The Hong Kong University of Science and Technology
MATH 333

Fall 2012
Solutions to Calculus, Multivariable Calculus, Linear Algebra
Review Problems
1. (a) Dierentiate both sides with respect to x, we get
4x3 + y 3 + 3xy 2
dy
dx
At (1,1),
=
dy
dy
ex
= ex )
=
dx
dx
4x3 y 3
3xy 2
e 5
.
3
(b) Dierentiate x with respect to y, we
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
Math 3033(Real Analysis)
Fall 2014
Homework Set #2
(due Wednesday, October 29, 1:30pm)
Make a copy of your homework before submitting the original! Write down your name as in
your student ID card and your tutorial session number on the homework!
1. Prove
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
Math 301 (Real Analysis) Lecture 1
Fall 20052006
Instructor: Dr. Kin Y. Li
Ofce: Room 3471
Ofce Phone: 2358 7420
Ofce Hours: Tu. 3:30pm  4:30pm (or by appointments)
email address: makyli@ust.hk
Prerequisite: Multivariable Calculus, Linear Algebra and I
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note 13
Measurable Function and Lebesgue Integral
Part 1: Measurable Function and Lebesgue Integral
Recall in Lebesgue Integration, to integrate , fxdx, we divide a, b into several
disjoint measurable sets E , E
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note 12
Lebesgue Measure of Sets (From intervals to general sets): Part 2
Starting with open intervals, we can measure the length of open sets, closed sets
(bounded and unbounded). Then we measure the length of o
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note 14
MCT and LDCT: Computation in Lebesgue Integration
In previous tutorial, we have briefly discussed some elementary operations of Lebesgue
integral: addition, subtraction, inequalities, interchange with Rie
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note #7
Sequence and Series of function 1:
Pointwise Convergence and Uniform Convergence
Part I: Pointwise Convergence
Definition of pointwise convergence:
A sequence of functions f1 , f2 , , fn , : E (where E is
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note 11
Lebesgue Measure of Sets (From intervals to general sets): Part 1
Recall in last tutorial, we developed a method to measure the length of two different
kinds of sets:
1. Open Sets
(An union of open interv
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note #10
Lebesgue Integration: An introduction
1. From Riemann Integration to Lebesgue Integration
In Riemann Integration, we compute the by dividing the interval , into finite
disjoint intervals , , , , , and es
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note 8
Sequence and Series of function 2:
Application of Uniform Convergence
To check the uniform convergence of complicated sequences or series of functions,
checking by definition can be messy. Here we have 2 b
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note #9
Sequence and Series of Functions 3: Power Series
Finally, we would like to discuss a special type of series power series. A power series is a
series of the form a x c .
Uniform Convergence of Power Series
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note #5
Limit Superior and Limit Inferior
(*Note: In the following, we will consider extended real number system ,
In MATH202, we study the limit of some sequences, we also see some theorems related to
limit. (S
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note #6
Technique in limit superior and limit inferior problems
Recall in the proof of root test, we have used a technique:
For limsup a  1, say limsup a  lim M 1
(where sup a ,
a ,
a , . . )
Pick such
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note #4
Inverse Function Theorem and Implicit Function Theorem
Part I: Inverse Function Theorem
Inverse Function Theorem:
If : is C near and det F 0,
then there is a ball B , such that
1) In , F has a C (hence di
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis
Tutorial Note #3
More Differentiation in Vectorvalued function:
Last time, we learn how to check the differentiability of a given vectorvalued function.
Recall a function F: is differentiable at a iff there is a linear
transformati
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note 1
Calculus, Multivariable Calculus, Linear Algebra Review
In this tutorial, we shall review some basic facts that you have learnt in Year 1. Those
facts are important in studying vectorvalued function (Chap
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Tutorial Note 2
Vectorvalued Functions: Limit, Continuity and Differentiability
In onevariable calculus, we know t lim if and only if lim  
0 a function fx is said to be continuous at point x a if lim
and differentia
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Final Review 1
Problem Solving: Measurable Functions
Part I: Proving some functions which are measurable or not measurable
Method 1: Prove by definition directly
Example 1
Let A be a measurable set and B be a nonmeasurab
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
MATH301 Real Analysis (2008 Fall)
Final Review 2
Problem Solving in Lebesgue Integration
Part 1: Simple Computation Problems
To start the discussion, lets compute some simple Lebesgue Integral
Example 1
Compute the integral
,
, \
where is Cantor Set
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
Definition and Theorem in Lebesgue Integration
Definition:
(Lebesgue Integral on unbounded functions)
Let : , 0, ( 0),
lim
,
sup
,
,
(Lebesgue integral on unbounded intervals)
Let : 0, ( 0)
lim
sup
,
(Lebesgue integral on general set)
Let :
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
Definition and Theorems of measurable functions
Definition: (Measurable Functions)
Let A be a measurable set, we say a function f: A is measurable
f a, b is measurable (Original Definition)
f a, b is measurable
f a, b is measurable
f a, b is measurabl
The Hong Kong University of Science and Technology
MATH 333

Fall 2012
mo. 39f<L§F maaSosRm am Empmsrrzw. ME 9 MP2: is. hr; xxx mxwoMo~ w WNW2.
I u .2 . v.25
in This. .15 +02%.me koxm n32 45623.3. _ . _; .
20.? T} Mix TX 4. 7.3 wPWMNxTXHHTOnO . We
domh? b §§mpa§P :4} an «v 593an n a .39, x3 q
. . u n 4 I m .> x 3 x .05. f