Hjem

1 Parametrized curves and surfaces
1.1 Curves
1.2 Surfaces
1.3 Graphs
1.4 Level sets
1.5 The implicit function theorem, two variables
1.6 The implicit function theorem, more variables
1.7 The implicit function theorem, more equations
1.8 Exercises

2 Tangents
2.1 Regular curves and tangent lines
2.2 The tangent line of a level set
2.3 The tangent plane of a regular surface
2.4 Curves on a surface
2.5 Reparametrization of curves
2.6 Reparametrization of surfaces
2.7 Invariance under reparametrization
2.8 The unit normal, orientation
2.9 Regular curves as graphs
2.10 The inverse function theorem
2.11 Regular surfaces as graphs
2.12 Exercises

3 The first fundamental form
3.1 Arc length
3.2 Lines as shortest curves
3.3 Unit speed parametrization
3.4 The first fundamental form
3.5 Introduction to areas and plane integrals
3.6 Null sets
3.7 Double integrals
3.8 Transformation of integrals
3.9 Surface area
3.10 Exercises

4 Curvature
4.1 Curvature of plane curves
4.2 Curvature of unit speed curves
4.3 The tangent angle
4.4 Curvature of space curves
4.5 Torsion
4.6 The osculating plane and the binormal vector
4.7 The Frenet formulas
4.8 Curvature of curves on a surface
4.9 Interpretation of normal curvature
4.10 Geodesics
4.11 Exercises

5 The second fundamental form
5.1 The shape operator
5.2 The second fundamental form
5.3 Coordinate expressions for the second fundamental form 80
5.4 A formula for the shape operator
5.5 Diagonalization of the second fundamental form
5.6 The graph of a quadratic form
5.7 The type of a surface at a point
5.8 Exercises

6 Teorema egregium
6.1 The Gaussian curvature
6.2 Intrinsic geometry
6.3 Christoffel symbols
6.4 The remarkable theorem of Gauss
6.5 Isometries
6.6 Exercises

7 Geodesics
7.1 The geodesic equations
7.2 Existence of geodesics
7.3 Geodesic coordinates
7.4 The first fundamental form of a geodesic coordinate system 113
7.5 Interpretation of the Gauss theorem
7.6 Exercises

Appendices
A Euclidean spaces
B Differentiable functions of several variables
C Normal vectors and cross products
D Diagonalization of symmetric matrices
E Hyperbolic functions

Notational index
Index