Math 117, Spring 2009 - Analysis With an Introduction to ProofMonday, Wednesday & Friday, 12:00-12:50pm, South Hall 6635

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We will have a ''@@color(red):Midterm exam on Friday, May 8th, 2009@@''. The ''@@color(red):Final exam will be on Tuesday, June 9th, 2009, from 12-3pm@@''.

Welcome to the Math 117 Wiki!

To navigate this page, simply follow the links on your left. 

# [[Instructor]]: Who I am, and how to contact me.
# [[Teaching Assistant]]: There will be one TA for this course. 
# [[Location]]: Where the class will meet.
# [[Syllabus]]: Information about the course.
# [[Weekly Readings]]: Here I will post the readings for each week, and the lesson objectives. Please take a look at it before the class.
# [[Homework]]: I will post the homework here. 
# [[Lecture Notes]]: I will post the notes from class here.
# [[Self-Assessment Questions]]: This will help you study for the midterm and the final exam.
# [[Exams]]: Information about the midterm and final exams.

Once you open a tiddler, you can close it by clicking on its link again. Alternatively you can use the close button over on the right. 

This page is experimental, and hopefully more services will be added as the course progresses, so come back often.

Homework will be assigned on Wednesday, and will be collected at the beginning of the class on the following Wednesday. ''@@color(red): Late homework will not be accepted. If you have a predictable absence, you will need to hand it in earlier@@''. 

''Assignments:''
#[[Homework 1|hw1.pdf]]: Due on April 8th, 2009. 
#[[Homework 2|hw2.pdf]]: Due on April 15th, 2009. 
#[[Homework 3|hw3.pdf]]: Due on April 22nd, 2009. 
#[[Homework 4|hw4.pdf]]: Due on April 29th, 2009. 
#[[Homework 5|hw5.pdf]]: Due on May 6th, 2009.
#[[Homework 6|hw6.pdf]]: Due on May 20th, 2009.
#[[Homework 7|hw7.pdf]]: Due on May 27th, 2009.
#[[Homework 8|hw8.pdf]]: Due on June 3rd, 2009.

!Carlos J. ~García-Cervera.
South Hall, Room 6707.
Office Phone: (805) 893 3681
Office Fax: (805) 893 2385
Office Hours: Mondays and Wednesdays, from 10-11:30am.
Main Website: http://www.math.ucsb.edu/~cgarcia

In this course we will study mostly second order linear ordinary differential equations of the form
ay' '+by'+cy = f(x).

Since the equation is linear, it turns out that the set of solutions forms a finite dimensional vector space, which explains why we study Linear Algebra in this course.

These are the notes from class:

#[[Logic and Proof|LN/Logic and Proof.pdf]].
#[[Sets and functions|LN/sets and functions.pdf]].
#[[The Real numbers|LN/The real numbers.pdf]].
#[[Topology of the Reals|LN/Topology of the Reals.pdf]].
#[[Compact Sets|LN/Compact Sets.pdf]].
#[[Sequences|LN/Sequences.pdf]].
#[[Subsequences|LN/Subsequences.pdf]].
#[[Limits and Continuity|LN/Limits and Continuity.pdf]].
#[[Differentiation|LN/Differentiation.pdf]].

Monday, Wednesday & Friday, 9:00-9:50am, South Hall [[6635|http://math.ucsb.edu/images/6thfloor/6635.gif]].

[[Instructor]]
[[Teaching Assistant]]
[[Location]]
[[Syllabus]]
[[Weekly Readings]]
[[Homework]]
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[[Self-Assessment Questions]]
[[Exams]]

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Try to answer the following questions without looking in the book. If you do not feel comfortable with them, please review the corresponding chapters in the book, and them attempt to solve them again.

#[[Self-Assessment 1|SelfAssessment1.pdf]].

Analysis With an Introduction to Proof
Monday, Wednesday & Friday, 12:00-12:50pm, South Hall 6635

Math 117, Spring 2009

http://www.math.ucsb.edu/~cgarcia/Courses/Math104B_S09/

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''Textbook:'' //Analysis With and Introduction to Proof//, by Steven R. Lay, 4th edition.

''Course description:'' This course represents an introduction to mathematical analysis, and mathematical proofs. We will study basic set operations, and foundational aspects of the set of real numbers. We will go on to study sequences of real numbers, limits, continuity, differentiation, integration, and series. Emphasis will be made on rigorous proofs. By the end of the course, the following will be expected:
# To be able to write a mathematical proof. 
# To be able to follow the logical arguments in a mathematical proof. 
# To be able to proof the theorems seen in class, as well as other results comparable to those seen in class, which could be extensions, or simple modifications of the ones studied during the course.
# Understand the foundations of set theory and the real number system.
# To be able to prove that a sequence of real numbers or functions converges, and to be able to compute the limit.
# Working knowledge of the concept of continuity and differentiation of a function, which includes understanding epsilon/delta-type arguments.
# What integration of a function means, and how to compute the integral (Fundamental Theorem of Calculus)
# To be able to determine whether a given infinite series of real numbers or functions is convergent or not.

