Math380_Summer07_Exam3 - D ). (a) In general, how is the...

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Name: Math 380 – Exam 3 August 1, 2007 50 points possible 1. If the density of a wire that lies along the planar curve σ ( t ) = (3 t,t 4 ), 0 t 1, is given by the function f ( x,y ) = xy , find the total mass of the wire. 2. Let D be a region appropriate for Green’s Theorem and let F ( x,y ) = ( M ( x,y ) ,N ( x,y )) be C 1 on D . (a) State the conclusion to Green’s Theorem. (b) Explain how Green’s Theorem is used to derive either Stokes’ Theorem for R 2 or the Divergence Theorem in the Plane. 2. Continued. Let F ( x,y ) = ( xy,y 2 ) and let D be the first quadrant region bounded by the line y = x and the parabola y = x 2 . (c) Parametrize the boundary of D . (d) Verify Green’s Theorem. 3. Let S be the surface defined by x 2 + y 2 + z = 4 and z 0. (a) Find a parametrization for the surface S . (b) Are there any points where the parametrization is not regular? Fully justify your answer. 4. Let Φ : D R 2 S R 3 be a smooth parametrized surface, where D is a bounded region. Let F ( x,y,z ) be C 1 whose domain includes the surface S = Φ(
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Unformatted text preview: D ). (a) In general, how is the integral ZZ S F · d ~ S related to ZZ Φ F · d ~ S . Clearly explain your answer. (b) Let Φ( s,t ) = ( s cos t,s sin t,t ), 0 ≤ s ≤ 1, 0 ≤ t ≤ 2 π . Determine if Φ is positively or negatively oriented. (c) Let F = ( x,y,z-2 y ). Evaluate ZZ Φ F · d ~ S . 5. (a) In Chapter 3, we defined the divergence of F at a point ( x,y,z ) as ∇ · F . In class, we demonstrated that this definition is actually a by-product of divergence’s original integral definition. State this definition and explain how the integral definition elucidates the physical significance of divergence. (b) State the conclusion of Gauss’s Theorem. (c) Calculate the surface integral ZZ S F · d ~ S ; where F = (3 y 2 z 3 , 9 x 2 yz 2 ,-4 xy 2 ) where S is the surface of the cube with vertices ( ± 1 , ± 1 , ± 1)....
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This note was uploaded on 02/15/2011 for the course MATH 380 taught by Professor Staff during the Summer '08 term at University of Illinois, Urbana Champaign.

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Math380_Summer07_Exam3 - D ). (a) In general, how is the...

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