Wedge product of differential forms

Let $M$ be a smooth manifold. The wedge product is the canonical bilinear product on the graded algebra of differential forms.

For $\alpha \in \Omega^k(M)$ and $\beta \in \Omega^\ell(M)$ , their wedge product $\alpha\wedge\beta \in \Omega^{k+\ell}(M)$ is defined pointwise by the alternating product of multilinear forms on each tangent space:

(\alpha\wedge\beta)_p(v_1,\dots,v_{k+\ell}) =\frac{1}{k!\,\ell!}\sum_{\sigma\in S_{k+\ell}}\operatorname{sgn}(\sigma)\, \alpha_p\!\bigl(v_{\sigma(1)},\dots,v_{\sigma(k)}\bigr)\, \beta_p\!\bigl(v_{\sigma(k+1)},\dots,v_{\sigma(k+\ell)}\bigr),

where $S_{k+\ell}$ is the set of permutations of $k+\ell$ symbols, $\operatorname{sgn}$ is the permutation sign, and $k!$ and $\ell!$ are factorials. The formula holds for all $p\in M$ and $v_1,\dots,v_{k+\ell}\in T_pM$ .

Equivalent characterizations

If alternation is normalized by

\operatorname{Alt}(T)(v_1,\ldots,v_r) =\frac1{r!}\sum_{\sigma\in S_r}\operatorname{sgn}(\sigma) T(v_{\sigma(1)},\ldots,v_{\sigma(r)}),

then the same convention gives

(\alpha\wedge\beta)_p =\frac{(k+\ell)!}{k!\ell!}\operatorname{Alt}(\alpha_p\otimes\beta_p).

Here the tensor product evaluates by multiplying the value of $\alpha_p$ on the first $k$ arguments by that of $\beta_p$ on the last $\ell$ .

Examples

Coordinate 1-forms on $\mathbb{R}^3$ . With standard coordinates $(x,y,z)$ , the 2-form $dx\wedge dy$ satisfies $(dx\wedge dy)(\partial_x,\partial_y)=1$ and changes sign when the vectors are swapped: $(dx\wedge dy)(\partial_y,\partial_x)=-1$ .

A wedge computation with functions. Let $\alpha = f\,dx + g\,dy$ and $\beta = h\,dz$ on $\mathbb{R}^3$ , where $f,g,h$ are smooth functions. Then $\alpha\wedge\beta = fh\,dx\wedge dz + gh\,dy\wedge dz.$

Wedge of a 1-form with itself is zero (odd degree). On any manifold, $dx\wedge dx=0$ . More generally, if $\eta$ is any 1-form then $\eta\wedge\eta=0$ by graded-commutativity with $k=\ell=1$ .

Properties

If $\alpha\in\Omega^k(M)$ , $\beta\in\Omega^\ell(M)$ , and $\gamma\in\Omega^m(M)$ , then:

Bilinearity: $\wedge$ is $\mathbb{R}$ -bilinear in each argument.
Associativity: $(\alpha\wedge\beta)\wedge\gamma=\alpha\wedge(\beta\wedge\gamma)$ .
Graded-commutativity: $\alpha\wedge\beta = (-1)^{k\ell}\,\beta\wedge\alpha.$ In particular, if $k$ is odd then $\alpha\wedge\alpha=0$ .
Compatibility with pullback: for any smooth map $F:M\to N$ , the pullback of forms satisfies $F^*(\alpha\wedge\beta)=F^*\alpha\wedge F^*\beta$ .
The wedge product is the product appearing in the graded Leibniz rule for the exterior derivative.