finalexam-practice1-solutions

# finalexam-practice1-solutions - Math 21a First Old Final...

This preview shows pages 1–3. Sign up to view the full content.

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 z i , so div(grad f ) = ∇ · ∇ f = f xx + f yy + f zz . For example, if f ( ) = x 2 + y 2 + z 2 , we get div(grad f ) = 6. T F If a nonempty quadric surface g ( ) = 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 ( ) = 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 . 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 ( ) = e - t sin x (b) u ( ) = e - 2 t sin 2 x (c) u ( ) = e t sin x (d) u ( ) = 3 (e) u ( ) = x 2 + 2 t Solution: (a), (d), and (e) are solutions.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Math 21a First Old Final Exam Solutions Spring, 2009 4 Which of the following line integrals below are path independent? (a) Z C (10 x - 7 y ) dx - (7 x - 2 y ) dy (b) Z C (45 x 4 y 2 - 6 y 6 + 3) dx + (18 x 5 y - 12 xy 5 + 7) dy (c) Z C (2 e y - ye x ) dx + (2 xe y - e x ) dy (d) Z C 4 y 2 cos( xy 2 ) dx + 8 x cos( xy 2 ) dy (e) Z C (sin y + y sin x ) dx + ( x cos y - cos x + 1) dy Solution: A line integral R C P dx + Qdy is path independent if f = P i + Q j for some function f . If P i + Q j is deﬁned on a simply connected open set, we need only check that P y = Q x . Thus, (a), (c), and (e) are path independent line integrals.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

finalexam-practice1-solutions - Math 21a First Old Final...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online