Applied Differential Geometry

Abstract

This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples.

Discrete Asymptotic Abstractions of Hybrid Systems

“…The stricter assumption we make on the (time-invariant) continuous part is the existence of a finite number of disjoint limit sets. Let Φ t (p) be the flow [4] of the vector field f ; that is, ∀p ∈ M , ∀t > 0, Φ t (p) = σ p (t). We assume that flows are ultimately bounded in the following sense: Assumption 1 For all p ∈ M , and for all t ≥ 0, sup t≥0 Φ t (p) < ∞.…”

A system of units compatible with geometry

“…The¯rst author showing physical quantities is such away was Schouten [1]. Then this has appeared also in other books [3,4,5].…”

“…The article is addressed to non specialists of covariance, multivectors and forms, therefore, I am listing in Section 2 the directed quantities and their relevant features what can be known from the books by Misner et al [3] and Burke [4], [5]. I present in Section 3 the reasoning allowing to establish the correspondence between the unit and the type of a directed physical quantity.…”

Maxwell's theory on a post-Riemannian spacetime and the equivalence principle

“…Let be given the odd (or twisted, see [17]) electric current three-form J. We assume that a (1 + 3)-foliation of spacetime holds locally.…”