Differential Forms 02: Wedge Product and Exterior Algebra
Differential forms / 02
Wedge product and exterior algebra
The wedge product is the multiplication rule that remembers orientation, dimension, and the cancellation caused by repeated directions.
Calculus root: substitution and signed area
The wedge product begins with the substitution rule. In one variable,
\[\int_a^b f(x)\,dx=\int_{\phi^{-1}(a)}^{\phi^{-1}(b)} f(\phi(u))\phi'(u)\,du.\]The derivative \(\phi'(u)\) measures signed stretching. In two variables, signed stretching is no longer one number from one derivative; it is the determinant
\[\det{\partial(x,y)\over\partial(u,v)}.\]The form identity
\[dx\wedge dy= \det{\partial(x,y)\over\partial(u,v)}\,du\wedge dv\]is the substitution rule written before integration.
Example: orientation reversal
The change \(x=u,\ y=-v\) gives
\[dx\wedge dy=du\wedge(-dv)=-du\wedge dv.\]The negative sign is the same sign one sees when reversing the limits of a one-variable integral.
Example: polar coordinates
For \(x=r\cos\theta,\ y=r\sin\theta\),
\[dx\wedge dy=r\,dr\wedge d\theta.\]The factor \(r\) is the familiar polar Jacobian. The wedge product remembers both area scaling and orientation.
Section 4.1 of the PDF uses the wedge product as the algebraic engine for determinants, area, volume, and exterior differentiation. The rule is compact:
\[\alpha\wedge\beta=(-1)^{pq}\beta\wedge\alpha \qquad (\alpha\in\Omega^p,\ \beta\in\Omega^q).\]For 1-forms this says
\[dx_i\wedge dx_j=-dx_j\wedge dx_i,\qquad dx_i\wedge dx_i=0.\]Determinants from wedges
If \(\alpha_1,\ldots,\alpha_k\) are 1-forms, then
\[(\alpha_1\wedge\cdots\wedge\alpha_k)(v_1,\ldots,v_k) =\det(\alpha_i(v_j))_{i,j}.\]This formula is the reason wedge products are not merely formal symbols: they compute signed volumes after the 1-forms measure vector components.
Example 1: signed area in the plane
Let \(v=(2,1)\) and \(w=(-1,3)\). Then
\[(dx\wedge dy)(v,w)= \det\begin{pmatrix}2&-1\\1&3\end{pmatrix}=7.\]The same parallelogram with reversed ordered basis gives
\((dx\wedge dy)(w,v)=-7.\)
Example 2: volume in three dimensions
For
\[u=(1,0,1),\quad v=(0,2,1),\quad w=(1,1,0),\]the form \(dx\wedge dy\wedge dz\) gives
\[\det \begin{pmatrix} 1&0&1\\ 0&2&1\\ 1&1&0 \end{pmatrix} =-3.\]The negative sign says the ordered frame \((u,v,w)\) has opposite orientation from \((\partial_x,\partial_y,\partial_z)\).
Expanding products
Let
\[\alpha=a\,dx+b\,dy,\qquad \beta=c\,dx+d\,dy.\]Then
\[\alpha\wedge\beta =(ad-bc)\,dx\wedge dy.\]The determinant has appeared again. In higher dimensions, the same principle gives minors. For instance, in \(\mathbb R^3\),
\[(P\,dx+Q\,dy+R\,dz)\wedge dx\wedge dz=-Q\,dx\wedge dy\wedge dz.\]Only the coefficient of the missing basis direction survives.
Common sign error
The identity \(dy\wedge dx=-dx\wedge dy\) is the most frequent source of wrong answers. In computations, move every term to the ordered basis \(dx_1\wedge\cdots\wedge dx_n\) before comparing coefficients.
Degree and vanishing
Forms vanish when their degree exceeds the dimension. On a surface,
\[\Omega^3=0.\]Thus every 2-form on a surface is automatically closed after applying one more exterior derivative for degree reasons. This is not a topological fact; it is linear algebra.
Example 3: automatic vanishing in two variables
For \(\omega=f(x,y)\,dx\wedge dy\),
\[d\omega=df\wedge dx\wedge dy =(f_x\,dx+f_y\,dy)\wedge dx\wedge dy=0.\]Both terms contain a repeated factor.
Example 4: non-automatic behavior in three variables
In \(\mathbb R^3\),
\[\omega=x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy\]is a 2-form, and its derivative is a 3-form:
\[d\omega=3\,dx\wedge dy\wedge dz.\]Top degree is not zero; degree larger than top degree is zero.
Pullback and the wedge
If \(F:V\to U\) is smooth, then pullback respects wedge products:
\[F^*(\alpha\wedge\beta)=F^*\alpha\wedge F^*\beta.\]For a parametrization \(F(s,t)=(u(s,t),v(s,t))\),
\[F^*(du\wedge dv)= (u_s\,ds+u_t\,dt)\wedge(v_s\,ds+v_t\,dt) =\det DF\,ds\wedge dt.\]This is the change-of-variables Jacobian written as an identity of 2-forms.