - 52:05
    would have the plane
    z equals 3 constant.
  • 52:05 - 52:08
    And then I'll say, I'm moving
    in circles, in circles,
  • 52:08 - 52:11
    in circles, in circles,
    with t as a real parameter,
  • 52:11 - 52:14
    and I'm not evolving.
  • 52:14 - 52:17
    But this is like, what, this
    like in in the avatar OK?
  • 52:17 - 52:22
    So I'm performing the circular
    motion, but at the same time
  • 52:22 - 52:25
    going on a different level.
  • 52:25 - 52:27
    Assume another life.
  • 52:27 - 52:31
    I'm starting another life
    on the next spiritual level.
  • 52:31 - 52:34
    OK, I have no religious
    beliefs in that area,
  • 52:34 - 52:36
    but it's a good physical
    example to give.
  • 52:36 - 52:38
    So I go circular.
  • 52:38 - 52:42
    Instead of going again
    circular and again circular,
  • 52:42 - 52:45
    I go, oh, I go up and
    up and up, and this 3t
  • 52:45 - 52:49
    tells me I should also
    evolve on the vertical.
  • 52:49 - 52:50
    Ah-hah.
  • 52:50 - 52:55
    So instead of circular motion
    I get a helicoidal motion.
  • 52:55 - 52:56
    This is a helix.
  • 52:56 - 52:59
  • 52:59 - 53:02
    Could somebody tell me how I'm
    going to draw such a helix?
  • 53:02 - 53:03
    Is it hard?
  • 53:03 - 53:04
    Is it easy?
  • 53:04 - 53:05
    This helix-- yes, sir.
  • 53:05 - 53:08
  • 53:08 - 53:09
    Yes.
  • 53:09 - 53:11
    STUDENT: [INAUDIBLE]
  • 53:11 - 53:12
    PROFESSOR: It's like a tornado.
  • 53:12 - 53:14
    It's like a tornado,
    hurricane, but how
  • 53:14 - 53:18
    do I draw the cylinder on
    which this helix exists?
  • 53:18 - 53:22
    I have to be a smart girl and
    remember what I learned before.
  • 53:22 - 53:25
    What is x squared
    plus y squared?
  • 53:25 - 53:29
    Suppose that z is not
    playing in the picture.
  • 53:29 - 53:33
    If I take Mr. x and Mr. y
    and I square them and I add
  • 53:33 - 53:35
    them together, what do I get?
  • 53:35 - 53:36
    STUDENT: It's the radius.
  • 53:36 - 53:38
    PROFESSOR: What is
    the radius squared?
  • 53:38 - 53:39
    4 squared.
  • 53:39 - 53:41
    I'm gonna write 4
    squared because it's
  • 53:41 - 53:43
    easier than writing 16.
  • 53:43 - 53:44
    Thank you for your help.
  • 53:44 - 53:51
    So I simply have to go ahead and
    draw the frame first, x, y, z,
  • 53:51 - 53:55
    and then I'll say, OK, smart.
  • 53:55 - 53:58
    R is 4.
  • 53:58 - 54:00
    The radius should be 4.
  • 54:00 - 54:02
    This is the cylinder
    where I'm at.
  • 54:02 - 54:07
    Where do I start
    my physical motion?
  • 54:07 - 54:10
    This bug is drunk,
    but sort of not.
  • 54:10 - 54:12
    I don't know.
  • 54:12 - 54:16
    It's a bug that can keep
    the same radius, which
  • 54:16 - 54:17
    is quite something.
  • 54:17 - 54:18
    STUDENT: It's tipsy.
  • 54:18 - 54:20
    PROFESSOR: Yeah,
    exactly, tipsy one.
  • 54:20 - 54:23
    So how about t equals 0.
  • 54:23 - 54:25
    Where do I start my motion?
  • 54:25 - 54:27
    At 4, 0, 0.
  • 54:27 - 54:29
    Where is 4, 0, 0?
