Abstract
We shall denote manifolds by M, N, X, Y,.. . Typical local coordinates on M are (x1,...,xn); sometimes we simply write(xi). The tangent space to M at the point p is Tp(M), and the tangent bundle is \( T(M) = {U_{p \in M}}\,{T_p}(M) \). We sometimes write points in T(M) as (p, v) where p∈M and v∈Tp(M). A local coordinate system (xi ) on M induces a local coordinate system \( \left( {{x^1}, \ldots ,{x^n};{{\dot x}^1}, \ldots ,{{\dot x}^n}} \right) = \left( {p,v} \right) \) on T(M) where p has coordinates (x1,...,xn) and \( v = {\dot x^i}\;\partial /\partial {x^i} \). Here, and throughout, we use the summation convention.
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© 1983 Springer Science+Business Media New York
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Griffiths, P.A. (1983). Preliminaries. In: Exterior Differential Systems and the Calculus of Variations. Progress in Mathematics, vol 25. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4615-8166-6_2
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DOI: https://doi.org/10.1007/978-1-4615-8166-6_2
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-3103-1
Online ISBN: 978-1-4615-8166-6
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