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Unformatted text preview: M 427L Second Exam Put your answers to all the questions on the scan-tron sheet. Be sure you put your eid on the scan-tron, and check to be sure you have entered it correctly (or the computer will not give you credit for the exam) 1. Let φ : R 3 → R 3 be a smooth function. Which of the following is the 3-form φ * ( dxdydz )? 2. Let φ : R 3 → R 3 be a smooth function. Suppose φ : Ω → M gives a parame- terization of the region M ⊂ n by the region Ω ⊂ R n (i.e. M = φ (Ω) and the function is one to one outside a set with zero volume). Assuming φ preserves orientation (i.e. D ( φ ) has non-negative determinant everywhere on Ω) what does integraldisplay Ω φ * ( dxdydz ) equal? (a) zero (by Stokes theorem) (b) The volume of the region M (this is the differential forms version of the change of variable formula) of the boundary of a given region Ω ⊂ R 3 (c) This is roughly the ratio of the volume of the image of a small box around a point to the volume of the box itself.a point to the volume of the box itself....
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This note was uploaded on 09/18/2011 for the course M 427L taught by Professor Staff during the Spring '11 term at Oklahoma State.
- Spring '11