Manifolds, Tensor Analysis, and Applications by Ralph Abraham

By Ralph Abraham

The objective of this e-book is to supply middle fabric in nonlinear research for mathematicians, physicists, engineers, and mathematical biologists. the most target is to supply a operating wisdom of manifolds, dynamical structures, tensors, and differential types. a few functions to Hamiltonian mechanics, fluid me­ chanics, electromagnetism, plasma dynamics and keep watch over thcory arc given in bankruptcy eight, utilizing either invariant and index notation. the present version of the ebook doesn't care for Riemannian geometry in a lot element, and it doesn't deal with Lie teams, significant bundles, or Morse concept. a few of this is often deliberate for a next version. in the meantime, the authors will make on hand to readers supplementary chapters on Lie teams and Differential Topology and invite reviews at the book's contents and improvement. in the course of the textual content supplementary issues are given, marked with the symbols ~ and {l:;J. This gadget allows the reader to bypass numerous themes with out anxious the most circulation of the textual content. a few of these supply extra history fabric meant for completeness, to lessen the need of consulting too many open air references. We deal with finite and infinite-dimensional manifolds concurrently. this can be partially for potency of exposition. with no complicated purposes, utilizing manifolds of mappings, the examine of infinite-dimensional manifolds may be not easy to motivate.

Show description

Read or Download Manifolds, Tensor Analysis, and Applications PDF

Best system theory books

Synergetics: an introduction

This booklet is an often-requested reprint of 2 vintage texts via H. Haken: "Synergetics. An creation" and "Advanced Synergetics". Synergetics, an interdisciplinary examine software initiated by way of H. Haken in 1969, offers with the systematic and methodological method of the swiftly transforming into box of complexity.

Robust Design: A Repertoire of Biological, Ecological, and Engineering Case Studies (Santa Fe Institute Studies on the Sciences of Complexity)

Strong layout brings jointly sixteen chapters by way of an eminent crew of authors in a variety of fields offering elements of robustness in organic, ecological, and computational structures. The volme is the 1st to deal with robustness in organic, ecological, and computational platforms. it truly is an outgrowth of a brand new study application on robustness on the Sante Fe Institute based via the David and Lucile Packard starting place.

Self-organized biological dynamics & nonlinear control

The becoming impression of nonlinear technology on biology and drugs is essentially altering our view of dwelling organisms and sickness approaches. This e-book introduces the applying to biomedicine of a wide diversity of techniques from nonlinear dynamics, reminiscent of self-organization, complexity, coherence, stochastic resonance, fractals, and chaos.

Semi-Autonomous Networks: Effective Control of Networked Systems through Protocols, Design, and Modeling

This thesis analyzes and explores the layout of managed networked dynamic platforms - dubbed semi-autonomous networks. The paintings ways the matter of potent keep watch over of semi-autonomous networks from 3 fronts: protocols that are run on person brokers within the community; the community interconnection topology layout; and effective modeling of those frequently large-scale networks.

Additional resources for Manifolds, Tensor Analysis, and Applications

Sample text

Proof Let e l , ... , en denote a basis of E, where n is the dimension of E. (i) A nonn on E is given, for example, by n lie II = Llail, i=1 where n e = L aiei i=1 (ii) Let II· II' be any other nonn on E. 1 Banach Spaces 47 is continuous with respect to the III· III -nonn on en. Since the set S = {x E en 1111 x 111= 1} is closed and bounded, it is compact. The restriction of this map to S is a continuous, strictly positive function, so it attains its minimum MI and maximum M2 on S; that is, for all (x I, ...

1 Definition A norm on a real (complex) vector space E is a mapping from E into the real numbers. II· II ; E ~ 1R; e ~ II e II. such that N1 II e II ~ 0 for all e E E and II e II =0 iff e =0 (positive definiteness); N2 II Ae II =I AI II e IIfor all e E E and A E 1R (homogeneity); N3 II e 1 + e 2 II :5 II e 1 II + II ~ II for all e 1• e2 E E (triangle inequality). 1 Banach Spaces The pair (E, II . II) is sometimes called a normed space. , II . liE' II . Jor the norm. The triangle inequality N3 has the following important consequence: which is proved in the following way: II e211 II e\ + (e2 - e\) II :5; II e\ II + II e\ - e211 , II e\ II = II e2 + (e\ - e2) II :5; II e211 + II e\ - e 2 II , = so that both II e211-11 e\ II and II e\ 11- II ~ II are smaller than or equal to II e\ - ~ II.

Show that for any continuous maps f, g : X -+ Y, the set {x E X I f(x) = g(x)} is closed. ) Thus, if f(x) = g(x) at all points of a dense subset of X, then f= g. 4D Define a topological manifold to be a space locally homeomorphic to JRn. Find a topological manifold that is not Hausdorff. 4E Show that a mapping

Download PDF sample

Rated 4.67 of 5 – based on 38 votes