Differential forms and determinants

Question

2-forms are defined as $du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}$

But what if I have two concret 1-forms in $R^{3}$? For example $(2dx-3dy+dz)\wedge (dx+2dy-dz)$ which gives

$(2dx-3dy+dz)\wedge (dx+2dy-dz)=-7dy \wedge dx +3dz \wedge dx - dy \wedge dz= 7 dx \wedge dy + 3 dz \wedge dx + dy \wedge dz$ I know this is the same as the vector product between $(2,-3,1)^{T}$ and $(1,2,-1)^{T}$.

What is the relationship with the determinant? Because when I calculate the wedge product between two 1-forms in $R^{2}$ then I get the value of the determinant $(2dx+4dx)\wedge (3dx+9dy) = -18 dx\wedge dy +12 dx \wedge dy = 6 dx \wedge dy$, but for 1-forms in $R^{3}$ I get the vector product. And is the interpretation right that parts of the area spanned by the 2-form above (the vector product) is that 7 get's projected onto the $xy$ plane, 3 onto $zx$ and 1 onto $yz$? Or is it in this case another coordinate system with planes $dxdy$, $dzdx$ and $dydz$? And since all the differential forms are functions of vectors, what happens when they come into the picture? Because doesn't $7 dx \wedge dy + 3 dz \wedge dx + dy \wedge dz$ looks like this?

$7 \begin{vmatrix} dx(v) & dx(w) \\ dy(v) & dy(w) \end{vmatrix} + 3 \begin{vmatrix} dz(v) & dz(w) \\ dx(v) & dx(w) \end{vmatrix} + \begin{vmatrix} dy(v) & dy(w) \\ dz(v) & dz(w) \end{vmatrix}$ For some vectors $v,w \in R^{3}$?

Answer 1

This is explained very nicely in Hubbard and Hubbard's book Vector Calculus, Linear Algebra, and Differential Forms, page 560 (Example 6.1.2) in the fourth edition.

Let's focus on a particular example: how the 2-form $dx\wedge dy$ on $\mathbb{R}^3$ is related to determinants. Well, given two three-dimensional vectors $\mathbf{u}=[u_1, u_2, u_3]^t$ and $\mathbf{v}=[v_1, v_2, v_3]^t$, we have

$$dx\wedge dy(\mathbf{u},\mathbf{v})=dx(\mathbf{u})dy(\mathbf{v})-dy(\mathbf{u})dx(\mathbf{v})=u_1v_2-u_2v_1=\det\begin{pmatrix}u_1 & v_1\\u_2 & v_2\end{pmatrix}$$

where the final matrix is the matrix we get by stacking $\mathbf{u}$ and $\mathbf{v}$ into a $3\times 2$ matrix and then selecting the rows corresponding to the $x$ and $y$ coordinates. We get a square matrix we can take the determinant of by selecting the appropriate rows.

More generally, consider a $k$-form $d_{i_1}\wedge\cdots\wedge d_{i_k}$ on $\mathbb{R}^n$. The value of this form on a $k$-tuple of $n$-dimensional vectors can be found first by stacking the $k$ vectors into an $n\times k$ matrix, then selecting the $k$ rows corresponding to the indices $i_1, \ldots, i_k$, and finally taking the determinant of the resulting $k\times k$ matrix.