Partial derivatives on Manifolds Let $F : A \times B \to C$ be a map of smooth manifolds.
Define the following maps ("partial derivatives"):
$E_1 F: TA \times B \to TC$
$E_1 F(a,v,b) = D_a F(-,b) v $ where $v \in T_a A$
$E_2 F: A \times TB \to TC$
$E_2 F(a,b,w) = D_b F(a,-) w $ where $w \in T_b B$
Composing $E_1$ and $E_2$ we get two maps $TA \times TB \to TTC$. An obvious question is to ask whether they are the same. I tried proving this, but it turned into a notational disaster. (I don't even see that the $E_iF$ are smooth.)
My questions are:


*

*Do people use this or a similar construction to define partial derivatives on manifolds?

*Does this construction have anything to do with $DF : T(A\times B) \to TC$?


My eventual goal is:
Let $G$ be a Lie group and $F : G \times G \to G$ defined as:
$F(g,h) = g^{-1}hg$
and I would like to differentiate $F$ in a way to get the Lie bracket on $T_e G$.
 A: You are right, that's the way abstract partial derivatives are defined in Differential Topology. Note however that you've only written the principal parts of them. The complete definition is
$$
E_1F(a,v,b)=(F(a,b),D_aF(-,b)v),
$$
and similarly for $E_2$. Both maps are fiber maps over $F:A\times B\to C$. The usual notation is however more classical, say $E_i=D_i$, even
$$
D_aF(-,b)v=\frac{\partial F}{\partial x}(a,b)v,\quad D_bF(a,-)w=\frac{\partial F}{\partial y}(a,b)w.
$$ 
Of course, 
$$
DF(a,b)(v,w)=\frac{\partial F}{\partial x}(a,b)v+\frac{\partial F}{\partial y}(a,b)w
$$
They are differentiable mappings and $E_1E_2=E_2=E_1$ (except maybe reordering components). And yes, computations are messy. The best way to proceed is by localization, as clearly the matter is local in $A, B, C$. Hence one can suppose these three manifolds are open sets in ${\mathbb R}^m, {\mathbb R}^n, {\mathbb R}^p$. Then after some work, one gets
$$
E_2E_1F(a,v,b,w)=\big(F(a,b),\frac{\partial F}{\partial x}(a,b)v,\frac{\partial F}{\partial y}(a,b)w, \frac{\partial^2 F}{\partial y\partial x}(a,b)(v,w)\big).
$$
If we compute $E_1E_2$ the difference might be the fourth term, but we have the Schwarz rule (for class 2 maps, otherwise there are counterexamples already in $A\times B=\mathbb R^2$). This fourth term is a kind of Hessian. Computed explicitely for $F=(F_1,\dots,F_p)$ (as we are supposing $C$ is en open set in ${\mathbb R}^p$) one gets the same algebraic expressions involving the sums
$$
\sum_{k,\ell}\frac{\partial^2 F_j}{\partial y_\ell\partial x_k}(a,b)w_\ell v_k
$$
for $E_2E_1$ and the sums
$$
\sum_{k,\ell}\frac{\partial^2 F_j}{\partial x_\ell\partial y_k}(a,b)v_\ell w_k
$$
for $E_1E_2$, which coincide by the Schwarz rule.
