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Final Math Exam Information
June, 2009
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Final exam Thursday, June 11, 8:30-10:20 in SWS B012 (our usual classroom). Bring your own paper to write on. Usual rules for notes. No take-home problem required for the exam. Extra Credit take-home problem for the nal. (a) (Worth up to 12% of the total number of points on the nal.) Compute the curvature in cm2 of one of my crocheted models of the hyperbolic plane. Write up an explanation of your method, the measurements you took, and the computations you did. Using loose stitching or tape, show your work on the model. (It may be helpful to have both markings on the model and a corresponding sketch on paper.) (b) (Worth up to and additional 8% of the points on the nal.) Repeat part (a) using a dierent method. (Because these results depend on measurements, you may very well get dierent values by your two methods.) Your solution to this extra credit should be turned in with my model directly to me before the in-class exam, with your in-class exam, or to my mailbox or oce between 11 and 2 on the day of the exam (June 11). ******************************************************************************* Oce hours during exam week: 1-2 PM MTuW, and by appointment. ******************************************************************************* Review Questions. It may also be helpful to review the midterm and the homework from this quarter. particular In review/include in your notes formulas for surfaces of revolution and Clairaut patches. 1. Let M be the paraboloid z = x2 + y 2 in R3 , oriented by the upward pointing normal. Fix a constant z0 > 0, and let B be the region of M below the plane z = z0 . Compute the total curvature B K dA of the region B by three dierent methods. (a) Choose a patch (it may omit a small subset of B , e.g. a curve) and compute the integral directly. (b) Use the Gauss map. (Remember to show the Gauss map is 1:1 and constant sign on B .) (c) Let C be the boundary of B , oriented counterclockwise as viewed from above, and use the Gauss-Bonnet formula. (Be careful about the sign of the geodesic curvature g of C with the given orientation.) 2. The proofs of the Gauss-Bonnet formula and theorem used only the intrinsic geometry, that is the R...