  • 54:29 - 54:29
    Over here.
  • 54:29 - 54:32
    So that's my first
    point where the bug
  • 54:32 - 54:33
    will start at t equals 0.
  • 54:33 - 54:34
    STUDENT: How'd you get 4, 0, 0?
  • 54:34 - 54:36
    PROFESSOR: Because I'm--
    very good question.
  • 54:36 - 54:39
    I'm on x, y, z axes.
  • 54:39 - 54:42
    4, y is 0, z is 0.
  • 54:42 - 54:47
    I plug in t, would be 0,
    and I get 4 times 1, 4 times
  • 54:47 - 54:51
    0, 3 times 0, so I
    know I'm starting here.
  • 54:51 - 54:56
    And when I move, I move
    along the cylinder like that.
  • 54:56 - 55:00
    Can somebody tell me at
    what time I'm gonna be here?
  • 55:00 - 55:04
    Not at 1:50, but what time am
    I going to be at this point?
  • 55:04 - 55:08
    And then I continue, and I go
    up, and I continue and I go up.
  • 55:08 - 55:10
    STUDENT: [INAUDIBLE]
  • 55:10 - 55:11
    PROFESSOR: Pi over 2.
  • 55:11 - 55:13
    Excellent.
  • 55:13 - 55:14
    And can you-- can
    you tell me what
  • 55:14 - 55:17
    point it is in space in R 3?
  • 55:17 - 55:18
    Plug in pi over 2.
  • 55:18 - 55:20
    You can do it faster than me.
  • 55:20 - 55:20
    STUDENT: 0.
  • 55:20 - 55:24
    PROFESSOR: 0, 4 and 3 pi over 2.
  • 55:24 - 55:26
    And I keep going.
  • 55:26 - 55:29
    So this is the helicoidal
    motion I'm talking about.
  • 55:29 - 55:32
    The unit vector-- is it easy
    to write it on the final?
  • 55:32 - 55:33
    Can do that in no time.
  • 55:33 - 55:39
    So we get like, let's say, 30%,
    30%, 30%, and 10% for drawing.
  • 55:39 - 55:41
    How about that?
  • 55:41 - 55:44
    That would be a typical
    grid for the problem.
  • 55:44 - 55:50
    So t will be minus 4 sine t.
  • 55:50 - 55:54
    If I make a mistake, are
    you gonna shout, please?
  • 55:54 - 55:59
    4 cosine t and 3
    divided by what?
  • 55:59 - 56:01
    What is the tangent unit vector?
  • 56:01 - 56:04
    At every point in
    space, I'm gonna
  • 56:04 - 56:06
    have this tangent unit vector.
  • 56:06 - 56:08
    It has to have
    length 1, and it has
  • 56:08 - 56:11
    to be tangent to my trajectory.
  • 56:11 - 56:12
    I'll draw him.
  • 56:12 - 56:16
    So he gives me a
    field, a vector field--
  • 56:16 - 56:19
    this is beautiful-- T
    of t is a vector field.
  • 56:19 - 56:21
    At every point of
    the trajectory,
  • 56:21 - 56:23
    I have only one such vector.
  • 56:23 - 56:27
    That's what we mean
    by vector field.
  • 56:27 - 56:30
    What's the magnitude?
  • 56:30 - 56:31
    It's buzzing.
  • 56:31 - 56:33
    It's buzzing.
  • 56:33 - 56:35
    How did you do it?
  • 56:35 - 56:40
    4, 16 times sine squared
    plus cosine squared.
  • 56:40 - 56:42
    16 plus 9 is 25.
  • 56:42 - 56:46
    Square root of 25 is 5.
  • 56:46 - 56:48
    Are you guys with me?
  • 56:48 - 56:50
    Do I have to write this down?
  • 56:50 - 56:52
    Are you guys sure?
  • 56:52 - 56:53
    STUDENT: You plugged in 0 for t?
