We have seen that the rank $p$ of differential forms in $\RR^3$ can be $0$, $1$, $2$, or $3$; the rank “counts” the number of times that “$d$” appears in each term. So 2-forms and 3-forms involve products of 1-forms; differential forms form an algebra, and can be multiplied. What are the rules?

We have been somewhat casual so far in our treatment of vector integration. In particular, we haven't yet worried about the orientation of the surfaces over which we integrate. It is clear however that the value of a vector line integral (“work”) changes sign if we traverse the curve backward. Similarly, a vector surface integral (“flux”) changes sign if we choose the opposite normal vector, that is, if we use the opposite orientation of $d\AA$. So it's not quite correct to say that \begin{equation} \FF\cdot d\AA = F_z\,dx\,dy \end{equation} as we did in the last section. The vector surface element $d\AA$ carries with it a choice of orientation. For instance, in the $xy$-plane, we must distinguish between $dx\,\xhat \times dy\,\yhat$ and $dy\,\yhat \times dx\,\xhat$, which differ by a sign, whereas $dx\,dy$ and $dy\,dx$ are normally regarded as being equal. 1) We therefore define multiplication of differential forms to be antisymmetric, and from now on use the symbol $\wedge$, read as “wedge”, to distinguish this product from ordinary multiplication. We have \begin{equation} dy\wedge dx = - dx\wedge dy \end{equation} by definition, and we conventionally define \begin{equation} \FF\cdot d\AA = F_x\,dy\wedge dz + F_y\,dz\wedge dx + F_z\,dx\wedge dy \end{equation}

Similarly, we regard volume integrals as being dependent on a choice of orientation (“right-handed” or “left-handed”), and regard $dV$ as representing the (standard, right-handed) orientation given by \begin{equation} dV = dx\wedge dy\wedge dz \end{equation}