Alternative definition of differential

Question

I've been thinking about idea of derivative, in particular about multivariable case. It seems that it is be pretty well-defined geometrically, so I thought maybe there is another definition, possibly topological one, which can make the definition of derivative coordinate-free. I mean there is topological definitions of limit and continuity and the point is that the last ones give better understanding of the idea then $\epsilon-\delta$ definitions. If there is such definition, then what are the differences between them, when the topological definition is more useful than standard one and vice versa? I'm not satisfied with standard definition because one can be lost in coordinates and it is so tedious too follow the proofs of related theorems that it's easy to lose the ideas which lie inside those theorems.

Answer 1

There is indeed a notion of derivative that we can make sense of without a need for coordinates. Really, the way we generalize the derivative is to define something called the 'differential' as a linear map with a quite a bit more information than what a derivative typically has (the term 'derivative' usually still hangs around though to refer to 'directional derivatives' and the special cases of these - the partial derivatives, all of which, in some sense, make up the differential). To define the differential, you first need an understanding of a coordinate-free definition of the tangent space of Euclidean space (and more generally of a manifold - a space that 'locally looks like Euclidean space' and on which we can therefore 'do calculus') at a point.

There are a couple of such definitions which can be seen to coincide. One of the most natural is for submanifolds in Euclidean space (in fact it is a theorem that every manifold can be realized as such). If you don't know what these are, you can think of them for the moment as something very like graphs of smooth functions. If $M \subset \mathbb{R}^n$ is such an object and $p \in M$ a point, then the tangent space $T_pM$ of $M$ at $p$ can be defined to be the set of equivalence classes of smooth curves $\gamma$ in $M$ through $p$ such that $\gamma_1 \sim \gamma_2$ iff $(\gamma_1')_p = (\gamma_2')_p$ (i.e. the curves have equal tangent vectors at $p$). These equivalence classes are then the 'tangent vectors' to $M$ at $p$ (you can think of the 'direction' of the tangent vector as the direction of the curves in the equivalence class at the point $p$). $T_pM$ can naturally be given the structure of a real vector space, and once we choose local coordinates on $M$ near $p$ we find that it is finite dimensional, with dimension equal to that of $M$ (as a manifold). It therefore easily coincides with the previous, more geometric notion of tangent space. Of course, to make calculations requires choosing coordinates locally, but this notion of tangent space is now invariant under such choices.

With this notion of tangent space in hand, say now $f$ is a real-valued smooth function on $M$ (you can think of $M = \mathbb{R}^n$ as a first example). Then $df_p$ (the differential of $f$ at $p\in M$) can be thought of as a linear map $df_p : T_pM \rightarrow T_{f(p)}\mathbb{R}$ which sends the equivalence class $[\gamma]$ of a curve $\gamma : (-\epsilon,\epsilon) \rightarrow M$ in $M$ to the equivalence class $[f\circ \gamma]$ of the curve $f\circ \gamma : (-\epsilon,\epsilon) \rightarrow M \rightarrow \mathbb{R}$ in $\mathbb{R}$. More generally, if $F : M\rightarrow N$ is a smooth map of manifolds, then $dF_p : T_pM \rightarrow T_{F(p)}N$ is a linear map between the tangent spaces, defined in an analogous way to that above.

In summary, the above explanation is certainly lacking in detail, and although the coordinate-free definitions I have given are workable there are even better ones (look up 'derivations'), but hopefully it gives a sense for how we can begin to make the notion of derivative (or, more importantly, the differential) coordinate free - we can use the coordinate-independent notion of curves (and their equivalence classes) as a substitute for directions typically indicated by an $n$-tuple of real numbers (which of course requires coordinates).

For a much better and fuller treatment of these ideas, some great references are Spivak's "A Comprehensive Introduction to Differential Geometry" and Lee's "Introduction to Smooth Manifolds".