book

Differential Forms, 2nd Edition

Name: Differential Forms, 2nd Edition
Author: Steven H. Weintraub
ISBN: 9780123946171

by Steven H. Weintraub

February 2014

Intermediate to advanced

408 pages

8h 30m

English

Academic Press

Read now

Unlock full access

Cover image
Title page
Table of Contents
Copyright
Dedication
Preface
1: Differential Forms in , I
1.0 Euclidean spaces, tangent spaces, and tangent vector fields 1.1 The algebra of differential forms 1.2 Exterior differentiation 1.3 The fundamental correspondence 1.4 The Converse of Poincaré’s Lemma, I 1.5 Exercises
2: Differential Forms in , II
2.1 -forms 2.2 -Forms 2.3 Orientation and signed volume 2.4 The converse of Poincaré’s Lemma, II 2.5 Exercises
3: Push-forwards and Pull-backs in
3.1 Tangent vectors 3.2 Points, tangent vectors, and push-forwards 3.3 Differential forms and pull-backs 3.4 Pull-backs, products, and exterior derivatives 3.5 Smooth homotopies and the Converse of Poincaré’s Lemma, III 3.6 Exercises
4: Smooth Manifolds
4.1 The notion of a smooth manifold 4.2 Tangent vectors and differential forms 4.3 Further constructions 4.4 Orientations of manifolds—intuitive discussion 4.5 Orientations of manifolds—careful development 4.6 Partitions of unity 4.7 Smooth homotopies and the Converse of Poincaré’s Lemma in general 4.8 Exercises

5: Vector Bundles and the Global Point of View
5.1 The definition of a vector bundle 5.2 The dual bundle, and related bundles 5.3 The tangent bundle of a smooth manifold, and related bundles 5.4 Exercises
6: Integration of Differential Forms
6.1 Definite integrals in 6.2 Definition of the integral in general 6.3 The integral of a -form over a point 6.4 The integral of a -form over a curve 6.5 The integral of a -form over a surface 6.6 The integral of a -form over a solid body 6.7 Chains and integration on chains 6.8 Exercises
7: The Generalized Stokes’s Theorem
7.1 Statement of the theorem 7.2 The fundamental theorem of calculus and its analog for line integrals 7.3 Cap independence 7.4 Green’s and Stokes’s theorems 7.5 Gauss’s theorem 7.6 Proof of the GST 7.7 The converse of the GST 7.8 Exercises
8: de Rham Cohomology
8.1 Linear and homological algebra constructions 8.2 Definition and basic properties 8.3 Computations of cohomology groups 8.4 Cohomology with compact supports 8.5 Exercises
Index

Overview

Differential forms are a powerful mathematical technique to help students, researchers, and engineers solve problems in geometry and analysis, and their applications. They both unify and simplify results in concrete settings, and allow them to be clearly and effectively generalized to more abstract settings. Differential Forms has gained high recognition in the mathematical and scientific community as a powerful computational tool in solving research problems and simplifying very abstract problems. Differential Forms, Second Edition, is a solid resource for students and professionals needing a general understanding of the mathematical theory and to be able to apply that theory into practice.

Provides a solid theoretical basis of how to develop and apply differential forms to real research problems
Includes computational methods to enable the reader to effectively use differential forms
Introduces theoretical concepts in an accessible manner

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.

Read now

Unlock full access

More than 5,000 organizations count on O’Reilly

O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.

Julian F.

Head of Cybersecurity

I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.

Addison B.

Field Engineer

I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.

Amir M.

Data Platform Tech Lead

I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.

Mark W.

Embedded Software Engineer

Publisher Resources

ISBN: 9780123944030

Cloud Computing

Data Engineering

Data Science

AI & ML

Programming Languages

Software Architecture

IT/Ops

Security

Design

Business

Soft Skills