Differential Geometry and Its Applications

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MAA, Sep 6, 2007 - Mathematics - 469 pages
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Differential geometry has a long, wonderful history it has found relevance in areas ranging from machinery design of the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole, it mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors, it is also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only “see” geodesics on surfaces, but they will also see the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. Students will also see how particles move under constraints. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.
  

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Contents

Surfaces
67
Curvatures
107
Constant Mean Curvature Surfaces
161
Geodesies Metrics and Isometries
209
Holonomy and the GaussBonnet Theorem
275
The Calculus of Variations and Geometry
311
A Glimpse at Higher Dimensions
397
A List of Examples
437
Examples in Chapter 3
438
Examples in Chapter 7
439
B Hints and Solutions to Selected Problems
441
Suggested Projects for Differential Geometry
453
Bibliography
455
Index
461
About the Author
469
Copyright

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About the author (2007)

John Oprea was born in Cleveland, Ohio and was educated at Case Western Reserve University and at Ohio State University. He received his PhD at OSU in 1982 and, after a post-doc at Purdue university, he began his tenure at Cleveland State in 1985. Oprea is a member of the Mathematical Association of America and the America Mathematical Society. He is an Associate Editor of the Journal of Geometry and Symmetry in Physics. In 1996, Oprea was awarded the MAA's Lester R. Ford award for his Monthly article, “Geometry and the Foucault Pendulum.” Besides various journal articles on topology and geometry, he is also the author of The Mathematics of Soap Films (AMS Student Math Library, volume 10), Symplectic Manifolds with no Kähler Structure (with A. Tralle, Springer Lecture Notes in Mathematics, volume 1661), Lusternik-Schnirelmann Category (with O. Cornea, G. Lupton and D. Tanré, AMS Mathematical Surveys and Monographs, volume 103) and the forthcoming Algebraic Models in Geometry (with Y. Félix and D. Tanré, for Oxford University Press).

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