The Operator '$d$' Apparently Having two Different Meanings in Differential Geometry.

Question

Given a smooth map $f:M\to N$ between smooth manifolds $M$ and $N$, we denote the global differential of $f$ by $df$.

Also, the letter '$d$' is used for denoting exterior derivative of a differential form. Given a differential form $\omega\in \Omega^k(M)$ on a smooth manifold $M$, $d\omega$ is a $k+1$-form on $M$.

The global differential, on the other hand, is not a form.

Can somebody explain why we use the same symbol for these two separate concepts?

Is there some connection between them?

Answer 1

In the case that $N=\Bbb R^n$, $df$ is a ($\Bbb R^n$-valued) $1$-form on $M$. As Willie suggests, some people eschew the $df$ notation for the map on tangent spaces; for example, Spivak writes $f_{*p}\colon T_pM\to T_{f(p)}N$.

Answer 2

This notation is only ambiguous in the case of $0$-forms on $M$, i.e. smooth real-valued functions. If $f \in C^{\infty}(M)$, then $df$ could either stand for the exterior derivative of $f$, thus being a $1$-form, or the differential of $f$. However, if we identify $T_x \mathbb{R} = \mathbb{R}$, then these objects could be considered the same; for as a $1$-form, $df$ would take a vector $X$ tangent to the point $p \in M$ and send it to $d_p f(X)$, just as the differential would.