Is it true this relation involving the change of coordinates? I'm trying to prove a proposition involving the change of coordinates.
In an open set $U$ of a differentiable manifold I have two sets of coordinates :$x_j^1,\dots, x_j^n$ and $x_k^1,\dots, x_k^n$.
I' m wondering if the following identity is true:
$$\sum_{\alpha,\beta} \frac{\partial x_j^\gamma}{\partial x_k^\alpha}\frac{\partial x_j^\delta}{\partial x_k^\beta}= \sum_{\alpha,\beta} \frac{\partial x_k^\alpha}{\partial x_j^\gamma}\frac{\partial x_k^\beta}{\partial x_j^\delta}$$
Is it true because the inverse of the jacobian $J_{j,k}$ is $J_{k,j}$?
I'm a little confused.Thanks for the help.
 A: A short answer is just no. You may check it by taking $U$ on $\mathbb{R}^2$.
The coordinates system can be just $(x,y)$ and $(r,\theta)$.
In general, if we have a pair of coordinates systems,
$(x_1, \cdots, x_n)$ and $(y_1, \cdots, y_n)$, then we have
$$\delta_{ij} = \frac{\partial x_i}{{\partial x_j}} 
= \sum\limits_{\alpha} \frac{\partial x_i}{{\partial y_\alpha}}
\frac{\partial y_\alpha}{{\partial x_j}}. $$
This equality is purely topological, and then in matrix form,
$$ I_n = J_{\bf x}({\bf y}) \cdot J_{\bf y}({\bf x}). $$
Note that 
$$ \frac{\partial}{{\partial x_i}} = 
\sum\limits_{\alpha}\frac{\partial y_\alpha}{{\partial x_j}}
\frac{\partial}{{\partial y_\alpha}}.$$
If we impose the metric $g$ on $U$, it gives us
$$ <\frac{\partial}{{\partial x_i}}, \frac{\partial}{{\partial x_j}}>=
\sum\limits_{\alpha, \beta}\frac{\partial y_\alpha}{{\partial x_i}}
\cdot \frac{\partial y_\beta}{{\partial x_j}} \cdot 
<\frac{\partial}{{\partial y_\alpha}}, \frac{\partial}{{\partial y_\beta}}>.$$
