[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki21\/archives\/294217#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki21\/archives\/294217","headline":"\u4f53\u7a4d\u8981\u7d20 – Wikipedia","name":"\u4f53\u7a4d\u8981\u7d20 – Wikipedia","description":"\u3053\u306e\u9805\u76ee\u300c\u4f53\u7a4d\u8981\u7d20\u300d\u306f\u7ffb\u8a33\u3055\u308c\u305f\u3070\u304b\u308a\u306e\u3082\u306e\u3067\u3059\u3002\u4e0d\u81ea\u7136\u3042\u308b\u3044\u306f\u66d6\u6627\u306a\u8868\u73fe\u306a\u3069\u304c\u542b\u307e\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u3001\u3053\u306e\u307e\u307e\u3067\u306f\u8aad\u307f\u3065\u3089\u3044\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002\uff08\u539f\u6587\uff1aen:Volume_element\uff09 \u4fee\u6b63\u3001\u52a0\u7b46\u306b\u5354\u529b\u3057\u3001\u73fe\u5728\u306e\u8868\u73fe\u3092\u3088\u308a\u81ea\u7136\u306a\u8868\u73fe\u306b\u3057\u3066\u4e0b\u3055\u308b\u65b9\u3092\u6c42\u3081\u3066\u3044\u307e\u3059\u3002\u30ce\u30fc\u30c8\u30da\u30fc\u30b8\u3084\u5c65\u6b74\u3082\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002\uff082017\u5e742\u6708\uff09 \u51fa\u5178\u306f\u5217\u6319\u3059\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u811a\u6ce8\u306a\u3069\u3092\u7528\u3044\u3066\u3069\u306e\u8a18\u8ff0\u306e\u60c5\u5831\u6e90\u3067\u3042\u308b\u304b\u3092\u660e\u8a18\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u3092\u304a\u9858\u3044\u3044\u305f\u3057\u307e\u3059\u3002\uff082017\u5e742\u6708\uff09 \u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u4f53\u7a4d\u8981\u7d20\uff08\u305f\u3044\u305b\u304d\u3088\u3046\u305d\u3001\u82f1: volume element\uff09\u3068\u306f\u3001\u95a2\u6570\u3092\u7403\u9762\u5ea7\u6a19\u7cfb\u3084\u5186\u67f1\u5ea7\u6a19\u7cfb\u306a\u3069\u69d8\u3005\u306a\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u4f53\u7a4d\u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u969b\u306b\u73fe\u308f\u308c\u308b\u6982\u5ff5\u3067\u3042\u308b\u3002\u6b21\u306e\u5f0f\u306b\u3088\u308a\u8868\u73fe\u3055\u308c\u308b: dV:=\u03c1(u1,u2,u3)du1du2du3.{displaystyle dV:=rho (u_{1},u_{2},u_{3}),du_{1},du_{2},du_{3}.} \u3053\u3053\u3067\u3001ui \u306f\u5ea7\u6a19\u3067\u3042\u308a\u3001\u4efb\u610f\u306e\u96c6\u5408 B \u306e\u4f53\u7a4d\u3092\u6b21\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u308b\u3082\u306e\u3068\u3059\u308b: Volume\u2061(B):=\u222bB\u03c1(u1,u2,u3)du1du2du3.{displaystyle operatorname {Volume} (B):=int _{B}rho (u_{1},u_{2},u_{3}),du_{1},du_{2},du_{3}.} \u305f\u3068\u3048\u3070\u3001\u7403\u9762\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u306fdV","datePublished":"2022-04-26","dateModified":"2022-04-26","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki21\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki21\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/2\/2a\/Translation_arrow.svg\/50px-Translation_arrow.