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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 twosided lettersize cheat sheet. No books, notes, or calculators allowed. From the lecture notes: problems 6.12, 6.13, 9.2, 9.3. 1. Let = n i = 1 f i dx i be a 1form on R n . (a) Find formulas for * , d * , * d * , and d * d * . (b) Find formulas for d , * d , d * d , and * d * d . (c) Finally compute d * d * + ( 1 ) n * d * d . Try to write the answer in terms of the Laplace operator , which is given by f = n i = 1 2 f / x 2 i . 2. Let : [ a , b ] R n be an embedding. Then M = ([ a , b ]) is a smooth 1manifold. Let us call the direction of the tangent vector ( t ) positive; this defines an orientation of M . Let be the element of arclength (volume element) of M ....
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