Given a real vector space V , an alternating k-form on V is a map ω : V k → R which is multilinear and alternating, i.e. ω(. . . , v + w, . . .) = ω(. . . , v, . . .) + ω(. . . , w, . . .); ω(. . . , cv, . . .) = cω(. . . , v, . . .); ω(. . . , v, w, . . .) = −ω(. . . , w, v, . . .). Note, that the alternating property implies that for σ ∈ Sk , ω(vσ(1) , . . . , vσ(k) ) = sign(σ)ω(v1 , . . . , vk ). The set of alternating k-forms on V is a real vector space Ak (V ) with (ω + η)(v1 , . . . , vk ) = ω(v1 , . . . , vk ) + η(v1 , . . . , vk ); (cω)(v1 , . . . , vk ) = cω(v1 , . . . , vk ). An example of an alternating form is the determinant, det ∈ An (Rn ). Given ω ∈ Ak (V ) and η ∈ Aj (V ) the wedge product ω ∧ η ∈ Ak+j (V ) is defined by X ω ∧ η(v1 , . . . , vk+j ) = sign(σ)ω(vσ(1) , . . . , vσ(j) )η(vσ(j+1) , . . . , vσ(j+k) ). σ∈Sk+j Corollary: The following properties hold for the wedge product. ω ∧ η = (−1)jk η ∧ ω; (ω ∧ η) ∧ ψ = ω ∧ (η ∧ ψ); (ω + α) ∧ η = ω ∧ η + α ∧ η; ω ∧ (η + β) = ω ∧ η + ω ∧ β; (cω) ∧ η = ω ∧ (cη) = c(ω ∧ η). Theorem: Let {e1 , . . . , en } be a basis for V . Let {e1 , . . . , en } be the dual basis for A1 (V ), i.e. linear maps V → R such that ei (ej ) = δji . Then {eI1 ∧ . . . ∧ eIk : 1 ≤ I1 < · · · < Ik ≤ n} is a basis for Ak (V ). Proof: ? A multi-index is a sequence I1 , . . . , Ik ∈ N of the form 1 ≤ I1 < · · · < Ik ≤ n. The set of such multi-indices is denoted I(n, k). Note that #I(n, k) = (nk ). For I ∈ I(n, k), ω I denotes ω I1 ∧ . . . ∧ ω Ik , so the basis above can be written more compactly as {eI : I ∈ I(n, k)}. Corollary: dim Ak (V ) = (nk ) where n = dim V . Given a linear map f : W → V , the pullback f ∗ ω ∈ Ak (W ) is defined by f ∗ ω(w1 , . . . , wn ) = ω(f (w1 ), . . . , f (wn )). 1 Corollary: The following properties hold for the pullback. f ∗ (ω + α) = f ∗ ω + f ∗ α; f ∗ (cω) = cf ∗ ω; f ∗ (ω ∧ η) = f ∗ ω ∧ f ∗ η. Given a manifold M , a differential k-form on M is a map ω : M → ∪p∈M Ak (Tp M ) with ω(p) ∈ Ak (Tp M ) such that for any smooth vector fields v1 , . . . , vn , the map M → R, p 7→ ω(p)(v1 (p), . . . , vn (p)) is smooth. The set of differential k-forms on M is denoted Ωk (M ) and is endowed with a real vector space structure and a wedge product inherited from the Ak (Tp M )’s. Also, Ω0 (M ) is the ring of scalar fields over which Ωk (M ) is a module. Scalar multiplication is both senses and wedge products are all compatible. The exterior derivative d : Ω0 (M ) → Ω1 (M ) is defined by dg(p) = Dp g. Theorem: Let U be an open subset of Rn and let xi ∈ Ω0 (U ) be the standard coordinate projections, i.e. P xi (p) = pi . Then there exists unique {ωI ∈ Ω0 (U ) : I ∈ I(n, k)} such that ω = I∈I(n,k) ωI dxI . Proof: The xi are restrictions of the linear maps ei which are dual to the standard basis for Rn , hence dxi = ei , so this follows from the basis theorem for Ak (V ) with V =P Rn . n Corollary: dg = i=1 ∂i gdxi . P The exterior derivative d : Ωk (U ) → Ωk+1 (U ) is defined by dω = I∈I(n,k) dωI ∧ dxI . Given a smooth map f : N → M , the pullback f ∗ ω ∈ Ωk (N ) is defined by ∗ f ω(p) = (Dp f )∗ ω(p). Theorem: For ω ∈ Ωk (M ) there exists a unique dω ∈ Ωk+1 (M ), the exterior derivative, such that for any parametrization f : U → M , df ∗ ω = f ∗ dω. Proof: ? Note, the three definitions of the exterior derivative are all compatible. Corollary: The following properties hold for the exterior derivative and the pullback. f ∗ (ω + η) = f ∗ ω + f ∗ η; f ∗ (cω) = cf ∗ ω; f ∗ (ω ∧ η) = f ∗ ω ∧ f ∗ η; d(ω + η) = dω + dη; d(cω) = cdω; d(ω ∧ η) = dω ∧ η + (−1)k ω ∧ dη; df ∗ ω = f ∗ dω; ddω = 0. Pn Proof: On U , ddg = d j=1 ∂j gdxj = i,j=1 ∂i ∂j gdxi ∧ dxj . Equality of mixed partials and antisymmetry of wedge product of 1-forms therefore implies P that ddg = −ddg. So, ddg = 0. So, ddω = I∈I(n,k) ddωI ∧ dxI = 0. On M , f ∗ ddω = ddf ∗ ω = 0, so ddω = 0 (why?). Pn 2

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