- #1

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x^{i} is the system of map coordinates \left ( U,\varphi \right ) of M and y^{i} are as

y= y^{i}\frac{\partial }{\partial x^{i}} , y \in T_{x}M .

Now if we take a new system of coordinates \left ( \tilde{x}^{i}, \tilde{y}^{i} \right )

on a map ( \pi^{-1}(V),\phi _{U} \right ) . \tilde{x}^{i} is the system of map coordinates \left ( V,\psi \right ) of M.

Then after the change of coordinates we have the following results :

( 1) \frac{\partial }{\partial \tilde{x}^{i}} = \frac{\partial x^{k}}{\partial \tilde{x}^{i}}\frac{\partial }{\partial x^{k}} .

(2) \tilde{y}^{j}= \frac{\partial x^{j}}{\partial \tilde{x}^{l}}y^{l}.

My question is this: It is clear from (1) that {x}^{i} depends \tilde{x}^{j} (and vice versa)

also {y}^{i} depends \tilde{y}^{j}. So \tilde{y}^{j} does it depend of {x}^{i} ?

What is the value then of \frac{\partial \tilde{y}^{j}}{\partial x^{i}} ?

In short, I seek the independence of variables \left ( x^{i},y^{i} \right ) and \left ( \tilde{x}^{i}, \tilde{y}^{i} \right ).