Math Interlude: A Quick Review of Smooth Manifolds And All That

Analytic Mechanics is most naturally formulated in the language of differential geometry. In this interlocutory chapter, we quickly review the basic facts and definitions of differential geometry mainly to fix notation and terminology. Topics reviewed: smooth manifolds, vector bundles, vector and tensor fields, differential forms and exterior algebra, Stokes theorem and applications, Lie derivative, Lie groups and algebras, Riemannian geometry and geodesics, and foliations and Frobenius theorem