# Lightman et al Question 3.10

1. Jul 15, 2008

### jdstokes

The question is show that given two coordinate transformation matrices, that their product is also a coorindate transformation.

$\Lambda^\alpha_\beta = \frac{\partial x'^\alpha}{\partial x_\beta}$

$\widetilde{\Lambda}^\alpha_\beta = \frac{\partial \widetilde{x}^\alpha}{\partial x_\beta}$

$\Lambda^\alpha_\gamma\widetilde{\Lambda}^\gamma_\beta = \frac{\partial x'^\alpha}{\partial x_\gamma}\frac{\partial \widetilde{x}^\gamma}{\partial x_\beta}$

Define $\bar{x}^\alpha = x'^\alpha (\widetilde{x}^\gamma(x^\beta))$. Then by the chain rule we obtain

$\frac{\partial \bar{x^\alpha}}{\partial x^\beta} = \frac{\partial x'^\alpha}{\partial \widetilde{x}^\gamma}\frac{\partial \widetilde{x}^\gamma}{\partial x^\beta}$.

This differs from the previous expression by a tilde. Lightman et al shrug this off by saing that it makes no difference what symbol is used to represent the argument variable of a partial derivative. I don't understand this claim. Is anybody able to clarify this?

Thanks