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--- math105-s22:hw:hw11 [2022/04/13 11:10]
pzhou [HW 11]
+++ math105-s22:hw:hw11 [2022/04/15 16:27] (current)
pzhou [HW 11]
@@ Line 7: / Line 7: @@
     * 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) \cos (2\pi t), \cos \pi s)  What is  $\int_\gamma \Omega_2$ ?
+    * 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): 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  $R \to \infty$ ?
+    * 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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