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 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

$$ <\frac{\partial}{{\partial x_i}}, \frac{\partial}{{\partial x_j}}>= \sum\limits_{\alpha, \beta}\frac{\partial y_\alpha}{{\partial x_i}} \cdot \frac{\partial y_\beta}{{\partial x_j}} \cdot <\frac{\partial}{{\partial y_\alpha}}, \frac{\partial}{{\partial y_\beta}}>.$$


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