Chapter 4: Differentation

Recall that $\bigwedge^0(M)$ is the set of functions on $M$, and that the differential $df$ of any function $f$ is a 1-form. Taking the differential, or “zapping a function with $d$”, is therefore a map $$d: \bigwedge\nolimits^0 \longmapsto \bigwedge\nolimits^1$$ What is this map? We have $$df = \Partial{f}{x^i}\,dx^i$$ a 1-form whose components are just the partial derivatives of $f$ — just like the gradient. We therefore identify $df$ with the gradient of $f$.