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/2\/2a\/Translation_arrow.svg\/50px-Translation_arrow.svg.png","height":"17","width":"50"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki21\/archives\/294217","about":["Wiki"],"wordCount":8405,"articleBody":"\u3053\u306e\u9805\u76ee\u300c\u4f53\u7a4d\u8981\u7d20\u300d\u306f\u7ffb\u8a33\u3055\u308c\u305f\u3070\u304b\u308a\u306e\u3082\u306e\u3067\u3059\u3002\u4e0d\u81ea\u7136\u3042\u308b\u3044\u306f\u66d6\u6627\u306a\u8868\u73fe\u306a\u3069\u304c\u542b\u307e\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u3001\u3053\u306e\u307e\u307e\u3067\u306f\u8aad\u307f\u3065\u3089\u3044\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002\uff08\u539f\u6587\uff1aen:Volume_element\uff09\u4fee\u6b63\u3001\u52a0\u7b46\u306b\u5354\u529b\u3057\u3001\u73fe\u5728\u306e\u8868\u73fe\u3092\u3088\u308a\u81ea\u7136\u306a\u8868\u73fe\u306b\u3057\u3066\u4e0b\u3055\u308b\u65b9\u3092\u6c42\u3081\u3066\u3044\u307e\u3059\u3002\u30ce\u30fc\u30c8\u30da\u30fc\u30b8\u3084\u5c65\u6b74\u3082\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002\uff082017\u5e742\u6708\uff09\u51fa\u5178\u306f\u5217\u6319\u3059\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u811a\u6ce8\u306a\u3069\u3092\u7528\u3044\u3066\u3069\u306e\u8a18\u8ff0\u306e\u60c5\u5831\u6e90\u3067\u3042\u308b\u304b\u3092\u660e\u8a18\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u3092\u304a\u9858\u3044\u3044\u305f\u3057\u307e\u3059\u3002\uff082017\u5e742\u6708\uff09\u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u4f53\u7a4d\u8981\u7d20\uff08\u305f\u3044\u305b\u304d\u3088\u3046\u305d\u3001\u82f1: volume element\uff09\u3068\u306f\u3001\u95a2\u6570\u3092\u7403\u9762\u5ea7\u6a19\u7cfb\u3084\u5186\u67f1\u5ea7\u6a19\u7cfb\u306a\u3069\u69d8\u3005\u306a\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u4f53\u7a4d\u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u969b\u306b\u73fe\u308f\u308c\u308b\u6982\u5ff5\u3067\u3042\u308b\u3002\u6b21\u306e\u5f0f\u306b\u3088\u308a\u8868\u73fe\u3055\u308c\u308b:dV:=\u03c1(u1,u2,u3)du1du2du3.{displaystyle dV:=rho (u_{1},u_{2},u_{3}),du_{1},du_{2},du_{3}.}  \u3053\u3053\u3067\u3001ui \u306f\u5ea7\u6a19\u3067\u3042\u308a\u3001\u4efb\u610f\u306e\u96c6\u5408 B \u306e\u4f53\u7a4d\u3092\u6b21\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u308b\u3082\u306e\u3068\u3059\u308b:  Volume\u2061(B):=\u222bB\u03c1(u1,u2,u3)du1du2du3.{displaystyle operatorname {Volume} (B):=int _{B}rho (u_{1},u_{2},u_{3}),du_{1},du_{2},du_{3}.