The University of Sydney
School of Mathematics and Statistics
MATH2061
Vector Calculus Quiz
29 May 2014f
Lecturers: D. Ivers, L. Poladian, A. Papanicolaou
Time Allowed: 40 Minutes
This quiz consists of 6 pages, numbered from 1 to 6.
There are 12 questions
The University of Sydney
School of Mathematics and Statistics
MATH2061
Vector Calculus Quiz
29 May 2014h
Lecturers: D. Ivers, L. Poladian, A. Papanicolaou
Time Allowed: 40 Minutes
This quiz consists of 6 pages, numbered from 1 to 6.
There are 12 questions
The University of Sydney
School of Mathematics and Statistics
Tutorial 7 (Week 8)
MATH2061: Vector Calculus
Semester 1
Revision questions (assumed knowledge)
1. Let u = 3i j k, v = i + 3j + k and w = 2i 3k. Find
(u v)w, u(v w), (u v) w, u (v w), (u v) w,
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 8 (Week 5)
MATH2061: Vector Calculus
Summer School
1. Find grad f if f (x, y) = x2 cos xy.
Solution:
grad f = f =
f
f
i+
j = (2x cos xy x2 y sin xy) i x3 sin xy j.
x
y
2.
The University of Sydney
School of Mathematics and Statistics
Tutorial 7 (Week 5)
MATH2061: Vector Calculus
Summer School
Revision questions (attempt before the tutorial)
1. Let u = 3i j k, v = i + 3j + k and w = 2i 3k. Find
(u v)w, u(v w), (u v) w, u (v
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 10 (Week 6)
MATH2061: Vector Calculus
Summer School
1. Find F in each of the following:
(a)
F = 2i + 3j + 4k
F = x2 yz i + xy 2 z j + xyz 2 k
(b)
Solution:
(a) F = 0
(b) F
The University of Sydney
School of Mathematics and Statistics
Tutorial 12 (Week 7)
MATH2061: Vector Calculus
Summer School
Preparatory question (attempt before the tutorial)
1. Compute F and F, and verify that ( F) = 0 for
(a) F = sin x i + cos y j;
(b) F
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 12 (Week 7)
MATH2061: Vector Calculus
Summer School
1. Compute F and F, and verify that ( F) = 0 for
(a) F = sin x i + cos y j;
Solution: F =
(sin x) (cos y)
+
= cos x sin
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 11 (Week 7)
MATH2061: Vector Calculus
Summer School
1. Using spherical coordinates, write equations which describe the surface of the hemisphere x2 + y 2 + z 2 = 1, z 0.
S
The University of Sydney
School of Mathematics and Statistics
Tutorial 10 (Week 6)
MATH2061: Vector Calculus
Summer School
Preparatory question (attempt before the tutorial)
1. Find F in each of the following:
(a)
F = 2i + 3j + 4k
F = x2 yz i + xy 2 z j +
The University of Sydney
School of Mathematics and Statistics
Tutorial 8 (Week 9)
MATH2061: Vector Calculus
Semester 1
Preparatory questions (attempt before the tutorial)
1. Find grad f if f (x, y) = x2 cos xy.
2. Find grad if (x, y, z) = 3x + 4y 8z.
3. F
The University of Sydney
School of Mathematics and Statistics
Tutorial 11 (Week 7)
MATH2061: Vector Calculus
Summer School
Preparatory questions (attempt before the tutorial)
1. Using spherical coordinates, write equations which describe the surface of th
AERO2703
Assignment 1
Table of Contents
Diagrams
1
Flight envelope(V-n diagram)
1
Specific Excess power(Ps) Contours1
Specific Energy (he) Contours
2
Range and Contours
2
Time to climb
3
Proof calculation
4
Ps and he Contours 4
V-n diagram 4
Stall Speed
4
AERO2703
Assignment 2
Yumeng WANG
SID:440003049
Table of Contents
Introduction about aircraf 1
Part A.
Turn performance
1
Maximum level turn load factor contours 1
Proof calculation
2
Maneuvers 2
Par B.
Weight and Balance 3
1. cg range chart 4
2. A tabula
AERO2703
Assignment 3
Yumeng WANG
SID:440003049
Table of Contents
The data given
1
Part 1.
Optimum Climb path to cruise condition: altitude 40,000ft, Mach No. = 2.0 2
Part 2.
Time to climb, fuel used in climb. 3
Proof calculation
4
Part 3.
Range at this a
Report Petrol Engine
Group members:
Anthony yomona-450311181
Lim Vincent salim-450605976
Ning zhou-450018246
Yumeng wang-440003049
Introduction
The engine used in this experiment will be the Villiers engine. The engine used are spark
ignition, four-stroke
The University of Sydney
School of Mathematics and Statistics
Tutorial 9 (Week 6)
MATH2061: Vector Calculus
Summer School
Preparatory question (attempt before the tutorial)
xy dxdy where R is the region in the xy plane bounded by y = x2 ,
1. Let I =
R
x =
The University of Sydney
School of Mathematics and Statistics
Tutorial 8 (Week 5)
MATH2061: Vector Calculus
Summer School
Preparatory questions (attempt before the tutorial)
1. Find grad f if f (x, y) = x2 cos xy.
2. Find grad if (x, y, z) = 3x + 4y 8z.
3
MATH2067 Assignment Solutions 2014
1. (a) Given all coeffs. of the d.e. are > 0, discriminant 72 4 12 > 0 overdamped.
OR (but this is partly solving): char. eqn. 2 + 7 + 12 = 0, roots: = 4, 3.
Both < 0 overdamped.
G.S.:
y = c1 e4x + c2 e3x
(b) Put y = e3x
Appendix C:
PRACTICE CLASS EXERCISES
93
PC2 Answers
(a) yc = c1 cos x + c2 sin x. Try yp = Ae2x . Subbing into d.e. gives A = 1/5.
GS: y = c1 cos x + c2 sin x + e2x /5.
(b) yc = c1 ex + c2 ex . First thought yp = Aex would fail since part of yc . So try y
8032A Semester 1
2014
MATH2067, Section A
Page 1 of 16
CONFIDENTIAL
SID:
SEAT NUMBER:
SURNAME:
Other Names:
ALL COPIES OF THIS PAPER MUST BE RETURNED TO THE EXAMINER.
PAPER NOT TO BE REMOVED FROM THE EXAMINATION ROOM.
University of Sydney, School of Mathe
Appendix C Practice Class
Exercises
Practice Class 1 2nd Order Linear Homogeneous O.D.E.s
Key points : Constant coefficient d.e. (4) on page 18:
Characteristic equation (5) on page 18:
y + ay + by = 0
2 + a + b = 0
General Solution (6) on page 19, and var
Page 1 of 2
MATH2067
Assignment
Differential Equations
2014
Due: 5pm Wednesday April 9
Please note that late assignments will not be accepted. Also, assignments will receive reduced marks if a completed Mathematics assignment cover sheet is not attached a
AMME1362: Materials 1
Unit of Study Overview
This Unit of Study requires the textbook: Materials Science and Engineering: An Introduction, 9th
Edition, Wiley 2013 by W. D. Callister, Jr.
In conjunction with the textbook, extra supplementary notes will als
440003049 Yumeng Wang
2067Assignment
1.(a)
!
+ $&'($
$
(b) Because of the x cannot equal to 0 when it is located denominator, and the diagram is
discontinued when x=0. The theorem requires that fy continuous on its defined interval.
!" =
Question (c),