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Differential Geometry A Geometric Introduction

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ISBN-10: 0135699630

ISBN-13: 9780135699638

Edition: 1st 1998

Authors: David W. Henderson, Daina Taimina

List price: $81.00
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Description:

Appropriate for introductory undergraduate courses in Differential Geometry with a prerequisite of multivariable calculus and linear algebra courses. This is the only text that introduces differential geometry by combining an intuitive geometric foundation, a rigorous connection with the standard formalisms, computer exercises with Maple, and a problems-based approach. Text has running theme of intrinsic vs. extrinsic ways of looking at curves and surfaces. Starting with basic geometric ideas and proceeding to the analytic and algebraic formalisms, this text provides a common and accessible foundation on which all of the various formalisms of differential geometry can be based and from which they can be assessed.
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Book details

List price: $81.00
Edition: 1st
Copyright year: 1998
Publisher: Prentice Hall PTR
Publication date: 7/24/1997
Binding: Paperback
Pages: 250
Size: 6.25" wide x 9.50" long x 0.75" tall
Weight: 0.990
Language: English

Preface
How to Use This Book
Surfaces and Straightness
When Do You Call a Straight Line?
How Do You Construct a Straight Line?
Local (and Infinitesimal) Straightness
Intrinsic Straight Lines on Cylinders
Geodesics on Cones
Is Shortest Always Straight?
Locally Isometric Surfaces
Local Coordinates for Cylinders and Cones
Geodesics in Local Coordinates
What Is Straight on a Sphere?
Intrinsic Curvature on a Sphere
Local Coordinates on a Sphere
Strakes, Augers, and Helicoids
Surfaces of Revolution
Hyperbolic Plane
Surface as Graph of a Function z=f(x,y)
Extrinsic Curves
Introduction
Give Examples of F.O.V.'s
Archimedian Property
Vectors and Affine Linear Space
Smoothness and Tangent Directions
Curvature of a Curve in Space
Curvature of the Graph of a Function
Osculating Circle
Strakes
When a Curve Does Not Lie in a Plane
Extrinsic Descriptions of Intrinsic Curvature
Smooth Surfaces and Tangent Planes
Extrinsic Curvature - Geodesics on Sphere
Intrinsic Curvature - Curves on Sphere
Intrinsic (Geodesic) Curvature
Geodesics on Surfacesthe Ribbon Test
Ruled Surfaces and the Converse of the Ribbon Test
Tangent Space, Metric, Directional Derivative
The Tangent Space
Mean Value Theorem: Curves, Surfaces
Natural Parametrizations of Curves
Riemannian Metric
Riemannian Metric in Local Coordinates on a Sphere
Riemannian Metric in Local Coordinates on a Strake
Vectors in Extrinsic Local Coordinates
Measuring Using the Riemannian Metric
Directional Derivatives
Directional Derivative in Local Coordinates
Differentiating a Metric
Expressing Normal Curvature
Geodesic Local Coordinates
Differential Operator
Metric in Geodesic Coordinates
Area, Parallel Transport, Intrinsic Curvature
The Area of a Triangle on a Sphere
Introducing Parallel Transport
The Holonomy of a Small Geodesic Triangle
Dissection of Polygons into Triangles
Gauss-Bonnet for Polygons on a Sphere
Parallel Fields and Intrinsic Curvature
Holonomy on Surfaces
Holonomy Explains Foucault's Pendulum
Intrinsic Curvature of a Surface
Gaussian Curvature Extrinsically Defined
Pep Talk to the Reader
Gaussian Curvature, Extrinsic Definition
Second Fundamental Form
The Gauss Map
Gauss-Bonnet and Intrinsic Curvature
Matrix of the Second Fundamental
Mean Curvature and Minimal Surfaces
Celebration of Our Hard Work
Applications of Gaussian Curvature
Gaussian Curvature in Local Coordinates
Curvature on Sphere, Strake, Catenoid
Circles, Polar Coordinates, and Curvature
Exponential Map and Shortest Is Straight
Ruled Surfaces and Ribbons
Surfaces with Constant Curvature
Curvature of the Hyperbolic Plane
Intrinsic Local Descriptions and Manifolds
Covariant Derivative and Connections
ManifoldsIntrinsic and Extrinsic
Christoffel Symbols, Intrinsic Descriptions
Intrinsic Curvature and Geodesics
Lie Brackets, Coordinate Vector Fields
Riemann Curvature Tensors
Calculation of Curvature Tensors in Local Coordinates
Intrinsic Calculations in Examples
Linear Algebraa Geometric Point of View
Where Do We Start? Geometric Affine Spaces
Vector Spaces
Inner ProductLengths and Angles
Linear Transformations and Operators
Areas, Cross Products, and Triple Products
Volumes, Orientation, and Determinants
Eigenvalues and Eigenvectors
Introduction to Tensors
Analysis from a Geometric Point of View
Smooth Functions
Invariance of Domain
Inverse Function Theorem
Implicit Function Theorem
Computer Scripts
Standard Functions
Strake
Surfaces of Revolution
Surfaces as Graph of a Function
Tangent Vectors to Curves
Curvature and Tangent Vectors
Osculating Planes
Osculating Circles
Frenet Frame
Tangent Planes to Surfaces
Curves on a Surface
Extrinisic Curvature Vectors
The Three Curvature Vectors
Ruled Surfaces
Non-dissectable Polyhedron
Sign of (Gaussian) Curvature
Mulitple Principle Directions
Gauss Map
Helicoid to Catenoid
Bibliography
Notation Index
Subject Index