McMaster University
Math 2XX3 (Term II, 2013-2014)
Course Information Sheet
Instructor: M. Wang
Office: HH 321
Extension: 23405
e-mail: [email protected]
Office Hours: M 14:00 - 15:30 ; W 13:00 14:30 or by appointment
Lectures:
MW Th 10:30 11:20 in HH/109
MATH 2XX3
Assignment 1
Due January 25, 2012
1. Let R be the region in the plane bounded by x = 0, y = 0, x + y = 1 and x + y = 2.
Calculate the integral R (x + y )1/2 dx dy by using the change of variables given by the
mapping T (u, v ) = (u uv, uv ).
2.
MATH 2XX3
Assignment 3
Due March 7, 2012
1. Consider the parametrized surface given by:
(u, v ) =
1
u(1
3
1
1
1
3 u2 + v 2 ), 3 v (1 1 v 2 + u2 ), 3 (u2 v 2 ) where 0 u 1, 0 v 1.
3
This is (a portion of) Ennepers surface (see http:/www.youtube.com/watch?
Information about Test 1
Date: February 5, 2015
Time: 15:30 16:20
Place: T28
Topics Covered:
Sections 6.1 6.4, 4.3, 4.4 in Marsden-Tromba except for the material on Fubinis
theorem on pp. 342-344
All practice problems from the above sections and problems
Information about Test 2
Date: March 12, 2015
Time: 15:30 16:20
Place: T28
Topics Covered:
Sections 7.1 7.6 in Marsden-Tromba except for
Supplement to 7.1 (p. 355-356)
All practice problems from the above sections and problems in Assignment 2, 3.
All mate
McMASTER UNIVERSITY FINAL EXAMINATION
MATH 2XX3, DAY CLASS
Dr. M. Wang
DATE: April, 2009.
DURATION OF EXAM: 3 hours
FAMILY NAME: _ FIRST NAME:
STUDENT NUMBER:
This exam paper contains 5 questions with parts on a total of 13 pages. You are
responsible for
MATH 2XX3 (Winter 2014) Practice Problems
Question 1. For parts (a) and (b) below, consider the parametrized surface
given by the formula
(u, ) = (eu + eu ) cos , (eu + eu ) sin , 2u)
r
(a) Find a largest possible range of values for and u so that the pa
MATH 2XX3 Assignment 2
Due Date: Before 12:30 pm, February 13, 2015
1. Let (t) denote the plane curve
c
(aebt cos t, aebt sin t)
where a, b are two positive constants and t 0.
a. Find the arclength function s(t) and evaluate limt+ s(t).
b. Reparametrize (
MATH 2XX3 Assignment 1
Due Date: Before 12:30 pm, January 23, 2015
x dxdy where D is the region in the rst quad1. Evaluate the integral
D
rant that lies between the circles x2 + y 2 = 4 and x2 + y 2 = 2x.
1
dxdy where R is the region in the
2. (Problem 1
MATH 2XX3 Assignment 4
Due Date: Before 12:30 pm, April 1, 2015
1. Let D be a region in the plane bounded by a simple closed curve C.
Suppose that the area of D is A. Show using Greens theorem that the
centroid (, y ) of D is given by
x
1
1
2
x dy, y =
MATH 2XX3 Assignment 3
Due Date: Before Noon, March 6, 2015
1. Find a parametrization for each of the surfaces below:
(i) the part of the surface z 2 x2 y 2 = 1 that lies below the rectangle
[1, 1] [3, 3] in the xy-plane.
(ii) the part of the plane z = x
LAST (Family) NAME:
Test # 2
Math 2XX3
FIRST NAME:
ID # :
March 20, 2017
Instructions: This exam consists of 4 questions in 6 pages. Indicate your answers clearly in the
appropriate places. Unless stated otherwise, justify your answers in order to receive
Math 2XX3
Hints for Practice Problems # 8 and 9
Practice Problems # 8
1+p2
1 (a) F (p, u) = u , so use the special integrated form of the EL equations,
1
C = pFp F = p
.
u 1 + (u0 )2
Solve for u0 , get u0 (x) = C1 u2 /u which is a separable ODE. Using tri
Math 2XX3
Practice Problems # 4
January 29, 2017
Not to be submitted, but should be prepared for tutorial Feb 37.
1. Consider the path ~c : R R2 defined by ~c(t) = (cos t , sin2 t).
(a) What curve is traced out by ~c(t) for t R?
