Unformatted text preview: Math 3210 final exam practice problems December 2009 On the final exam you may use anything that has been given in class, in the homework, or in the lecture notes , as long as you show clearly what you are using. You may bring a two-sided letter-size cheat sheet. No books, notes, or calculators allowed. From the lecture notes: problems 6.12, 6.13, 9.2, 9.3. 1. Let α = ∑ n i = 1 f i dx i be a 1-form on R n . (a) Find formulas for * α , d * α , * d * α , and d * d * α . (b) Find formulas for d α , * d α , d * d α , and * d * d α . (c) Finally compute d * d * α + (- 1 ) n * d * d α . Try to write the answer in terms of the Laplace operator Δ , which is given by Δ f = ∑ n i = 1 ∂ 2 f / ∂ x 2 i . 2. Let ψ : [ a , b ] → R n be an embedding. Then M = ψ ([ a , b ]) is a smooth 1-manifold. Let us call the direction of the tangent vector ψ ( t ) positive; this defines an orientation of M . Let μ be the element of arclength (“volume element”) of M ....
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- Math, Trigraph, Manifold, two-sided letter-size cheat