This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 TrueFalse 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 crosssections 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 halfcone opening upward. Similarly, if π 2 < α < π , then φ = α is a halfcone opening downward. If α = 0 or α = π , then φ = α is a halfline (the positive or negative zaxis). That leaves φ = π 2 , which is the xyplane. 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?...
View
Full
Document
 Spring '10
 George

Click to edit the document details