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By George D. Birkhoff, Ralph Beatley

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1 (Curl of a Vector) Consider a one-form φ = exterior derivative of φ is a two-form dφ = dφν ∧ d x ν = φμ d x μ in IR 3 . The ∂φν μ dx ∧ dxν ∂xμ which has three independent components, so that it is equivalent to some one-form, which is associated with a vector v = φ μ (∂/∂ x μ ), with components given by dφ = ∂φ1 ∂φ2 − 1 2 ∂x ∂x , ∂φ3 ∂φ1 − 1 3 ∂x ∂x , ∂φ3 ∂φ2 − 2 3 ∂x ∂x In other words, the exterior derivative of φ is equivalent to the rotational ∇ × V .

Taking Consequently {d x μ ∧d xμν p the dimension of the manifold M to be n, the alternate condition gives a total of n(n − 1)/2 independent elements in that basis, so that it has n(n − 1)/2 dimensions. 2 (k-Forms) The concept of two-forms extends naturally to k-forms in M as a map ξ : T p M × T p M × · · · × T p M → IR (k-factors) such that it is k-linear and alternate. The alternate condition means that for any set of indices {μ1 , μ2 , . . , μk }, we have ,k ξ(eμ1 , . . , eμk ) = εμ1 1,...

Likewise, a linear form or one-form in an n-dimensional manifold M is just a map φ p : T p M → IR such that it is linear on T p M φ p (av p + bw p ) = aφ p (v p ) + bφ p (w p ) We may define a sum of linear forms φ p and ψ p and a product of linear forms by numbers a and b as follows: (aφ p + bψ p )(v p ) = aφ(v p ) + bψ(v p ) Then, it follows from these definitions that the set of all linear forms at a point of M generates a vector space denoted by T p∗ M , called the dual tangent space T p M .

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