I need help about understand the definition and the using of basis vectors
$\{e_{i}=\frac{\partial}{\partial x^{i}}\}$ and its dual basis $\{e^{i}=dx^{i}\}$ (with $x_{i}$ curvilinear coordinates).
Indeed, we have the following relation, taking $\vec{OM}$ the position vector :
$$\vec{e_{i}}=\dfrac{\partial \vec{OM}}{\partial x^{i}}\quad\quad(equation 1)$$
Actually, this expression above is not used, into differential geometry courses, I often see $\vec{e_{i}}$ defined by :
$$e_{i}=\frac{\partial}{\partial x^{i}}\quad\quad(equation 2)$$
This expression above seems to look as a differential operator and not a "classic vector" like on (equation 1)
With this difference, I get the following issue about the demonstration of $\nabla_{\frac{\partial}{\partial x^i}} \left(\vec{V}\right)$ :
$$\nabla_{\frac{\partial}{\partial x^i}} \left(\vec{V}\right)=\nabla_{\frac{\partial}{\partial x^i}} \left( V^j \frac{\partial}{\partial x^j} \right)$$ $$ = \frac{\partial V^j}{\partial x^i} \frac{\partial}{\partial x^j} + V^j \nabla_{\frac{\partial}{\partial x^i}} \left( \frac{\partial}{\partial x^j} \right) = \frac{\partial V^j}{\partial x^i} \frac{\partial}{\partial x^j} + V^j \Gamma_{ij}^k \frac{\partial}{\partial x^k}\quad\quad(equation 3)$$.
So, by multiplying $(equation3)$ by $\vec{OM}$, I get the following expression :
$$\left(\nabla_i \frac{\partial V^j}{\partial x^i} \frac{\partial}{\partial x^j}\right)\vec{OM}= \big(\frac{\partial V^k}{\partial x^i} + V^j \Gamma_{ij}^{k}\big)\,\vec{e_{k}}$$
FIRST QUESTION : As this point, can I write :
$$\left(\nabla_i \left(\frac{\partial V^j}{\partial x^i} \frac{\partial}{\partial x^j}\right)\right)\vec{OM}=\left(\nabla_i \left(\frac{\partial V^j}{\partial x^i}\vec{e_{j}}\right)\right)$$
???
As you can see in the expression $(equation3)$, I have taken the expression $\vec{e_{i}}=\dfrac{\partial}{\partial x^{i}}$ instead of $\vec{e_{i}}=\dfrac{\partial \vec{OM}}{\partial x^{i}}$.
As a consequence of this difference, I don't know how to start from :
$$\nabla_{\frac{\partial}{\partial x^i}} \left(\vec{V}\right)=\nabla_{i} \left( V^j \frac{\partial\vec{OM}}{\partial x^j} \right) $$
and find the expression of $(equation 3)$, especially can we write :
$$\nabla_{i} \left( V^j \dfrac{\partial\vec{OM}}{\partial x^j} \right) =$$ $$ = \left(\nabla_{i}(V^j)\right)\frac{\partial\vec{OM}}{\partial x^j} + V^j \nabla_{i} \left( \frac{\partial\vec{OM}}{\partial x^j} \right)\quad\quad(equation 4) $$
?? (I suppose Leibniz derivation rule is valid for covariant derivative).
SECOND QUESTION :
Anyone could help me to continue the calculation of the (equation4) expression in order to get the $(equation 5)$ below which expresses the $k-th$ component of $\nabla_i(\vec{V})$
$$\left(\nabla_i \vec{V}\right)^{k}= \frac{\partial V^k}{\partial x^i} + V^j \Gamma_{ij}^{k} \quad\quad(equation 5)$$.
If someone could see the subtleties between these 2 notations for basis vectors $\vec{e_{i}}$, one defined as an operator and the other defined in a classic way and as a real vector, this would be nice to tell it.
Thanks for your help
You'll have to go read the literature yourself for all the details, but your confusion lies in understanding the answer to this question,
If $p \in M$ where $M$ is a abstract manifold. Here we simply mean that we don't think of $M$ as a subset of euclidean space. Then charts on $M$ are of the form $(U, \phi) = (U, x^1,...,x^n)$ and $\phi(U) \subset \mathbb{R}^n$. So what do elements in $T_pM$ look like?
Immediately you see this problem is much different than that which would of been easy to answer if $M \subset \mathbb{R}^n$ and $\phi: U \subset \mathbb{R}^n \to M$. You would of simply said, oh it's spanned by $\partial \phi/ \partial x^j$ where $j = 1,...,n$. However now we "don't initially have a coordinate system about $p$'', so what do we do? That's what $\partial/ \partial x^j$ is! But since this is just notation, we must find a way to link it to $(U, \phi)$ which is a chart that contains $p$.
I hope this makes sense. I really wanted to motivate the idea and have you read it for yourself without telling you everything in one setting. You can read the relevant material in Tu's Introduction to Manifolds, John Lee Smooth Manifolds or I have notes here, notes. It starts on page $6$, good luck :)