# Differential Forms and Applications

Textbook

Part of the Universitext book series (UTX)

1. Front Matter
Pages I-IX
2. Manfredo P. do Carmo
Pages 1-16
3. Manfredo P. do Carmo
Pages 17-31
4. Manfredo P. do Carmo
Pages 33-54
5. Manfredo P. do Carmo
Pages 55-75
6. Manfredo P. do Carmo
Pages 77-98
7. Manfredo P. do Carmo
Pages 99-113
8. Back Matter
Pages 115-120

### Introduction

This is a free translation of a set of notes published originally in Portuguese in 1971. They were translated for a course in the College of Differential Geome­ try, ICTP, Trieste, 1989. In the English translation we omitted a chapter on the Frobenius theorem and an appendix on the nonexistence of a complete hyperbolic plane in euclidean 3-space (Hilbert's theorem). For the present edition, we introduced a chapter on line integrals. In Chapter 1 we introduce the differential forms in Rn. We only assume an elementary knowledge of calculus, and the chapter can be used as a basis for a course on differential forms for "users" of Mathematics. In Chapter 2 we start integrating differential forms of degree one along curves in Rn. This already allows some applications of the ideas of Chapter 1. This material is not used in the rest of the book. In Chapter 3 we present the basic notions of differentiable manifolds. It is useful (but not essential) that the reader be familiar with the notion of a regular surface in R3. In Chapter 4 we introduce the notion of manifold with boundary and prove Stokes theorem and Poincare's lemma. Starting from this basic material, we could follow any of the possi­ ble routes for applications: Topology, Differential Geometry, Mechanics, Lie Groups, etc. We have chosen Differential Geometry. For simplicity, we re­ stricted ourselves to surfaces.

### Keywords

Diferential forms Differentialformen begleitendes Dreibein differential geometry differential geometry of surfaces differential manifolds differenzierbare Mannigfaltigkeit eingebettete Flächen immersed surfaces intrinsic geeometry of surfaces intrinsische Geometrie von Flächen manifold moving frames

#### Authors and affiliations

1. 1.Instituto de Matematica Pura e Aplicada (IMPA)Rio de JaneiroBrazil

### Bibliographic information

• Book Title Differential Forms and Applications
• Authors Manfredo P. Do Carmo
• Series Title Universitext
• DOI https://doi.org/10.1007/978-3-642-57951-6
• Copyright Information Springer-Verlag Berlin Heidelberg 1994
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Softcover ISBN 978-3-540-57618-1
• eBook ISBN 978-3-642-57951-6
• Series ISSN 0172-5939
• Series E-ISSN 2191-6675
• Edition Number 1
• Number of Pages X, 118
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour