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Unformatted text preview: Math 21a First Old Final Exam Solutions Spring, 2009 Part I: Multiple choice. Each problem has a unique correct answer. You do not need to justify your answers in this part of the exam. 1 True-False questions. Circle the correct letter. No justifications are required. T F The divergence of the gradient of any f ( x,y,z ) is always zero. Solution: This is False . In general grad f = ∇ f = h f x ,f y ,f z i , so div(grad f ) = ∇ · ∇ f = f xx + f yy + f zz . For example, if f ( x,y,z ) = x 2 + y 2 + z 2 , we get div(grad f ) = 6. T F If a nonempty quadric surface g ( x,y,z ) = ax 2 + by 2 + cz 2 = 5 can be contained inside a finite box, then a,b,c ≥ 0. Solution: This is True . If one of a , b , and c is zero, then the surface is a cylinder or an infinite space whose cross-sections are all hyperbolas. If one or two of a , b , and c is negative, then the surface is a hyperboloid (of one or two sheets). None of these surfaces fits in a finite box. T F If F is a vector field in space, then the flux of F through any closed surface S is 0. Solution: This is False . This is only true if div F = 0, since the Divergence Theorem says that RR S F ˙ d S = RRR E div F dV , where E is the solid enclosed by S . This is zero for any closed surface S only when div F = 0. T F The flux of the vector field F ( x,y,z ) = h y + z,y,- z i through the boundary of a solid region E is equal to the volume of E . Solution: This is False . Again we refer to the Divergence Theorem, which says RR S F ˙ d S = RRR E div F dV , where S is the boundary of E . In this case div F = 0 + 1- 1 = 0, so the flux through S is zero, not the volume of E . T F If in spherical coordinates the equation φ = α (with a constant α ) defines a plane, then α = π/ 2. Solution: This is True . If 0 < α < π 2 , then φ = α is a half-cone opening upward. Similarly, if π 2 < α < π , then φ = α is a half-cone opening downward. If α = 0 or α = π , then φ = α is a half-line (the positive or negative z-axis). That leaves φ = π 2 , which is the xy-plane. 2 Which of the following is the cosine of the angle between the two planes- 5 x- 3 y- 4 z = 0 and x + y + z = 1? (a) 6 √ 2 5 (b) 2 √ 5 6 (c) 2 √ 6 5 (d) 5 √ 3 2 (e) None of the above. Solution: The correct answer is (c). Normal vectors to the two planes are a = h 5 , 3 , 4 i and b = h 1 , 1 , 1 i , and cos θ = a · b | a | | b | = 2 √ 6 5 , where θ is the angle between the two planes. 3 Which of the functions u ( x,t ) below are solutions to the partial differential equation ∂u ∂t = ∂ 2 u ∂x 2 ? (a) u ( x,t ) = e- t sin x (b) u ( x,t ) = e- 2 t sin2 x (c) u ( x,t ) = e t sin x (d) u ( x,t ) = 3 (e) u ( x,t ) = x 2 + 2 t Solution: (a), (d), and (e) are solutions. Math 21a First Old Final Exam Solutions Spring, 2009 4 Which of the following line integrals below are path independent?...
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- Spring '10