
\magnification=1200
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\centerline{\bf TOPICS FOR MATHEMATICS 138A, WINTER  2004}
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\centerline{\bf I.  Classical differential geometry of curves}
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1. Cross products (do Carmo, \S 1--4)

2. Parametrized curves  (do Carmo, \S 1--2)

3. Arc length and reparametrization  (do Carmo, \S 1--3)

4. Curvature and torsion (do Carmo, \S\S 1--5, 1--6)

5. Frenet-Serret Formulas (do Carmo, \S\S 1--5, 1--6, 4--Appendix)
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\centerline{\bf II.  Closed curves as boundaries}
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1. Regions, limits and continuity (do Carmo, 2--Appendix A,
5--Appendix)

2. Smooth mappings (do Carmo, 2--Appendix B)

3. Inverse and implicit function theorems  (do Carmo, 2--Appendix B)

4. Global properties of plane curves (do Carmo, \S 1--7)
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\centerline{\bf  III.  Surfaces in 3-dimensional space}
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1. Mathematical descriptions of surfaces (do Carmo, \S\S 2--2, 2--3)

2. Parametrizations of surfaces (do Carmo, \S\S 2--2, 2--3)

3. Tangent planes (do Carmo, \S 2--4)

4. The First Fundamental Form  (do Carmo, \S 2--5)

5. Surface area (do Carmo, \S\S 2--5, 2--8)

6. Curves as surface intersections (do Carmo, \S 2--3)
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\centerline{\bf IV.  Oriented surfaces}
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1. Normal directions and Gauss maps (do Carmo, \S\S 2--6, 3--2)

2. The Second Fundamental Form (do Carmo, \S\S 3--2, 3--3)

3. Quadratic forms and adjoint transformations (do Carmo, 3--Appendix)

4. Normal, Gaussian and mean curvature (do Carmo, \S\S 3--2, 3--3)

5. Special classes of surfaces (do Carmo, \S 3--5)

6. Compatibility equations,  {\sl Theorema Egregium} (do Carmo, \S 4--3)

7. Fundamental Theorem of Local Surface Theory (do Carmo, \S 4--3, 
4--Appendix)

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