Notation for tensors: $\partial_i,\rm dx^i$ vs. $e_i, e^i$ vs. $Z_i, Z^i$ This is not a question about raising and lowering indices as suggested.

This is proof that I still don't understand the topic well, but here it goes: Pavel Grinfeld in his youtube videos uses $\bf Z_i$ for the covariant basis vectors (or on the blackboard $\vec Z_i$), which he describes as the derivative of the position vector (assumes Euclidean straight lines) along coordinate systems. So he also uses notation such as $\Gamma^k_{ij}\bf Z^k=\frac{\partial\bf Z_i}{\partial Z^i}$ to note the partial derivative of the covariant basis with respect to each of the coordinates, and introduce the Christoffel symbols.
I am sure this is perfect, and to his credit, he does mention "a lot of Z's here" at some point. But I would like to know what he has in mind to use these "Z" all over, and how to reconcile this with what I see elsewhere on this topic, i.e. $\bf e_i, e^i$ or $\partial_i,\rm dx^i.$
I see that the $\bf e$'s may be too close to the Cartesian system for his "coordinate-free" thinking, perhaps, but I have no clue if this is the motivation. Likewise, perhaps the $\partial$ and differential $\rm d$ are to physics-centered, but again, no clue if this is the motivation.
So what are the different notations for different occasions or disciplines to denote covariant and contravariant basis vectors in tensor algebra and calculus?
NB: I am not asking about the orthogonality of vectors and covectors (and the upper/lower index position). I am asking about where and why a letter (Z or e) is expected, and where and why, in other spots, the calculus notation is used.
 A: In typical differential/Riemannian geometry books, we start with a smooth manifold $M$ and typically introduce the following objects:


*

*a coordinate chart $(U,x=(x^1,\dots, x^n))$, i.e each $x^i:U\to\Bbb{R}$ is a smooth function.

*This coordinate chart induces a basis of vector fields on $U$, denoted $\left\{\frac{\partial}{\partial x^i}\right\}_{i=1}^n$. This may be abbreviated to $\partial_i$ in some cases. THis is fine if you're only dealing with a single coordinate chart, but if there are two that you're talking about simultaneously, it's better to be more explicit.

*A basis of covector fields $\{dx^i\}_{i=1}^n$. These are the exterior derivatives of the functions $x^i$, and they are the dual to the basis above. Alternatively, you can define a-priori $\{dx^i\}_{i=1}^n$ to be the symbolic expression for the dual basis, and then prove that they are indeed the exterior derivative of the coordinate functions $x^i$.

Next, if we suppose that we have a Riemannian metric $g$ (this is the classical notation for the metric tensor). Then, we introduce the notation


*$g_{ij}=g\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right)$, i.e take the inner product with respect to $g$ of the coordinate-induced basis vector fields.


*We can store the above functions $g_{ij}:U\to\Bbb{R}$ in a matrix $[g_{ij}]$. This is an invertible matrix, and its inverse matrix is denoted by $[g^{ij}]$, so $g^{ij}$ refers to the $(i,j)$-entry of the inverse matrix to $[g_{ij}]$. A more abstract definition of $g^{ij}$ is as follows. Since $g$ is a metric tensor, in particular it means that the mapping $g^{\flat}:TM\to T^*M$, $g^{\flat}(v)=g(v,\cdot)$ is a vector-bundle isomorphism; its inverse is denoted $g^{\sharp}:T^*M\to TM$. Thus, it allows us to "transfer" the $(0,2)$ tensor field $g$ from $TM$ to $T^*M$, i.e we get a $(2,0)$ tensor field $\tilde{g}(\cdot,\cdot)= g(g^{\sharp}(\cdot),g^{\sharp}(\cdot))$. Then, the components $\tilde{g}(dx^i,dx^j)$ are precisely the functions $g^{ij}$ I defined above.


