Copies of the classnotes are on the internet in PDF format as given below. The notes and supplements may contain hyperlinks to posted webpages; the links appear in red fonts. The "Proofs of Theorems" files were prepared in Beamer and they contain proofs of the results from the class notes. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

0. Chapter 0. Fundamental Not(at)ions.
1. Chapter I. Real Vector Spaces.
2. Chapter II. Affine Spaces.
3. Chapter III. Dual Spaces.
4. Chapter IV. Metric Vector Spaces.
5. Chapter V. Tensors and Multilinear Forms.
6. Chapter VI. Topological Vector Spaces.
7. Chapter VII. Differentiation and Manifolds.
8. Chapter VIII. Connections and Covariant Differentiation.
9. Chapter IX. Geodesics.
10. Chapter X. Curvature.
11. Chapter XI. Special Relativity.
12. Chapter XII. General Relativity.

Chapter II. Affine Spaces.

• Section II.1. Spaces. Section II.1 notes
• Supplement. Proofs of Theorems in Section II.1. Beamer file of Section II.1 proofs of theorems
• Supplement. Printout of the Proofs of Theorems in Section II.1. Print file of Section II.1 proofs of theorems
• Section II.2. Combinations of Points. Section II.2 notes
• Section II.3. Maps. Section II.3 notes
• Study Guide II

Chapter III. Dual Spaces.

• Section III.1. Contours, Covariance, Contravariance, Dual Basis. Section III.1 notes
• Supplement. Proofs of Theorems in Section III.1. Beamer file of Section III.1 proofs of theorems
• Supplement. Printout of the Proofs of Theorems in Section III.1. Print file of Section III.1 proofs of theorems
• Study Guide III

Chapter IV. Metric Vector Spaces.

• Section IV.1. Metrics. Section IV.1 notes
• Supplement. Proofs of Theorems in Section IV.1. Beamer file of Section IV.1 proofs of theorems
• Supplement. Printout of the Proofs of Theorems in Section IV.1. Print file of Section IV.1 proofs of theorems
• Section IV.2. Maps. Section IV.2 notes
• Supplement. Proofs of Theorems in Section IV.2. Beamer file of Section IV.2 proofs of theorems
• Supplement. Printout of the Proofs of Theorems in Section IV.2. Print file of Section IV.2 proofs of theorems
• Section IV.3. Coordinates. Section IV.3 notes
• Supplement. Proofs of Theorems in Section IV.3. Beamer file of Section IV.3 proofs of theorems
• Supplement. Printout of the Proofs of Theorems in Section IV.3. Print file of Section IV.3 proofs of theorems
• Section IV.4. Diagonalizing Symmetric Operators. Section IV.4 notes
• Supplement. Proofs of Theorems in Section IV.4. Beamer file of Section IV.4 proofs of theorems
• Supplement. Printout of the Proofs of Theorems in Section IV.4. Print file of Section IV.4 proofs of theorems
• Study Guide IV

Chapter V. Tensors and Multilinear Forms.

• Section V.1. Multilinear Forms. Section V.1 notes
• Supplement. Proofs of Theorems in Section V.1. Beamer file of Section V.1 proofs of theorems
• Supplement. Printout of the Proofs of Theorems in Section V.1. Print file of Section V.1 proofs of theorems
• Study Guide V

Chapter VI. Topological Vector Spaces.

• Section VI.1. Continuity.
• Section VI.2. Limits.
• Section VI.3. The Usual Topology.
• Section VI.4. Compactness and Completeness.
• Study Guide VI.

Chapter VII. Differentiation and Manifolds.

• Section VII.1. Differentiation. (Partial) Section VII.1 notes
• Section VII.2. Manifolds. Section VII.2 notes
• Section VII.3. Bundles and Fields.
• Section VII.4. Components.
• Section VII.5. Curves.
• Section VII.6. Vector Fields and Flows.
• Section VII.7. Lie Brackets.
• Study Guide VII.

Chapter IX. Geodesics.

• Section IX.1. Local Characterisation.
• Section IX.2. Geodesics from a Point.
• Section IX.3. Global Characterisation.
• Section IX.4. Maxima, Minima, Uniqueness.
• Section IX.5. Geodesics in Embedded Manifolds.
• Section IX.6. An Example of Lie Group Geometry.
• Study Guide IX.

Chapter X. Curvature.

• Section X.1. Flat Spaces.
• Section X.2. The Curvature Tensor.
• Section X.3. Curved Surfaces.
• Section X.4. Geodesic Deviation.
• Section X.5. Sectional Curvature.
• Section X.6. Ricci and Einstein Tensors.
• Section X.7. The Weyl Tensor.
• Study Guide X.

Other Notes on Differential Geometry and Relativity

Differential Geometry

Differential Geometry (in preparation)

General Relativity (in preparation)

General Relativity (in preparation)

Return to Bob Gardner's home page