}\u305f\u3068\u3048\u3070\u3001\u7403\u9762\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u306fdV = u12 sin u2 du1 du2 du3 \u3067\u3042\u308a\u3001\u5f93\u3063\u3066 dV = u12 sin u2 \u3067\u3042\u308b\u3002\u4f53\u7a4d\u8981\u7d20\u3068\u3044\u3046\u6982\u5ff5\u306f\u4e09\u6b21\u5143\u306b\u7559\u307e\u308b\u3082\u306e\u3067\u306f\u306a\u3044\u3002\u4e8c\u6b21\u5143\u3067\u306f\u9762\u7a4d\u8981\u7d20\uff08\u3081\u3093\u305b\u304d\u3088\u3046\u305d\u3001area element\uff09\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u3082\u591a\u304f\u3001\u9762\u7a4d\u5206\u3092\u884c\u3046\u969b\u306b\u6709\u7528\u3067\u3042\u308b\u3002\u5ea7\u6a19\u5909\u63db\u306e\u969b\u3001\uff08\u5909\u6570\u5909\u63db\u516c\u5f0f\u306b\u3088\u308a\uff09\u4f53\u7a4d\u8981\u7d20\u306f\u5ea7\u6a19\u5909\u63db\u306e\u30e4\u30b3\u30d3\u884c\u5217\u306e\u884c\u5217\u5f0f\u306e\u7d76\u5bfe\u5024\u3060\u3051\u5909\u5316\u3059\u308b\u3002\u3053\u306e\u4e8b\u5b9f\u304b\u3089\u3001\u4f53\u7a4d\u8981\u7d20\u306f\u591a\u69d8\u4f53\u306e\u4e00\u7a2e\u306e\u6e2c\u5ea6\u3068\u3057\u3066\u5b9a\u7fa9\u3067\u304d\u308b\u3053\u3068\u304c\u5f93\u3046\u3002\u5411\u304d\u4ed8\u3051\u53ef\u80fd\u306a\u53ef\u5fae\u5206\u591a\u69d8\u4f53\u306b\u304a\u3044\u3066\u306f\u3001\u5178\u578b\u7684\u306b\u306f\u4f53\u7a4d\u8981\u7d20\u306f\u4f53\u7a4d\u5f62\u5f0f\u3001\u3059\u306a\u308f\u3061\u6700\u9ad8\u6b21\u306e\u5fae\u5206\u5f62\u5f0f\u304b\u3089\u5c0e\u304b\u308c\u308b\u3002\u5411\u304d\u4ed8\u3051\u4e0d\u53ef\u80fd\u306a\u591a\u69d8\u4f53\u306b\u304a\u3044\u3066\u306f\u3001\u5178\u578b\u7684\u306b\u306f\u4f53\u7a4d\u8981\u7d20\u306f\uff08\u5c40\u6240\u7684\u306b\u5b9a\u7fa9\u3055\u308c\u308b\uff09\u4f53\u7a4d\u8981\u7d20\u306e\u7d76\u5bfe\u5024\u3067\u3042\u308a\u30011-\u5bc6\u5ea6\uff08\u82f1\u8a9e\u7248\uff09\u3092\u5b9a\u7fa9\u3059\u308b\u3002Table of Contents\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u304a\u3051\u308b\u4f53\u7a4d\u8981\u7d20[\u7de8\u96c6]\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u306b\u304a\u3051\u308b\u4f53\u7a4d\u8981\u7d20[\u7de8\u96c6]\u591a\u69d8\u4f53\u306e\u4f53\u7a4d\u8981\u7d20[\u7de8\u96c6]\u66f2\u9762\u306e\u9762\u7a4d\u8981\u7d20[\u7de8\u96c6]\u4f8b: \u7403[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u304a\u3051\u308b\u4f53\u7a4d\u8981\u7d20[\u7de8\u96c6]\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u304a\u3044\u3066\u306f\u3001\u4f53\u7a4d\u8981\u7d20\u306f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u306b\u6cbf\u3063\u305f\u5fae\u5206\u306e\u7a4d\u306b\u3088\u308a\u4e0e\u3048\u3089\u308c\u308b\u3002  dV=dxdydz{displaystyle mathrm {d} V=mathrm {d} x,mathrm {d} y,mathrm {d} z}\u4ed6\u306e\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u306f\u3001x = x(u1, u2, u3), y = y(u1, u2, u3), z = z(u1, u2, u3) \u3068\u3059\u308b\u3068\u30e4\u30b3\u30d3\u884c\u5217\u3092\u7528\u3044\u3066\u4f53\u7a4d\u8981\u7d20\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u308b\u3002dV=|\u2202(x,y,z)\u2202(u1,u2,u3)|du1du2du3{displaystyle mathrm {d} V=left|{frac {partial (x,y,z)}{partial (u_{1},u_{2},u_{3})}}right|,mathrm {d} u_{1},mathrm {d} u_{2},mathrm {d} u_{3}}\u305f\u3068\u3048\u3070\u3001\u7403\u9762\u5ea7\u6a19\u7cfb\u3067\u306fx=\u03c1cos\u2061\u03b8sin\u2061\u03d5y=\u03c1sin\u2061\u03b8sin\u2061\u03d5z=\u03c1cos\u2061\u03d5{displaystyle {begin{aligned}x&=rho cos theta sin phi \\y&=rho sin theta sin phi \\z&=rho cos phi end{aligned}}}\u3067\u3042\u308b\u304b\u3089\u30e4\u30b3\u30d3\u30a2\u30f3\u306f|\u2202(x,y,z)\u2202(\u03c1,\u03b8,\u03d5)|=\u03c12sin\u2061\u03d5{displaystyle left|{frac {partial (x,y,z)}{partial (rho ,theta ,phi )}}right|=rho ^{2}sin phi }\u3068\u306a\u308a\u3001\u3057\u305f\u304c\u3063\u3066\u4f53\u7a4d\u8981\u7d20\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002dV=\u03c12sin\u2061\u03d5d\u03c1d\u03b8d\u03d5.{displaystyle mathrm {d} V=rho ^{2}sin phi ,mathrm {d} rho ,mathrm {d} theta ,mathrm {d} phi .}\u3053\u306e\u3053\u3068\u306f\u5fae\u5206\u5f62\u5f0f\u304c\u5f15\u304d\u623b\u3057\uff08\u82f1\u8a9e\u7248\uff09 F* \u306b\u3088\u308a\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u63db\u3059\u308b\u3053\u3068\u306e\u4f8b\u3068\u898b\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002F\u2217(udy1\u2227\u22ef\u2227dyn)=(u\u2218F)det(\u2202Fj\u2202xi)dx1\u2227\u22ef\u2227dxn{displaystyle F^{*}(u;mathrm {d} y^{1}wedge cdots wedge mathrm {d} y^{n})=(ucirc F)det left({frac {partial F^{j}}{partial x^{i}}}right)mathrm {d} x^{1}wedge cdots wedge mathrm {d} x^{n}}\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u306b\u304a\u3051\u308b\u4f53\u7a4d\u8981\u7d20[\u7de8\u96c6]n-\u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593 Rn \u306e\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u304c\u6b21\u306e\u7dda\u5f62\u72ec\u7acb\u306a\u30d9\u30af\u30c8\u30eb\u306b\u3088\u308a\u5f35\u3089\u308c\u308b\u3082\u306e\u3068\u3059\u308b\u3002X1,\u2026,Xk{displaystyle X_{1},dots ,X_{k}}\u3053\u306e\u90e8\u5206\u7a7a\u9593\u306b\u304a\u3051\u308b\u4f53\u7a4d\u8981\u7d20\u3092\u8a08\u7b97\u3059\u308b\u5834\u5408\u3001Xi \u306e\u5f35\u308b\u5e73\u884c\u591a\u80de\u4f53[\u8a33\u8a9e\u7591\u554f\u70b9]\u304c\u7dda\u5f62\u5e7e\u4f55\u5b66\u304b\u3089 Xi \u306e\u30b0\u30e9\u30e0\u884c\u5217\u306e\u884c\u5217\u5f0f\u306e\u5e73\u65b9\u6839\u306b\u3088\u308a\u4e0e\u3048\u3089\u308c\u308b\u3053\u3068\u3092\u77e5\u3063\u3066\u304a\u304f\u3068\u4fbf\u5229\u3067\u3042\u308b\u3002det(Xi\u22c5Xj)i,j=1\u2026k{displaystyle {sqrt {det(X_{i}cdot X_{j})_{i,j=1dots k}}}}\u3053\u306e\u90e8\u5206\u7a7a\u9593\u4e0a\u306e\u4efb\u610f\u306e\u70b9 p \u306f\u3042\u308b\u5ea7\u6a19 (u1, u2, …, uk) \u306b\u3088\u308a\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u308f\u3055\u308c\u308b\u3002p=u1X1+\u22ef+ukXk{displaystyle p=u_{1}X_{1}+cdots +u_{k}X_{k}}\u70b9 p \u306b\u304a\u3044\u3066\u3001\u8fba\u3092 dui \u3068\u3059\u308b\u5fae\u5c0f\u5e73\u884c\u591a\u80de\u4f53\u3092\u4f5c\u308b\u3068\u3001\u305d\u306e\u4f53\u7a4d\u306f\u30b0\u30e9\u30e0\u884c\u5217\u306e\u884c\u5217\u5f0f\u306e\u5e73\u65b9\u6839\u306b\u3088\u308a\u4e0e\u3048\u3089\u308c\u308b\u3002det((duiXi)\u22c5(dujXj))i,j=1\u2026k=det(Xi\u22c5Xj)i,j=1\u2026kdu1du2\u22efduk{displaystyle {sqrt {det left((mathrm {d} u_{i}X_{i})cdot (mathrm {d} u_{j}X_{j})right)_{i,j=1dots k}}}={sqrt {det(X_{i}cdot X_{j})_{i,j=1dots k}}};mathrm {d} u_{1},mathrm {d} u_{2},cdots ,mathrm {d} u_{k}}\u3053\u308c\u306b\u3088\u308a\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u306b\u304a\u3051\u308b\u4f53\u7a4d\u5f62\u5f0f\u3092\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u591a\u69d8\u4f53\u306e\u4f53\u7a4d\u8981\u7d20[\u7de8\u96c6]\u5411\u304d\u4ed8\u3051\u53ef\u80fd\u306a\u6b21\u5143 n \u306e\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u306b\u304a\u3051\u308b\u4f53\u7a4d\u8981\u7d20\u306f\u5b9a\u6570\u95a2\u6570\u306e f(x) = 1 \u306e\u30db\u30c3\u30b8\u53cc\u5bfe\u306b\u7b49\u3057\u3044\u3002\u03c9=\u22c61{displaystyle omega =star 1}\u3053\u308c\u3068\u7b49\u4fa1\u306b\u3001\u4f53\u7a4d\u8981\u7d20\u306f\u6b63\u78ba\u306b\u30ec\u30f4\u30a3\uff1d\u30c1\u30f4\u30a3\u30bf\u30c6\u30f3\u30bd\u30eb \u03b5 \u3068\u6b63\u78ba\u306b\u4e00\u81f4\u3059\u308b[1]\u3002\u5ea7\u6a19\u3092\u7528\u3044\u3066\u66f8\u3051\u3070\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002\u03c9=\u03f5=|detg|dx1\u2227\u22ef\u2227dxn{displaystyle omega =epsilon ={sqrt {|det g|}},mathrm {d} x^{1}wedge cdots wedge mathrm {d} x^{n}}\u3053\u3053\u3067\u00a0det g \u306f\u305d\u306e\u5ea7\u6a19\u7cfb\u306b\u304a\u3051\u308b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb g \u306e\u884c\u5217\u5f0f\u3067\u3042\u308b\u3002\u66f2\u9762\u306e\u9762\u7a4d\u8981\u7d20[\u7de8\u96c6]n-\u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u57cb\u3081\u8fbc\u307e\u308c\u305f\u4e8c\u6b21\u5143\u66f2\u9762\u3092\u8003\u3048\u308b\u3053\u3068\u3067\u3001\u4f53\u7a4d\u8981\u7d20\u306e\u5358\u7d14\u306a\u4f8b\u3092\u8003\u5bdf\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u306e\u5834\u5408\u3001\u4f53\u7a4d\u8981\u7d20\u306f\u9762\u7a4d\u8981\u7d20\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u3082\u3042\u308b\u3002\u90e8\u5206\u96c6\u5408 U \u2282 R2 \u3068\u5199\u50cf\u03c6:U\u2192Rn{displaystyle varphi :Uto mathbf {R} ^{n}}\u3092\u8003\u3048\u308b\u3053\u3068\u306b\u3088\u308a\u3001Rn \u306b\u57cb\u3081\u8fbc\u307e\u308c\u305f\u66f2\u9762\u3092\u5b9a\u7fa9\u3059\u308b\u3002\u4e8c\u6b21\u5143\u3067\u306f\u4f53\u7a4d\u306f\u9762\u7a4d\u3067\u3042\u308a\u3001\u4f53\u7a4d\u8981\u7d20\u306f\u66f2\u9762\u306e\u4efb\u610f\u306e\u90e8\u5206\u306e\u9762\u7a4d\u3092\u6c7a\u5b9a\u3059\u308b\u65b9\u6cd5\u3092\u4e0e\u3048\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4f53\u7a4d\u8981\u7d20\u306f\u6b21\u306e\u5f62\u5f0f\u3092\u3068\u308b\u3002f(u1,u2)du1du2{displaystyle f(u_{1},u_{2}),mathrm {d} u_{1},mathrm {d} u_{2}}\u3053\u308c\u306b\u3088\u308a\u66f2\u9762\u4e0a\u306e\u96c6\u5408 B \u306e\u9762\u7a4d\u3092\u7a4d\u5206\u3092\u7528\u3044\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u308b\u3002Area\u2061(B)=\u222bBf(u1,u2)du1du2{displaystyle