  • 56:53 - 56:56
    Is that what you did
    when you [INAUDIBLE]
  • 56:56 - 56:59
    PROFESSOR: No, I plugged
    0 for t when I started.
  • 56:59 - 57:02
    But when I'm computing,
    I don't plug anything,
  • 57:02 - 57:04
    I just do it in general.
  • 57:04 - 57:08
    I said 16 sine squared
    plus 16 cosine squared
  • 57:08 - 57:10
    is 16 times 1 plus 9.
  • 57:10 - 57:13
    My son would know this
    one and he's 10, right?
  • 57:13 - 57:16
    16 plus 9 square root of 25.
  • 57:16 - 57:18
    And I taught him
    about square roots.
  • 57:18 - 57:21
    So square root of 25,
    he knows that's 5.
  • 57:21 - 57:22
    And if he knows
    that's 5, then you
  • 57:22 - 57:24
    should do that in a
    minute-- in a second.
  • 57:24 - 57:25
    All right.
  • 57:25 - 57:32
    So t will simply be-- if you
    don't simplify 1/5 minus 4 sine
  • 57:32 - 57:37
    t 4 cosine t 3 in the final,
    it wouldn't be a big deal,
  • 57:37 - 57:39
    I would give you
    still partial credit,
  • 57:39 - 57:42
    but what if we raise this
    as a multiple choice?
  • 57:42 - 57:47
    Then you have to be able
    to find where the 5 is.
  • 57:47 - 57:47
    What is the speed?
  • 57:47 - 57:49
    Was that hard for you to find?
  • 57:49 - 57:51
    Where is the speed hiding?
  • 57:51 - 57:54
    It's exactly the
    denominator of R.
  • 57:54 - 57:57
    This is the speed
    of the curve in t.
  • 57:57 - 57:59
    And that was 5.
  • 57:59 - 58:01
    You told me the speed was
    5, and I'm very happy.
  • 58:01 - 58:08
    So you got 30%, 30%, 10% from
    the picture-- no, this picture.
  • 58:08 - 58:09
    This picture's no good.
  • 58:09 - 58:13
    STUDENT: What does the
    first word of c say?
  • 58:13 - 58:15
    Question c, what does
    the first word say?
  • 58:15 - 58:16
    PROFESSOR: The first what?
  • 58:16 - 58:18
    STUDENT: The word.
  • 58:18 - 58:19
    PROFESSOR: Reparametrize.
  • 58:19 - 58:23
    Reparametrize this
    curve in arclength.
  • 58:23 - 58:26
    Oh my God, so according
    to that chain rule,
  • 58:26 - 58:31
    could you guys remember-- if you
    remember, what is the s of t?
  • 58:31 - 58:39
    If I want to reparametrize
    in arclength integral from 0
  • 58:39 - 58:46
    to t of the speed, how
    is the speed defined?
  • 58:46 - 58:49
    Absolute value of r prime of t.
  • 58:49 - 58:54
    dt, but I don't like t,
    I write-- I write tau.
  • 58:54 - 58:57
    Like Dr. [? Solinger, ?]
    you know him,
  • 58:57 - 58:59
    he's one of my colleagues,
    calls that-- that's
  • 58:59 - 59:01
    the dummy dummy variable.
  • 59:01 - 59:04
    In many books, tau is
    the dummy variable.
  • 59:04 - 59:08
    Or you can-- some people even
    put t by inclusive notation.
  • 59:08 - 59:10
    All right?
  • 59:10 - 59:13
    So in my case, what is s of t?
  • 59:13 - 59:14
    It should be easy.
  • 59:14 - 59:19
    Because although this
    not a circular motion,
  • 59:19 - 59:21
    I still have constant speed.
  • 59:21 - 59:24
    So who is that special speed?
  • 59:24 - 59:24
    5.
  • 59:24 - 59:31
    Integral from 0 to t5 d tau,
    and that is 5t, am I right?
  • 59:31 - 59:32
    5t.