[Careful! | sin t|, | cos
Math 2XX3
Practice Problems # 3
January 22, 2017
Not to be submitted, but should be prepared for tutorial Jan 27-31.
1. Let F : Rn R be a differentiable function, and define the level set
S = cfw_~x Rn : F (~x) = 0
Let ~c(t) a differentiable curve lying o
Math 2XX3
Practice Problems # 7
February 27, 2017
Not to be submitted, but should be prepared for tutorial Mar 37.
1. Assume f : Rn ! R is C 3 , ~a is a critical point of f , and the Hessian is strictly negative
definite, ~h D2 f (~a) ~h < 0 for all ~h 2
Math 2XX3
Practice Problems # 11
March 29, 2017
Not to be submitted, but should be prepared for the Final Exam, Thursday April 13 at
9:00am.
1. (a) Calculate the Fourier Series of f (x) = |x|,
< x < .
(b) Draw the 2-periodic function which the series con
Math 2XX3
Homework Assignment # 1
January 15, 2017
Due Thursday Jan 26, 4:00 pm. Submit your assignment in the appropriate locker slot
in the basement of Hamilton Hall.
1. For the subset U = cfw_(x1 , x2 ) R2 : 1 < x2 1:
(a) Find all interior points of U
Math 2XX3
Homework # 3
March 5, 2017
Due by 4:00pm on Wednesday March 15, in the appropriate locker slot in the basement
of Hamilton Hall.
1.
For f (x, y, z) = 3y + x2 + z 2
y 3 + x2 z 2 + z 4 , find all critical points and use the Hessian
to classify the
Math 2XX3
Hints for Practice Problems # 10 and 11
Practice Problems # 10
1. For powers and product of sin and cos, you get Fourier Series by trig identity: sin x cos x =
1
2
sin(2x), which is a Fourier Series with ak = 0 k = 0, 1, 2, . . . , bk = 0 for al
Math 2XX3
Homework # 4
March 21, 2017
Due by 4:00pm on Thursday March 30, in the appropriate locker slot in the basement of
Hamilton Hall.
1.
(a) Find the Fourier Series S(x) of f (x) = cos(x/2),
< x < .
3 < x < 3. Does S(x) represent a continuous functi
Some Facts about Functions
revised September 9, 2013
1
Basic denitions
Denition 1.1. Suppose X and Y are two sets. Then their Cartesian product X Y is dened as
X Y := cfw_(x, y)| x X, y Y .
Example 1.2. If X = R , Y = R, then X Y = cfw_(x, y), z)| (x, y)
MATH 2XX3 Assignment 3
Due Date: Before Noon, March 6, 2015
1. Find a parametrization for each of the surfaces below:
(i) the part of the surface z 2 x2 y 2 = 1 that lies below the rectangle
[1, 1] [3, 3] in the xy-plane.
(ii) the part of the plane z = x
MATH 2XX3 Assignment 1
Due Date: Before 12:30 pm, January 23, 2015
1. Evaluate the integral
x dxdy where D is the region in the rst quadD
rant that lies between the circles x2 + y 2 = 4 and x2 + y 2 = 2x.
1
2. (Problem 10, 6.2) Calculate
dxdy where R is
Test 3 / Answers
1. (a) Review the definition 8.2 in the book.
(b) When x > 0 (from the right), |f (x) f (0)| = | x cos(1/x) 0| x < provided 0 < x <
with = 2 .
(c) f is not continuous at x = 0, since lim f (x) does not exist.
x0
2. Review the definition
Test 1 / Math 2XX3 / Solutions
1. S = cfw_q Q | 0 < q 3 R.
(a) sup S = 3. S does not have a maximum element since 3 6 Q
(b) First, if x > 0, either 0 < x 3 or x > 3. If
0 < x 3, between
any two real
numbers there is a rational number q Q with 0 < q < x
Math 2XX3 / Test 2 / Short answers
1. [See Exercise 5.21 (1)]
(a) The definition of a Cauchy sequence (see 5.16 in the book):
> 0 N N so that |xk xm | < , k, m N .
(b) Since the right hand side is a telescoping sum, xn xm =
n
X
(xk xk1 ).
k=m+1
Using the
Test # 1/ Math 2XX3
NAME:
-2ID #:
1. [8 pts.] Consider the surface x4 + (x2 + y 2 )z + z 3 = 1.
(a) Determine for which points on the surface it is possible to solve for z = g(x, y) locally.
(b) Find
@z @z
,
at the point (x, y, z) = ( 1, 2, 0).
@x @y
Use