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May be this is a stupid question but... in Definition $10$ of O'Neill's book Semi-Riemannian Geometry With Applications to Relativity, it is stated that the partial derivative of a function $f\in \mathfrak{F}(M)$ with respect to the coordinate $x^i$ of a coordinate system $\xi = (x^1, \ldots, x^n)$ at $p\in M$ is $$ \frac{\partial f}{\partial x^i}(p) = \frac{\partial (f\circ \xi^{-1})}{\partial u^i} (\xi p) \tag{1}, $$ where $u^1, \ldots, u^n$ are the natural coordinate functions on $\mathbb{R}^n$. My question is, why are the variables $u^i$ in the r.h.s of $(1)$. Shouldn't it be just $$ \frac{\partial f}{\partial x^i}(p) = \frac{\partial (f\circ \xi^{-1})}{\partial x^i} (\xi p) , $$ because, in fact, the function $(f\circ \xi^{-1})$ doesn't depend on the variables $u^i$ but on the variables $x^i$. Is it a typo?

Gary
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Salvador
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  • Sure, it's just a typo. – Qi Zhu Mar 10 '22 at 09:24
  • It all depends, of course, on the coordinates, but, nevertheless, to be sure, can you clarify the title of the book? – zkutch Mar 10 '22 at 09:33
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    No, its not a typo. That is just the usual notation. The function $f\circ\xi^{-1}$ is a function defined on $\mathbb{R}^{n}$ and as usual, for functions $f:\mathbb{R}^{n}\to\mathbb{R}$ you denote the partial derivative with respect to the natural coordinates $u^{1},\dots,u^{n}$. So, thats just the standard notation for the partial derivative. – G. Blaickner Mar 10 '22 at 09:35
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    More precisely, if you want, $\frac{\partial(f\circ\xi^{-1})}{\partial u^{i}}(\xi p):=\lim_{h\to 0}\frac{(f\circ\xi^{-1})(\xi p+u^{i}h)-(f\circ\xi^{-1})(\xi p)}{h}$. Now, it should be clear why you need $u^{i}$ and not $x^{i}$... – G. Blaickner Mar 10 '22 at 09:40
  • Many thanks to all of you. Yes,@zkutch, it's Semi-Riemannian geometry. With applications to relativity. Ok, @G. Blaickner, I see, in fact that's the formula that appears in Spivak's book (A comprehensive introduction to differential calculus. v1), but I didn't interpret it that way. However, I have a question then, how can we sum two different coordinates? I mean, suppose polar and natural coordinates in some region of $\mathbb{R}^2$, with $\xi(p) = (r_p, \theta_p)$ and $u(p) = (x_p, y_p)$. I can't sum, $\xi(p) + h u^1 = (r_p, \theta_p) + (h, 0)$, isn't it? Or am I completely mistaken? – Salvador Mar 10 '22 at 13:05
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    Oh sorry, I don't know what I was thinking. G. Blaickner is right, the $\frac{\partial}{\partial u^1}, \dots, \frac{\partial}{\partial u^n}$ denote the usual partial derivatives on $\mathbb{R}^n$. – Qi Zhu Mar 10 '22 at 13:27

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