  • 59:32 - 59:37
    So-- so if I want to
    reparametrize this helix,
  • 59:37 - 59:42
    keeping in mind
    that s is simply 5t,
  • 59:42 - 59:47
    what do I have to do to
    get 100% on this problem?
  • 59:47 - 59:58
    All I have to do is say little r
    of s, which represents actually
  • 59:58 - 60:01
    big R of t of s.
  • 60:01 - 60:02
    Are you guys with me?
  • 60:02 - 60:04
    Do you have to write
    all this story down?
  • 60:04 - 60:05
    No.
  • 60:05 - 60:08
    But that will remind
    you of the diagram.
  • 60:08 - 60:12
    So I have R of t of s.
  • 60:12 - 60:13
    Or alpha of s.
  • 60:13 - 60:15
    And this is t of s.
  • 60:15 - 60:16
    t of s.
  • 60:16 - 60:20
    R of t of s is R of s, right?
  • 60:20 - 60:21
    Do you have to remind me?
  • 60:21 - 60:22
    No.
  • 60:22 - 60:23
    The heck with the diagram.
  • 60:23 - 60:27
    As long as you understood
    it was about a composition
  • 60:27 - 60:28
    of functions.
  • 60:28 - 60:31
    And then R of s
    will simply be what?
  • 60:31 - 60:33
    How do we do that fast?
  • 60:33 - 60:37
    We replaced t by s over 5.
  • 60:37 - 60:39
    Where from?
  • 60:39 - 60:42
    Little s equals 5t,
    we just computed it.
  • 60:42 - 60:44
    Little s equals 5t.
  • 60:44 - 60:45
    That's all you need to do.
  • 60:45 - 60:49
    To pull out t, replace
    the third sub s.
  • 60:49 - 60:53
    So what is the function
    t in terms of s?
  • 60:53 - 60:55
    It's s over 5.
  • 60:55 - 61:00
    What is the function t, what's
    the parameter t, in terms of s?
  • 61:00 - 61:01
    s over 5.
  • 61:01 - 61:07
    And finally, at the end, 3
    times what is the stinking t?
  • 61:07 - 61:09
    s over 5.
  • 61:09 - 61:11
    I'm done.
  • 61:11 - 61:16
    I got 100% I don't want
    to say how much time it's
  • 61:16 - 61:18
    gonna take me to
    do it, but I think
  • 61:18 - 61:20
    I can do it in like, 2
    or 3 minutes, 5 minutes.
  • 61:20 - 61:24
    If I know the problem I'll
    do it in a few minutes.
  • 61:24 - 61:27
    If I waste too
    much time thinking,
  • 61:27 - 61:29
    I'm not gonna do it at all.
  • 61:29 - 61:30
    So what do you have to remember?
  • 61:30 - 61:35
    You have to remember the
    formula that says s of t,
  • 61:35 - 61:41
    the arclength parameter--
    the arclength parameter
  • 61:41 - 61:47
    equals integral from 0 to
    t is 0 to t of the speed.
  • 61:47 - 61:53
    Does this element of information
    remind you of something?
  • 61:53 - 61:56
    Of course, s will be the
    arclength, practically.
  • 61:56 - 61:58
    What kind of parameter is that?
  • 61:58 - 62:04
    Is you're measuring how
    big-- how much you travel.
  • 62:04 - 62:07
    s of t is the time you
    travel-- the distance
  • 62:07 - 62:11
    you travel in time t.
  • 62:11 - 62:16
  • 62:16 - 62:20
    So it's a space-time continuum.
  • 62:20 - 62:24
    It's a space-time relationship.
  • 62:24 - 62:27
    So it's the space you
    travel in times t.
  • 62:27 - 62:30
    Now, if I drive to Amarillo
    at 60 miles an hour,
  • 62:30 - 62:35
    I'm happy and sassy, and I
    say OK, it's gonna be s of t.
  • 62:35 - 62:38
    My displacement to
    Amarillo is given
  • 62:38 - 62:42
    by this linear law, 60 times t.
