Geometry and analysis
4 points * Second semester * Clayton * Prerequisites: MAP2011, MAT2010, MAT2020 * MAP3071 is recommended
Objectives On the completion of this subject students will be able to understand basic concepts in nonlinear analysis within a geometric context and apply these in areas such as differential topology, differential geometry and mathematical physics. Students will acquire thorough familiarity with geometric concepts such as surfaces, tangent planes, tensors and curvature. They will also be able to carry out explicit calculations in specific situations and design proofs of basic mathematical statements. Furthermore, they will have learned to appreciate the role of these fundamental ideas in active research areas of modern mathematics such as minimal surface theory and general relativity.
Synopsis Brief review of several variables, surfaces in Euclidean space given as level sets of functions, tangent planes, normal vectors and vector-fields on surfaces, geodesics (curves of shortest length), parallel transport, curvature, surfaces given by coordinate systems (parametrized surfaces), local equivalence between parametrized surfaces and level sets (implicit function theorem), surface integrals, basics on differential forms and tensors, Stokes' theorem; further topics: minimal surfaces, Gauss-Bonnet theorem.
Assessment Examination (1.5 hours): 70% * Assignments: 30%
Recommended texts
Thorpe J A Elementary topics in differential geometry Springer, 1985
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