§ Derivatives in diffgeo

• A function of the form $f: \mathbb R^i \to \mathbb R^o$ has derivative specified by an $(o \times i)$ matrix, one which says how each output varies with each input.
• Now consider a vector field $V$ on the surface of the sphere, and another vector field $D$. Why is $W \equiv \nabla_D V$another vector field? Aren't we differentiating a thing with 3 coordinates with another thing with 3 coordinates?
• Well, suppose we consider the previous function $f: \mathbb R^i \to \mathbb R^o$, and we then consider a curve $c: (-1, 1) \to \mathbb R^i$. Then the combined function $(f \circ c): (-1, 1) \to \mathbb R^o$ needs only $o$ numbers to specify the derivative, since there's only one parameter to the curve (time).
• So what's going on in the above example? Well, though the full function we're defining is from $\mathbb R^i$ to $\mathbb R^o$, composing with $c$ "limits our attention" to a 1D input slice. In this 1D input slice, the output is also a vector.
• This should be intuitive, since for example, we draw a circle parameterized by arc length, and then draw its tangents as vectors, and then we draw the normal as vectors to the tangents! Why does that work? In both cases (position -> vel, vel -> accel) we have a single parameter, time. So in both cases, we get vector fields!
• That's somehow magical, that the derivative of a thing needs the same "degrees of freedom" as the thing in itself. Or is it magical? Well, we're used to it working for functions from $\mathbb R$ to $\mathbb R$. It's a little disconcerting to see it work for functions from $\mathbb R$to $\mathbb R^n$.
• But how does this make sense in the case of diffgeo? We start with a manifold $M$. We take some curve $c: (-1, 1) \to M$. It's derivative must live as $c': (-1, 1) \to TM$. Now what about $c''$? According to our earlier explanation, this too should be a vector! Well... it is and it isn't, right? but how? I don't understand this well.
• Looping back to the original question, $W \equiv \nabla_D V$ is a vector field because the value of $W(p)$ is defined as taking $D(p) \in T_p M$, treating it as a curve $d_p: [-1, 1] \to M$ such that $d_p(0) = p$ and $d_p'(0) = D(p)$, and then finally taking $V()$.