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HW 12

  1. Following Thursday's class, can you verify explicitly that any closed differential 2-form in $\R^3$ is exact?
  1. (optional) read about Brouer's fixed point theorem in Pugh, and try Pugh Exercise 71, hairy ball theorem.
2022/04/16 09:23 · pzhou

HW 11

This weeks material is mostly conceptual, although the statement and proof in Pugh are all based on concrete formula.

  • Consider the generalized angular forms $\Omega_{n-1}$ defined on $\R^n \RM 0$
    • For $n=2$, we define $\Omega_1 = |x|^{-2} (x_1 dx_2 - x_2 dx_1)$
    • For $n=3$, we define $\Omega_2 = |x|^{-3} (x_1 dx_2 \wedge dx_3 - x_2 dx_1 \wedge dx_3 + x_3 dx_1 \wedge dx_2)$
    • Can you prove that $d \Omega_1=0$, $d \Omega_2=0$?
    • Can you write down the expression for the general $n$? Or just prove the general case?
    • Consider the following 2-cell in $\R^3$ (it parametrized the unit sphere), $$\gamma: [0,1]^2 \to \R^3, \quad (s,t) \mapsto (\sin (\pi s) \cos (2\pi t), \sin (\pi s) \sin (2\pi t), \cos \pi s) $$ What is $\int_\gamma \Omega_2$?
    • Suppose we use a different parametrization of $S^2$, the stereographic projection $$ \gamma: \R^2 \mapsto \R^3, \quad (a,b) \mapsto (\frac{2a}{1+a^2+b^2}, \frac{2b}{1+a^2+b^2}, \frac{-1+a^2+b^2}{1+a^2+b^2}) $$ Can you explain why $\int_{\gamma} \Omega_2$ is the same as the previous one?
  • (Optional) Let $\gamma_1, \gamma_2, [0,1] \to \R^3$ be two smooth loops, i.e. $\gamma_i(0)=\gamma_i(1)$ and $\gamma_i'(0) = \gamma_i'(1)$. Suppose they have disjoint images. Define a 2-cell $\phi: [0,1]^2 \to \R^3$ by $\phi(s,t) = \gamma_1(s) - \gamma_2(t)$. Prove that $(4\pi)^{-1} \int_\phi \Omega_2$ is an integer (hence insensitive to small perturbation of $\gamma_1, \gamma_2$). This is called the linking number of two knots, and is a topological invariant of links. Can you compute some examples? Can you see its topological meaning?
    • Here is one example (maybe a bit degenerate): imagine two loops with very large radiu, one is lying on the xy plane, one is lying on the xz plane. Say, $\gamma_1$ is given by $z=0, (x+R-1)^2 + y^2 = R^2$; and $\gamma_2$ is given by $y=0, (x-R+1)^2+z^2 = R^2$. Can you compute the linking number? Can you picture what happens to the two circles if $R \to \infty$?
    • That wasn't very easy to compute. Here is a simpler one: $\gamma_1$ has the image of a loop $z=0, x^2 + y^2 = r^2$, with $r$ very small. And $\gamma_2$ is the circle $y=0, (x-R)^2 + z^2 = R^2$, with $R$ very large.
2022/04/08 14:12 · pzhou

HW 10

Rudin Ch 9,

  • 12 (a,b,c)
  • 13
  • 19

Pugh Ch 5. Ex 14, 24

2022/04/02 12:58 · pzhou

HW 9

  1. Rudin Ex 8.6
  2. Rudin Ex 8.7 (Rudin's $D_i f = \partial f / \partial x_i$)
  3. Show that, for any closed subset $E \In \R^2$, there is a continuous function $f: \R^2 \to \R$, such that $f^{-1}(0) = E$. (bonus, can you make $f$ a smooth function?)
  4. For the implicit function theorem, take $n=m=1$, and interpret it graphically and intuitively.
2022/03/17 22:16 · pzhou

HW 8

1. Read appendix F about Littlewood's three principles, and write some comments about it in your webpage (for example, a summary of what this is about, or questions)

2. Do Pugh Ex 83

3. Let $(\R^n, | \cdot |_{1})$ be the normed vector space where $|(x_1, \cdots, x_n)|_{1}: = \sum_i |x_i| $. Let $T: \R^n \to \R^n$ be a linear operator, given by the matrix $T_{ij}$, that sends $(x_i)$ to $(y_j)$, where $y_i = \sum_j T_{ij} x_j$. How to compute $\|T \|$?

  • optional: if we use $\| - \|_{max}$ norm on $\R^n$, how to compute the operator norm $\|T\|$?

4. Read about Hölder inequality and Minkowski inequality. In the simplest setting, we have

  • (Hölder inequality), for $p,q \geq 1$ that $1/q+1/p=1$, we have

$$ (\sum_{i=1}^n |x_i y_i|) \leq (\sum_i |x_i|^p)^{1/p} (\sum_i |y_i|^q)^{1/q} $$

  • (Minkowski inequality) for any $p\geq 1$, $$(\sum_{i=1}^n |x_i + y_i|^p)^{1/p} \leq (\sum_i |x_i|^p)^{1/p} + (\sum_i |y_i|^p)^{1/p} $$

Read about the proof (in wiki, or any textbook about functional analysis, say Folland). Why it works?

2022/03/11 20:46 · pzhou
math105-s22/hw/start.txt · Last modified: 2022/01/20 11:49 by pzhou