*Next, on a Riemannian manifold, we can talk about covariant derivatives with respect to the Levi-Civita connection $\nabla$. With this, we introduce the Christoffel symbols via the equation $\nabla_{\frac{\partial}{\partial x^i}}\left(\frac{\partial}{\partial x^j}\right)= \Gamma^{k}_{ij}\frac{\partial}{\partial x^k}$, or more explicitly, $\Gamma^k_{ij}=dx^k\left(\nabla_{\frac{\partial}{\partial x^i}}\left(\frac{\partial}{\partial x^j}\right)\right)$.
Now, I typically reserve the notation $\mathbf{e}_i$ to mean the normalized version of $\frac{\partial}{\partial x^i}$, so
\begin{align}
\mathbf{e}_i:=\frac{\frac{\partial}{\partial x^i}}{\left\|\frac{\partial}{\partial x^i}\right\|}=
\frac{\frac{\partial}{\partial x^i}}{\sqrt{g\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^i}\right)}}=\frac{1}{\sqrt{g_{ii}}}\frac{\partial}{\partial x^i}.
\end{align}
Sometimes, I might use $\{\mathbf{e}_i\}_{i=1}^n$ to just mean an arbitrary local basis of vector fields (possibly orthonormal, but not necessarily; another term for this is a moving (tangent) frame). Actually, I personally would drop the boldface, and just say "let $\{e_i\}_{i=1}^n$ be a local basis of vector fields for the tangent bundle", or I also like the letter $\xi$, so I might say "let $\{\xi_i\}_{i=1}^n$ be a local frame for $TM$". However, some people may use $\mathbf{e}_i$ synonymously with $\frac{\partial}{\partial x^i}$, so always read the definition provided by the author first, and decide the meaning based on context.

Now I've only taken a cursory look at Pavel Grinfeld's notation, and here's what I gather (I'll try to match up each point above with a cooresponding one below). Also,  he seems to work only in $\Bbb{R}^n$, so there's always a Riemannian metric present, and he seems to like the letter $Z$ alot:


*

*' He uses $(Z^1,\dots, Z^n)$ to denote coordinates instead of $(x^1,\dots, x^n)$.

*' He uses the notation $\mathbf{Z}_i$ for what I've called $\frac{\partial}{\partial x^i}$ above.

*' I haven't seen him introduce an analogue to $dx^i$, but if he did, he'd just denote it as $dZ^i$.

*' He writes $Z_{ij}$ in place of $g_{ij}$.

*' He writes $Z^{ij}$ in place of $g^{ij}$.

*' He writes $\frac{\partial \mathbf{Z}_i}{\partial Z^j}=\Gamma^k_{ij}\mathbf{Z}_k$ for the definition of the Christoffel symbols. Note this only makes sense because he's working in $\Bbb{R}^n$, which is why we can take partial derivatives of vector fields.

Lastly, he introduces a "contravariant basis" $\mathbf{Z}^i$, defined as $\mathbf{Z}^i= g^{ij}\mathbf{Z}_j$. In the usual notation, it would be written as $g^{ij}\frac{\partial}{\partial x^j}$. In my opinion, this is a confusing step because it is artifically forcing one to work within the tangent space and avoid venturing out into the dual space, which is a perfectly natural place to go. I suggest taking a look at this answer of mine where I give more details.


