the wedge product
/ WEJ /
The wedge product is the multiplication of differential forms. It takes a p-form alpha and a q-form beta and builds a (p+q)-form written alpha ^ beta. Geometrically, if alpha measures p-dimensional signed volume and beta measures q-dimensional signed volume in complementary directions, then alpha ^ beta measures the (p+q)-dimensional signed volume of the combined block. It is the engine that lets you grow forms in degree and the tool that turns de Rham cohomology into a ring.
Its defining properties are: bilinearity (linear in each factor), associativity ((alpha^beta)^gamma = alpha^(beta^gamma)), and graded anticommutativity alpha ^ beta = (-1)^{pq} beta ^ alpha, where p = deg alpha and q = deg beta. From the sign rule, the wedge of any odd-degree form with itself is zero (so dx ^ dx = 0), but an even-degree form can wedge with itself nontrivially (a symplectic 2-form omega has omega ^ omega nonzero). In coordinates you compute mechanically: distribute, and reorder factors using the sign rule, killing any term with a repeated dx^i.
The wedge product is why forms, not arbitrary tensors, are the right objects for calculus on manifolds. It descends to de Rham cohomology (because d is an antiderivation for it) and makes H*_dR(M) a graded-commutative ring, encoding how submanifolds intersect. A frequent slip: the wedge is graded-commutative, not anticommutative outright. Two 1-forms anticommute (alpha^beta = -beta^alpha), but a 1-form and a 2-form COMMUTE, since (-1)^{1*2} = +1. Always track the degrees.
In R^3 let alpha = x dx + dy and beta = dz. Then alpha ^ beta = x dx^dz + dy^dz. Reordering dx^dz = -dz^dx. And (dx^dy) ^ dz = dx^dy^dz is the standard volume form, while (dx^dy) ^ (dx^dy) = 0 because the dx^dx factor vanishes. But on R^4 with omega = dx_1^dx_2 + dx_3^dx_4, the self-wedge omega ^ omega = 2 dx_1^dx_2^dx_3^dx_4 is nonzero — an even form need not square to zero.
Mechanical computation: distribute, reorder by the sign rule, drop repeated differentials. Note an even form can have nonzero square.
Some books put a combinatorial factor in front of the wedge (the Alt convention with k!/(p!q!)), others do not (the 'determinant' convention). The two differ by constants and change formulas like the volume element. Both are standard; just be consistent within one document.