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A 2.5 hour closed book examination for mathematics 227, advanced calculus ii, at the university of british columbia, held in april 2006. The examination covers various topics in advanced calculus, including vector calculus, surface integrals, and simply connected regions. It consists of 8 questions, with a total of 100 marks.
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Marks
[12] 1. A bug is flying through space so that its coordinates at time t are x(t) = (t^2 + t, t^2 − t, t^3 ).
(a) (6 marks) Find the bug’s velocity and acceleration for all t, and the curvature of its trajectory at time t = 0.
(b) (6 marks) Find all values of t at which the osculating plane is perpendicular to the xz-plane.
[10] 2. Evaluate the integral
D
x + y x − 2 y
dA, if D is the region in R^2 enclosed by the lines y = x/2, y = 0, and x + y = 1.
[8] 4. Evaluate
x F^ ·^ ds, if^ F^ = sin^ yi^ + (x^ cos^ y^ −^ cos^ z)j^ +^ y^ sin^ zk^ and^ x^ is the parametrized curve (
π 2
sin
πt 2
, πt^2 , πt^3 ), 0 ≤ t ≤ 1. (Hint: this can be done without any complicated calculations.)
(a) (6 marks) Evaluate
S 1 F^ ·^ dS, if^ F^ =^ e
x+y (^) i − ex+y (^) j + 2zk and S 1 is the disc x (^2) + y (^2) ≤ 9, z = 3, oriented so that the normal vector points upward.
(b) (8 marks) Evaluate
S 2 F^ ·^ dS, if^ F^ is as in (a) and^ S^2 is the part of the sphere^ x
(^2) + y (^2) + (z − 3)^2 = 9 that lies above the plane z = 3, oriented so that the normal vector points upward. (Hint: use the Divergence Theorem.)
[18] 7. Let X be the parametrized surface X(s, t) = (st, s + t, s − t), s^2 + t^2 ≤ 1.
(a) (6 marks) Find the surface area of X.
(b) (6 marks) Find
X F^ ·^ dS, if^ F^ = (y^ +^ z)
(^2) i + yj + zk.
(continued on next page)
(c) (6 marks) Find
X ω, if^ ω^ = (y^ −^ z)dx^ ∧^ dz^ +^ xdz^ ∧^ dy.
[6] 8. Decide whether the following regions are simply connected. (3 marks for each).
(a) The complement of the line segment from (− 1 , 0) to (1, 0) in R^2.
(b) {(x, y, z) ∈ R^3 : 1 ≤ x^2 + y^2 + z^2 ≤ 9 }
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