{"id":5225,"date":"2019-08-30T07:46:02","date_gmt":"2019-08-29T22:46:02","guid":{"rendered":"https:\/\/pandanote.info\/?p=5225"},"modified":"2022-08-07T12:41:33","modified_gmt":"2022-08-07T03:41:33","slug":"%e9%ab%98%e6%ac%a1%e5%85%83%e3%81%aejacobian2-%e3%83%a4%e3%82%b3%e3%83%93%e8%a1%8c%e5%88%97%e5%bc%8f%e3%82%92%e8%a8%88%e7%ae%97%e3%81%99%e3%82%8b%e3%80%82","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=5225","title":{"rendered":"\u9ad8\u6b21\u5143\u306eJacobian(2): \u30e4\u30b3\u30d3\u884c\u5217\u5f0f\u3092\u8a08\u7b97\u3059\u308b\u3002"},"content":{"rendered":"<h2>\u306f\u3058\u3081\u306b<\/h2>\n<p><a href=\"https:\/\/pandanote.info\/?p=5190\">\u524d\u306e\u8a18\u4e8b<\/a>\u3067\u3001$n$\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u304b\u3089\u7403\u5ea7\u6a19\u7cfb\u3078\u306e\u5909\u6570\u5909\u63db\u3092\u884c\u3046\u305f\u3081\u306e\u30e4\u30b3\u30d3\u884c\u5217\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u305f\u3002<\/p>\n<p>\u3053\u306e\u8a18\u4e8b\u3067\u306f\u30e4\u30b3\u30d3\u884c\u5217\u304b\u3089\u30e4\u30b3\u30d3\u884c\u5217\u5f0f\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<h2>\u8a08\u7b97\u306e\u65b9\u91dd\u3092\u8003\u3048\u308b\u3002<\/h2>\n<p><a href=\"https:\/\/pandanote.info\/?p=5190\">\u524d\u306e\u8a18\u4e8b<\/a>\u3088\u308a\u3001$n$\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u304b\u3089\u7403\u5ea7\u6a19\u7cfb\u3078\u306e\u5909\u6570\u5909\u63db\u3092\u884c\u3046\u305f\u3081\u306e\u30e4\u30b3\u30d3\u884c\u5217\u306f(\\ref{eq:jacobian_second})\u5f0f\u3067\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\n\\dfrac{\\partial(x_1, x_2, \\cdots, x_n)}{\\partial(r, \\phi_1, \\cdots, \\phi_{n-1})} &#038;= \\left(<br \/>\n\\begin{array}{ccc|c}<br \/>\n\\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i &#038; r \\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-1}\\sin\\phi_i &#038; \\cdots &#038; r \\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i \\cr<br \/>\n\\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i &#038; r \\cos\\phi_{n-1}\\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-2}\\sin\\phi_i &#038; \\cdots &#038; -r \\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i \\cr \\hline<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; {\\bf 0} \\cr<br \/>\n\\hline<br \/>\n\\cos\\phi_1 &#038; -r \\sin\\phi_1 &#038; \\cdots &#038; 0<br \/>\n\\end{array}<br \/>\n\\right) \\label{eq:jacobian_second}<br \/>\n\\end{align}<\/p>\n<p>\u307e\u305f\u3001$\\dfrac{\\partial(x_1, x_2, \\cdots, x_n)}{\\partial(r, \\phi_1, \\cdots, \\phi_{n-1})} = A_n$\u3068\u304a\u304d\u307e\u3059\u3002<\/p>\n<p>(\\ref{eq:jacobian_second})\u5f0f\u306e\u53f3\u8fba\u306e\u884c\u5217\u3092\u3088\u30fc\u304f\u89b3\u5bdf\u3059\u308b\u3068\u3001$\\phi_{n-1}$\u3092\u542b\u3080\u9805\u3092\u6301\u3064\u6210\u5206\u306f\u7b2c1\u884c\u53ca\u3073\u7b2c2\u884c\u306b\u306e\u307f\u73fe\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u307e\u305f\u3001\u7b2c$n$\u5217\u306b\u3064\u3044\u3066\u306f\u7b2c1\u884c\u53ca\u3073\u7b2c2\u884c\u4ee5\u5916\u306e\u6210\u5206\u306f0\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