We now have two quite different interpretations of $df$:

• $df$ represents a small change in $f$ (infinitesimals);
• $df(\vv)=\grad f\cdot\vv$ (linear map on vectors).

The first interpretation comes from calculus; $df$ is a differential. The second comes from linear algebra; $df$ is a tensor. 1) We will use both of these interpretations: The calculus of differentials allows us to relate differentials of different functions, and tensor algebra tells us how to pair vectors and 1-forms (so that, for example, $dx$ and $\xhat$ contain the same physical information). Both interpretations contribute to the pictures in the previous section.

You may wonder how $dx$ can be small (an infinitesimal) and large (equivalent to a unit vector) at the same time. The answer comes from studying Figures 5 and 6 in the previous section. Calculus is about linearization, the process of zooming in so that functions become linear. The 1-forms shown in these figures are represented by a stack at each point. They are small, because they live at a single point, which is as zoomed in as possible. But they are also large, because at that point they behave like unit vectors.

Differential geometers often carry this analogy even further. We introduced the action $\vv(f) = \vv\cdot\grad f$ of a vector on a function, which is just the directional derivative of $f$ along $\vv$. In particular, we have \begin{equation} \xhat(f) = \xhat\cdot\grad f = \Partial{f}{x} \end{equation} which allows us to identify basis vectors such as $\xhat$ with partial derivatives in coordinate directions. For this reason, differential geometers not only use $\{dx^i\}$ as a basis for the 1-forms on $\RR^n$, they also use $\bigl\{\Partial{}{x^i}\bigr\}$ as a basis for the vector fields on $\RR^n$.