VISUAL DIFFERENTIAL GEOMETRY and FORMS
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VISUAL DIFFERENTIAL GEOMETRY and FORMS
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VISUAL DIFFERENTIAL GEOMETRY and FORMS a mathematical drama in five acts
The Book
Prologue
Acknowledgements
Act I The Nature of Space
Chapter 1 Euclidean and non-Euclidean Geometry
Chapter 2 Gaussian Curvature
Chapter 3 Exercises for Prologue and Act I
Act II The Metric
Chapter 4 Mapping Surfaces: The Metric
Chapter 5 The Pseudosphere and the Hyperbolic Plane
Chapter 6 Isometries and Complex Numbers
Act III Curvature
Chapter 8 Curvature of Plane Curves
8.1 Introduction
8.2 The Circle of Curvature
8.3 Newton's Curvature Formula
8.4 Curvature as Rate of Turning
8.5 Example: Newton's Tractrix
Act IV
Act V
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General Information about the Forum
General Discussion
Princeton University link to book, errata etc
Official Errata
News about the book
Useful links
GaryVasco
Prologue (discussion and suggested exercises)
Act I The Nature of Space
Chapter 1 Euclidean and non-Euclidean Geometry
Chapter 2 Gaussian Curvature
2.1 Introduction
2.2 The circumference and Area of a Circle
2.3 The Local Gauss-Bonnet Theorem
Chapter 3 Exercises for Prologue and Act I
Exercises 1-4 Prologue: Newtonian Ultimate Equality $(\asymp)$
Exercises 5-11 Euclidean and Non-Euclidean Geometry
Exercises 12-14 Gaussian Curvature
Act II The Metric
Chapter 4 Mapping Surfaces: The Metric
Chapter 5 The Pseudosphere and the Hyperbolic Plane
Chapter 6 Isometries and Complex Numbers
Chapter 7 Exercises for Act II
Exercises 1-13 Mapping Surfaces: The Metric
Exercises 14-23 The Pseudosphere and the Hyperbolic Plane
Exercise 24-34 Isometries and Complex Numbers
Act III Curvature
Chapter 8 Curvature of Plane Curves
8.1 Introduction
8.2 The Circle of Curvature
8.3 Newton's Curvature Formula
8.4 Curvature as Rate of Turning
8.5 Example: Newton's Tractrix
Chapter 9 Curves in 3-Space
Chapter 10 The Principal Curvatures of a Surface
10.1 Euler's Curvature Formula
10.2 Proof of Euler's Curvature Formula
10.3 Surfaces of Revolution
Chapter 11 Geodesics and Geodesic Curvature
11.1 Geodesic Curvature and Normal Curvature
11.2 Meusnier's Theorem
11.3 Geodesics are "Straight"
11.4 Intrinsic Measurement of Geodesic Curvature
11.5 A simple Extrinsic Way to Measure Geodesic Curvature
11.6 A New Explanation of the Sticky-Tape Construction of Geodesics
11.7 Geodesics on Surfaces of Revolution
11.7.1 Clairot's Theorem on the Sphere
11.7.2 Kepler's Second Law
11.7.3 Newton's Geometrical Explanation of Kepler's Law
11.7.4 Dynamical Proof of Clairot's Theorem
11.7.5 Application: Geodesics in the Hyperbolic Plane (Revisited)
Chapter 12 The Extrinsic Curvature of a Surface
Chapter 13 Gauss's Theorema Egregium
Chapter 14 The Curvature of a Spike
Chapter 15 The Shape Operator
Chapter 16 Introduction to the Global Gauss-Bonnet Theorem
Chapter 17 First (Heuristic ) Proof of the Global Gauss-Bonnet Theorem
Chapter 18 Second (Angular Excess) Proof of the Global Gauss-Bonnet Theorem
Chapter 19 Third (Vector Field) Proof of the Global Gauss-Bonnet Theorem
19.1 Introduction
19.2 Vector fields in the Plane
19.3 The Index of a Singular Point
19.4 The Archetypal Singular Points: Complex Powers
19.5 Vector Fields on Surfaces
19.6 The Poincaré-Hopf Theorem
19.7 Vector Field Proof of the Global Gauss'Bonnet Theorem
19.8 The Road Ahead
Chapter 20 Exercises for Act III
Exercise 1 Curvature of Plane Curves
Exercises 2-6 Curves in 3-Space
Exercises 7-10 The Principal Curvatures of a Surface
Exercise 11 Gaus's Theorema Egregium
Exercises 12-20 The Shape Operator
Exercises 21-22 Introduction to the Global Gauss-Bonnet Theorem
