Multivariable calculus question

  • Thread starter evilcman
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  • #1
evilcman
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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?
 

Answers and Replies

  • #2
lurflurf
Homework Helper
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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:

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