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I don't understand the following statement in Landau & Lifshitz, Classical Theory of Fields, p.5:

$ds$ and $ds'$ are infinitesimals of same order. [...] It follows that $ds^2$ and $ds'^2$ must be proportional to each other: $$ds^2 = a \, ds'^2.$$.

I don't get why the proportionality applies, and why does it apply to the squares of the infinitesimals.

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It would be useful to provide the context. –  leongz Jan 4 '13 at 2:15
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up vote 3 down vote accepted

If $\lim_{x\rightarrow 0}\frac{\alpha(x)}{\beta(x)}=A$ ($A$ is a number different from zero), then the functions $\alpha(x)$ and $\beta(x)$ are called infinitesimals of the same order [1].

The proportionality at $x\rightarrow 0$ should be obvious from this.

[1] http://www.math24.net/infinitesimals.html

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I still don't understand why "$ds$ and $ds'$ are infinitesimals of the same order". –  becko Mar 2 '13 at 19:41
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