Multivariable calculus question

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

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

Answers and Replies

lurflurf
Homework Helper
by the chain rule
$$\frac{\partial f}{\partial x_i} = \sum_j \frac{\partial y_j}{\partial x_i} \frac{\partial f}{\partial y_j}= \sum_j Q_{ij} \frac{\partial f}{\partial y_j}$$
so
$$Q_{ij}=\frac{\partial y_j}{\partial x_i}$$

"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.

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