Exercise 23 First (Heuristic) Proof of GGB
Exercises 24-28 Second (Angular Excess) Proof of GGB
Exercises 29-33 Third (Vector Field) Proof of GGB
Act IV Parallel Transport
Chapter 21 An Historical Puzzle
Chapter 22 Extrinsic Constructions
22.1 Project into the Surface as You Go
22.2 Geodesics and Parallel Transport
22.3 Potato-Peeler Transport
Chapter 23 Intrinsic Constructions
23.1 Parallel Transport via Geodesics
23.2 The Intrinsic (aka, "Covariant") Derivative
Chapter 24 Holonomy
24.1 Example: The Sphere
24.2 Holonomy of a General Geodesic Triangle
24.3 holonomy Is Additive
24.4 Example: The Hyperbolic Plane
Chapter 25 An Intuitive Geometric Proof of the Theorema Egregium
25.1 Introduction
25.2 Some Notation and Reminders of Definitions
25.3 The Story So Far
25.4 The Spherical Map Preserves Parallel Transport
25.5 The Beautiful Theorem and Theorema Egregium Explained
Chapter 26
26.1 Introduction
26.2 Holonomy Along an Open Curve?
26.3 Hopf's Intrinsic Proof of the Global Gauss-Bonnet Theorem
Chapter 27
27.1 Introduction
27.2 The Circulation of a Vector Field Around a Loop
27.3 Dry Run: Holonomy in the Flat Plane
27.4 Holonomy as the Circulation of a Metric-Induced Vector Field in the Map
27.5 Geometric Proof of the Metric Curvature Formula
Errata for Chapter 27
Chapter 28
Chapter 29
Chapter 30
Chapter 31 Exercises for Act IV
Exercises 1-3 Extrinsic Constructions
Exercise 4 Intrinsic Constructions
Exercises 5-6 Holonomy
Exercises 7-9 Curvature as a Force between Neighbouring Geodesics
Exercises 10-12 Riemann Curvature
Exercises 13-15 Einstein's Curved Spacetime
Act V Forms
VISUAL DIFFERENTIAL GEOMETRY and FORMS a mathematical drama in five acts
The Book
Prologue
Acknowledgements
Act I The Nature of Space
Chapter 1 Euclidean and non-Euclidean Geometry
Chapter 2 Gaussian Curvature
Chapter 3 Exercises for Prologue and Act I
Act II The Metric
Chapter 4 Mapping Surfaces: The Metric
Chapter 5 The Pseudosphere and the Hyperbolic Plane
Chapter 6 Isometries and Complex Numbers
Act III Curvature
Chapter 8 Curvature of Plane Curves
8.1 Introduction
8.2 The Circle of Curvature
8.3 Newton's Curvature Formula
8.4 Curvature as Rate of Turning
8.5 Example: Newton's Tractrix
Act IV
Act V
General Information about the Forum
General Discussion
Princeton University link to book, errata etc
Official Errata
News about the book
Useful links
GaryVasco
Prologue (discussion and suggested exercises)
Act I The Nature of Space
Chapter 1 Euclidean and non-Euclidean Geometry
Chapter 2 Gaussian Curvature
2.1 Introduction
2.2 The circumference and Area of a Circle
2.3 The Local Gauss-Bonnet Theorem
Chapter 3 Exercises for Prologue and Act I
Exercises 1-4 Prologue: Newtonian Ultimate Equality $(\asymp)$
Exercises 5-11 Euclidean and Non-Euclidean Geometry
Exercises 12-14 Gaussian Curvature
Act II The Metric
Chapter 4 Mapping Surfaces: The Metric
Chapter 5 The Pseudosphere and the Hyperbolic Plane
Chapter 6 Isometries and Complex Numbers
Chapter 7 Exercises for Act II
Exercises 1-13 Mapping Surfaces: The Metric
Exercises 14-23 The Pseudosphere and the Hyperbolic Plane
Exercise 24-34 Isometries and Complex Numbers
Act III Curvature
Chapter 8 Curvature of Plane Curves
8.1 Introduction
8.2 The Circle of Curvature
8.3 Newton's Curvature Formula
8.4 Curvature as Rate of Turning
8.5 Example: Newton's Tractrix
Chapter 9 Curves in 3-Space
Chapter 10 The Principal Curvatures of a Surface
10.1 Euler's Curvature Formula
10.2 Proof of Euler's Curvature Formula
10.3 Surfaces of Revolution
Chapter 11 Geodesics and Geodesic Curvature
11.1 Geodesic Curvature and Normal Curvature
11.2 Meusnier's Theorem
11.3 Geodesics are "Straight"
11.4 Intrinsic Measurement of Geodesic Curvature
11.5 A simple Extrinsic Way to Measure Geodesic Curvature
