Mathematical Analysis, Written Project Description
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Mathematical Analysis, Written Project Description
Goal: Make your own extra problem set with 7 problems from different topics in this course. Write solutions with clear explanations.
You can either 1) do problems similar to the existing HW; 2) choose more problems from our textbook; 3) choose new problems from the recommended reading list; 4) learn further topics related to our course.
Counts towards 10% of the total course grade.
|
Excellent |
Good |
Satisfactory |
Unsatisfactory |
No Submission |
|
9-10 |
7.5-8.5 |
6-7 |
4 |
0 |
Requirements:
1. Problems and solutions needs to be typed and not hand-written. Do not plagerize (0 points). Do not copy solutions directly from textbooks (4 points).
2. You may form study-group of 2-3 students, but submit problems and solutions independently.
2. Theory and explanations. Include key definitions, calculations, and explanations, that shows your understanding of the theory part of this course. Be clear and precise. For longer, technical proofs, focus on the key ideas and how key results are being used, and break them up into smaller sub-problems.
3. [Optional] Numerical implementations. Create at least 1-2 original graphics, ideally by writing codes in Matlab/Python/C++ etc, or at the very least using Desmos or Geogebra. Optionally, provide 1-2 example where a calculation is done using computer software. Briefly explain your code.
What to submit: pdf notes. We suggest learning LaTex.
A convenient brouswer Markdown+Latex editor is https://upmath.me/
[Optional:] Code for computer graphics, or at the very least, screenshots of software/websites such as Desmos, Geogebra.
Deadline: Last day of class.
Examples of Possible topics：
Basic: For students who don't want to learn new material.
1. Core material of the course: most important formulae, definitions, examples, and results in mathematical analysis and/or vector calculus, together with some simplest calculations and examples.
[Emphesis: Create a short notes that cover some essential basics of the course.]
Standard: Summarize and review main material, with a little extra topics.
2. Vector calculus and physics. For example,
. Classical mechanics: line integral vs work; gradient of potential vs force; conservative fields vs convervation of energy.
. Electromagnetism: curl, divergence, (Stokes'/divergence theorem) and Maxwell's equation.
. Divergence of gradient of temperature, vs heat equation; divergence of velocity field of a fluid, conservation of mass.
[Emphesis: 1. Review basic materials in vector calculus. 2. Give examples of how these theory are applied in physical situations.]
3. More analysis topics and problems
. See “Mathematical Analysis - hidden topics” .
Advanced: Requires more programming skills and deeper understanding of theory.
4. Visuallizing classic strange examples in analysis.
. [Short] Thomae's function: t(x) on [0; 1], continuous at every irrational number, but discontinuous at every rational number..?
. [Short] Dirichlet's function: takes the value of 0 or 1, but discontinuous everywhere..?
. [Medium-long] Cantor set (3.1) and Cantor Function (Ex 6.2.12): a continuous, increasing function on [0; 1] such that f(0) = 0; f(1) = 1; but f 0 (x) = 0 on an open set of “total length 1” ..?
. [Long] A nowhere-differentiable continuous function (chapter 5.4)..?
. [Long] A non-integrable derivative (chapter 7.6): a function f differentiable on (a ; b)
but Ra(b)f 0dx does not exist..?
[Emphesis: 1. Create computer graphics for the sequence/series of functions. 2. Explain the theory behind why these examples work, eventhough they are counter-intuitive.]
5. Weierstrass Approximation Theorem (chapter 6.7): for any continuous function f(x) on [a ; b], there exists a sequence of polynomial functions gn (x) on [a ; b] that “converges uniformly” to f(x). There are at least 3 different algorithms and proofs:
. Using Bernstein polynomials: [a ; b] = [0; 1], Bn ; k (x) = 
k(n))xk (1 ¡ x)k ; gn (x) =
Σk(n)=0 f 
Bn ; k (x):
. Using piecewise linear/quadratic functions (see textbook 6.7).
. Using convolutions: [a ; b] = [ ¡ 1; 1]; Qn (x) = cn (1 ¡ x2)n ; gn (x) = (Qn?f)(x). (See Rudin, Principles of Mathematical Analysis, Theorem 7.26.)
Choose 1 approach and a few continuous functions to plot the first few approximation polynomials, and plot how the error " depends on either p (degree of polynomial) or h (step size for piecewise functions). Understand and explain the proof of why they actually converge uniformly.
2023-12-23