Understanding the definition of derivative

[lriuui0x0]

As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate $Y$ in the direction of $X$, since each $X_p$ is itself a derivation operator, we can already do differentiation?

Consider the following definition for differentiate $Y$ along $X$. Write $Y$ in components $Y = (y^1, \dots, y^n)$, each $y^i$ is a scalar function and differentiate it along $X$ we get $Xy^i$. Do it for all components we get $XY$.

I believe there's problem with this definition, which might relate to coordinate dependence. Can somebody explain to me why is this an ill definition? With many different concepts in differential geometry, it gets confusing on what is coordinate dependent and what is not.

 

[fresh_42]

As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed.

Differentiation is always a local procedure in one direction. You have to choose this direction, e.g. along a flow.

If we want to differentiate $Y$ in the direction of $X$, since each $X_p$ is itself a derivation operator, we can already do differentiation?

This is too sloppy. In direction of an entire vector field? You have to specify one specific direction, e.g. $X_p$. The ordinary product $XY$ is not necessarily a left invariant (or right invariant, depends on your convention) vector field again. What you can do is walk along $X$ and then along $Y$ but you will get different starting points.

Consider the following definition for differentiate $Y$ along $X$. Write $Y$ in components $Y = (y^1, \dots, y^n)$, each $y^i$ is a scalar function ...

No, the $y^i$ are scalars, the coefficients for the basis vectors, which are one forms. What you have is $Y=\sum_k y^k\dfrac{\partial}{\partial x_k}$ if the $\{\,x_k\,\}$ are your standard variables.

... and differentiate it along $X$ we get $Xy^i$. Do it for all components we get $XY$.

No, you don't. It isn't even defined. You can talk about the "matrix entries" of $XY$ but that doesn't give them meaning.

I believe there's problem with this definition, which might relate to coordinate dependence. Can somebody explain to me why is this an ill definition? With many different concepts in differential geometry, it gets confusing on what is coordinate dependent and what is not.

Yes, there is a problem. See above.
Here is an example (under section "B")
https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/