11.6 A New Explanation of the Sticky-Tape Construction of Geodesics
11.7 Geodesics on Surfaces of Revolution
11.7.1 Clairot's Theorem on the Sphere
11.7.2 Kepler's Second Law
11.7.3 Newton's Geometrical Explanation of Kepler's Law
11.7.4 Dynamical Proof of Clairot's Theorem
11.7.5 Application: Geodesics in the Hyperbolic Plane (Revisited)
Chapter 12 The Extrinsic Curvature of a Surface
Chapter 13 Gauss's Theorema Egregium
Chapter 14 The Curvature of a Spike
Chapter 15 The Shape Operator
Chapter 16 Introduction to the Global Gauss-Bonnet Theorem
Chapter 17 First (Heuristic ) Proof of the Global Gauss-Bonnet Theorem
Chapter 18 Second (Angular Excess) Proof of the Global Gauss-Bonnet Theorem
Chapter 19 Third (Vector Field) Proof of the Global Gauss-Bonnet Theorem
19.1 Introduction
19.2 Vector fields in the Plane
19.3 The Index of a Singular Point
19.4 The Archetypal Singular Points: Complex Powers
19.5 Vector Fields on Surfaces
19.6 The Poincaré-Hopf Theorem
19.7 Vector Field Proof of the Global Gauss'Bonnet Theorem
19.8 The Road Ahead
Chapter 20 Exercises for Act III
Exercise 1 Curvature of Plane Curves
Exercises 2-6 Curves in 3-Space
Exercises 7-10 The Principal Curvatures of a Surface
Exercise 11 Gaus's Theorema Egregium
Exercises 12-20 The Shape Operator
Exercises 21-22 Introduction to the Global Gauss-Bonnet Theorem
Exercise 23 First (Heuristic) Proof of GGB
Exercises 24-28 Second (Angular Excess) Proof of GGB
Exercises 29-33 Third (Vector Field) Proof of GGB
Act IV Parallel Transport
Chapter 21 An Historical Puzzle
Chapter 22 Extrinsic Constructions
22.1 Project into the Surface as You Go
22.2 Geodesics and Parallel Transport
22.3 Potato-Peeler Transport
Chapter 23 Intrinsic Constructions
23.1 Parallel Transport via Geodesics
23.2 The Intrinsic (aka, "Covariant") Derivative
Chapter 24 Holonomy
24.1 Example: The Sphere
24.2 Holonomy of a General Geodesic Triangle
24.3 holonomy Is Additive
24.4 Example: The Hyperbolic Plane
Chapter 25 An Intuitive Geometric Proof of the Theorema Egregium
25.1 Introduction
25.2 Some Notation and Reminders of Definitions
25.3 The Story So Far
25.4 The Spherical Map Preserves Parallel Transport
25.5 The Beautiful Theorem and Theorema Egregium Explained
Chapter 26
26.1 Introduction
26.2 Holonomy Along an Open Curve?
26.3 Hopf's Intrinsic Proof of the Global Gauss-Bonnet Theorem
Chapter 27
27.1 Introduction
27.2 The Circulation of a Vector Field Around a Loop
27.3 Dry Run: Holonomy in the Flat Plane
27.4 Holonomy as the Circulation of a Metric-Induced Vector Field in the Map
27.5 Geometric Proof of the Metric Curvature Formula
Errata for Chapter 27
Chapter 28
Chapter 29
Chapter 30
Chapter 31 Exercises for Act IV
Exercises 1-3 Extrinsic Constructions
Exercise 4 Intrinsic Constructions
Exercises 5-6 Holonomy
Exercises 7-9 Curvature as a Force between Neighbouring Geodesics
Exercises 10-12 Riemann Curvature
Exercises 13-15 Einstein's Curved Spacetime
Act V Forms
VISUAL DIFFERENTIAL GEOMETRY and FORMS a mathematical drama in five acts
The Book
Prologue
Acknowledgements
Act I The Nature of Space
Chapter 1 Euclidean and non-Euclidean Geometry
Chapter 2 Gaussian Curvature
Chapter 3 Exercises for Prologue and Act I
Act II The Metric
Chapter 4 Mapping Surfaces: The Metric
Chapter 5 The Pseudosphere and the Hyperbolic Plane
Chapter 6 Isometries and Complex Numbers
Act III Curvature
Chapter 8 Curvature of Plane Curves
8.1 Introduction
8.2 The Circle of Curvature
8.3 Newton's Curvature Formula
8.4 Curvature as Rate of Turning
8.5 Example: Newton's Tractrix
Act IV
Act V
General Information about the Forum
VISUAL DIFFERENTIAL GEOMETRY and FORMS a mathematical drama in five acts
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