PHZ 3113 Fall 2011 — Practice Exam 2
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Instructions: Attempt all five questions. The maximum possible credit for each question
is shown in square brackets. Please try to write your solution neatly and legibly.
You will receive credit only for knowledge and understanding that you demonstrate in
your written solutions. It is in your best interest to write down something relevant for every
question, even if you can’t provide a complete answer. To maximize your score, you should
briefly explain your reasoning and show all working. Give all final algebraic answers in terms
of variables defined in the problem.
During this exam, you may use up to two formula sheets. You are not permitted (a)
to consult any other books, notes, or papers, (b) to use any electronic device, or (c) to
communicate with anyone other than the proctor. In accordance with the UF Honor Code,
by turning in this exam to be graded, you affirm the following pledge: On my honor, I have
neither given nor received unauthorized aid in doing this assignment.
Print your name where indicated below, and sign to confirm that you have read and
understood these instructions. Please do not write anything else below the line.
Name (printed):
Signature:
Question
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Total
Score
Throughout the questions below, x̂, ŷ, and ẑ are unit vectors along the Cartesian x, y,
and z axes, respectively.
R
1. [20 points] Calculate the outward flux A F·da of the vector field F = 3yzx̂+2x2 z ŷ+z 3 ẑ
over the surface of a triangular prism with corners at ±ẑ, 4x̂ ± ẑ, and 3ŷ ± ẑ.
2. [20 points] ConsiderH the vector field F = (x2 − y 2 )x̂ + 2xyŷ. Use Stokes’ theorem to
find the circulation F·dr around the perimeter of the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 3,
following a path that heads first from (x, y) = (0, 0) to (2, 0).
√
3. [15 points] If z1 = 2eiπ/4 and z2 = (3 − i)eiπ , evaluate
(a) Re (z1 z2 ),
(b) (z1 + z2 )1/3
(list all distinct answers).
4. [15 points] Suppose that f (z) = u(x, y) + iv(x, y), where z = x + iy and u, v, x, and
y are all real. Given that f (z) is analytic in some region and v(x, y) = e−y sin x, find
f (z). (Express your answer as a function of z, not of x and y.)
5. [30 points] Use contour integration to evaluate
Z ∞
Z 2π
x sin πx
sin θ
(a)
dx,
(b)
dθ.
2
5 + 3 sin θ
−∞ 1 + x
0
The following are problems of similar difficulty to those above and might have appeared
instead.
6. [15 points] Determine whether or not F = (ln y + 2xy 3 )x̂ + (3x2 y 2 + x/y)ŷ is a conservative field on the half plane y > 0. If it is, find a function f (x, y) such that F = ∇f .
7. [20 points] Evaluate
Z ∞
(a)
sin x δ(2x − π) dx,
Z
(b)
−∞
8. [15 points] Given z1 = 2eiπ/6 and z2 = 1 +
Note: cos(π/6) =
√
√
π
(1 + sin2 θ) δ(cos θ) dθ.
−π
3 i, evaluate
(b) (z1 − z2 )1+i .
(a) |z1 /z2 |,
3/2, sin(π/6) = 1/2.
9. [20 points] A resistor R and an inductor L are connected in series with one another.
This pair of elements are then connected in parallel with a capacitor C.
(a) Find the total impedance of this combination of three elements when connected
to a voltage source varying (co)sinusoidally at angular frequency ω.
(b) Find the angular frequency ω at which the impedance is real.
Acknowledgment: Some of the questions on this practice exam were adapted from ones
provided by Prof. Peter Hirschfeld.
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