Math 247
Assignment 7
due Monday, November 17, 2014
1. Suppose that f : O Rn Rn , O open and convex, f C 1 (O), and the symmetric
(Df )(x) + (Df )T (x)
part of the Jacobian matrix,
, is positive definite for all x O.
2
Prove that f is 1-1 on O. Hint: supp
Math 247, Spring Term 2012
Homework Assignment 6 ~ Solutions
Problem 1. Let A1, . . . ,AT be nonempty bounded subsets of R such that A,- Aj = (Z)
for z' 75 j. Suppose that for every 1 S 2' g 7 we are given a function f,- E Intb(A,~,lR) (that
is, f,- : A,-
Math 247, Fall Term 2011
Homework Assignment 8 Solutions
Problem 1. Let b be a positive real number, and let D be the punctured disk of radius
17 in R2:
D = cfw_(s,t) e R2 10 < \/82+t2 g b.
By using polar coordinates, calculate the integral fD e(s2+2) d(s
Math 247, Spring Term 2012
Homework Assignment 10 Solutions
Problem 1. Let f : R > R be a 01 function (that is, f is differentiable and f : R > R
is continuous). Let F : R2 > R be the function dened by by the formula F(s,t) :2
f(st), V(s,t) E R2. Prove th
Math 247, Spring Term 2012
Homework assignment 4 Solutions
Problem 1. Consider the following equivalence which was stated in class (in Lecture
4, Proposition 4.4): given A Q R", a function f : A -) Rm and a point ('1 E A, one has that
f respects the seque
Math 247, Fall Term 2012
Homework Assignment 1 Solutions
In Problem 1 we consider the CauchySchwarz inequality that was proved in class:
(C-S) 1(f,37)lSllfll-llll-
Problem 1. Let f and g be two vectors in R, such that if, 37 # 6. Prove the following
equiv
Math 247, Spring Term 2012
Homework Assignment 10
Posted on Thursday, July 19; due on Wednesday, July 25
Problem 1. Let f : R R be a C 1 function (that is, f is differentiable and f 0 : R R
is continuous). Let F : R2 R be the function defined by by the fo
1
Math 247, Spring Term 2012
Homework Assignment 1
Posted on Wednesday, May 2; due on Wednesday, May 9
In Problem 1, let ~x and ~y be two non-zero vectors in Rn . We consider the CauchySchwarz inequality that was proved in class (Proposition 1.3):
| h~x ,
Math 247, Spring Term 2012
Homework assignment 5
Posted on Thursday, May 31; due on Friday, June 8
Problems 1-3 are about the oscillation of a function on a set. Recall that if f : A R
is a bounded function and if B is a non-empty subset of A, then we den
1
Math 247, Spring Term 2012
Homework Assignment 3
Posted on Thursday, May 17; due on Wednesday, May 23
Problem 1. (a) Let (Di )iI be a collection of open subsets of Rn (where I is some
index set, finite or infinite). Consider the union
U = iI Di := cfw_~
Math 247, Spring Term 2012
Homework Assignment 5 Solutions
Problems 1-3 are about the oscillation of a function on a set, Recall that if f : A A R
is a bounded function and if B is a nonempty subset of A, then we denote for short
sup (f) := supcfw_f(a':)
Math 247, Spring Term 2012
Homework Assignment 1 Solutions
In Problem 1, let :3 and 37 be two non-zero vectors in R". We consider the Cauchy-
Schwarz inequality that was proved in class (Proposition 1.3):
(0-3) Mir/H SllfH-llll-
Problem 1. In the conditio
Math 247
Assignment 5
Due Friday, October 31, 2014
1. Let f : Rn \ cfw_0 R be homogeneous of degree p: f (tx) = tp f (x) for all x Rn \ cfw_0
and for all t > 0, where p is a constant.
(a) For f C 1 (Rn \ cfw_0), prove that x f (x) = pf (x).
n X
n
X
2
n
(b
Math 247
Assignment 6
Due Friday, November 7, 2014
1. Let f : O Rn R, f C 2 (O), O an open convex set. Assume that D2 f (x) is
positive semi-definite x O. Such f are said to be convex functions.