u3088\u3063\u3066\u3001(\\ref{eq:jacobian_second})\u5f0f\u306e\u53f3\u8fba\u306e\u884c\u5217\u5f0f\u3001\u3059\u306a\u308f\u3061\u30e4\u30b3\u30d3\u884c\u5217\u5f0f\u3092\u8a08\u7b97\u3059\u308b\u969b\u306b\u7b2c$n$\u5217\u306b\u6cbf\u3063\u3066\u4f59\u56e0\u5b50\u5c55\u958b\u3059\u308b\u3068\u8a08\u7b97\u304c\u5bb9\u6613\u306b\u306a\u308b\u306e\u3067\u306f\u306a\u3044\u304b\u3068\u4e88\u60f3\u3067\u304d\u307e\u3059\u3002<\/p>\n<h2>\u4f59\u56e0\u5b50\u5c55\u958b\u3057\u3066\u307f\u308b\u3002<\/h2>\n<p>\u2026\u3068\u3044\u3046\u308f\u3051\u3067\u4f59\u56e0\u5b50\u5c55\u958b\u3092\u884c\u3044\u307e\u3059\u304c\u3001\u305d\u306e\u524d\u306b\u3053\u306e\u5f8c\u306e\u8b70\u8ad6\u306e\u5c55\u958b\u306e\u90fd\u5408\u4e0a\u3001(\\ref{eq:jacobian_second})\u5f0f\u306e\u7b2c3\u884c\u53ca\u3073\u7b2c$n-1$\u5217\u306e\u6210\u5206\u306e\u5177\u4f53\u7684\u306a\u5024\u306b\u3064\u3044\u3066\u3082\u78ba\u8a8d\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n<p>\u884c\u5217$A_n$\u306e\u7b2c3\u884c\u53ca\u3073\u7b2c$n-1$\u5217\u306e\u6210\u5206\u306e\u5177\u4f53\u7684\u306a\u5024\u3092(\\ref{eq:jacobian_second})\u5f0f\u306b\u8ffd\u52a0\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e(\\ref{eq:jacobian_third})\u5f0f\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\begin{align}<br \/>\nA_n &#038;= \\left(<br \/>\n\\begin{array}{ccc|c|c}<br \/>\n\\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i &#038; r \\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-1}\\sin\\phi_i &#038; \\cdots &#038; r\\cos\\phi_{n-2}\\displaystyle\\prod_{i=1, i \\ne n-2}^{n-1}\\sin\\phi_i &#038; r \\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i \\cr<br \/>\n\\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i &#038; r \\cos\\phi_{n-1}\\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-2}\\sin\\phi_i &#038; \\cdots &#038; r \\cos\\phi_{n-1}\\cos\\phi_{n-2}\\displaystyle\\prod_{i=1}^{n-3}\\sin\\phi_i &#038; -r \\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i \\cr<br \/>\n\\cos\\phi_{n-2}\\displaystyle\\prod_{i=1}^{n-3}\\sin\\phi_i &#038; r \\cos\\phi_{n-2}\\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-3}\\sin\\phi_i &#038; \\cdots &#038; -r \\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i &#038; 0 \\cr<br \/>\n\\hline<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; {\\bf 0} &#038; {\\bf 0} \\cr<br \/>\n\\hline<br \/>\n\\cos\\phi_1 &#038; -r \\sin\\phi_1 &#038; \\cdots &#038; 0 &#038; 0<br \/>\n\\end{array}<br \/>\n\\right) \\label{eq:jacobian_third}<br \/>\n\\end{align}<\/p>\n<p>\u7b2c$n-1$\u5217\u306e\u6210\u5206\u306e\u5177\u4f53\u7684\u306a\u5024\u304c\u78ba\u8a8d\u3067\u304d\u305f\u3068\u3053\u308d\u3067\u3001(\\ref{eq:jacobian_third})\u5f0f\u306e\u53f3\u8fba\u306e\u884c\u5217\u5f0f$|A_n|$\u3092\u7b2c$n$\u5217\u306b\u305d\u3063\u3066\u4f59\u56e0\u5b50\u5c55\u958b\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u3001$(1,n)$\u6210\u5206\u53ca\u3073$(2,n)$\u6210\u5206\u4ee5\u5916\u7b2c$n$\u5217\u306e\u6210\u5206\u306f\u3059\u3079\u30660\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001$|A_n|$\u306f$(1,n)$\u6210\u5206\u53ca\u3073$(2,n)$\u6210\u5206\u306b\u3064\u3044\u3066\u306e\u5c0f\u884c\u5217\u5f0f$|\\Delta_{1,n}|$\u53ca\u3073$|\\Delta_{2,n}|$\u3092\u7528\u3044\u3066\u4ee5\u4e0b\u306e\u5f0f\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\n|A_n| &#038;= (-1)^{n+1}|\\Delta_{1,n}|\\cdot