CC1458 Semester 1, 2016
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CONFIDENTIAL
THE UNIVERSITY OF SYDNEY
FACULTIES OF ARTS, ECONOMICS, EDUCATION,
ENGINEERING AND SCIENCE
MATH2961: Linear Algebra and Vector Calculus (Advanced)
Lecturers: Daniel Daners and Ruibin Zhang
Time allowed: 2 Hours
This booklet contains 4 pages.
• No books, notes or additional paper.
• University approved non-programmable calculators are permitted.
• Show all necessary work.
CC1458 Semester 1, 2016 page 2 of 4
Vector Calculus
1. (a) Write down an equation for the tangent plane to the graph of the function 3 Marks
푧 = cos(푥 + 푦) − 푥푦
at the point (푥, 푦, 푧) = (1,−1, 2).
(b) Find all critical points of the function 4 Marks
푓 (푥, 푦) = 2푥3 − 6푥푦 + 3푦2
and use the second derivatives test to classify each one of them as a local maximum, a
local minimum, or a saddle point.
(c) Let 퐷 = {(푥, 푦) ∈ ℝ2 ∣ 0 ≤ 푥 ≤ 1, 푥2 ≤ 푦 ≤ 1}. Calculate the double integral 3 Marks
∬퐷
√
1 + 푦3∕2 푑푥푑푦.
2. (a) Let 퐶 be the straight line segment inℝ3 starting at (0, 0, 0) and ending at (1, 2, 1). Com- 2 Marks
pute the integral
∫퐶(푥푦 − 푦푧 + 푥푧) 푑푠.
(b) Let 퐷 = {(푥, 푦) ∈ ℝ2 ∣ 0 ≤ 푥 ≤ 1, 0 ≤ 푦 ≤ 1} and let 휕퐷 be its boundary oriented 4 Marks
anticlockwise. Using Green’s Theorem, or otherwise, compute the integral
∫휕퐷(cos(푥
2) − 푦3) 푑푥 + (푥푦2 + sin(푦2)) 푑푦.
(c) Let  be the parallelogram in the (푥, 푦)-plane spanned by the vectors (2,−3) and (3, 2). 4 Marks
Using the transformation formula for double integrals, or otherwise, compute
∬ sin(2푥 − 3푦) 푑푥 푑푦.
turn to page 3
CC1458 Semester 1, 2016 page 3 of 4
3. (a) Consider the vector field 풇 (푥, 푦, 푧) = (−푦, 푥, 푧 − 푥 + 푦) defined in ℝ3.
(i) Calculate curl풇 . 1 Mark
(ii) Let the curve 퐶 in ℝ3 be the intersection of the cylinder 푥2 + 푦2 = 1 with the 4 Marks
plane 푥 + 푦 + 푧 = 6. The curve 퐶 is oriented anticlockwise when viewed from
above. Calculate
∫퐶 풇 ⋅ 푑풙.
(b) Consider the vector field
품(푥, 푦, 푧) =
(
푥
(푥2 + 푦2 + 푧2)3∕2
,
푦
(푥2 + 푦2 + 푧2)3∕2
, 푧
(푥2 + 푦2 + 푧2)3∕2
)
.
defined on ℝ3 ⧵ {(0, 0, 0)}.
(i) Calculate div 품. 1 Mark
(ii) Let푅 be a solid region ofℝ3 with (0, 0, 0) in its interior. Let the boundary surface 4 Marks
푆 = 휕푅 of 푅 by smooth with outward pointing unit normal vector 풏. Determine
the value of the integral
∬푆 품 ⋅ 풏 푑푆.
Linear Algebra
4. (a) Prove that the family of functions (푒3푡, 푡푒3푡) on ℝ is linearly independent. 3 Marks
(b) Find the solutions of 3 Marks⎡⎢⎢⎣
1 0 1
1 1 0
0 1 1
⎤⎥⎥⎦
⎡⎢⎢⎣
푥
푦
푧
⎤⎥⎥⎦ =
⎡⎢⎢⎣
1
1
0
⎤⎥⎥⎦
as a system of equations in the two element field ℤ2.
(c) Consider the family (cos 푡, sin 푡) of functions on ℝ. Let 푉 ∶= span(cos 푡, sin 푡) be the
vector space spanned over ℝ. You may assume that (cos 푡, sin 푡) is linearly indepen-
dent.
(i) Show that 푉 = {푐 cos(푡 − 푠)∶ 푐, 푠 ∈ ℝ}. 2 Marks
(ii) Determine the coordinate vector of 2 cos(푡 − 휋∕3) ∈ 푉 with respect to the basis 2 Marks
(cos 푡, sin 푡).
turn to page 4
CC1458 Semester 1, 2016 page 4 of 4
5. (a) Consider the 4 × 5 matrix 퐴 =
⎡⎢⎢⎢⎣
1 2 7 4 4
5 3 0 0 19
3 5 16 −5 26
0 1 5 4 −1
⎤⎥⎥⎥⎦ .
The reduced row echelon form of 퐴 is given by 퐵 =
⎡⎢⎢⎢⎣
1 0 −3 0 2
0 1 5 0 3
0 0 0 1 −1
0 0 0 0 0
⎤⎥⎥⎥⎦ .
(i) Determine a basis of the image of 퐴 as a map from ℝ5 into ℝ4. Briefly justify 2 Marks
your answer.
(ii) Determine a basis of the kernel of 퐴 as a map from ℝ5 into ℝ4. 3 Marks
(b) Let the matrix 퐴 with entries in ℂ be given by
퐴 =
[
−1 2 + 푖
2 − 푖 3
]
.
(i) Show that 퐴 is a Hermitian matrix. 1 Mark
(ii) Find a diagonal matrix 퐷 and a unitary matrix 푆 such that 퐷 = 푆̄푇퐴푆. 4 Marks
6. (a) Denote by ℝ푛[푋] the vector space of polynomials with real coefficients of order 푛 orless. Consider the map 푇 ∶ ℝ3[푋]→ ℝ2[푋] given by
푇 (푎0 + 푎1푋 + 푎2푋2 + 푎3푋3) ∶= 푎1 − 푎0 + (푎2 − 푎1)푋 + (푎3 − 푎2)푋2
(i) Show that 푇 is linear. 2 Marks
(ii) Determine the matrix representing 푇 with respect to the bases (1, 푋,푋2, 푋3) in 3 Marks
ℝ3[푋] and (1, 푋,푋2) in 푅2[푋].
(b) Let 푛 ≥ 1. The functions 1, cos 푡, cos 2푡,… , cos 푛푡 form a linearly independent family of 2 Marks
functions on [−휋, 휋]. Let 푉 be the vector spanned by this family. Define a linear map
푆 ∶ 푉 → ℝ2 by 푇푓 ∶= (푓 (0), 푓 ′(휋∕2)).
Using the rank-nullity theorem, or otherwise, determine the dimension of the subspace
푈 ∶= {푓 ∈ 푉 ∶ 푓 (0) = 0 and 푓 ′(휋∕2) = 0}.
(c) Let 푉 be a vector space overℝ. Suppose that (풖1,… , 풖푛) is a linearly independent family 3 Marksin 푉 . Assume that 풗 ∈ 푉 is such that (풖1 + 풗, 풖2 + 풗,… , 풖푛 + 풗) is linearly dependent.
Show that 풗 ∈ span(풖1,… , 풖푛).
This is the end of the examination paper