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math105-s22:hw:hw11 [2022/04/08 14:12] pzhou created |
math105-s22:hw:hw11 [2022/04/15 16:27] (current) pzhou [HW 11] |
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* Consider the generalized angular forms $\Omega_{n-1}$ defined on $\R^n \RM 0$ | * 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 $n=2$, 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_1 dx_2 \wedge | + | * 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 |
* Can you prove that $d \Omega_1=0$, | * Can you prove that $d \Omega_1=0$, | ||
* 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), | + | * Consider the following 2-cell in $\R^3$ (it parametrized the unit sphere), |
- | * Suppose we use a different parametrization of $S^2$ $$ \gamma: \R^2 \mapsto \R^3, \quad (a,b) \mapsto (\frac{2a}{1+a^2+b^2}, | + | * Suppose we use a different parametrization of $S^2$, |
- | * 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' | + | * (Optional) |
+ | * Here is one example (maybe a bit degenerate): | ||
+ | * 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. | ||