TITLE: Differentiation: Vectors or scalars
QUESTION [6 upvotes]: As you know, we can geometrically interpret the set of all reals $\Bbb R$: every point on the line can be uniquely expressed by a real number x (it's coordinate). However, we could go a step further and use the line to geometrically express operations between real numbers, such as addition. In order for us to do this we introduce vectors and now every point on the line and every coordinate associated with that point represent a vector starting from the origin and ending at the given point. Thus every real number either represents a point on the line or a vector on the line. Now, in the definition of the derivative$$ \frac{{\rm d} }{{\rm d}x} y(x) =\lim_{h\rightarrow0} \frac{1}{h}( y(x+h)-y(x)) $$ we have x+h, which only makes sense geometrically if x and h are both vectors (addition of points on the line makes no sense). My question is: In order to have a meaningful definiton of the derivative, should the function y(x) be geometrically interpreted as mapping vectors to vectors?

REPLY [2 votes]: In a nitpicking sense, the derivative
$$
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
$$
of a function $f$ is indeed a scaling map that specifies how a (small) displacement (vector) $h$ at a point $x$ gets stretched into a displacement vector $f'(x)\, h$ at $y = f(x)$.
Commonly this relationship is expressed in terms of points as $f(x + h) \approx f(x) + f'(x)\, h$. The derivative itself, $f'(x)$, is neither a point nor a vector.
In more general settings (such as coordinate-free formulations of mechanics), "points" live in a space $M$ (a "manifold"). To formulate differential calculus, one introduces an entirely new space $TM$ (the "tangent bundle"), whose elements may be viewed as pairs $(x, h)$ consisting of a point $x$ of $M$ and a displacement $h$ at $x$.
In this setting, if $f:M \to N$ is a differentiable mapping, its "(total) derivative" is a mapping $f_{*}:TM \to TN$ defined by
$$
f_{*}(x, h) = \bigl(f(x), f'(x)\, h\bigr).
$$
The derivative $f'(x):T_{x} M \to T_{f(x)} N$ is a "linear transformation" between tangent spaces, a generalization of a scaling map.
The conceptual asymmetry between points and vectors becomes starkly visible in this setting. Adding points makes no sense at all. The sum "$x + h$" of a point and a vector cannot be viewed as a point of $M$. It can be viewed as a point of $TM$, but doing so doesn't have much use. Instead, one "probes the infinitesimal structure of $M$" with "paths through $x$ whose velocity is $h$".