(a) Prove that f (x) f (a) + f (a) (x a) x O, a O. Interpret
Math 247
Assignment 3
Due Friday, October 10, 2014
1. Let S Rn be both open and closed. Show that S = Rn or S = .
2. For a Rn , r > 0, n 2, show that Er (a) = cfw_x Rn : |xa| > r is path-connected
and hence connected.
3. Let S R be non-empty and connected
Math 247
Assignment 2
Due Friday, October 3, 2014
n
1. Suppose the sequence cfw_xk
k=1 R satisfies |xk+2 xk+1 | |xk+1 xk | for all
k 1, with 0 < < 1. Show that cfw_xk is a Cauchy sequence and hence converges.
2. Let A Rn be closed and let f : A Rm be co
Math 247
Assignment 4
Due Friday, October 24, 2014
1. Let
x5
if (x, y) 6= (0, 0)
f (x, y) =
(y x2 )2 + x8
0
if (x, y) = (0, 0).
(a) Show that Du f (0, 0) exists for every unit vector u.
(b) Show that f is not continuous at (0, 0).
2. Consider A = cfw_(x,
Math 247, Spring Term 2012
Homework Assignment 7 Solutions
Problem 1. Let A be the bounded subset of R2 dened as follows:
A :2 cfw_(s,t) 6 (0,1] X (0,1] ls S t.
Consider the function f : A % R dened by f( (s,t) := 6t2, for (3, t) E A.
(a) Prove that bd(A)
Math 247, Fall Term 2012
Homework Assignment 3 Solutions
For Problem 1, recall that a function f : R" > Rm is said to be linear when it respects
vector space operations:
f(0153+)= af(fl+f(37), VCYHB 6 R and 93,376 R"-
We denote by 51,.,'n the special basi
Math 247 , Spring Term 2012
Homework Assignment 2 Solutions
Problem 1. Let (:Ek >211 and 55 be vectors in R", and considerthe following condition
which they may satisfy:
(EV-CT) 31% E N such that \75 > 0 one has H55;c &'H < a for all k: 2 k0.
Prove that
3
Math 247, Spring Term 2012
Homework assignment 9 Solutions
Problem 1. Consider the function f : R2 + IR dened by
_ s2t/(32+t2) if (s,t)7$(0,0)
f(s,t)-cfw_ 0 if (s,t)=(0,0).
This is the same function as in Problem 4 of assignment 8. We accept the following
1
Math 247, Spring Term 2012
Homework Assignment 2
Posted on Thursday, May 10; due on Wednesday, May 16
Problem 1. Let ( ~xk )
a be vectors in Rn , and consider the following condition
k=1 and ~
which they may satisfy:
ko N such that > 0 one has | ~xk ~a
Math 247, Fall Term 2011
Homework assignment 8
Posted on Thursday, July 5; due on Wednesday, July 11
Problem 1. Let b be a positive real number, and let D be the punctured disk of radius
b in R2 :
p
D = cfw_(s, t) R2 | 0 < s2 + t2 b.
R
2
2
By using polar
Math 247, Spring Term 2012
Information on the Final Exam
The final exam of Math 247 is scheduled on
Wednesday, August 8, from 9 to 11:30 am.
The exam will be written in the Area 6 of the PAC.
Types of questions. The questions on the final exam will be of
Assumptions 1 Steady operating conditions exist. 2 The thermal properties of the wire are constant.
Properties The properties of aluminum are given to be
= 2702 kg/m3 and cp = 0.896 kJ/kg. C.
Analysis The mass flow rate of the extruded wire through the ai
Assumptions 1 The kinetic and potential energy changes are negligible. 2 The cylinder is well-insulated
and thus heat transfer is negligible. 3 The thermal energy stored in the cylinder itself is negligible. 4 The
process is stated to be reversible.
Analy
Assumptions
1 The thermal properties of the balls are constant. 2 There are no changes in kinetic and potential
energies. 3 The balls are at a uniform temperature at the end of the process
Properties The density and specific heat of the balls are given to
Assumptions The heat pump operates steadily.
Analysis Since the heat pump is completely reversible, the combination of the coefficient of performance
expression, first Law, and thermodynamic temperature scale gives
1
COPHP,rev
1
1 (283 K ) /( 294 K )
1 TL
Assumptions The two air conditioners are comparable in all aspects other than the initial cost and the
efficiency.
Analysis The unit that will cost less during its lifetime is a better buy. The total cost of a system during
its lifetime (the initial, oper