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Let's say I have a set of $m$ functions depending on $n$ variables $$x^{i} = x^{i}(s^{1}, ... , s^{n})$$ I use latin indices to denote $(1, ..., m)$ and greek letters to denote $(1, ..., n)$.

I know $$\frac{\partial x^{i}}{\partial x^{j}} = \delta^{i}_{j}$$ $$ \implies \frac{\partial^{2} x^{i}}{\partial s^{\alpha} \partial x^{j}} = \frac{\partial (\delta^{i}_{j})}{\partial s^{\alpha}} = 0$$

What I'd like to know is whether commutativity of partial derivatives allows me to do this:

$$\frac{\partial^{2} x^{i}}{\partial x^{j} \partial s^{\alpha}} = \frac{\partial^2 x^{i}}{\partial s^{\alpha} \partial x^{j}} = 0$$

And if I'm not allowed to do so, why?

Thanks a lot, and I'm sorry if it is a silly question.

Edit:

For example, I get troubles when I'm in the simplest case, $m=1, n=1$. Let's say $x=s^2$ $$\frac{dx}{dx}=1$$ $$\implies \frac{d}{ds}\left( \frac{dx}{dx} \right) = 0$$

But when I do it in the opposite order:

$$\frac{dx}{ds}=2s$$ $$\implies \frac{d}{dx}\left( \frac{dx}{ds} \right) = \frac{d}{dx}\left( 2s \right)$$ substituting $s = \sqrt{x}$ $$\frac{d(2s)}{dx} = \frac{d(2\sqrt{x})}{dx} = \frac{1}{\sqrt{x}} = \frac{1}{s}$$

What am I doing wrong?

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  • $\begingroup$ Your notation is making this a bit less readable, but Schwarz theorem holds whenever the function is $C^2$. $\endgroup$ – Panda Nov 22 '17 at 18:03
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Thats not necessarily true. Take a counterexample $$ x = r \cos \theta, \quad y= r \sin \theta $$ With $x^1=x$, $x^2=y$ and $s^1=r$, $s^2=\theta$, you'll have $$ \frac{\partial}{\partial y}\frac{\partial x}{\partial r} = \frac{\partial}{\partial y}(\cos \theta) = \Bigg(\frac{\partial r}{\partial y} \frac{\partial}{\partial r}+ \frac{\partial \theta}{\partial y} \frac{\partial}{\partial \theta}\Bigg) (\cos \theta) = 0 + \frac{\partial \theta}{\partial y} (-\sin \theta) $$ whereas $$ \frac{\partial}{\partial r}\frac{\partial x}{\partial y} = 0. $$ By Schwarz's theorem, the commutativity of partial derivatives only apply to variables on the same coordinates.

In your case, when you compare these two expression $$\frac{\partial }{\partial x^{j}} \frac{\partial x^{i}}{\partial s^{\alpha}} \quad \text{and} \quad \frac{\partial}{\partial s^{\alpha}} \frac{\partial x^{i}}{\partial x^{j}}, $$ we know that the second expression will always zero, but the first is not. Because if you try to expand as below \begin{align} \frac{\partial }{\partial x^{j}} \frac{\partial x^{i}}{\partial s^{\alpha}} &= \frac{\partial s^{\beta}}{\partial x^j} \frac{\partial }{\partial s^{\beta}} \Bigg( \frac{\partial x^{i}}{\partial s^{\alpha}} \Bigg) = \frac{\partial s^{\beta}}{\partial x^j} \frac{\partial }{\partial s^{\alpha}} \Bigg( \frac{\partial x^{i}}{\partial s^{\beta}} \Bigg) = \frac{\partial}{\partial s^{\alpha}} \Bigg( \frac{\partial s^{\beta}}{\partial x^j} \frac{\partial x^{i}}{\partial s^{\beta}}\Bigg) - \frac{\partial}{\partial s^{\alpha}} \frac{\partial s^{\beta}}{\partial x^j} \\ &= \frac{\partial}{\partial s^{\alpha}} \frac{\partial x^i}{\partial x^j} - \frac{\partial}{\partial s^{\alpha}} \frac{\partial s^{\beta}}{\partial x^j} , \end{align} the last equality immidiately tells us that the expression is not always equal to $\frac{\partial}{\partial s^{\alpha}} \frac{\partial x^i}{\partial x^j}$.

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  • $\begingroup$ Yes!, that's what I mean. I guess the change of variable I introduce in my example is, in a sense, a new cordinate system, right? Could you provide me with a textbook where can I find more information on this topic? Thank you very much. $\endgroup$ – Jackozee Hakkiuz Nov 22 '17 at 18:27
  • $\begingroup$ Usually its not explicitly written in the text (in fact its not at all in math text). This kind of thing uses very often in physics espescially in GR text. You may look at Inverno's book or Schultz or Carroll for more example. $\endgroup$ – Sou Nov 22 '17 at 18:38
  • $\begingroup$ Yes, actually I came across this problem when I was trying to inherit a connection from a riemannian manifold to an immersion. I'll take a look on the literature you suggested. Thanks a lot. $\endgroup$ – Jackozee Hakkiuz Nov 22 '17 at 19:07
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As long as $x_i$ varies with $s_i$ you cannot switch the partial derivative variables. That is why the result does not make sense to you. According to Schwarz's theorem the commutative property of partial derivative is valid if $x_i$ are independent variables (e.g. $x,y,z$ in Cartesian coordinate systems).

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