Lecture Notes

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====== 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 $S^2$,[[https://en.wikipedia.org/wiki/Stereographic_projection |  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 [[https://en.wikipedia.org/wiki/Linking_number | 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): <del>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 $R \to \infty$?</del>
    * 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.
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