Perhaps one final thing I should mention is why we use the partial differential notation $\frac{\partial}{\partial x^i}(p)$ to mean an element of the tangent space $T_pM$. One way of viewing it is that the definition of a tangent vector is simply as a derivation (a linear map acting on smooth functions, which satisfies the product rule). This is an abstract definition, but it's technically pretty smooth-sailing. A slightly more geometric way of looking at things is by looking at (equivalence classes of) smooth curves passing through the point $p$. If you're given a coordinate chart $(U,x)$ on the manifold about $p$, and if we let $\{e_i\}_{i=1}^n$ denote the standard basis on $\Bbb{R}^n$, then one example of a curve is
\begin{align}
\gamma_i(t)=x^{-1}(x(p)+te_i).
\end{align}
In words, we have a point $p$ in the manifold, and we can look at it's coordinate representation $x(p)\in\Bbb{R}^n$. We can no move along the $i^{th}$-direction, which yields the line $t\mapsto x(p)+te_i$, and finally we can use $x^{-1}$ to map this back to the manifold. So, the curve $\gamma_i$ is like "moving in the manifold while only varying $x^i$". Now, given this curve, there is a natural way in which we can allow it to "act" on smooth functions, namely for any smooth function $f:M\to\Bbb{R}$, consider $(f\circ\gamma_i)'(0)$, but this is just
\begin{align}
(f\circ\gamma_i)'(0)=\frac{d}{dt}\bigg|_{t=0}f(x^{-1}(x(p)+te_i))=
D(f\circ x^{-1})_{x(p)}(e_i)=D_i(f\circ x^{-1})(x(p)),
\end{align}
where $D_i$ means partial derivative with respect to the $i^{th}$ variable. The last term is what we simply write as $\frac{\partial f}{\partial x^i}(p)$. Because of $f$ being arbitrary, it makes sense to 'identify' the curve $\gamma_i$ with the operator $\frac{\partial}{\partial x^i}(p)$, thereby obtaining an identification between the geometric definition of tangent vectors via curves, and the algebraic version via derivations.
A: The notations $\mathbf Z_i$ or $\mathbf e_i$ are completely equivalent (just a different letter is used). $\mathbf e^i$ with the upper index is used for a basis of covectors (a basis of the cotangent space, instead of the tangent space). This is usually the dual basis of $\mathbf e_i$, so we have the relation $\mathbf e^i (\mathbf e_j) = \delta^i_j$ where $\delta$ is the Kronecker symbol.
Those notations can be used for any set of (co)vector fields which are pointwise a basis of the (co)tangent space. On the other hand $\partial_i$ and $\text dx^i$ are specifically the basis of the tangent and cotangent spaces which arise from local coordinates.
Edit : one difference between a general local frame $\mathbf e_i$ (a set of local vector fields which are a basis of the tangent space at each point) and the one arising from local coordinates $x_i$ (ie the $\partial_i$) can be seen when computing Lie brackets of vector fields. We always have $[\partial_i, \partial_j] = 0$, while for a general frame $[\mathbf e_i, \mathbf e_j]$ can be non zero. In fact, this is a necessary and sufficient condition : if $\mathbf e_i$ is a frame whose Lie brackets vanish, then there are local coordinates $x^i$ such that $\mathbf e_i = \partial_i$.
Edit 2 It was not clear from OP that Pavel Grinfelds uses the notation $Z^i$ for coordinates (on top of $\mathbf Z_i$ for the tangent vectors). Taking this into account, $\mathbf Z_i$ is completely equivalent to $\partial_i$, while $\mathbf e_i$ can be used, depending on the context, for a frame which might not come from coordinates.
A: I have linked a post already. If that doesn't give a full pic, lemme summarize what's going on here. So, the way Pavel Grinfeld is doing it is that he has managed to find a way to squeeze the dual space of a vector space into the vector space itself using the metric tensor and then a canonical isomorphism between dual and regular space which works due to two reason:

*

*All vector space of same dim is isomorphic


*Dual vector space of a vector space is of same dim as vector space
Now this is pretty convoluted to do in a rigorous conceptual sense as peek-a-boo(zebra pfp) has pointed out very correctly in the linked answer that they wrote.
Now, what is the relation between $\partial_i$ and $e_i$ thing? Well the idea is they can be identified with each other and the exact details of how that is done relates to a topic known as derivations.
Finally $dx^i$ or the differential forms, in context of what all I said above, a way one could see them is that you identify $e_i$ with $\partial_i$ then you see the dual of $\partial_i$. This dual space has a basis of $dx^i$ for all the index of $i$ as $\partial_i$ having property that:
$$ dx^i(\partial_j) = \delta_j^i$$
Hope this help you fit in all the ideas together.
