I am reading Tu's book and I am new in smooth manifold so please ignore my ignorance. I am reading a section which states:
Given a smooth manifold $\mathcal{M}$ and a chart $\phi = (x^1, x^2, \dots, x^n): U \rightarrow \mathbb{R}^n$, a function $f\in C^{\infty}(\mathcal{M}, \mathbb{R})$ and $x^i=r^i\circ \phi$ where $r^i$ coordinate function, for $p\in U$ the partial derivative of $f$ w.r.t $x^i$ is given by
$$\frac{\partial }{\partial x^i}\bigg|_{p}f:=\frac{\partial f}{\partial x^i}(p):=\frac{\partial (f \circ\phi^{-1}) }{\partial r^i}(\phi(p)):=\frac{\partial }{\partial r^i}\bigg|_{\phi(p)}(f\circ\phi^{-1})$$
For me it is not so obvious why the partial derivative is changed from $x^i$ to $r^i$. Could you please give a mathematical explanation (or intuition) on this change?
Thanks!
Best Answer
Note that what you wrote is a definition. For an intuition think that the $x_i$ live on the manifold, whereas the $r_i$ live on $\mathbb{R}$. You want to define the partial derivative on a Manifold by going back to the already defined partial derivative on $\mathbb{R}^n$. Therefore you need to change the coordinates from coordinates on $M$ to coordinates on $\mathbb{R}$ and at the same time the function needs to be in $\mathbb{R}^n$, so naturally you consider $f\circ \phi^{-1}$.