  • 62:42 - 62:43
    Suppose I'm on cruise control.
  • 62:43 - 62:44
    But I've never on
    cruise control.
  • 62:44 - 62:47
  • 62:47 - 62:51
    So this is going to
    be very variable.
  • 62:51 - 62:55
    And the only way you can compute
    this displacement or distance
  • 62:55 - 62:57
    traveled, it'll
    be as an integral.
  • 62:57 - 63:01
    From time 0, when I start
    driving, to time t of my speed,
  • 63:01 - 63:02
    and that's it.
  • 63:02 - 63:04
    That's all you have to remember.
  • 63:04 - 63:08
    It's actually-- mathematics
    should not be memorized.
  • 63:08 - 63:12
    It should be sort of
    understood, just like physics.
  • 63:12 - 63:15
    What if you take your
    first test, quiz,
  • 63:15 - 63:19
    whatever, on WeBWorK or in
    person, and you freak out.
  • 63:19 - 63:23
    You get such a
    problem, and you blank.
  • 63:23 - 63:25
    You just blank.
  • 63:25 - 63:28
    What do you do?
  • 63:28 - 63:32
    You sort of know this,
    but you have a blank.
  • 63:32 - 63:34
    Always tell me, right?
  • 63:34 - 63:36
    Always email, say I'm
    freaking out here.
  • 63:36 - 63:39
    I don't know what's
    the matter with me.
  • 63:39 - 63:46
    Don't cut our correspondence,
    either by speaking or by email.
  • 63:46 - 63:49
    Very few of you email me.
  • 63:49 - 63:52
    I'd like you to be
    more like my friends,
  • 63:52 - 63:53
    and I would be more
    like your tutor,
  • 63:53 - 63:55
    and when you
    encounter an obstacle,
  • 63:55 - 63:58
    you email me and
    I email you back.
  • 63:58 - 64:01
    This is what I want.
  • 64:01 - 64:04
    The WeBWorK, this is what I
    want our model of interaction
  • 64:04 - 64:06
    to become.
  • 64:06 - 64:07
    Don't be shy.
  • 64:07 - 64:11
    Many of you are shy even to
    ask questions in the classroom.
  • 64:11 - 64:12
    And I'm not going
    to let you be shy.
  • 64:12 - 64:17
    At 2 o'clock I'm going to let
    you ask all the questions you
  • 64:17 - 64:20
    have about homework,
    and we will do
  • 64:20 - 64:21
    more homework-like questions.
  • 64:21 - 64:24
    I want to imitate some
    WeBWorK questions.
  • 64:24 - 64:28
    And we will work them out.
  • 64:28 - 64:32
    So any questions right now?
  • 64:32 - 64:33
    Yes, sir.
  • 64:33 - 64:36
    STUDENT: You emailed-- did
    you email us this weekend
  • 64:36 - 64:38
    the numbers for WeBWorK?
  • 64:38 - 64:41
    PROFESSOR: I emailed you the
    WeBWorK assignment completely.
  • 64:41 - 64:45
    I mean, the link-- you
    get in and you of see it.
  • 64:45 - 64:48
    STUDENT: Which email
    did you send that to?
  • 64:48 - 64:50
    PROFESSOR: To your TTU.
  • 64:50 - 64:52
    All the emails go to your TTU.
  • 64:52 - 64:56
    You have one week
    starting yesterday until,
  • 64:56 - 64:58
    was it the 2nd?
  • 64:58 - 65:00
    I gave you a little
    bit more time.
  • 65:00 - 65:03
    So it's due on the
    2nd of February at,
  • 65:03 - 65:04
    I forgot what time.
  • 65:04 - 65:05
    1 o'clock or something.
  • 65:05 - 65:06
    Yes, sir.
  • 65:06 - 65:08
    STUDENT: [INAUDIBLE]
    I was confused
  • 65:08 - 65:10
    at the beginning where you got
    x squared plus y squared equals
  • 65:10 - 65:11
    4 squared.