r \\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i+(-1)^{n+2}|\\Delta_{2,n}|\\left(-r \\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i\\right) \\nonumber\\cr<br \/>\n&#038;= (-1)^{n+1}|\\Delta_{1,n}|\\cdot r \\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i+(-1)^{n+3}|\\Delta_{2,n}|\\left(r \\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i\\right) \\nonumber\\cr<br \/>\n&#038;= (-1)^{n+1}r\\left(|\\Delta_{1,n}|\\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i+|\\Delta_{2,n}|\\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i\\right)\\label{eq:cofactor}<br \/>\n\\end{align}<br \/>\n(\\ref{eq:cofactor})\u5f0f\u306e\u53f3\u8fba\u306e\u7b2c1\u9805\u53ca\u3073\u7b2c2\u9805\u306e\u7dcf\u4e57\u306e\u8a08\u7b97\u6642\u306e$i$\u306e\u5024\u306e\u7bc4\u56f2\u304c\u5fae\u5999\u306b\u7570\u306a\u308b\u3053\u3068\u306b\u306f\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059\u3002<\/p>\n<p>\u5f8c\u306e\u5f0f\u5909\u5f62\u3092\u8003\u3048\u308b\u3068\u53f3\u8fba\u7b2c2\u9805\u306e$\\sin\\phi_{n-1}$\u306f\u7dcf\u4e57\u306e\u5916\u306e\u65b9\u304c\u826f\u3044\u3088\u3046\u306a\u6c17\u3082\u3057\u307e\u3059\u304c\u3001\u7121\u7406\u3084\u308a\u307e\u3068\u3081\u3066\u3042\u308a\u307e\u3059\u3002\ud83d\udc3c<\/p>\n<p>\u305d\u3053\u3067\u3001$(1,n)$\u6210\u5206\u53ca\u3073$(2,n)$\u6210\u5206\u306b\u3064\u3044\u3066\u306e\u5c0f\u884c\u5217\u5f0f\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n<h3>$(1,n)$\u6210\u5206\u306b\u3064\u3044\u3066\u306e\u5c0f\u884c\u5217\u5f0f<\/h3>\n<p>(\\ref{eq:jacobian_third})\u5f0f\u3088\u308a$(1,n)$\u6210\u5206\u306b\u3064\u3044\u3066\u306e\u90e8\u5206\u884c\u5217$\\Delta_{1,n}$\u306f\u3001(\\ref{eq:cofactor_one_n})\u5f0f\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\\begin{align}<br \/>\n\\Delta_{1,n} &#038;= \\left(<br \/>\n\\begin{array}{ccc|c}<br \/>\n\\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i &#038; r \\cos\\phi_{n-1}\\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-2}\\sin\\phi_i &#038; \\cdots &#038; r \\cos\\phi_{n-1}\\cos\\phi_{n-2}\\displaystyle\\prod_{i=1}^{n-3}\\sin\\phi_i \\cr<br \/>\n\\cos\\phi_{n-2}\\displaystyle\\prod_{i=1}^{n-3}\\sin\\phi_i &#038; r \\cos\\phi_{n-2}\\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-3}\\sin\\phi_i &#038; \\cdots &#038; -r \\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i \\cr<br \/>\n\\hline<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; {\\bf 0} \\cr<br \/>\n\\hline<br \/>\n\\cos\\phi_1 &#038; -r \\sin\\phi_1 &#038; \\cdots &#038; 0<br \/>\n\\end{array}<br \/>\n\\right) \\label{eq:cofactor_one_n}<br \/>\n\\end{align}<\/p>\n<p>\u3053\u3053\u3067\u3001$\\phi_{n-1}$\u3092\u542b\u3080\u5f0f\u306f\u7b2c1\u884c\u306e\u6210\u5206\u306e\u307f\u306b\u3042\u308a\u3001\u304b\u3064\u3001\u7b2c1\u884c\u306e\u6210\u5206\u306f\u3059\u3079\u3066$\\cos\\phi_{n-1}$\u304c\u56e0\u6570\u306b\u542b\u307e\u308c\u3066\u3044\u308b\u3053\u3068\u306b\u7740\u76ee\u3057\u307e\u3059\u3002<\/p>\n<p>\u305d\u3053\u3067\u3001$\\Delta_{1,n}$\u306e\u884c\u5217\u5f0f\u3092\u6c42\u3081\u308b\u969b\u306b\u4ee5\u4e0b\u306e(\\ref{eq:cofactor_one_n_determinant})\u5f0f\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\\begin{align}<br \/>\n|\\Delta_{1,n}| &#038;= \\left|<br \/>\n\\begin{array}{ccc|c}<br \/>\n\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i &#038; r \\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-2}\\sin\\phi_i &#038; \\cdots &#038; r \\cos\\phi_{n-2}\\displaystyle\\prod_{i=1}^{n-3}\\sin\\phi_i \\cr<br \/>\n\\cos\\phi_{n-2}\\displaystyle\\prod_{i=1}^{n-3}\\sin\\phi_i &#038; r \\cos\\phi_{n-2}\\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-3}\\sin\\phi_i &#038; \\cdots &#038; -r \\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i \\cr<br \/>\n\\hline<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; {\\bf 0} \\cr<br \/>\n\\hline<br \/>\n\\cos\\phi_1 &#038; -r \\sin\\phi_1 &#038; \\cdots &#038; 0<br \/>\n\\end{array}<br \/>\n\\right|\\cos\\phi_{n-1}\\label{eq:cofactor_one_n_determinant}<br \/>\n\\end{align}<\/p>\n<p>(\\ref{eq:cofactor_one_n_determinant})\u5f0f\u306e\u53f3\u8fba\u306e\u884c\u5217\u5f0f\u306e\u8a08\u7b97\u306e\u5bfe\u8c61\u3068\u306a\u3063\u3066\u3044\u308b\u884c\u5217\u306e\u6210\u5206\u304b\u3089$\\phi_{n-1}$\u3092\u884c\u5217\u5f0f\u306e\u5916\u306b\u62ec\u308a\u51fa\u3057\u305f\u3068\u3053\u308d\u3067\u3001<a href=\"https:\/\/pandanote.info\/?p=5190\">\u524d\u306e\u8a18\u4e8b<\/a>\u306e(6),(7),(8)\u5f0f\u306e$n$\u3092$n-1$\u3068\u7f6e\u304d\u63db\u3048\u3066\u307f\u305f\u308a\u3057\u306a\u304c\u3089\u6539\u3081\u3066\u884c\u5217\u3092\u3088\u30fc\u304f\u898b\u308b\u3068\u3001$A_{n-1}$\u306b\u7b49\u3057\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p>\u305d\u3053\u3067\u3001(\\ref{eq:cofactor_one_n_determinant})\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\n|\\Delta_{1,n}| &#038;= |A_{n-1}|\\cos\\phi_{n-1} \\label{eq:cofactor_one_n_determinant_second}<br \/>\n\\end{align}<\/p>\n<h3>$(2,n)$\u6210\u5206\u306b\u3064\u3044\u3066\u306e\u5c0f\u884c\u5217\u5f0f<\/h3>\n<p>\u6b21\u306b\u3001$(2,n)$\u6210\u5206\u306b\u3064\u3044\u3066\u306e\u5c0f\u884c\u5217\u5f0f$|\\Delta_{2,n}|$\u306b\u3064\u3044\u3066\u3082\u524d\u9805\u3068\u540c\u69d8\u306e\u5909\u5f62\u304c\u3067\u304d\u306a\u3044\u304b\u8003\u3048\u3066\u307f\u307e\u3059\u3002<\/p>\n<p>\u307e\u305a\u3001$\\Delta_{2,n}$\u306f\u4ee5\u4e0b\u306e(\\ref{eq:cofactor_two_n})\u5f0f\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\\begin{align}<br \/>\n\\Delta_{2,n} &#038;= \\left(<br \/>\n\\begin{array}{ccc|c}<br \/>\n\\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i &#038; r \\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-1}\\sin\\phi_i &#038; \\cdots &#038; r\\cos\\phi_{n-2}\\displaystyle\\prod_{i=1, i \\ne n-2}^{n-1}\\sin\\phi_i \\cr<br \/>\n\\cos\\phi_{n-2}\\displaystyle\\prod_{i=1}^{n-3}\\sin\\phi_i &#038; r \\cos\\phi_{n-2}\\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-3}\\sin\\phi_i &#038; \\cdots &#038; -r \\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i \\cr<br \/>\n\\hline<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; {\\bf 0} \\cr<br \/>\n\\hline<br \/>\n\\cos\\phi_1 &#038; -r \\sin\\phi_1 &#038; \\cdots &#038; 0<br \/>\n\\end{array}<br \/>\n\\right) \\label{eq:cofactor_two_n}<br \/>\n\\end{align}<\/p>\n<p>\u3053\u3053\u3067\u3001$\\phi_{n-1}$\u3092\u542b\u3080\u5f0f\u306f(\\ref{eq:cofactor_two_n})\u5f0f\u53f3\u8fba\u306e\u884c\u5217\u306e\u7b2c1\u884c\u306e\u6210\u5206\u306e\u307f\u306b\u3042\u308a\u3001\u304b\u3064\u3001\u7b2c1\u884c\u306e\u6210\u5206\u306f\u3059\u3079\u3066$\\sin\\phi_{n-1}$\u304c\u56e0\u6570\u306b\u542b\u307e\u308c\u3066\u3044\u308b\u3053\u3068\u306b\u7740\u76ee\u3057\u307e\u3059\u3002<\/p>\n<p>\u305d\u3053\u3067\u3001$|\\Delta_{2,n}|$\u3092\u6c42\u3081\u308b\u969b\u306b\u4ee5\u4e0b\u306e(\\ref{eq:cofactor_two_n_determinant})\u5f0f\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\\begin{align}<br \/>\n|\\Delta_{2,n}| &#038;= \\left|<br \/>\n\\begin{array}{ccc|c}<br \/>\n\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i &#038; r \\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-2}\\sin\\phi_i &#038; \\cdots &#038; r\\cos\\phi_{n-2}\\displaystyle\\prod_{i=1}^{n-3}\\sin\\phi_i \\cr<br \/>\n\\cos\\phi_{n-2}\\displaystyle\\prod_{i=1}^{n-3}\\sin\\phi_i &#038; r \\cos\\phi_{n-2}\\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-3}\\sin\\phi_i &#038; \\cdots &#038; -r \\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i \\cr<br \/>\n\\hline<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; {\\bf 0} \\cr<br \/>\n\\hline<br \/>\n\\cos\\phi_1 &#038; -r \\sin\\phi_1 &#038; \\cdots &#038; 0<br \/>\n\\end{array}<br \/>\n\\right|\\sin\\phi_{n-1} \\label{eq:cofactor_two_n_determinant}<br \/>\n\\end{align}<\/p>\n<p>\u3059\u308b\u3068\u3001(\\ref{eq:cofactor_two_n_determinant})\u5f0f\u306e\u53f3\u8fba\u306e\u884c\u5217\u5f0f\u306e\u5185\u90e8\u306e\u884c\u5217\u306f$A_{n-1}$\u306b\u7b49\u3057\u304f\u306a\u308b\u306e\u3067\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\\begin{align}<br \/>\n|\\Delta_{2,n}| &#038;= |A_{n-1}|\\sin\\phi_{n-1} \\label{eq:cofactor_two_n_determinant_second}<br \/>\n\\end{align}<\/p>\n<h2>\u6f38\u5316\u5f0f\u306e\u5c0e\u51fa\u3002<\/h2>\n<p>(\\ref{eq:cofactor_one_n_determinant_second})\u5f0f\u53ca\u3073(\\ref{eq:cofactor_two_n_determinant_second})\u5f0f\u3092(\\ref{eq:cofactor})\u5f0f\u3078\u4ee3\u5165\u3059\u308b\u3068\u2026<br \/>\n\\begin{align}<br \/>\n|A_n| &#038;= (-1)^{n+1}r\\left(|\\Delta_{1,n}|\\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i+|\\Delta_{2,n}|\\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i\\right) \\nonumber\\cr<br \/>\n&#038;= (-1)^{n+1}r\\left(|A_{n-1}|\\cos\\phi_{n-1}\\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i+|A_{n-1}|\\sin\\phi_{n-1}\\sin\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i\\right) \\nonumber\\cr<br \/>\n&#038;= (-1)^{n-1} (-1)^2 r|A_{n-1}|\\left(\\cos^2\\phi_{n-1}+\\sin^2\\phi_{n-1}\\right)\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i \\nonumber\\cr<br \/>\n&#038;= (-1)^{n-1}r|A_{n-1}|\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i\\label{eq:recurrence}<br \/>\n\\end{align}<br \/>\n\u3068\u5909\u5f62\u3067\u304d\u3001$|A_n|$\u306b\u3064\u3044\u3066\u306e\u6f38\u5316\u5f0f\u304c\u5c0e\u51fa\u3067\u304d\u307e\u3059\u3002<\/p>\n<h2>\u30e4\u30b3\u30d3\u884c\u5217\u5f0f\u306e\u5c0e\u51fa\u3002<\/h2>\n<p>$n=2$\u306e\u5834\u5408\u306b\u306f\u3001<br \/>\n\\begin{align}<br \/>\n\\begin{pmatrix}<br \/>\nx_1\\cr<br \/>\nx_2<br \/>\n\\end{pmatrix} &#038;= \\begin{pmatrix}<br \/>\nr\\sin\\phi_1\\cr<br \/>\nr\\cos\\phi_2<br \/>\n\\end{pmatrix}\\label{eq:twodimension}<br \/>\n\\end{align}<br \/>\n\u3068\u304a\u3044\u3066\u3044\u308b\u3053\u3068\u304b\u3089\u3001$|A_2| = -r$\u306b\u306a\u308b\u3053\u3068\u306b\u7559\u610f\u3057\u3064\u3064\u524d\u7bc0\u306e\u6f38\u5316\u5f0f\u3092\u7e70\u308a\u8fd4\u3057\u9069\u7528\u3059\u308b\u3068\u3001$n \\gt 2$\u306e\u5834\u5408\u306b\u306f$|A_n|$\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u305b\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\n|A_n| &#038;= |A_2|\\displaystyle\\prod_{k=3}^{n}\\left[(-1)^{k-1}r\\left(\\displaystyle\\prod_{i=1}^{k-2}\\sin\\phi_i\\right)\\right] \\nonumber\\cr<br \/>\n&#038;= (-1)^{\\left[\\frac{(n-2)(n+1)}{2}+1\\right]}r^{n-1}\\left[\\displaystyle\\prod_{k=3}^{n}\\left(\\displaystyle\\prod_{i=1}^{k-2}\\sin\\phi_i\\right)\\right] \\nonumber\\cr<br \/>\n&#038;= (-1)^{\\frac{n(n+1)}{2}}r^{n-1}\\left[\\displaystyle\\prod_{k=3}^{n}\\left(\\displaystyle\\prod_{i=1}^{k-2}\\sin\\phi_i\\right)\\right] \\label{eq:an_first}<br \/>\n\\end{align}<br \/>\n\u3053\u3053\u3067\u3001(\\ref{eq:an_first})\u5f0f\u53f3\u8fba\u306e\u7dcf\u4e57\u306e\u4e2d\u306b$\\phi_i$\u304c\u767b\u5834\u3059\u308b\u56de\u6570\u3092\u691c\u8a0e\u3059\u308b\u305f\u3081\u306b$i$\u3092\u4e00\u65e6\u56fa\u5b9a\u3057\u3001\u3053\u308c\u3092$l$\u3068\u7f6e\u304f\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n<p>\u3059\u308b\u3068\u3001\u5185\u5074\u306e\u7dcf\u4e57\u306e\u56e0\u6570\u306b$\\phi_l$\u3092\u542b\u3080\u3082\u306e\u304c\u767b\u5834\u3059\u308b\u6761\u4ef6\u306f$l \\le k-2$\u3068\u306a\u308b\u3053\u3068\u3067\u3042\u308a\u3001\u3055\u3089\u306b\u3001$k$\u306f$3$\u304b\u3089$n$\u307e\u3067\u306e\u6574\u6570\u5024\u3092\u3068\u308a\u307e\u3059\u306e\u3067\u3001\u5185\u5074\u306e\u7dcf\u4e57\u306e\u56e0\u6570\u306b$\\phi_l$\u3092\u542b\u3080\u3082\u306e\u304c\u767b\u5834\u3059\u308b\u500b\u6570\u306f$n-2-(l-1)=n-l-1$\u500b\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p>\u3088\u3063\u3066\u3001(\\ref{eq:an_first})\u5f0f\u306e\u53f3\u8fba\u306f$\\sin^{n-l-1}\\phi_l$\u306e\u7a4d\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u3066\u3001$l$\u306f1\u304b\u3089$n-2$\u307e\u3067\u306e\u6574\u6570\u5024\u3092\u3068\u308a\u307e\u3059\u306e\u3067\u2026<br \/>\n\\begin{align}<br \/>\n|A_n| &#038;= (-1)^{\\frac{n(n+1)}{2}}r^{n-1}\\displaystyle\\prod_{l=1}^{n-2}\\sin^{n-l-1}\\phi_l \\label{eq:an_second}<br \/>\n\\end{align}<br \/>\n$l$\u3092$i$\u306b\u66f8\u304d\u63db\u3048\u308b\u3068\u2026<br \/>\n\\begin{align}<br \/>\n|A_n| &#038;= (-1)^{\\frac{n(n+1)}{2}}r^{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin^{n-i-1}\\phi_i \\label{eq:an_third}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u7b26\u53f7\u306f\u5c11\u3005\u6c17\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u305d\u3053\u305d\u3053\u7c21\u6f54\u306a\u5f0f\u306b\u306a\u308a\u307e\u3057\u305f\u3002\ud83d\ude00<\/p>\n<h2>\u30e4\u30b3\u30d3\u884c\u5217\u5f0f\u3068\u7a4d\u5206\u8a08\u7b97\u306b\u304a\u3051\u308b\u5909\u6570\u5909\u63db<\/h2>\n<p>\u3053\u3053\u307e\u3067\u8a08\u7b97\u3057\u3066\u304d\u305f\u30e4\u30b3\u30d3\u884c\u5217\u5f0f\u3067\u3059\u304c\u3001$n$\u6b21\u5143\u306e\u7403(\u8d85\u7403)\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u308b\u969b\u306b\u4f7f\u3048\u307e\u3059\u3002<\/p>\n<p>$n$\u6b21\u5143\u306e\u7403(\u8d85\u7403)\u306e\u4f53\u7a4d\u306f$n$\u6b21\u5143\u306e\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u306b\u304a\u3051\u308b\u5fae\u5c0f\u4f53\u7a4d\u8981\u7d20$dV$\u3092$n$\u6b21\u5143\u306e\u7403(\u8d85\u7403)$S$<br \/>\n\\begin{align}<br \/>\n\\sum_{i=1}^n x_i^2 &#038;\\le R^2 \\label{eq:supersphere}<br \/>\n\\end{align}<br \/>\n\u306b\u308f\u305f\u3063\u3066<br \/>\n\\begin{align}<br \/>\nV &#038;= \\int_S dV \\label{eq:ints}<br \/>\n\\end{align}<br \/>\n\u3063\u3066\u306a\u611f\u3058\u3067\u7a4d\u5206\u3057\u305f\u3044\u3068\u3053\u308d\u3067\u3059\u304c\u3001<a href=\"https:\/\/pandanote.info\/?p=5190\">\u524d\u306e\u8a18\u4e8b<\/a>\u306e\u65b9\u6cd5\u3067\u3001$n$\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u304b\u3089\u7403\u5ea7\u6a19\u7cfb\u3078\u306e\u5909\u6570\u5909\u63db\u3092\u884c\u3046\u3068\u3001\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u306b\u304a\u3051\u308b\u9818\u57df$S$\u306f\u7403\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u306f$n$\u6b21\u5143\u306e\u8d85\u7acb\u65b9\u4f53\u306e\u3088\u3046\u306a\u3082\u306e\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001\u3053\u306e\u8d85\u7acb\u65b9\u4f53\u306e\u3088\u3046\u306a\u3082\u306e\u306e\u5fae\u5c0f\u4f53\u7a4d\u8981\u7d20$d\\phi_1 d\\phi_2 \\cdots d\\phi_{n-1} dr$\u3092\u3068\u308b\u3053\u3068\u304c\u3067\u304d\u305d\u3046\u3067\u3059\u3002<\/p>\n<p>\u3057\u304b\u3057\u3001$dV$\u3068$d\\phi_1 d\\phi_2 \\cdots d\\phi_{n-1} dr$\u306f\u540c\u3058\u3082\u306e\u3067\u306f\u306a\u3044\u306e\u3067\u3001\u4f53\u7a4d\u306e\u6bd4\u3092\u5b9a\u3081\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n<p>\u3053\u3053\u3067\u98af\u723d\u3068\u767b\u5834\u3059\u308b\u306e\u304c\u30e4\u30b3\u30d3\u884c\u5217\u5f0f\u306e\u7d76\u5bfe\u5024\u3067\u3001$dV$\u3068$d\\phi_1 d\\phi_2 \\cdots d\\phi_{n-1} dr$\u306f(\\ref{eq:an_third})\u5f0f\u306e\u7d50\u679c\u3092\u7528\u3044\u308b\u3068\u4ee5\u4e0b\u306e\u95a2\u4fc2\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\ndV &#038;= \\left| (-1)^{\\frac{n(n+1)}{2}}r^{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin^{n-i-1}\\phi_i \\right|d\\phi_1 d\\phi_2 \\cdots d\\phi_{n-1} dr \\nonumber \\cr<br \/>\n&#038;= \\left| \\displaystyle\\prod_{i=1}^{n-2}\\sin^{n-i-1}\\phi_i \\right|d\\phi_1 d\\phi_2 \\cdots d\\phi_{n-1} dr\\label{eq:transform_with_jacobian_first}<br \/>\n\\end{align}<br \/>\n(\\ref{eq:transform_with_jacobian_first})\u5f0f\u3067\u306f-1\u306e\u51aa\u3092\u6700\u521d\u306b\u6d88\u53bb\u3057\u307e\u3057\u305f\u304c\u3001$1 \\le i \\le n-2$\u306b\u304a\u3044\u3066\u306f$\\phi_i \\in [0,\\pi]$\u3067\u3042\u308b\u3053\u3068\u304b\u3089$\\sin\\phi_i \\ge 0$\u3068\u306a\u308a\u307e\u3059($\\phi_{n-1} \\in [0,2\\pi]$\u3067\u3059\u304c\u3001(\\ref{eq:transform_with_jacobian_first})\u5f0f\u306e\u7d76\u5bfe\u5024\u306e\u5185\u90e8\u306b\u306f\u767b\u5834\u3057\u3066\u3044\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3002)\u306e\u3067\u3001(\\ref{eq:transform_with_jacobian_first})\u5f0f\u53f3\u8fba\u306e\u7d76\u5bfe\u5024\u306e\u5185\u90e8\u306e\u5024\u3082\u3059\u3079\u3066\u6b63\u306e\u5024\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u3088\u3063\u3066\u3001\u7d76\u5bfe\u5024\u306e\u8a18\u53f7\u306f\u5916\u3059\u3053\u3068\u304c\u3067\u304d\u3066\u3001<br \/>\n\\begin{align}<br \/>\ndV &#038;= \\left(\\displaystyle\\prod_{i=1}^{n-2}\\sin^{n-i-1}\\right)\\phi_i d\\phi_1 d\\phi_2 \\cdots d\\phi_{n-1} dr \\label{eq:transform_with_jacobian_final}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u3002$\\blacksquare$<\/p>\n<h2>\u307e\u3068\u3081<\/h2>\n<p>\u6b21\u306f(\\ref{eq:transform_with_jacobian_final})\u5f0f\u3092\u7a4d\u5206\u2026 \u3068\u884c\u304d\u305f\u3044\u3068\u3053\u308d\u3067\u3057\u305f\u304c\u3001\u5177\u4f53\u7684\u306a\u65b9\u6cd5\u306fWikipedia\u306b\u8f09\u3063\u3066\u3044\u305f\u308a\u3057\u307e\u3059\u306e\u3067\u3001\u5225\u9014\u305d\u3061\u3089\u3092\u3054\u53c2\u7167\u3044\u305f\u3060\u3051\u308b\u3068\u5e78\u3044\u3067\u3059\u3002<\/p>\n<p>\u306a\u304a\u3001\u884c\u5217\u307e\u305f\u306f\u884c\u5217\u5f0f\u304c\u9762\u7a4d\u3092\u53d6\u308a\u3059\u304e\u3066\u3044\u308b\u305f\u3081\u306b\u6a2a\u30b9\u30af\u30ed\u30fc\u30eb\u304c\u5fc5\u9808\u3068\u306a\u3063\u3066\u3057\u307e\u3044\u3001\u30b9\u30de\u30db\u3067\u306f\u304b\u306a\u308a\u898b\u8f9b\u304f\u306a\u3063\u3066\u3044\u308b\u70b9\u306b\u3064\u304d\u307e\u3057\u3066\u306f\u6df1\u304f\u304a\u8a6b\u3073\u7533\u3057\u4e0a\u3052\u307e\u3059\u3002<\/p>\n<p>\u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u306f\u3058\u3081\u306b \u524d\u306e\u8a18\u4e8b\u3067\u3001$n$\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u304b\u3089\u7403\u5ea7\u6a19\u7cfb\u3078\u306e\u5909\u6570\u5909\u63db\u3092\u884c\u3046\u305f\u3081\u306e\u30e4\u30b3\u30d3\u884c\u5217\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u305f\u3002 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u30e4\u30b3\u30d3\u884c\u5217\u304b\u3089\u30e4\u30b3\u30d3\u884c\u5217\u5f0f\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002 \u8a08\u7b97\u306e\u65b9\u91dd\u3092\u8003\u3048\u308b\u3002 \u524d\u306e\u8a18\u4e8b\u3088\u308a\u3001$n$\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u304b\u3089\u7403\u5ea7\u6a19\u7cfb\u3078\u306e\u5909\u6570\u5909\u63db\u3092\u884c\u3046\u305f\u3081\u306e\u30e4\u30b3\u30d3\u884c\u5217\u306f(\\ref{eq:jacobian_second})\u5f0f\u3067\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 \\begin{align}\u2026 <span class=\"read-more\"><a href=\"https:\/\/pandanote.info\/?p=5225\">Read More 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