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Multivariable calculus question

  1. Oct 28, 2011 #1
    I have a scalar function f dependent on a few variables $x_i$, and I would like to change variables, so that [tex]y_i = \sum_j {M_{ij} x_j},[/tex] where M is an invertible matrix independent of the x_i-s, and compute:
    [tex]
    \frac{\partial f}{\partial x_i} = \frac{\partial f}{\partial \left( \sum_j {M^{-1}_{ij} y_j} \right)} = \sum_j Q_{ij} \frac{\partial f}{\partial y_j}
    [/tex]

    I suggest that the last identity is true for some matrix Q. Is there a general formula for Q?
     
  2. jcsd
  3. Oct 28, 2011 #2

    lurflurf

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    Homework Helper

    by the chain rule
    [tex]
    \frac{\partial f}{\partial x_i} = \sum_j \frac{\partial y_j}{\partial x_i} \frac{\partial f}{\partial y_j}= \sum_j Q_{ij} \frac{\partial f}{\partial y_j}
    [/tex]
    so
    [tex]Q_{ij}=\frac{\partial y_j}{\partial x_i}[/tex]

    "I suggest that the last identity is true for some matrix Q. Is there a general formula for Q?"

    It is, use the chain rule to compute it. It is obvious.
     
    Last edited: Oct 28, 2011
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