remember:
derivative is a function $df\from TM \to TN$ covering $f\from M \to N$
(FIXME: commutative diagram)

it's -weird- that for functions $f\from \bR \to \bR$ we have
$f'\from \bR \to \bR$.

Indeed, we also have given $f\from \bR \to \bR^n$
an $f'\from \bR \to \bR^n$ by taking coordinates;
the point is that products have a categorical interpretation in the codomain.

What's much trickier (as it doesn't have a categorical interpretation)
is understanding $f\from \bR^m \to \bR$, and this leads
to linear algebra: instead of a tangent line, you get a tangent plane.

Directional derivatives and vector fields.
However, given $f\from \bR^m \to \bR$ and $v\in T_p\bR^m$, we can find the
\Def{directional derivative} of $f$ in the direction $v$ (at $p$).
Informally, you can take a path $\gamma\from (-\epsilon,\epsilon) \to \bR^m$ 
such that $\gamma(0)=p$ and $\dot \gamma(0) = v$, and then pull back $f$ to
$f\gamma\from \bR \to \bR$, and then take a single-variable derivative.
However, you need to check that this is independent of choice of $\gamma$.

Formally, this is because the codomain has canonically trivialized tangent
bundle, so $df\from T_p\bR^m \to T_f(p) = f(p) \times \bR = \bR$,
so $df(v) \in \bR$ naturally.

I.e., $df$ is a section of the cotangent bundle.

More generally, given a vector field $\xi$ on a manifold $M$ and $f\from M \to
\bR$ ($f \in \Diff(M)$, i.e., differentiable real-valued functions on $M$),
we can differentiate in the $\xi$ direction (at each point $p$, differentiate
in the direction $\xi_p$), and get a function $\xi f \in \Diff(M)$, i.e.,
$\xi$ is a differential operator on the algebra $\Diff(M)$.

All differential operators on $\Diff(M)$ are of this form for some
vector field, so in particular $\xi\psi - \psi\xi$ is some vector field,
(careful: $\xi\psi$ is an operator, but no longer a differential operator
 I don't think; this is what we mean by saying that we've embedded a lie
 algebra in an associative algebra)
which yields the Poisson bracket $\left\{\xi,\psi\right\}$ on these operators,
and this leads to a lie algebra structure on vector fields.

Indeed, in Zimmer's class we \emph{defined} the cotangent bundle as the
algebra of differential operators on $\Diff(M)$.

--------------

Staying on track, this allows us to deepen our understanding of what we mean
by ``differentiating a function'' or ``integrating a function''.

