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Multivariable Calculus - chain rule on vectors

  1. Sep 10, 2013 #1
    Hi! I do not understand the math used in the beginning of this video:

    In example 1 (4 minutes in the video), why is it wrong to simply solve the problem like this:

    [tex]\vec{V} = [x,-y] \Rightarrow \frac{d\vec{V}}{dt} = [\frac{dx}{dt},-\frac{dy}{dt}] = \vec{a} = [V_x,-V_y][/tex], where V_x and V_y are the velocity-components in the x and y directions, respectively.

    I thought you'd only use the chain-rule on non-vector multivariable functions??

    EDIT: I'm farily sure the guy did some mistakes.. did he not? Look at his work 5:00 minutes in.
    Last edited by a moderator: Sep 25, 2014
  2. jcsd
  3. Sep 10, 2013 #2


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    Perhaps you have not learned Calculus on vector valued functions? (It is typically introduced in "Calculus III" but there are whole books on the subject.) If (f(t), g(t), h(t)) is a (three dimensional) vector valued function of the variable t, then differentiation is defined "component-wise". That is, the derivative with respect to t is <df/dt, dg/dt, dh/dt>. That is precisely what is being done in your example (though I see NO use of the "chain rule" in what you give).

    (I looked at the you-tube from 4:45 to 5:15 and saw no error or anything unusual.)
  4. Sep 10, 2013 #3
    I've been through multivariable calculus, but they never talked much about vector-differentiation. It was kinda obvious that you just differentiate term by term. So I've never heard about using the chain-rule directly on a vector. Though I suppose there is nothing wrong in doing that.

    Anyway, concerning the video, if you look closely at his calculations from 1:00 to 5:00, you will see that the guy uses the chain rule (wrongly) and ends up with: [tex] \vec{a} = \frac{d}{dt}(x*\vec{i}-y*\vec{j}) = [x,y][/tex] Which is obviously wrong. He does everything, including his use of the chain-rule, wrong...

    Duno why I wasted those 10 minutes on that guy's videos. Need to find some decent lecture-videos on fluid mechanics.
    Last edited: Sep 10, 2013
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