  • 65:11 - 65:14
    Where did you get that?
  • 65:14 - 65:14
    PROFESSOR: Oh.
  • 65:14 - 65:15
    OK.
  • 65:15 - 65:19
    I eliminated the t between
    the first two guys.
  • 65:19 - 65:25
    This is called eliminating a
    parameter, which was the time
  • 65:25 - 65:28
    parameter between x and y.
  • 65:28 - 65:32
    When I do that, I get a
    beautiful equation which
  • 65:32 - 65:37
    is x squared plus y squared
    equals 16, which tells me, hey,
  • 65:37 - 65:40
    your curve sits on
    the surface x squared
  • 65:40 - 65:42
    plus y squared equals 16.
  • 65:42 - 65:44
    It's not the same
    with the surface,
  • 65:44 - 65:48
    because you have additional
    constraints on the z.
  • 65:48 - 65:52
    So the z is constrained
    to follow this thing.
  • 65:52 - 66:00
    Now, could anybody tell me how
    I'm gonna write eventually--
  • 66:00 - 66:02
    this is a harder
    task, OK, but I'm
  • 66:02 - 66:09
    glad you asked because I
    wanted to discuss that.
  • 66:09 - 66:13
    How do I express t
    in terms of x and y?
  • 66:13 - 66:17
    I mean, I'm going to have an
    intersection of two surfaces.
  • 66:17 - 66:18
    How?
  • 66:18 - 66:21
    This is just practically
    differential geometry
  • 66:21 - 66:24
    or advanced calculus
    at the same time.
  • 66:24 - 66:28
    x squared plus y squared
    equals our first surface
  • 66:28 - 66:32
    that I'm thinking about, which
    I'm sitting with my curve.
  • 66:32 - 66:35
    But I also have my curve
    to be at the intersection
  • 66:35 - 66:39
    between the cylinder
    and something else.
  • 66:39 - 66:45
    And it's hard to figure out how
    I'm going to do the other one.
  • 66:45 - 66:49
    Can anybody figure
    out how another
  • 66:49 - 66:51
    surface-- what is the surface?
  • 66:51 - 66:56
    A surface will have an implicit
    equation of the type f of x, y,
  • 66:56 - 66:58
    z equals a constant.
  • 66:58 - 67:01
    So you have to sort of
    eliminate your parameter t.
  • 67:01 - 67:02
    The heck with the time.
  • 67:02 - 67:05
    We don't care about time,
    we only care about space.
  • 67:05 - 67:07
    So is there any other
    way to eliminate
  • 67:07 - 67:10
    t between the equations?
  • 67:10 - 67:14
    I have to use the information
    that I haven't used yet.
  • 67:14 - 67:15
    All right.
  • 67:15 - 67:20
    Now my question is
    that, how can I do that?
  • 67:20 - 67:23
    z is beautiful.
  • 67:23 - 67:24
    3 is beautiful.
  • 67:24 - 67:26
    t drives me nuts.
  • 67:26 - 67:30
    How do I get the t out of
    the first two equations?
  • 67:30 - 67:33
    [INTERPOSING VOICES]
  • 67:33 - 67:36
    Yeah, I divide them
    one to the other one.
  • 67:36 - 67:40
    So if I-- for example,
    I go y over x.
  • 67:40 - 67:43
    What is y over x?
  • 67:43 - 67:45
    It's tangent of t.
  • 67:45 - 67:49
    How do I pull Mr. t out?
  • 67:49 - 67:52
    Say t, get out.
  • 67:52 - 67:55
    Well, I have to think about
    if I'm not losing anything.
  • 67:55 - 67:58
    But in principle, t would
    be arctangent of y over x.
  • 67:58 - 68:02
  • 68:02 - 68:02
    OK?
  • 68:02 - 68:06
    So, I'm having two
    equations of this type.
