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With its hearty history and its many triumphs in the study of everything from the fundamentals of nature to DNA, differential geometry is not an ordinary branch of mathematics. Oprea (mathematics, Cleveland State U.) argues that differential geometry combines features of geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations and ideas taken from the sciences, and makes his case for taking his point of view with this undergraduate text that works with computer algebra programs to achieve visible results. Along the way he covers the geometry of curves, surfaces, curvatures, constant mean curvature surfaces, geodesics, metrics, isometries, holonomy and the Gauss-Bonnet theorem, the calculations of variations and geometry, and higher dimensions, just for fun. Annotation ©2007 Book News, Inc., Portland, OR (booknews.com)

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 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.

Publisher:
Washington, D.C. : Mathematical Association of America, Ă2007

Edition:
2nd ed

ISBN:
9781614446088

1614446083

0883857480

9780883857489

1614446083

0883857480

9780883857489

Characteristics:
1 online resource (xxi, 469 pages) : illustrations

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