THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
JUNE 2018
MATH2069
MATHEMATICS 2A
(1) TIME ALLOWED – 2 hours
(2) TOTAL NUMBER OF QUESTIONS – 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QUESTION IN A SEPARATE BOOK
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
(7) ONLY CALCULATORSWITH ANAFFIXED “UNSWAPPROVED” STICKER
MAY BE USED
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
JUNE 2018 MATH2069 Page 2
Use a separate book clearly labelled Question 1
1. [20 marks]
i) [5 marks]
a) Find where the curve
r(t) = ti+ t2(j+ k)
intersects the plane
x+ 7y + 8z = 0.
b) Write a parametric vector equation for the tangent line to the curve
at each point of intersection.
c) Find the cosine of the angle between the curve and the normal direc-
tion to the plane at each point of intersection.
ii) [5 marks]
a) Find an equation for the tangent plane to the sphere
x2 + y2 + z2 2y 4z + 2 = 0
at the point (1, 2, 1).
b) Show that the normal to the sphere is perpendicular to the normal
to the paraboloid
3x2 2y + 2z2 = 1
at their point of intersection (1, 2, 1).
iii) [5 marks] Let f(x, y) = x2 2y3.
a) Find a unit vector in the xy-plane which points in the direction of
greatest increase of f at the point (1, 2).
b) Find the directional derivative of f at the point (1, 2) in the direction
of the vector i j.
iv) [5 marks] Use the method of Lagrange multipliers (or any other method
that works) to find the maximum and minimum values of the function
xy2 on the circle x2 + y2 = 1.
Please see over . . .
JUNE 2018 MATH2069 Page 3
Use a separate book clearly labelled Question 2
2. [20 marks]
i) [6 marks] Find the volume of the region bounded by the cone
z =
p
3(x2 + y2)
and the sphere
x2 + y2 + z2 = 5.
(Hint: you may use spherical coordinates)
ii) [8 marks] Let F be the vector field
F(x, y, z) = (yzexyz)i+ xzexyzj+ (xyexyz + z)k.
a) Show that F is conservative by computing its curl.
b) Find a scalar potential function with the property r = F.
c) Hence, or otherwise, calculateZ
C
F(r) · dr,
where C is the curve
r(t) = t5i t3j+ tk, 0  t  1.
iii) [6 marks] Use Gauss’ Divergence Theorem (or any other method that
works) to find the flux of the vector field
F(x, y, z) = xi+ ex+zj+ sin(x y)k,
out of the solid bounded by the paraboloid z = 4 x2 y2 and the
xy-plane.
Please see over . . .
JUNE 2018 MATH2069 Page 4
Use a separate book clearly labelled Question 3
3. [20 marks]
i) [4 marks] Let
f(x+ iy) = x5 iy5
a) Determine the set of points where f is di↵erentiable.
b) Where is f analytic? Give a reason for your answer.
c) Find f 0(x+ iy) where it exists.
ii) [6 marks] Given that the function u : R2 ! R defined by
u(x, y) = cosh x cos y sinh x sin y + x2 y2.
is harmonic (you do not need to prove this):
a) Find a harmonic conjugate v for u.
b) Let f(x + iy) = u(x, y) + iv(x, y) for all x, y 2 R, for the function v
found in the previous part. Find f(z) as a function of z alone.
iii) [3 marks] Find all values of the expression
i2i
in Cartesian form. Which of these is the principal value?
iv) [7 marks] Suppose that
g(z) =
cos(⇡2 z)
(z + 1)(z 1)2 .
Let denote the circle with centre at 0 and radius 3, traversed in the
anticlockwise direction.
a) Show that the function g has a removable singularity at z = 1.
Find and classify any other singularity of g.
b) Determine the residue of g at z = 1.
c) Hence, or otherwise, calculate the integralZ

g(z)dz.
Please see over . . .
JUNE 2018 MATH2069 Page 5
Use a separate book clearly labelled Question 4
4. [20 marks]
i) [6 marks] Evaluate the contour integralZ

z¯ + |z|2 dz,
where
a) is the straight line from 1 i to 1 + i;
b) is the upper semicircle of unit radius and center 0, traversed anti-
clockwise.
ii) [7 marks] Suppose that
f(z) =
6
(z + 1)(z 5) .
a) Write down the three (maximal) regions with centre 1 on which f(z)
has a convergent Laurent (or Taylor) expansion in powers of z 1.
b) Find the Laurent series expansion of f in powers of z 1 which is
convergent at the point z = 4.
iii) [7 marks] Use complex analysis methods to find
I1 =
Z 1
1
cos x
x2 2x+ 2 dx
and
I2 =
Z 1
1
sin x
x2 2x+ 2 dx .
End of Exam

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