Download Advanced Calculus IV Exam - December 2011 and more Exams Mathematics in PDF only on Docsity!
Marks
[18] 1. Short answers. Answer each question below. Read and think carefully. For this question only, no explanation or justification is needed, and no credit will be given for an incorrect answer. (There are no typos in this question. If something looks incorrect, you should say so in your answer.)
(a) (3 marks) True or false? If r(t) is the position at time t of an object moving in R^3 , and r(t) is twice differentiable, then |r′′(t)| is the tangential component of its acceleration.
(b) (3 marks) Let r(t) is a smooth curve in R^3 with unit tangent, normal and binormal vectors T(t), N(t), B(t). Which two of these vectors span the plane normal to the curve at r(t)?
(c) (3 marks) True or false? If F = P i + Qj + Rk is a vector field on R^3 such that P, Q, R have continuous first order derivatives, and if curl F = 0 everywhere on R^3 , then F is conservative.
(d) (3 marks) True or false? If F = P i + Qj + Rk is a vector field on R^3 such that P, Q, R have continuous second order derivatives, then curl(div F) = 0.
(e) (3 marks) True or false? If F is a vector field on R^3 such that |F(x, y, z)| = 1 for all x, y, z, and if S is the sphere x^2 + y^2 + z^2 = 1, then
S F^ ·^ dS^ = 4π.
(f) (3 marks) True or false? Every closed surface S in R^3 is orientable. (Recall that S is closed if it is the boundary of a solid region E.)
[10] 3. Let C be the curve in R^2 given by the graph of the function y =
x^3 3
. Let κ(x) be the curvature of C at the point (x, x^3 /3). Find all points where κ(x) attains its maximal values, or else explain why such points do not exist. What are the limits of κ(x) as x → ∞ and x → −∞?
[16] 4. Evaluate the line integrals below. (Use any method you like.)
(a) (8 marks)
C (x
(^2) + y)dx + xdy, where C is the arc of the parabola y = 9 − x (^2) from (− 3 , 0) to (3, 0).
(b) (8 marks)
C F^ ·^ n^ ds, where^ F(x, y) = 2x
(^2) i + yexj, C is the boundary of the square 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1, and n is the unit normal vector pointing outward.
[10] 6. Let F = (2y + 2)i be a vector field on R^2. Find an oriented curve C from (0, 0) to (2, 0) such that
C F^ ·^ dr^ = 8.
[24] 7. Evaluate the surface integrals. (Use any method you like.)
(a) (8 marks)
S
z^2 dS, if S is the part of the cone x^2 + y^2 = 4z^2 where 0 ≤ x ≤ y and 0 ≤ z ≤ 1.
(c) (8 marks)
S
F · dS, where F = (y − z^2 )i + (z − x^2 )j + z^2 k and S is the boundary surface of the box 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 3, with the normal vector pointing outward.
(Use this blank page if you need more space.)
The End