  • 68:06 - 68:08
    I'm eliminating t
    between the two.
  • 68:08 - 68:10
    I don't care about
    the other one.
  • 68:10 - 68:14
    I only cared for you
    to draw the cylinder.
  • 68:14 - 68:17
    So we can draw point
    by point the helix.
  • 68:17 - 68:19
    I don't draw many points.
  • 68:19 - 68:23
    I draw only t equals 0,
    where I'm starting over here,
  • 68:23 - 68:25
    t equals pi over 2, which
    [INAUDIBLE] gave me,
  • 68:25 - 68:27
    then what was it?
  • 68:27 - 68:30
    At pi I'm here, and so on.
  • 68:30 - 68:34
    So I move-- when
    I move one time,
  • 68:34 - 68:39
    so let's say from 0 to
    2 pi, I should be smart.
  • 68:39 - 68:49
    Pi over 2, pi, 3 pi over 2,
    2 pi just on top of that.
  • 68:49 - 68:52
    It has to be on the same line.
  • 68:52 - 68:55
    On top of that--
    on the cylinder.
  • 68:55 - 68:56
    They are all on the cylinder.
  • 68:56 - 68:59
    I'm not good enough to draw
    them as being on the cylinder.
  • 68:59 - 69:03
    So I'm coming where I started
    from, but on the higher
  • 69:03 - 69:08
    level of intelligence-- no, on
    a higher level of experience.
  • 69:08 - 69:09
    Right?
  • 69:09 - 69:14
    That's kind of the idea
    of evolving on the helix?
  • 69:14 - 69:17
    Any other questions?
  • 69:17 - 69:18
    Yes, sir.
  • 69:18 - 69:20
    STUDENT: So that
    capital R of t is
  • 69:20 - 69:24
    you position vector, but what's
    little r of t? [INAUDIBLE]
  • 69:24 - 69:26
    PROFESSOR: It's also
    a position vector.
  • 69:26 - 69:32
    So practically it depends on
    the type of parametrization
  • 69:32 - 69:33
    you are using.
  • 69:33 - 69:36
  • 69:36 - 69:40
    The dependence of
    time is crucial.
  • 69:40 - 69:43
    The dependence of the
    time parameter is crucial.
  • 69:43 - 69:51
    So when you draw
    this diagram, r of s
  • 69:51 - 69:59
    will practically be the same
    as R of s of t-- R of t of s,
  • 69:59 - 70:00
    I'm sorry.
  • 70:00 - 70:02
    R of t of s.
  • 70:02 - 70:06
    So practically it's telling
    me it's a combination.
  • 70:06 - 70:12
    Physically, it's the same
    thing, but at a different time.
  • 70:12 - 70:20
    So you look at one vector
    at time-- time is t here,
  • 70:20 - 70:23
    but s was 5t.
  • 70:23 - 70:26
    So I'm gonna be-- let
    me give you an example.
  • 70:26 - 70:30
    So we had s was 5t, right?
  • 70:30 - 70:33
    I don't remember how it went.
  • 70:33 - 70:36
    So when I have
    little r of s, that
  • 70:36 - 70:43
    means the same as
    little r of 5t,
  • 70:43 - 70:47
    which means this kind of guy.
  • 70:47 - 70:58
    Now assume that I have something
    like cosine 5t, sine 5t, and 0.
  • 70:58 - 71:01
    And what does this mean?
  • 71:01 - 71:10
    It means that R of 2 pi over
    5 is the same as little r of 2
  • 71:10 - 71:16
    pi where R of t is cosine
    of 5t, and little r of s
  • 71:16 - 71:21
    is cosine of s, sine s, 0.
  • 71:21 - 71:24
    So Mr. t says, I'm
    running, I'm time.
  • 71:24 - 71:29
    I'm running from 0 to 2 pi
    over 5, and that's when I stop.
  • 71:29 - 71:31
    And little s says,
    I'm running too.
