Differential Forms and Applications Project Topics and materials : Current School News

Differential Forms and Applications

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Differential Forms and Applications

TABLE OF CONTENTS

1 MANIFOLDS AND FORMS . . . . . . . . . . . . . . . . . . . . . . . .2
1.1 Submanifolds of Rn without boundary . . . . . . . . . . . . . 2
1.2 Notions of forms and elds . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 forms and vector elds on Rn . . . . . . . . . . . . . . 10
1.2.3 Integration over cubes and chains . . . . . . . . . . . . 15
1.3 Classical theorems of Green and Stokes . . . . . . . . . . . . . 18
1.3.1 Orientable Manifolds . . . . . . . . . . . . . . . . . . . 20
1.3.2 Riemannian Manifolds . . . . . . . . . . . . . . . . . . 27

2 EXAMPLES OF DIFFERENTIAL FORMS ON RIEMAN- NIAN MANIFOLDS 28
2.1 Winding form and volume element associated to ellipsoids in R2 and in R3 .  28
2.1.1 Di erential forms on the 1-dimensional ellipsoid . . . . 28
2.1.2 Di erential forms on the 2-dimensional ellipsoid . . . . 33
2.2 Other quantities associated to R3 ellipsoid derived from Riemannian structure, geodesics of R3 ellipsoid. . . 39
2.2.1 The shape operator . . . . . . . . . . . . . . . . . . . . 39
2.2.2 Geodesics of the 2-dimensional ellipsoid . . . . . . . . . 42
2.3 Manifolds in higher dimensions: volume element, geodesics . . 48
2.3.1 Higher dimensional volume forms . . . . . . . . . . . . 48
2.3.2 Higher dimensional geodesics . . . . . . . . . . . . . . 52
Bibliography 58

INTRODUCTION

This body of work introduces exterior calculus in Euclidean spaces and subsequently implements classical results from standard Riemannian geometry to analyze certain differential forms on a manifold of reference, which here is a symmetric ellipsoid in Rn.

We focus on the foundations of the theory of differential forms in a pro- gressive approach to present the relevant classical theorems of Green and Stokes and establish volume (length, area or volume) formulas.

Furthermore, we introduce the notion of geodesics and show how to obtain them with respect to the reference manifold.

BIBLIOGRAPHY

CALCULUS ON MANIFOLDS – Michael Spivak, copyright 1965, W.A. Benjamin Inc, New York, USA.

GLOBAL ANALYSIS: DIFFERENTIAL FORMS IN ANALYSIS, GEOMETRY AND PHYSICS – Ilka Agricola, Thomas Friedrich, copyright 2002 by the American Mathematical Society.

GEODESICS ON AN ELLIPSOID – PITTMAN’S METHOD Rod Daekin, http://rmit.academia.edu/RodDeakin/Papers/137514/Geodesics on an ellipsoid – Pittmans method.

DIFFERENTIAL GEOMETRY : CURVES – SURFACES – MANIFOLDS – Wolfgang Kuhnel, copyright 2003 by the American Mathematical Society.

CALCULUS OF VARIATIONS – Jurgen Jost and Xiangqing Li-Jost, copyright 1998 by the Cambridge University Press.

CSN Team.

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