Twoforms

Twoforms, or two-forms, are mathematical objects that generalize oriented area elements. In linear algebra, a 2-form on a vector space V is a completely antisymmetric bilinear form β: V × V → R. Equivalently, it is an element of Λ2(V*), the second exterior power of the dual space. On a smooth manifold M, a 2-form is a smooth section of Λ2(T*M). For vectors X and Y, a 2-form assigns a scalar β(X, Y) that changes sign when X and Y are swapped.

In coordinates, a 2-form ω can be written as ω = Σi<j ωij dxi ∧ dxj, where the coefficients satisfy ωij = −ωji. The space of all 2-forms in n dimensions has dimension n(n−1)/2. A 2-form is linear in each argument and encodes an oriented infinitesimal parallelogram area spanned by two tangent directions.

Exteriors and operations: The wedge product combines forms to higher degrees, so α ∧ β increases degree, and the exterior derivative d maps a 2-form to a 3-form. A 2-form is closed if dω = 0 and exact if ω = dη for some 1-form η. With a choice of basis, a 2-form corresponds to an antisymmetric matrix (ωij).

In three dimensions, the Hodge dual provides a correspondence between 2-forms and vectors: a form a dy∧dz + b dz∧dx + c dx∧dy can be associated with the vector (a, b, c). Physically, many field theories express fields as 2-forms; for example, the electromagnetic field tensor F is a 2-form, and integrating a 2-form over a surface yields a flux.