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. |
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. |
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, RolleÕs 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 |
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. |
|
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11 M 2/15 |
|
Proof of Thm: Continuous fn on compact set is integrable. |
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12 W 2/17 |
|
Fundamental Theorem of Calculus Ch. 16 |
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13 F 2/19 |
|
Numerical Riemann Integration with Mathematica; Tayor polynomials, sequences of functions (Ch 17). Mathematica handout |
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14/ M 2/22 |
|
Pointwise and uniform limits of functions (Ch 17). Introductory worksheet on limits of functions. |
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15 W 2/24 |
|
HW review: uniform continuity via Mean Value Theorem; existence of Riemann Integral for a function that is discontinuous |
|
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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: |
|
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 |
|
22 F 3/19 |
|
Review of Sequences of functions for exam. Updated list of proofs that might be on exam. |
|
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23 M 3/22 |
|
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|
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 Òthe sameÓ. 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. |
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28 F 4/2 |
|
Local attracting and repelling points. Finding multiple fixed points and characterizing the type. |
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29 M 4/5 |
|
Periodic points, attracting/repelling periodic pts; Carrying capacity, change of variables for quadratic map |
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30 W 4/7 |
|
Base 3 expansions |
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31 F 4/9 |
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Cantor Set and Base 3; Cantor set is uncountable. |
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32 M 4/12 |
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Introduction to Series (Ch. 19): what does 1-1+1 -1 + É equal? |
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32 W 4/14 |
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Geometric Series |
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33 F 4/17 |
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Nth term test, Harmonic Series, Integral test |
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34 M 4/20 |
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Comparision Test, Absolutely convergent, Conditionally convergent (Ch. 20) |
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35 W 4/22 |
|
Cauchy convergence criteria, Series of functions, Weierstrass M-Test (Ch. 21) |
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36 F 4/24 |
|
Presentation 1: Bifurcations and Dynamical Systems. |
HW #16, due Friday 4/30. |
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