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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] |
| * 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? |
| * (Optional) Let γ1,γ2,[0,1]→R3 be two smooth loops, i.e. γi(0)=γi(1) and $\gamma_i'(0) = \gamma_i'(1).Supposetheyhavedisjointimages.Definea2−cell\phi: [0,1]^2 \to \R^3by\phi(s,t) = \gamma_1(s) - \gamma_2(t).Provethat(4\pi)^{-1} \int_\phi \Omega_2isaninteger(henceinsensitivetosmallperturbationof\gamma_1, \gamma_2$). 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 γ1,γ2,[0,1]→R3 be two smooth loops, i.e. γi(0)=γi(1) and $\gamma_i'(0) = \gamma_i'(1).Supposetheyhavedisjointimages.Definea2−cell\phi: [0,1]^2 \to \R^3by\phi(s,t) = \gamma_1(s) - \gamma_2(t).Provethat(4\pi)^{-1} \int_\phi \Omega_2isaninteger(henceinsensitivetosmallperturbationof\gamma_1, \gamma_2$). 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? |