operatorname {Area} (B)=int _{B}f(u_{1},u_{2}),mathrm {d} u_{1},mathrm {d} u_{2}}\u3053\u3053\u3067\u3001\u901a\u5e38\u306e\u610f\u5473\u3067\u306e\u9762\u7a4d\u3092\u4e0e\u3048\u308b\u3088\u3046\u306a\u4f53\u7a4d\u8981\u7d20\u3092\u6c7a\u5b9a\u3057\u305f\u3044\u3002\u5199\u50cf\u306e\u30e4\u30b3\u30d6\u884c\u5217\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002\u03bbij=\u2202\u03c6i\u2202uj{displaystyle lambda _{ij}={frac {partial varphi _{i}}{partial u_{j}}}}\u3053\u3053\u3067\u6dfb\u5b57 i \u306f 1 \u304b\u3089 n \u3092\u3001j \u306f 1 \u304b\u3089 2 \u3092\u8d70\u308b\u3002n-\u6b21\u5143\u7a7a\u9593\u306e\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u8a08\u91cf\u304b\u3089\u3001\u96c6\u5408 U \u4e0a\u306e\u8a08\u91cf g = \u03bbT\u03bb \u3092\u8a08\u7b97\u3067\u304d\u3001\u305d\u306e\u884c\u5217\u8981\u7d20\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u4e0e\u3048\u3089\u308c\u308b\u3002gij=\u2211k=1n\u03bbki\u03bbkj=\u2211k=1n\u2202\u03c6k\u2202ui\u2202\u03c6k\u2202uj{displaystyle g_{ij}=sum _{k=1}^{n}lambda _{ki}lambda _{kj}=sum _{k=1}^{n}{frac {partial varphi _{k}}{partial u_{i}}}{frac {partial varphi _{k}}{partial u_{j}}}}\u3053\u306e\u8a08\u91cf\u306e\u884c\u5217\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u3002detg=|\u2202\u03c6\u2202u1\u2227\u2202\u03c6\u2202u2|2=det(\u03bbT\u03bb){displaystyle det g=left|{frac {partial varphi }{partial u_{1}}}wedge {frac {partial varphi }{partial u_{2}}}right|^{2}=det(lambda ^{mathrm {T} }lambda )}\u6b63\u5247\u66f2\u9762\u306b\u304a\u3044\u3066\u306f\u3001\u3053\u306e\u884c\u5217\u5f0f\u306f\u3044\u305f\u308b\u3068\u3053\u308d\u975e\u96f6\u3001\u3059\u306a\u308f\u3061\u30e4\u30b3\u30d6\u884c\u5217\u306e\u30e9\u30f3\u30af\u306f\u3044\u305f\u308b\u3068\u3053\u308d\u3067 2 \u3067\u3042\u308b\u3002\u3053\u3053\u3067\u3001U \u4e0a\u306e\u5ea7\u6a19\u3092\u5fae\u5206\u540c\u76f8\u5199\u50cf\u306b\u3088\u308a\u5909\u63db\u3057\u3001\u5ea7\u6a19 (u1, u2) \u304c (v1, v2) \u306b\u79fb\u308b\u3082\u306e\u3068\u3059\u308b\u3002\u3059\u306a\u308f\u3061\u3001 (u1, u2) = f(v1, v2) \u3068\u306a\u308b\u3002\u3053\u306e\u5909\u63db\u306e\u30e4\u30b3\u30d6\u884c\u5217\u306f\u6b21\u306e\u3088\u3046\u306b\u4e0e\u3048\u3089\u308c\u308b\u3002Fij=\u2202fi\u2202vj{displaystyle F_{ij}={frac {partial f_{i}}{partial v_{j}}}}\u3053\u306e\u65b0\u3057\u3044\u5ea7\u6a19\u3067\u306f\u3001\u2202\u03c6i\u2202vj=\u2211k=12\u2202\u03c6i\u2202uk\u2202fk\u2202vj{displaystyle {frac {partial varphi _{i}}{partial v_{j}}}=sum _{k=1}^{2}{frac {partial varphi _{i}}{partial u_{k}}}{frac {partial f_{k}}{partial v_{j}}}}\u3068\u306a\u308a\u3001\u8a08\u91cf\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u63db\u3055\u308c\u308b\u3002g~=FTgF{displaystyle {tilde {g}}=F^{mathrm {T} }gF}\u3053\u3053\u3067\u3001 ~g \u306f v \u5ea7\u6a19\u7cfb\u306b\u5f15\u304d\u623b\u3057\u305f\u8a08\u91cf\u3067\u3042\u308b\u3002\u3053\u306e\u884c\u5217\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u3002detg~=detg(detF)2{displaystyle det {tilde {g}}=det g(det F)^{2}}\u4ee5\u4e0a\u304b\u3089\u3001\u4f53\u7a4d\u8981\u7d20\u304c\u5411\u304d\u3092\u4fdd\u5b58\u3059\u308b\u5ea7\u6a19\u5909\u63db\u306e\u4e0b\u306b\u4e0d\u5909\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u4e8c\u6b21\u5143\u306b\u304a\u3044\u3066\u306f\u3001\u4f53\u7a4d\u306f\u9762\u7a4d\u3067\u3042\u308b\u3002\u90e8\u5206\u96c6\u5408 B \u2282 U \u306e\u9762\u7a4d\u306f\u6b21\u306e\u3088\u3046\u306b\u7a4d\u5206\u3067\u5f97\u3089\u308c\u308b\u3002Area(B)=\u222cBdetgdu1du2=\u222cBdetg|detF|dv1dv2=\u222cBdetg~dv1dv2{displaystyle {begin{aligned}{mbox{Area}}(B)&=iint _{B}{sqrt {det g}};mathrm {d} u_{1};mathrm {d} u_{2}\\&=iint _{B}{sqrt {det g}};|det F|;mathrm {d} v_{1};mathrm {d} v_{2}\\&=iint _{B}{sqrt {det {tilde {g}}}};mathrm {d} v_{1};mathrm {d} v_{2}end{aligned}}}\u3057\u305f\u304c\u3063\u3066\u3001\u3069\u3061\u3089\u306e\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u3082\u4f53\u7a4d\u8981\u7d20\u306f\u540c\u4e00\u306e\u5f62\u5f0f\u3092\u6301\u3061\u3001\u3059\u306a\u308f\u3061\u5ea7\u6a19\u5909\u63db\u3067\u4e0d\u5909\u306b\u4fdd\u305f\u308c\u308b\u3002\u4ee5\u4e0a\u306e\u8b70\u8ad6\u306b\u4e8c\u6b21\u5143\u7279\u6709\u306e\u3082\u306e\u306f\u4e00\u5207\u306a\u3044\u306e\u3067\u3001\u4efb\u610f\u306e\u6b21\u5143\u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3064\u3002\u4f8b: \u7403[\u7de8\u96c6]\u4f8b\u3068\u3057\u3066\u3001\u534a\u5f84 r \u306e\u4e2d\u5fc3\u3092\u539f\u70b9\u3068\u3059\u308b R3 \u4e0a\u306e\u7403\u3092\u8003\u3048\u308b\u3002\u3053\u306e\u7403\u306f\u7403\u9762\u5ea7\u6a19\u7cfb\u3092\u7528\u3044\u3066\u6b21\u306e\u5199\u50cf\u3067\u8868\u73fe\u3055\u308c\u308b\u3002\u03d5(u1,u2)=(rcos\u2061u1sin\u2061u2,rsin\u2061u1sin\u2061u2,rcos\u2061u2){displaystyle phi (u_{1},u_{2})=(rcos u_{1}sin u_{2},rsin u_{1}sin u_{2},rcos u_{2})}\u3057\u305f\u304c\u3063\u3066\u3001g=(r2sin2\u2061u200r2){displaystyle g={begin{pmatrix}r^{2}sin ^{2}u_{2}&0\\0&r^{2}end{pmatrix}}}\u3068\u306a\u308a\u3001\u9762\u7a4d\u8981\u7d20\u306f\u6b21\u306e\u3088\u3046\u306b\u5f97\u3089\u308c\u308b\u3002\u03c9=detgdu1du2=r2sin\u2061u2du1du2{displaystyle omega ={sqrt {det g}};mathrm {d} u_{1}mathrm {d} u_{2}=r^{2}sin u_{2},mathrm {d} u_{1}mathrm {d} u_{2}}\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]^ Carroll, Sean. Spacetime and Geometry. Addison Wesley, 2004, p. 90\u53c2\u8003\u6587\u732e[\u7de8\u96c6]Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: Springer-Verlag, pp.\u00a0xii+510, ISBN\u00a0978-3-540-15279-8\u00a0"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki21\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki21\/archives\/294217#breadcrumbitem","name":"\u4f53\u7a4d\u8981\u7d20 – Wikipedia"}}]}]