  • 71:31 - 71:35
    I'm also time, but I'm
    a special kind of time,
  • 71:35 - 71:38
    and I'm running from 0 to
    2 pi, and I stop at 2 pi
  • 71:38 - 71:41
    where the circle will stop.
  • 71:41 - 71:44
    Then physically,
    the two vectors,
  • 71:44 - 71:49
    at two different moments
    in time, are the same.
  • 71:49 - 71:51
    Where-- why-- why is that?
  • 71:51 - 71:53
    So I start here.
  • 71:53 - 71:56
    And I end here.
  • 71:56 - 72:01
    So physically, these two guys
    have the same, the red vector,
  • 72:01 - 72:05
    but they are there at
    different moments in time.
  • 72:05 - 72:07
    All right?
  • 72:07 - 72:13
    So imagine that you have sister.
  • 72:13 - 72:18
    And she is five times faster
    than you in a competition.
  • 72:18 - 72:21
    It's a math competition,
    athletic, it doesn't matter.
  • 72:21 - 72:26
    You both get there, but you
    get there in different times,
  • 72:26 - 72:27
    in different amounts of time.
  • 72:27 - 72:31
    And unfortunately, this is--
    I will do philosophy still
  • 72:31 - 72:36
    in mathematics-- this is the
    situation with many of us
  • 72:36 - 72:39
    when it comes to
    understanding a material,
  • 72:39 - 72:42
    like calculus or advanced
    calculus or geometry.
  • 72:42 - 72:47
    We get to the understanding
    in different times.
  • 72:47 - 72:51
    In my class-- I was
    talking to my old--
  • 72:51 - 72:56
    they are all old now,
    all in their 40s--
  • 72:56 - 72:59
    when did you
    understand this helix
  • 72:59 - 73:02
    thing being on a cylinder?
  • 73:02 - 73:04
    Because I think I
    understood it when
  • 73:04 - 73:08
    I was in third-- like a
    junior level, sophomore level,
  • 73:08 - 73:10
    and I understood nothing
    of this kind of stuff
  • 73:10 - 73:15
    in my freshman [INAUDIBLE]
    And one of my colleagues
  • 73:15 - 73:18
    who was really smart,
    had a big background,
  • 73:18 - 73:21
    was in a Math
    Olympiad, said, I think
  • 73:21 - 73:23
    I understood it as a freshman.
  • 73:23 - 73:25
    So then the other two that
    I was talking-- actually
  • 73:25 - 73:28
    I never understood it.
  • 73:28 - 73:32
    So we all eventually get to
    that point, that position,
  • 73:32 - 73:35
    but at a different
    moment in time.
  • 73:35 - 73:39
    And it's also unfortunate it
    happens about relationships.
  • 73:39 - 73:42
    You are in a relationship
    with somebody,
  • 73:42 - 73:45
    and one is faster
    than the other one.
  • 73:45 - 73:47
    One grows faster
    than the other one.
  • 73:47 - 73:50
    Eventually both get to the
    same level of understanding,
  • 73:50 - 73:53
    but since it's at
    different moments in time,
  • 73:53 - 73:56
    the relationship could
    break by the time
  • 73:56 - 73:59
    both reach that level
    of understanding.
  • 73:59 - 74:03
    So physical phenomena,
    really tricky.
  • 74:03 - 74:05
    It's-- physically you
    see where everything is,
  • 74:05 - 74:08
    but you have to think
    dynamically, in time.
  • 74:08 - 74:11
    Everything evolves in time.
  • 74:11 - 74:15
    Any other questions?
  • 74:15 - 74:18
    I'm gonna do problems
    with you next time,
  • 74:18 - 74:23
    but you need a break because
    your brain is overheated.
  • 74:23 - 74:28
    And so, we will take a
    break of 10-12 minutes.
  • 74:28 - 74:31
Title:
TTU Math2450 Calculus3 Sec 10.1
Description:

Vector Value Functions

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Video Language:
English

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