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\chapter{Higher differential forms}
\label{cha:higher-forms}

\begin{notextbook}Your textbook probably defines the \emph{double integral} of a two-variable function $f$ over a region $R$, using the notation\end{notextbook}
\begin{stewart}In section 12.1, the textbook defines the \emph{double integral} of a two-variable function $f$ over a rectangle $R$, using the notation\end{stewart}
\[ \iint_R f(x,y) \,\mathrm{d}A. \]
We would like to interpret this as an integral of a sort of differential form ``$f(x,y) \,\mathrm{d}A$''.
However, as we did for the length element $\dl$, we will change the notation slightly and write $\dA$ instead of $\mathrm{d}A$, since as we will see, it is not the differential of anything.

\section{2-forms}
\label{sec:2-forms}

What sort of a thing might $\dA$ be?
Recall that in a one-variable integral $\int_a^b f(x)\,\dx$, the differential form $f(x)\,\dx$ is a function that takes as input a point $x$ and a vector $\dx$ based at that point.
In the Riemann sums that define a double integral, the role of the lengths $\Delta x$ is played by the \emph{areas} $\Delta x \,\Delta y$.
Each of these can be interpreted as the length of a vector: one which points in the $x$ direction, and one which points in the $y$ direction.
Thus, we consider $\dA$ to be a function which takes \emph{two} vectors as input.

\begin{defn}
  A \textbf{differential 2-form} is a function whose input is a point $\ptx$ and two vectors based at $\ptx$.
  We denote the vectors by ??.
\end{defn}