wedge product
/ WEJ product /
How do you build a 2-form (which measures area) out of two 1-forms (which measure length)? You multiply them — but with a twist that remembers orientation. The wedge product, written with the symbol ^ (a small caret read 'wedge'), is the multiplication that combines a p-form and a q-form into a (p+q)-form. Its whole personality is one rule: it is antisymmetric on 1-forms, meaning dx^dy = -dy^dx, and therefore dx^dx = 0.
That antisymmetry is not a quirk; it is the algebra of oriented area. The 2-form dx^dy fed two vectors returns the signed area of their projection onto the x-y plane — a determinant — and determinants flip sign when you swap two columns, which is exactly dx^dy = -dy^dx. The vanishing dx^dx = 0 says a degenerate parallelogram with two equal edges has no area. For mixed degrees the sign rule generalizes: if alpha is a p-form and beta is a q-form then alpha^beta = (-1)^(pq) beta^alpha, so two 1-forms anticommute, but a 2-form and a 1-form commute, and so on. The wedge is also associative and distributes over addition, so you compute with it almost like ordinary multiplication — you just keep careful track of signs when reordering, and kill any term that repeats a basic 1-form.
The wedge product is the engine that makes higher forms exist at all. It is how dx, dy, dz generate the basic 2-forms and the volume form dx^dy^dz; it is hiding inside the cross product (u cross v corresponds to the 2-form built from the 1-forms dual to u and v) and inside every change-of-variables Jacobian. In physics it assembles the electromagnetic field 2-form out of the potential 1-form and underlies the symplectic 2-form of Hamiltonian mechanics.
Wedge alpha = 2 dx + dy with beta = dx - 3 dy. Expand and use dx^dx = dy^dy = 0 and dy^dx = -dx^dy: alpha^beta = 2 dx^(-3 dy) + dy^dx = -6 dx^dy - dx^dy = -7 dx^dy. The single coefficient -7 is exactly the determinant of the matrix [2, 1; 1, -3] of coefficients.
Wedging two 1-forms in the plane produces a single multiple of dx^dy whose coefficient is a 2-by-2 determinant.
The sign rule depends on degree, not on whether the forms commute as numbers. A form always wedges to zero with itself when it has odd degree (alpha^alpha = 0 for odd alpha), but an even-degree form need not: a 2-form can wedge nontrivially with itself, as happens for the symplectic form.