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Chain rule when taking vector derivatives

  1. Feb 19, 2015 #1
    Consider a function of several variables ##T=T(x_{1},....,x_{3N})## Let's say I have N vectors of the form ##\vec{r_{1}}=(x_1,x_{2},x_{3})## and ##x_j=x_j(q_1,...,q_n)##. Awkward inex usage but the point is just that the each variable is contained in exactly 1 vector.

    Is it correct to in general use the chain rule in this way? :

    ##\frac{\partial T}{\partial q_j} = \sum\limits_{k=1}^N \frac{\partial T}{\partial \vec{r_k}}.\frac{\partial \vec{r_k}}{\partial q_j}##

    Where the notation ##\frac{\partial T}{\partial \vec{r_k}}## is just ##(\frac{\partial T}{\partial x_k},...., \frac{\partial T}{\partial x_{k+2}})##
     
    Last edited: Feb 19, 2015
  2. jcsd
  3. Feb 19, 2015 #2

    jedishrfu

    Staff: Mentor

    There's something wrong with your Latex, please edit your post to fix it.
     
  4. Feb 19, 2015 #3
    Done
     
  5. Feb 19, 2015 #4

    PeroK

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    The way you've written this, I'd infer that the x's are all independent variables, so it makes no sense to differentiate one wrt another.
     
  6. Feb 19, 2015 #5
    Yeah totally right. I've edited it, correctly this time I hope.
     
  7. Feb 19, 2015 #6

    Ray Vickson

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    Do you mean that you have ##\vec{r}_1 = (x_1,x_2,x_3)##, ##\vec{r}_2 = (x_4, x_5, x_6)##, etc? Using the notation ##\xi_i(q_1,q_2, \ldots, q_n)## instead of ##x_i(q_1, q_2, \ldots, q_n)## (just so as to keep straight the distinction between the variable ##x_i## and the function ##\xi_i## that delivers you the value of ##x_i##), it seems you are asking for the derivative of
    [tex] T = {\cal T}(\xi_1(q_1, q_2, \ldots, q_n), \xi_2(q_1, q_2, \ldots, q_n) , \dots, \xi_{3N}(q_1, q_2, \ldots, q_n)), [/tex]
    again being careful to distinguish between the variable ##T## and the function that delivers you ##T## (which I call ##{\cal T}##).
     
  8. Feb 19, 2015 #7
    Yeah. Maybe I should have included the physics context since this is a physics forum. Would have messed up less and wasted less of people's time by converting it into math format wrongly.

    Basically T is supposed to be a kinetic energy function of a scleronomic system of particles. Where q's represent generalized coordinates and the r-vectors represent the positions of the particles in a carthesian frame.

    I was wondering how to formulate the chain rule this way when taking the partial derivative of T wrt. of one of the generalized coordinates. In case you are not familiar with Lagrangian mechanics just nevermind what I said and yes you seem to have stated my problem very well.
     
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