twoform

A twoform, or differential 2-form, is a mathematical object used in differential geometry and related fields. It is a smooth section of the second exterior power of the cotangent bundle, denoted Λ^2 T^*M, on a differentiable manifold M. At each point p in M, a twoform ω_p is a skew-symmetric bilinear form on the tangent space T_pM, meaning ω_p(v,w) = −ω_p(w,v) for tangent vectors v and w, and ω_p(v,v) = 0.

Locally, a twoform can be written in coordinates as ω = ∑_{i<j} ω_{ij} dx^i ∧ dx^j, where dx^i are coordinate 1-forms and the coefficients satisfy ω_{ij} = −ω_{ji}. The wedge product of differential forms provides a natural operation: if α is a p-form and β is a q-form, then α ∧ β is a (p+q)-form. The exterior derivative d maps p-forms to (p+1)-forms, with the property d^2 = 0. A 2-form is closed if dω = 0 and exact if ω = dα for some 1-form α. By the Poincaré lemma, closed 2-forms are locally exact on contractible domains.

Twoforms have wide-ranging applications. In physics, the electromagnetic field strength is modeled as a 2-form on spacetime. In geometry, nondegenerate closed 2-forms define symplectic structures, giving a framework for classical mechanics. There is also a relationship to bivectors: under a metric, a twoform can be related to a skew-symmetric rank-2 tensor or a bivector via isomorphisms such as the Hodge dual in orientable, metric spaces. Examples include the standard area form dx ∧ dy on R^2 and, in higher dimensions, similar coordinate area forms.