1 The Genesis of Differential Methods

1.1 The Static Approach to Curves

1.2 The Dynamic Approach to Curves

1.3 Cartesian Versus Parametric

1.4 Singularities and Multiplicities

1.5 Chasing the Tangents

1.6 Tangent：The Differential Approach

1.7 Rectification of a Curve

1.8 Length Versus Curve Integral

1.9 Clocks,Cycloids and Envelopes

1.10 Radius of Curvature and Evolute

1.11 Curvature and Normality

1.12 Curve Squaring

1.13 Skew Curves

1.14 Problems

1.15 Exercises

2 Plane Curves

2.1 Parametric Representations

2.2 Regular Representations

2.3 The Cartesian Equation of a Curve

2.4 Tangents

2.5 Asymptotes

2.6 Envelopes

2.7 The Length of an Arc of a Curve

2.8 Normal Representation

2.9 Curvature

2.10 Osculating Circle

2.11 Evolutes and Involutes

2.12 Intrinsic Equation of a Plane Curve

2.13 Closed Curves

2.14 Piecewise Regular Curves

2.15 Simple Closed Curves

2.16 Convex Curves

2.17 Vertices of a Plane Curve

2.18 Problems

2.19 Exercises

3 A Museum of Curves

3.1 Some Terminology

3.2 The Circle

3.3 The Ellipse

3.4 The Hyperbola

3.5 The Parabola

3.6 The Cycloid

3.7 The Cardioid

3.8 The Nephroid

3.9 The Astroid

3.10 The Deltoid

3.11 The Lima？on of Pascal

3.12 The Lemniscate of Bernoulli

3.13 The Conchoid of Nicomedes

3.14 The Cissoid of Diocles

3.15 The Right Strophoid

3.16 The Tractrix

3.17 The Catenary

3.18 The Spiral of Archimedes

3.19 The Logarithmic Spiral

3.20 The Spiral of Cornu

4 Skew Curves

4.1 Regular Skew Curves

4.2 Normal Representations

4.3 Curvature

4.4 The Frenet Trihedron

4.5 Torsion

4.6 Intrinsic Equations

4.7 Problems

4.8 Exercises

5 The Local Theory of Surfaces

5.1 Parametric Representation of a Surface

5.2 Regular Surfaces

5.3 Cartesian Equation

5.4 Curves on a Surface

5.5 The Tangent Plane

5.6 Tangent Vector Fields

5.7 Orientation of a Surface

5.8 Normal Curvature

5.9 Umbilical Points

5.10 Pfincipal Directions

5.11 The Case of Quadrics

5.12 Approximation by a Quadric

5.13 The Rodrigues Formula

5.14 Lines of Curvature

5.15 Gauss'Approach to Total Cuivature

5.16 Gaussian Curvature

5.17 Problems

5.18 Exercises

6 Towards Riemannian Geometry

6.1 What Is Riemannian Geometry？

6.2 The Metric Tensor

6.3 Curves on a Riemann Patch

6.4 Vector Fields Along a Curve

6.5 The Normal Vector Field to a Curve

6.6 The Christoffel Symbols

6.7 Covariant Derivative

6.8 Parallel Transport

6.9 Geodesic Cuivature

6.10 Geodesics

6.11 The Riemann Tensor

6.12 What Is a Tensor？

6.13 Systems of Geodesic Coordinates

6.14 Cuivature in Geodesic Coordinates

6.15 The Poincaré Half Plane

6.16 Embeddable Riemann Patches

6.17 What Is a Riemann Surface？

6.18 Problems

6.19 Exercises

7 Elements of the Global Theory of Surfaces

7.1 Surfaces of Revolution

7.2 Ruled Surfaces

7.3 Applicability of Surfaces

7.4 Surfaces with Zero Curvature

7.5 Developable Surfaces

7.6 Classification of Developable Surfaces

7.7 Surfaces with Constant Curvature

7.8 The Sphere

7.9 A Counterexample

7.10 Rotation Numbers

7.11 Polygonal Domains

7.12 Polygonal Decompositions

7.13 The Gauss-Bonnet Theorem

7.14 Geodesic Triangles

7.15 The Euler-Poincaré Characteristic

7.16 Problems

7.17 Exercises

Appendix A Topology

A.1 Open Subsets in Real Spaces

A.2 Closed Subsets in Real Spaces

A.3 Compact Subsets in Real Spaces

A.4 Continuous Mappings of Real Spaces

A.5 Topological Spaces

A.6 Closure and Density

A.7 Compactness

A.8 Continuous Mappings

A.9 Homeomorphisms

A.10 Connectedness

Appendix B Differential Equations

B.1 First Order Differential Equations

B.2 Second Order Differential Equations

B.3 Variable Initial Conditions

B.4 Systems of Partial Differential Equations

References and Further Reading

Index