How do we differentiate differential forms?

It's pretty clear how to differentiate $0$-forms (functions): Take the gradient. The 1-form associated with $\grad f$ is \begin{equation} df = \grad f\cdot d\rr \end{equation} which we sometimes refer to as the Master Formula in vector calculus.

Our remaining vector derivative operators are the curl, which takes vector fields to vector fields, and the divergence, which takes vector fields to scalar fields. We would like to express these operators in terms of differential forms. Since adding a “$d$” takes a $p$-form to a $(p+1)$-form, a reasonable choice is to define differentiation of $1$- and $2$-forms via \begin{align} d\left(\FF\cdot d\rr\right) &= \left(\grad\times\FF\right)\cdot d\AA \\ d\left(\FF\cdot d\AA\right) &= \left(\grad\cdot\FF\right)\, dV \end{align} Using the relationship between vector fields $\GG$ and $1$-forms $\GG\cdot d\rr$, it is now easy to check that $*dF$ is the 1-form associated with $\grad\times\FF$, and $*d{*}F$ is the 0-form $\grad\cdot\FF$. Explicitly, we have \begin{align} F &= \FF\cdot d\rr \\ dF &= \grad\times\FF\cdot d\AA \\ {*}dF &= \grad\times\FF\cdot d\rr \end{align} and \begin{align} {*}F &= \FF\cdot d\AA \\ d{*}F &= \grad\cdot\FF\>dV \\ {*}d{*}F &= \grad\cdot\FF \end{align} We can regard these relations as the definitions of the curl and divergence of a $1$-form, namely \begin{align} *dF &= \hbox{curl}(F) \\ *d{*}F &= \hbox{div}(F) \end{align} and of course we have \begin{equation} df = \hbox{grad}(f) \end{equation} for $0$-forms. 1)

Why have we rewritten vector calculus in this new language? Because it unifies the results. By convention, $d$ acting on a $3$-form is zero; the result should be a $4$-form, but there are no (nonzero) $4$-forms in three dimensions. Both second derivative identities therefore reduce to the single statement that \begin{equation} d^2 = 0 \end{equation} when acting on $p$-forms for any $p$. Furthermore, all of the integral theorems of vector calculus can be combined into the single statement \begin{equation} \int_D d\alpha = \int_{\partial D} \alpha \end{equation} where $D$ is a $p$-dimensional region in $\RR^3$, $\alpha$ is a $(p-1)$-form, and $\partial D$ denotes the boundary of $D$.

The remainder of this book seeks to generalize this language to other dimensions (and signatures).