Math 302

Spring 2010

Prof. Donnay

 

Class Materials and Assignments

 

Class#,

Date

Section

Class Material

Assignment

 

1

W 1/20

a.        Ch. 27

b.       Metric Spaces: Taxi Cab metric, C[0,1].

Due Friday 1/22: Prove that Taxi Cab Metric satisfies 3 properties of metric. Complete review worksheet of Analysis definitions.

2

F

1/22 

Ch. 28

Sup metric satisfies 3 properties of metric. Analysis concepts defined on a metric space.

Week 1 quiz review.

 Quiz #1 is available on shelf outside my office – hand in on class on Wed.

HW #1-2 for Wed 1/27 and Fri 1/29

3

M

1/25 

 

NO CLASS  as Prof Donnay is out of town.

 

4

W

1/27 

 

Function spaces, Cn[a,b], example of C1 but not C2 function; Map/Function between function spaces (Differentiation, Integration).

Start of Differentiation Ch. 14.

 

5

F

1/29

 

More on positivity property of metric spaces. In a metric space, if a sequence converges, the limit is unique and the sequence is bounded.

Wk 2 quiz review

Quiz 2 avaiable outside my office; due by 4pm on Monday

Hw #3-4 for Wed Feb 3 and Fri Feb 5.

6

M

2/1

 

Local extrema Theorem, Rolle’s Theorem,

Mean Value Theorem Ch 14

 

7 W

2/3

 

Implications of Mean Value Theorem, relation to linear approximation and then to Taylor Polynomials,

 

8 F

2/5

 

Differentiable -> Continuous

Intro to Uniform Continuity Ch. 11

Wk 3 quiz review

Hw #5-6 for Wed Feb 10 and Fri Feb 12. Material from Stewart on Riemann Sums: 1, 2, 3

9 M

2/8

 

Uniform Continuity (Cont fn on Compact Set)

Intro to integration via Riemann Sums Ch. 15

 

10 F

2/12

 

More on (Riemann) integration; existence of non-integrable function. Riemann Sum Worksheet.

Wk 4 quiz review

HW #6-7

11 M

2/15

 

Proof of Thm: Continuous fn on compact set is integrable.

 

12 W

2/17

 

Fundamental  Theorem of Calculus Ch. 16

 

13 F

2/19

 

Numerical Riemann Integration with Mathematica; Tayor polynomials, sequences of functions (Ch 17). Mathematica handout

Wk 5 quiz review

HW #8-9

14/ M 2/22

 

Pointwise and uniform limits of functions (Ch 17). Introductory worksheet on limits of functions.

Mathematica illustration of sequences of functions.

 

15 W 2/24

 

HW review: uniform continuity via Mean Value Theorem; existence of Riemann Integral for a function that is discontinuous

 

16 M 3/1

 

Uniform convergence of continuous functions and associated theorems (Ch. 17)

HW #10-11 .Slight change in HW. Do the revised HW assignment instead:

HW Revised #10-11.

18 W 3/3

 

Uniform convergence allows interchange of limit and integral.

 

19 F 3/5

 

Uniform convergence is the same as convergence in the sup metric of the metric space C[a,b]. Mathematica and the command Manipulate (Mathematica file) , (pdf file)to visualize a sequence of functions converging to a limit function.

 

20 M

3/15

 

 

 Projects: information and list of potential projects.

Information about test prep

Dynamical Systems via Staircase method, plots of various f(x) functions.

Choice of group projects: due Monday 3/22

21 W 3/17

 

Relation of staircase diagram to hopping diagram. Classifying dynamics of linear functions.  Varying the initial conditions.

Quiz 7 : due Friday 3/19

Wk 6-7 quiz Review

22 F 3/19

 

Review of Sequences of functions for exam.

Updated list of proofs that might be on exam.

 

23 M 3/22

 

Instructions for Midterm.

 

24 W

3/24

 

Parameter space diagram for the linear maps: m values in R. Various regions grouped by type of dynamics.

 

25 F 3/26

 

Special types: m = +1, m = - 1, m = 0. Bifurcation/tipping point – small change in parameter value can lead to major change in the dynamical. Stable systems – can change parameter and the dynamics do not change.

 

26 M 3/28

 

Topologically conjugacy – rigoruous definition of “the same”. Equivalence relation.

 

27 W 3/30

 

Invariants of conjugacy. If two systems are the top. conj., then this property must be the same for both systems. A way to determine that two systems are not the same.  

 

28 F 4/2

 

Local attracting and repelling points. Finding multiple fixed points and characterizing the type.

 

29 M

4/5

 

Periodic points, attracting/repelling periodic pts; Carrying capacity, change of variables for quadratic map

 

30 W 4/7

 

Base 3 expansions

 

31 F 4/9

 

Cantor Set and Base 3; Cantor set is uncountable.

HW #14, due Friday 4/16

32 M 4/12

 

Introduction to Series (Ch. 19): what does

1-1+1 -1 + … equal?

 

32 W 4/14

 

Geometric Series

 

33 F 4/17

 

Nth term test, Harmonic Series, Integral test

HW #15, due Friday 4/23

34 M 4/20

 

Comparision Test, Absolutely convergent, Conditionally convergent (Ch. 20)

 

35 W 4/22

 

Cauchy convergence criteria, Series of functions, Weierstrass M-Test (Ch. 21)

 

36 F 4/24

 

Presentation 1: Bifurcations and Dynamical Systems.

-           Choices for Final Work

-           Information on writing paper.

HW #16, due Friday 4/30.