{"id":869,"date":"2021-07-28T08:22:17","date_gmt":"2021-07-28T08:22:17","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=869"},"modified":"2021-07-29T04:12:50","modified_gmt":"2021-07-29T04:12:50","slug":"oushiduliangzaiqiuzuobiaoxiadebiaodashi","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=869","title":{"rendered":"\u6b27\u6c0f\u5ea6\u91cf\u3001\u6807\u51c6\u7403\u5ea6\u91cf\u5728\u6781\u5750\u6807\u4e0b\u7684\u8868\u8fbe\u5f0f"},"content":{"rendered":"

1. \u6b27\u6c0f\u5ea6\u91cf\u5728\u6781\u5750\u6807\u4e0b\u7684\u8868\u8fbe\u5f0f<\/a><\/span>\n\n\u5047\u8bbe
\n \\[
\n \\Psi: \\mathbb{R}^n\\to \\mathbb{R}^2,\\quad
\n x=(x^1,\\ldots,x^n)\\mapsto (r,\\theta=(\\phi_1,\\ldots,\\phi_{n-1})
\n \\]
\n \u662f\u6b27\u6c0f\u7a7a\u95f4\u4e2d\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u5230\u7403\u5750\u6807\u7cfb\u7684\u53d8\u6362, \u5219\u6b27\u6c0f\u5ea6\u91cf
\n \\[
\n ds^2=\\sum_{i=1}^n(dx^i)^2,
\n \\]
\n \u5728\u6781\u5750\u6807\u7cfb
\n \\[
\n d\\tilde{s}^2=(\\Psi^{-1})^*ds^2
\n \\]
\n \u4e0b\u7684\u8868\u793a\u4e3a
\n \\[
\n d\\tilde{s}^2=dr^2+r^2d\\sigma^2,
\n \\]
\n \u5176\u4e2d$d\\sigma^2$\u4e3a$\\mathbb{R}^n$(\u6807\u51c6\u6b27\u6c0f\u7a7a\u95f4)\u4e2d\u7684\u5355\u4f4d\u7403\u7684\u6807\u51c6\u5ea6\u91cf.
\n <\/p>\n

\u6ce8\u610f\u5230
\n \\[
\n \\Psi^{-1}:(r,\\theta)\\mapsto x,
\n \\]
\n \u53ef\u8868\u793a\u4e3a
\n \\[
\n \\begin{cases}
\n x_n=r\\cos\\phi_{n-1},\\\\
\n x_{n-1}=r\\sin\\phi_{n-1}\\cos\\phi_{n-2},\\\\
\n x_{n-2}=r\\sin\\phi_{n-1}\\sin\\phi_{n-2}\\cos\\phi_{n-3},\\\\
\n \\cdots\\\\
\n x_{2}=r\\sin\\phi_{n-1}\\sin\\phi_{n-2}\\cdots\\sin\\phi_{2}\\cos\\phi_{1},\\\\
\n x_{1}=r\\sin\\phi_{n-1}\\sin\\phi_{n-2}\\cdots\\sin\\phi_{2}\\sin\\phi_{1},
\n \\end{cases}
\n \\]
\n \u4ece\u800c, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230$d\\sigma^2$\u7684\u5177\u4f53\u8868\u8fbe\u5f0f. \u4f8b\u5982, \u5bf9$n=2,3,4$\u6211\u4eec\u5206\u522b\u6709
\n \\begin{align*}
\n d\\sigma_2^2&=d\\phi_1^2,\\\\
\n d\\sigma_3^2&=\\sin^2\\phi_2d\\phi_1^2+d\\phi_2^2,\\\\
\n d\\sigma_4^2&=\\sin^2\\phi_3\\left( \\sin^2\\phi_2d\\phi_1^2+d\\phi_2^2 \\right)+d\\phi_3^2.
\n \\end{align*}
\n \u7531\u6b64, \u4e0d\u96be\u8bc1\u660e,
\n \\[
\n d\\sigma_{k+1}^2=\\sin_{k}^2d\\sigma_k^2+d\\phi_k^2,\\quad k=1,2,\\ldots,n-1.
\n \\]
\n2. \u6807\u51c6\u7403\u5ea6\u91cf\u5728\u6781\u5750\u6807\u4e0b\u7684\u8868\u8fbe\u5f0f<\/a><\/span>\n\n\u8003\u5bdf\u534a\u5f84\u4e3a$R$\u7684\u7403\u9762\u4e0a\u7684\u6781\u5750\u6807
\n\\[
\n \\Psi:S_R^{n-1}\\mapsto \\mathbb{R}^{n},\\quad
\n (r,\\phi_1,\\phi_2,\\ldots,\\phi_{n-2})\\mapsto (x_1,x_2,\\ldots,x_n),
\n\\]
\n\u5176\u4e2d
\n\\[
\n \\begin{cases}
\n x_n=R\\cos(r\/R),\\\\
\n x_{n-1}=R\\sin(r\/R)\\cos\\phi_{n-2},\\\\
\n x_{n-2}=R\\sin(r\/R)\\sin\\phi_{n-2}\\cos\\phi_{n-3},\\\\
\n \\cdots\\\\
\n x_{2}=R\\sin(r\/R)\\sin\\phi_{n-2}\\cdots\\sin\\phi_{2}\\cos\\phi_{1},\\\\
\n x_{1}=R\\sin(r\/R)\\sin\\phi_{n-2}\\cdots\\sin\\phi_{2}\\sin\\phi_{1}.
\n \\end{cases}
\n\\]
\n\u5bb9\u6613\u6c42\u5f97, \u5bf9$n=2,3,4$, \u6211\u4eec\u6709
\n\\begin{align*}
\n d\\bar\\sigma_1^2&=dr^2,\\\\
\n d\\bar\\sigma_2^2&=dr^2+R^2\\sin^2(r\/R)d\\phi_1^2,\\\\
\n d\\bar\\sigma_3^2&=dr^2+R^2\\sin^2(r\/R)\\left( \\sin^2\\phi_2d\\phi_1^2+d\\phi_2^2 \\right)
\n\\end{align*}
\n\u7531\u6b64, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u9012\u63a8\u5173\u7cfb
\n\\[
\n d\\bar\\sigma_k^2=dr^2+R^2\\sin^2(r\/R)d\\sigma_{k-1}^2, k=2,\\ldots,n-1.
\n\\]<\/p>\n

\u6700\u540e, \u6211\u7ed9\u51fa\u4f7f\u7528MMA, \u51e0\u79cd\u60c5\u51b5\u4e0b\u8ba1\u7b97$d\\sigma^2$\u7684\u7a0b\u5e8f
\n