Differential Forms

Jacques Lafontaine
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Jacques Lafontaine: Université Montpellier 2, Département de Mathématiques

Chapter Chapter 5 in An Introduction to Differential Manifolds, 2015, pp 185-233 from Springer

Abstract: Abstract Does there exist a theory of integration – first for p-dimensional submanifolds of Euclidean space –, and more generally for manifolds? We can start with what we call line integrals, which is to say the circulation of a vector field V along a curve. This is classically defined as the integral ∫ a b V c ( t ) , c ′ ( t ) d t . $$\displaystyle{\int _{a}^{b}\big\langle V _{ c(t)},c^{{\prime}}(t)\big\rangle \,dt.}$$ Here, c : [ a , b ] → R n $$c:\, [a,b] \rightarrow \mathbf{R}^{n}$$ is a curve parametrization (in fact a piece of the parametrization as we restricted the parameter to the interval [a, b]) and ⟨ , ⟩ $$\langle \;,\,\rangle$$ is an inner product on R n $$\mathbf{R}^{n}$$ . Replacing the vectors V c(t) by linear forms α c(t) has the advantage of no longer requiring the inner product. We can then integrate curves on any manifold X, the “field of linear forms” x ↦ α x , for all x ∈ X, where α x is a linear form on the tangent space T x X, by writing ∫ c α = ∫ a b α c ( t ) c ′ ( t ) d t . $$\displaystyle{\int _{c}\alpha =\int _{ a}^{b}\alpha _{ c(t)}\big(c^{{\prime}}(t)\big)\,dt.}$$

Keywords: Differential Form; Vector Field; Star-shaped Open Subset; Darboux; Lorentz Basis (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-20735-3_5

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