Exploring the Dependence of Variables in the Triplet (TM,M,$\pi$)

[math6]

The triplet \left ( TM,M,\pi \right ) is a vector bundle called the tangent bundle TM such that M is its manifold basis, \pi : TM \rightarrow M the canonical projection. \left ( x^{i},y^{i} \right ) is a local coordinate system on a map \left ( \pi^{-1}(U),\varphi_{U} \right ).
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 ).

 

[Fredrik]

Please put itex or tex tags around the LaTeX expressions. Use the preview feature to verify that they images look the way you intended. You need to refresh and resend after each preview (due to a bug).

 

[arkajad]

Yes, use the features. Now, the whole point is that coordinates $x^i$ in a natural way coordinates on the tangent bundle. And normally this is how the tangent bundle is described. I am not sure from your post whether [itex]y^i[/tex] are the induced coordinates in this way, or "any coordinates", for instance referring to some selected "moving frame".

 

[Kevin_Axion]

The triplet $$\left ( TM,M,\pi \right )$$ is a vector bundle called the tangent bundle $$TM$$ such that $$M$$ is its manifold basis, $$\pi : TM \rightarrow M$$ the canonical projection. $$\left ( x^{i},y^{i} \right )$$ is a local coordinate system on a map $$\left ( \pi^{-1}(U),\varphi_{U} \right )$$.
$$x^{i}$$ 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 $$\left ( \tilde{x}^{i}, \tilde{y}^{i} \right )$$

Although I don't have an answer to your question, this might help you get more responses by those that are more knowledgeable.

Kevin

 

[blue_raver22]

I would first like to commend you on your thorough understanding of the concept of the tangent bundle and its coordinate systems. Your question about the independence of variables is a valid one and can be answered by exploring the mathematical relationships given in your statements.

From (1), we can see that the partial derivative of \tilde{x}^{i} with respect to x^{j} is equal to the Jacobian matrix \frac{\partial x^{k}}{\partial \tilde{x}^{i}}. This matrix represents the change of coordinates from the \tilde{x}^{i} system to the x^{j} system. Similarly, from (2), we can see that the partial derivative of \tilde{y}^{j} with respect to x^{i} is equal to the Jacobian matrix \frac{\partial x^{j}}{\partial \tilde{x}^{l}}. This matrix represents the change of coordinates from the \tilde{y}^{j} system to the y^{l} system.

Therefore, the value of \frac{\partial \tilde{y}^{j}}{\partial x^{i}} can be calculated by taking the product of these two Jacobian matrices, which represents the change of coordinates from the \tilde{y}^{j} system to the x^{i} system. In other words, it represents the dependence of \tilde{y}^{j} on x^{i}.

In conclusion, the variables \left ( x^{i},y^{i} \right ) and \left ( \tilde{x}^{i}, \tilde{y}^{i} \right ) are not independent as they are connected through the coordinate transformations given in (1) and (2). The value of \frac{\partial \tilde{y}^{j}}{\partial x^{i}} represents the dependence of \tilde{y}^{j} on x^{i}.