Differential geometry and its applications / John Oprea.

Format:

Book

Author/Creator:

Oprea, John.

Series:

Classroom resource materials (Unnumbered)

Classroom resource materials

Language:

English

Subjects (All):

Geometry, Differential.

Physical Description:

1 online resource (501 p.)

Edition:

1st ed.

Place of Publication:

Washington, D.C. : Mathematical Association of America, c2007.

Language Note:

English

Summary:

Differential geometry has a long, wonderful history and has found relevance in many areas. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the standard university curriculum to a type of mathematics that is a unified whole, by mixing 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, but 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. 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.

Contents:

""cover ""; ""copyright page ""; ""title page ""; ""Contents""; ""Preface""; ""The Point of this Book""; ""Projects""; ""Prerequisites""; ""Book Features""; ""Elliptic Functions and Maple Note""; ""Thanks""; ""For Users of Previous Editions""; ""Maple 8 to 9""; ""Note to Students""; ""1 The Geometry of Curves""; ""1.1 Introduction""; ""1.2 Arclength Parametrization""; ""1.3 Frenet Formulas""; ""1.4 Non-Unit Speed Curves""; ""1.5 Some Implications of Curvature and Torsion""; ""1.6 Green�s Theorem and the Isoperimetric Inequality""; ""1.7 The Geometry of Curves and Maple""; ""2 Surfaces""

""2.1 Introduction""""Examples of Patches (or Parametrizations) on Surfaces""; ""2.2 The Geometry of Surfaces""; ""2.3 The Linear Algebra of Surfaces""; ""2.4 Normal Curvature""; ""2.5 Surfaces and Maple""; ""3 Curvatures""; ""3.1 Introduction""; ""3.2 Calculating Curvature""; ""3.3 Surfaces of Revolution""; ""3.4 A Formula for Gauss Curvature""; ""3.5 Some Effects of Curvature(s)""; ""3.6 Surfaces of Delaunay""; ""3.7 Elliptic Functions, Maple and Geometry""; ""3.8 Calculating Curvature with Maple""; ""4 Constant Mean Curvature Surfaces""; ""4.1 Introduction""

""4.2 First Notions in Minimal Surfaces""""4.3 Area Minimization""; ""4.4 Constant Mean Curvature""; ""4.5 Harmonic Functions""; ""4.6 Complex Variables""; ""4.7 Isothermal Coordinates""; ""4.8 The Weierstrass-Enneper Representations""; ""4.9 Maple and Minimal Surfaces""; ""4.9.1 Minimal Surface Plots""; ""4.9.2 The Minimal Surface Equation""; ""4.9.3 A Geometric Condition: Minimal Surfaces of Revolution""; ""4.9.4 An Algebraic Condition""; ""4.9.5 Maple and Area Minimization""; ""4.9.6 Maple and the Weierstrass- Enneper Representation""; ""Special color pages ""

""A geodesic on an ellipsoid""""A closed geodesic on an unduloid""; ""A non-closed geodesic on an unduloid""; ""Catalan�s surface""; ""A perturbed Boy�s surface""; ""Enneper�s surface""; ""A helicoid""; ""Henneberg�s surface""; ""A planar lines of curvature surface""; ""A twisted cylinder""; ""Scherk�s fifth surface""; ""Another view of the Bat""; ""5 Geodesics, Metrics and Isometries""; ""5.1 Introduction""; ""5.2 The Geodesic Equations and the Clairaut Relation""; ""5.3 A Brief Digression on Completeness""; ""5.4 Surfaces not in R^3""; ""5.5 Isometries and Conformal Maps""

""5.6 Geodesics and Maple""""5.6.1 Plotting Geodesics""; ""5.6.2 Geodesics on the Cone""; ""5.6.3 Geodesics on the Cylinder""; ""5.6.4 Geodesics on the Unduloid""; ""5.6.5 Geodesics on Surfaces not in R^3""; ""5.6.6 Stereographic and Mercator Projections""; ""5.7 An Industrial Application""; ""6 Holonomy and the Gauss-Bonnet Theorem""; ""6.1 Introduction""; ""6.2 The Covariant Derivative Revisited""; ""6.3 Parallel Vector Fields and Holonomy""; ""6.4 Foucault�s Pendulum""; ""6.5 The Angle Excess Theorem""; ""6.6 The Gauss-Bonnet Theorem""; ""6.7 Applications of Gauss-Bonnet""

""6.8 Geodesic Polar Coordinates""

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 455-459) and index.

ISBN:

1-61444-608-3

OCLC:

929120371