What's the standard notation for the differential of $f:M\rightarrow N$? What's the standard notation for the differential of a function $f:M\rightarrow N$ between manifolds?
If $N=\mathbb{R}$, it's written $df:TM\rightarrow T\mathbb{R}$. Is this used for multidimensional case?
 A: I've seen:


*

*$df$

*$Df$

*$J_f$

*$\operatorname{Jac}(f)$

*Attack ships on fire off the shoulder of Orion

*$\frac{\partial f_i}{\partial x_j}$ (in coordinates)

*C-beams glitter in the dark near Tannhäuser Gate


What I use depends on context. If I'm thinking in coordinate-free terms, I like $df$. If I'm thinking in terms of coordinates, I like "$J_f$" to emphasize that this is a Jacobian matrix of $f$.
A: I would write $Tf : TM \to TN$ which is a morphism of vector bundles. This makes transparent that we get a functor $T$ from the category of smooth manifolds and smooth 
maps to the category of smooth vector bundles over smooth manifolds and smooth bundle morphisms.
$Tf$ may be considered as the collection of linear maps $T_p f : T_p M \to T_f(p) N$, $p \in M$.
If the tangent bundle $TN$ is trivial (that is, there exists a smooth bundle isomorphism $h : TN \to N \times \mathbb{R}^n$), you can "identify" $Tf$ with the smooth map
$$T^h f : TM \stackrel{Tf}{\rightarrow} TN \stackrel{h}{\rightarrow} N \times \mathbb{R}^n \stackrel{p}{\rightarrow} \mathbb{R}^n .$$
In fact, $Tf$ can be recovered from $T^h f$ via $T_p f = h_{f(p)}^{-1} \circ T^h_p f$. Note however, that $h$ is not uniquely determined, so $T^h f$ involves a choice. In case $N = \mathbb{R}$ there is a canonical choice.
A: Just a discussion:
If $M$ and $N$ are two smooth manifolds and $f:M\rightarrow N$ is differentiable at $p\in M$ then the differential map is denoted and defined as $D_pf:T_pM\rightarrow T_{f(p)}N$, where $T_pM$ and $T_{f(p)}N$ are the tangent spaces at $p$ and $f(p)$ of $M$ and $N$ respectively.
