Consider $\bigwedge^1(\RR^3)$. Since all 3-dimensional vector spaces are isomorphic, we can identify the 1-form \begin{equation} v = v_x\,dx + v_y\,dy + v_z\,dz \end{equation} with the ordinary vector (field) \begin{equation} \vv = v_x\,\xhat + v_y\,\yhat + v_z\,\zhat \end{equation} and similarly for \begin{equation} w = w_x\,dx + w_y\,dy + w_z\,dz \end{equation} In other words, we identify the basis vectors as follows: \begin{align} \xhat &\longleftrightarrow dx \\ \yhat &\longleftrightarrow dy \\ \zhat &\longleftrightarrow dz \end{align} This identification turns vectors into things we can integrate, and can be thought of as the map from vectors to 1-forms given by \begin{equation} v = \vv\cdot d\rr \end{equation} for any vector $\vv$.

Direct computation shows that $v\wedge w$ looks quite a bit like $\vv\times\ww$. However, $v\wedge w$ is a 2-form, whereas $\vv\times\ww$ is again a vector. But the space of 2-forms is also 3-dimensional, so we can identify it, too, with ordinary vectors. However, care now needs to be taken to identify as follows: \begin{align} \xhat &\longleftrightarrow dy\wedge dz \\ \yhat &\longleftrightarrow dz\wedge dx \\ \zhat &\longleftrightarrow dx\wedge dy \end{align} This identification also turns vectors into things we can integrate, but in this case using surface integrals, rather than line integrals; we are now mapping $\vv$ to the 2-form $\vv\cdot d\AA$, representing a flux in the $\vv$ direction.

This demonstrates several important properties of the wedge and cross products. First of all, the cross product doesn't really take vectors to vectors! Recall the geometric definition: The magnitude of the cross product is the area spanned by the two vectors. Thus, if each vector has the dimensions of length, the cross product has the dimensions of area and is therefore a different object, living in a different vector space.

Another way to see this distinction is to consider a parity reversal, or space inversion, in which every point in space is taken to its opposite. Thus, each vector $\vv$ is taken to $-\vv$. The cross product of two vectors then transforms with two minus signs, and remains unchanged! Historically, such vectors were called pseudovectors. In the language of differential forms, the “cross product” of two 1-forms is indeed a different object: a 2-form.

This comparison also sheds some light on how the wedge product can be associative, while the cross product is not. Associativity requires the multiplication of three objects, and one must keep track of where the various products live.

A byproduct of our analysis of the cross product is that we have implicitly established a map between the 1-forms and 2-forms on $\RR^3$, as also discussed in § Relationships between Differential Forms. Such identifications between differential forms of different ranks will be discussed further in Chapter 3.