What operation corresponds to spatial derivatives on smooth vector fields, if any? Let $\pi:E\to M$ be a smooth vector bundle on a $d$ dimensional manifold.
By choosing some local coordinates $x^\mu=x^\mu(p)=(x_1(p),\ldots,x_d(p))$ on a chart $U$ and some section $v:M\to E$, every $v(p)\in \pi^{-1}(\{p\})=V_p$ can be described locally by $v(x^\mu(p))$ if I'm not mistaken.
My question is wether we are allowed to do
$$\lim_{h\to 0}\left(\frac1h(v{(x_1,\ldots,x_i+h,\ldots,x_d)}-v{(x_1,\ldots,x_i,\ldots,x_d)})\right).$$
If $V$ is the vector field on $U$ defined by the restriction of the section $v$ on $U$, what operation does it define? Is it $\dfrac{\partial v({x^\mu})}{\partial x^i}$, $(\nabla_i V)_p$ or something else?
My guess would be that we can parallel transport $v(p)$ from $x^\mu$ to $x^{\mu+h}=(x_1,\ldots,x_i+h,\ldots,x_d)$ and then take the usual derivative on $V_{x^{\mu+h}}$, so I think that it is well defined. The problem is I don't know what operation it is equivalent to. Any light would be greatly appreciated. Thanks.
EDIT:
Couldn't we just define $v(x+h)=L_h(v(x))$ where $L_h\circ L_{h'}=L_{h+h'}$ linear iso and take the derivative of $L$ at $0$ giving us that the result is the connection applied to $v(p)$? Isn't parallel transport defined this way giving us a sense of defining the above?
 A: Given a choice of coordinates, you can take a derivative of a vector field in this manner, but the result will depend on your choice of coordinates. If, however, you want to take such a derivative in a coordinate independent manner, it isn't really possible to define such a derivative canonically.
The problem is that "subtracting" vectors in different fibers has no intrinsic meaning, and defining things in coordinates as you do will not be consistent. This is precisely what a connection is: an additional piece of data which makes these kind of derivatives unambiguous. This extra data is generally necessary.
For a vector bundle $\pi:E\to M$, a common starting point is to define an operator $\nabla:\Gamma(TM)\times\Gamma(E)\to\Gamma(E)$ (satisfying a product rule) as this additional piece of data, and define this to be the derivative for vector fields. In general, a given vector bundle can have many different connections.
By choosing a coordinate chart and defining a derivative as you do, you are essentially defining a (local, flat) connection such that fields with constant coordinates in that chart have vanishing derivative.
