120A. Differential Geometry. (4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, 131A. Curves in 3-space, Frenet formulas, surfaces in 3-space, normal curvature, Gaussian curvature, congruence of curves and surfaces, intrinsic geometry of surfaces, isometries, geodesics, Gauss/Bonnet theorem. P/NP or letter grading.
Class meeting time: Monday, Wednesday and Friday , 1-1:50 pm in MS 5118
Textbook: R. Millman and G. Parker, Elements of Differential Geometry, Prentice-Hall Inc.
| Exam 1 | 20% |
| Exam 2 | 20% |
| Homework | 10% |
| Final Exam | 50% |
There will be no make up midterm exams . If you miss a midterm exam for any reason, your missed midterm grade will be interpolated (half final exam , half other midterm) from other exams.
Check here often for assignments and other information.
Lecture |
Topics |
Introduction; review
of lines, planes, vectors, spheres HW #1 --Click here, then print. Cauchy Schwarz and other stuff---some lecture notes. Click here . |
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Plane
curves; speed, arc-length, tangent and normal vectors Notes about Curves, Unit Tangents ,and Reparametrizations |
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Curvature;
Frenet formulas for plane curves; geometric and physical interpretations
of curvature |
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Space
curves; tangent, normal, and binormal vectors; Frenet formulas for space
curves Uniqueness of Curves with specified curvature and torsion Click here for HW #2--in problem C, part a, you should prove these statements. More notes on Curves, Reparametrization of Curves, Normal, Binormal . |
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Convex curves; curvature
and convexity |
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Here are some notes on the Isoperimetric Inequality ( by REG ) Isoperimetric Inequality Proof(Hurwitz) Isoperimetric inequality |
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Introduction to
surfaces; coordinate patches; regular surfaces |
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The first fundamental
form; element of area; arc-length of curves; unit normal vector to a surface HW 5 . Click here Due date to be announced . |
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Normal curvature
and Meusnier's Theorem; the second fundamental form Basic Information on Quadratic Forms in Two Dimensions. (Notes from REG ) Integration on Surfaces and the Gauss Bonnet Theorem Outline Proof of Gauss Bonnet theorem from Gauss Bonnet Formula Notes on Scaling of Surfaces and Gauss Curvature Gauss Bonnet Example |
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Examples; surfaces
of revolution Sample Problems Summary of Surface Theory Example of Gauss Curvature Formula ( handed out earlier in class but posted Dec 1 )  Intrinsic Gauss Curvature example (handed out earlier in class but posted Dec 1)  Here is HW #6 . Due on Friday, November 14. Summary of Surface Theory (things to know for Midterm II ) Sample problems for midterm II Postive Gauss Curvature --Homework Sequence Poincare Unit Disc  |
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Geodesic curvature;
the Christoffel symbols |
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Euler's equations
for a geodesic; examples |
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Sign of the Gaussian
curvature; elliptic, parabolic, and hyperbolic points |
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Gauss's Theorema
Egregium ("Gaussian curvature is intrinsic") |
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Consequences of
the Theorema Egregium; non-existence of accurate flat maps of the world;
statement of the Gauss-Bonnet theorem |