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math105-s22:hw:hw11 [2022/04/08 14:16]
pzhou |
math105-s22:hw:hw11 [2022/04/15 16:27] (current)
pzhou [HW 11] |
| * Consider the generalized angular forms Ωn−1 defined on Rn\0 | * Consider the generalized angular forms Ωn−1 defined on Rn\0 |
| * For n=2, we define Ω1=∣x∣−2(x1dx2−x2dx1) | * For n=2, we define Ω1=∣x∣−2(x1dx2−x2dx1) |
| * For n=3, we define $\Omega_2 = |x|^{-3} (x_1 dx_2 \wedge dx_3 - x_2 dx_1 \wedge dx_3 + x_1 dx_2 \wedge dx_3)$ | * 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 dΩ1=0, dΩ2=0? | * Can you prove that dΩ1=0, dΩ2=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 R3 (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 ∫γΩ2? | * Consider the following 2-cell in R3 (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 ∫γΩ2? |
| * Suppose we use a different parametrization of S2,[[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 ∫γΩ2 is the same as the previous one? | * Suppose we use a different parametrization of S2,[[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 ∫γΩ2 is the same as the previous one? |
| * Let γ1,γ2,[0,1]→R3 be two smooth loops, i.e. γi(0)=γi(1) and $\gamma_i'(0) = \gamma_i'(1).Supposetheyhavedisjointimages.Definea2−cell\phi: [0,1]^2 \to \R^3by\phi(s,t) = \gamma_1(s) - \gamma_2(t).Provethat\int_\phi \Omega_2isinsensitivetosmallperturbationof\gamma_1, \gamma_2$. This is called the Gauss linking number, and is a topological invariant of links. Can you compute some examples? Can you see its topological meaning? [[https://en.wikipedia.org/wiki/Linking_number | (wiki page)]] | * (Optional) Let γ1,γ2,[0,1]→R3 be two smooth loops, i.e. γi(0)=γi(1) and $\gamma_i'(0) = \gamma_i'(1).Supposetheyhavedisjointimages.Definea2−cell\phi: [0,1]^2 \to \R^3by\phi(s,t) = \gamma_1(s) - \gamma_2(t).Provethat(4\pi)^{-1} \int_\phi \Omega_2$ is an integer (hence insensitive to small perturbation of γ1,γ2). 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, γ1 is given by z=0,(x+R−1)2+y2=R2; and γ2 is given by y=0,(x−R+1)2+z2=R2. Can you compute the linking number? Can you picture what happens to the two circles if R→∞?</del> |
| | * That wasn't very easy to compute. Here is a simpler one: γ1 has the image of a loop z=0,x2+y2=r2, with r very small. And γ2 is the circle $y=0, (x-R)^2 + z^2 = R^2,withR$ very large. |
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