Lecture Notes

User Tools

Site Tools

You are here: Course Notes » Math 105: A second course to real analysis (2022 Spring) » Homeworks » HW 11
math105-s22:hw:hw11

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
math105-s22:hw:hw11 [2022/04/13 17:27]
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 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?      * 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?      * 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?    * (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? 
math105-s22/hw/hw11.1649896069.txt.gz · Last modified: 2022/04/13 17:27 by pzhou

Page Tools

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
CC Attribution-Share Alike 4.0 International Donate Powered by PHP Valid HTML5 Valid CSS Driven by DokuWiki