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.

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.

Rudin Ch 9,

• 12 (a,b,c)
• 13
• 19

Pugh Ch 5. Ex 14, 24

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.

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?