I will not hand out //practice exams// or anything like that. As part of the course, you should develop the necessary skills to study this subject. In order to help you study, I will periodically post [[Self-Assessment|Self-Assessment Questions]] questions. You do not need to hand them in. Those questions will help you know whether you are studying effectively or not, and whether you need to review some parts of the material covered in class.

''Prerequisites:'' Math 8.

''Assignments and Grading Policy'': Homework will be assigned on Wednesday, and will be collected at the beginning of the class on the following Wednesday. ''@@color(red): Late homework will not be accepted. If you have a predictable absence, you will need to hand it in earlier@@''. 

There will be a midterm and a final exam. Your final grade for the course will be given by the formula 

Final Grade = 20% Homework + 30% Midterm + 50% Final.

''@@color(red):You must obtain 50% or more in the Final exam@@''.

''Syllabus:'' (Available also in [[pdf|syllabus.pdf]]). During this course we will try to follow the following schedule. However, much like everything said earlier, this is subject to change.

!!!Week 1: Logic and Proof.
#Logical Connectives.
#Quantifiers.
#Techniques of Proof: I.
!!!Week 2:Logic and Proof, and Set and Functions.
#Techniques of Proof: II.
#Basic Set Operations.
#Relations.
!!!Week 3: Set and Functions, and The Real Numbers.
#Functions.
#Cardinality.
#Natural Numbers and Induction.
!!!Week 4: The Real Numbers.
#Ordered fields.
#The completeness axiom.
#Topology of the Reals.
!!!Week 5:  The Real Numbers and Sequences.
#Compact Sets.
#Convergence.
#Limit theorems.
!!!Week 6: Sequences and Limits and Continuity.
#Monotone sequences and Cauchy sequences.
#Subsequences.
#Limits of Functions. 
!!!Week 7: Limits and Continuity.
#Continuous functions.
#Properties of continuous functions.
#Uniform continuity.
!!!Week 8: Differentiation.
#The derivative.
#The Mean Value Theorem.
#L'Hospital's Rule.
!!!Week 9: Differentiation and Integration.
#Taylor's theorem.
#The Riemann integral.
#Properties of the Riemann integral.
!!!Week 10: Integration and Infinite series.
#The Fundamental Theorem of Calculus.
#Convergence of infinite series.
#Convergence tests.
!!!Week 11: Infinite series and Sequences and Series of Functions.
#Power series.
#Pointwise and uniform convergence.
!!!Week 12: Infinite series and Sequences and Series of Functions.
#Applications of uniform convergence.
#Uniform convergence of power series.

!!Martin Harrison.
South Hall, Room 6432J (on the pink side). 
Office Hours: Tuesday and Thursday 11am-12pm; "Mathlab PM" from 5-7pm on Tuesday.

# Week 1: ''Sections 1, 2, 3, and 4''.
## Objectives for Chapter 1:
### To develop a solid understanding of logic, and how known facts can be combined to prove new facts. This is essential in order to understand mathematics, and mathematical arguments.
### Learn the mathematical notation that we will be using.
# Week 2: ''Sections 5, 6, 7, and 8''.
## Objectives for Chapter 2:
### Introduction to set theory, and connection between set operations and logical connectives.
### Equivalence relations and how the induce a partition of a set. Set theory provides a good introduction to proof, and logical arguments.
### Functions: injective, surjective, and bijective functions; composition of functions, and the inverse. 
### Cardinality, in particular, the different 'infinites', countability, and the continuum hypothesis.
# Week 3: ''Sections 10, 11, 12''.
## Objectives for Chapter 3:
### Definition of the natural numbers and the principle of induction.
### Algebraic structure of the real numbers: Fields.
### Order in the real numbers.
### The completeness axiom, and the definition of supremum and infimum.
# Week 4 & 5: ''Sections 13, 14, and 16''.
## Objectives:
### Definition of neighborhoods, open and closed sets, accumulation points.
### Compactness, and the ~Heine-Borel theorem.
### Convergence of a sequence of real numbers.
# Week 6: ''Sections 17, 18, 19, and 20''.
## Objectives:
### Properties of limits of sequences.
### Special sequences: Monotonicity and Cauchy sequences.
### Subsequences.
### Limits of functions.
# Week 7: ''Sections 21, 22, and 23''.
## Objectives:
### Continuity of functions.
### Uniform continuity.
# Week 8: ''Sections 25, 26, 27, and 28''.
## Objectives:
### Definition of derivative of  function.
### Rules of differentiation.
### L'Hospital rule.
### Taylor's theorem.