[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/3851#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/3851","headline":"\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7 – Wikipedia","name":"\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7 – Wikipedia","description":"\u30ea\u30fc\u30de\u30f3\u5e7e\u4f55\u5b66\u306b\u304a\u3044\u3066\u3001\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\uff08\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u304d\u3054\u3046\u3001\u82f1: Christoffel symbols\uff09\u307e\u305f\u306f\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u306e\u4e09\u6dfb\u5b57\u8a18\u53f7\uff08\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u306e\u3055\u3093\u305d\u3048\u3058\u304d\u3054\u3046\u3001\u82f1: Christoffel three index symbols\uff09\u3068\u306f\u3001\u6e2c\u5730\u7dda\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u8868\u3059\u306b\u3042\u305f\u3063\u3066\u30d6\u30eb\u30fc\u30ce\u30fb\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb (1829\u20131900) \u306b\u3088\u3063\u3066\u5c0e\u5165\u3055\u308c\u305f\u8a18\u53f7\u3092\u8a00\u3046[1]\u3002 \u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306b\u306f\u7b2c\u4e00\u7a2e\u8a18\u53f7 [jk,a]{displaystyle left[j;k,aright]} \u3068\u7b2c\u4e8c\u7a2e\u8a18\u53f7 {ijk}{displaystyle left{{{i} atop {j;k}}right}} \u306e\u4e8c\u7a2e\u985e\u304c\u3042\u308b[2]\u304c\u3001\u57fa\u672c\u7684\u306b\u306f\u7b2c\u4e8c\u7a2e\u8a18\u53f7\u306e\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002 \u30ea\u30fc\u30de\u30f3\u5e7e\u4f55\u5b66\u306b\u304a\u3044\u3066\u306f\u3001n \u6b21\u5143\u591a\u69d8\u4f53\u3068\u547c\u3070\u308c\u308b\u7a7a\u9593\u4e0a\u306b\u3042\u308b\u66f2\u7dda","datePublished":"2020-09-29","dateModified":"2020-09-29","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki7\/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\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2acdd7487f41a0cd299a9dd1a3af50ce163ae49f","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2acdd7487f41a0cd299a9dd1a3af50ce163ae49f","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/3851","about":["Wiki"],"wordCount":12618,"articleBody":"\u30ea\u30fc\u30de\u30f3\u5e7e\u4f55\u5b66\u306b\u304a\u3044\u3066\u3001\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\uff08\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u304d\u3054\u3046\u3001\u82f1: Christoffel symbols\uff09\u307e\u305f\u306f\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u306e\u4e09\u6dfb\u5b57\u8a18\u53f7\uff08\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u306e\u3055\u3093\u305d\u3048\u3058\u304d\u3054\u3046\u3001\u82f1: Christoffel three index symbols\uff09\u3068\u306f\u3001\u6e2c\u5730\u7dda\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u8868\u3059\u306b\u3042\u305f\u3063\u3066\u30d6\u30eb\u30fc\u30ce\u30fb\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb (1829\u20131900) \u306b\u3088\u3063\u3066\u5c0e\u5165\u3055\u308c\u305f\u8a18\u53f7\u3092\u8a00\u3046[1]\u3002\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306b\u306f\u7b2c\u4e00\u7a2e\u8a18\u53f7 [jk,a]{displaystyle left[j;k,aright]} \u3068\u7b2c\u4e8c\u7a2e\u8a18\u53f7 {ijk}{displaystyle left{{{i} atop {j;k}}right}} \u306e\u4e8c\u7a2e\u985e\u304c\u3042\u308b[2]\u304c\u3001\u57fa\u672c\u7684\u306b\u306f\u7b2c\u4e8c\u7a2e\u8a18\u53f7\u306e\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002\u30ea\u30fc\u30de\u30f3\u5e7e\u4f55\u5b66\u306b\u304a\u3044\u3066\u306f\u3001n \u6b21\u5143\u591a\u69d8\u4f53\u3068\u547c\u3070\u308c\u308b\u7a7a\u9593\u4e0a\u306b\u3042\u308b\u66f2\u7ddaxh = xh (t), t1 \u2266 t \u2266 t2\u306e\u9577\u3055\u3092\u3001\u7a4d\u5206I=\u222bt1t2\u2211i,jgij(x)dxidtdxjdtdt{displaystyle I=int _{t_{1}}^{t_{2}}{sqrt {sum _{i,j}g_{ij}(x){frac {mathrm {d} x^{i}}{mathrm {d} t}}{frac {mathrm {d} x^{j}}{mathrm {d} t}}}},mathrm {d} t}\u3067\u8a08\u7b97\u3067\u304d\u308b\u3088\u3046\u306b\u3001\u5404\u5ea7\u6a19\u8fd1\u508d\u5185\u306bgij(x) = gji(x)\u3068\u3044\u3046\u95a2\u6570\uff08\u57fa\u672c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u3068\u547c\u3070\u308c\u308b\uff09\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3002\u3053\u306e\u7a4d\u5206\u306e\u7b2c\u4e00\u5909\u5206 \u03b4I \u3092 0 \u3068\u304a\u304f\u3053\u3068\u3067\u5f97\u3089\u308c\u308b\u30aa\u30a4\u30e9\u30fc\u30fb\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\u3001\u6e2c\u5730\u7dda\u306b\u6cbf\u3063\u3066\u306e\u5b64\u306e\u9577\u3055\u3092\u5a92\u4ecb\u5909\u6570\u306b\u3068\u308c\u3070\u3001d2xhds2+\u2211j,i{hji}dxjdsdxids=0{displaystyle {frac {mathrm {d} ^{2}x^{h}}{mathrm {d} s^{2}}}+sum _{j,i}left{{{h} atop {j;i}}right}{frac {mathrm {d} x^{j}}{mathrm {d} s}}{frac {mathrm {d} x^{i}}{mathrm {d} s}}=0}\u3068\u306a\u308b[\u6ce8\u91c8 1]\u3002\u3053\u308c\u3092\u6e2c\u5730\u7dda\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u547c\u3076\u3002\u306a\u304a\u3001\u3053\u3053\u3067{hji}=\u2211a12gha(\u2202gia\u2202xj+\u2202gja\u2202xi\u2212\u2202gji\u2202xa){displaystyle left{{{h} atop {j;i}}right}=sum _{a}{frac {1}{2}}g^{ha}left({frac {partial g_{ia}}{partial x^{j}}}+{frac {partial g_{ja}}{partial x^{i}}}-{frac {partial g_{ji}}{partial x^{a}}}right)}\u3067\u3042\u308a\u3001\u3053\u308c\u3092\uff08\u7b2c\u4e8c\u7a2e\uff09\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\uff08Christoffel symbol (of the second kind)\uff09\u3068\u547c\u3076\u3002\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f\u3001\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u304b\u3089\u5c0e\u304b\u308c\u305f\u30ec\u30f4\u30a3\u30fb\u30c1\u30f4\u30a3\u30bf\u63a5\u7d9a\u306b\u5bfe\u3059\u308b\u3001\u5ea7\u6a19\u7a7a\u9593\u3067\u306e\u8868\u793a\u5f0f\u3067\u3042\u308b[\u6ce8\u91c8 2][\u6ce8\u91c8 3][\u6ce8\u91c8 4][\u6ce8\u91c8 5]\u3002n\u6b21\u5143\u5fae\u5206\u591a\u69d8\u4f53 M \u4e0a\u306e\u5404\u70b9\u8fd1\u508d\u306b\u5b9a\u307e\u308b\u5ea7\u6a19\u7cfb\u3092 xh (h = 1, 2, …, n) \u3068\u3059\u308b\u3002\u3055\u3089\u306b\u5404\u5ea7\u6a19\u8fd1\u508d\u5185\u306b\u57fa\u672c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb[\u6ce8\u91c8 6]gij(x)=gji(x){displaystyle g_{ij}(x)=g_{ji}(x)}\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3082\u306e\u3068\u3059\u308b[\u6ce8\u91c8 7]\u3002\u306a\u304a\u4ee5\u4e0b\u306b\u304a\u3044\u3066\u306f\u3001\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u306e\u548c\u306e\u898f\u7d04\u3092\u7528\u3044\u308b\u3002Table of Contents\u7b2c\u4e00\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7[\u7de8\u96c6]\u7b2c\u4e8c\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7[\u7de8\u96c6]\u5171\u5909\u5fae\u5206\u306b\u3088\u308b\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u5c0e\u51fa[\u7de8\u96c6]\u7b2c\u4e8c\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f\u4e0b\u306e\u6dfb\u5b57\u306b\u3064\u3044\u3066\u5bfe\u79f0[\u7de8\u96c6]\u7b2c\u4e8c\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f\u30c6\u30f3\u30bd\u30eb\u3067\u306f\u306a\u3044[\u7de8\u96c6]\u6e2c\u5730\u5ea7\u6a19\u7cfb\u306e\u3042\u308b\u4e00\u70b9\u306b\u304a\u3044\u3066\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f 0[\u7de8\u96c6]\u6e2c\u5730\u5ea7\u6a19\u7cfb\u306e\u539f\u70b9\u306b\u304a\u3044\u3066\u5171\u5909\u5fae\u5206\u306f\u901a\u5e38\u306e\u5fae\u5206\u3068\u4e00\u81f4\u3059\u308b[\u7de8\u96c6]\u57fa\u672c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u884c\u5217\u5f0f\u306b\u3088\u308b\u8868\u793a[\u7de8\u96c6]\u4e00\u822c\u76f8\u5bfe\u8ad6\u3078\u306e\u5fdc\u7528[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u6ce8\u91c8[\u7de8\u96c6]\u51fa\u5178[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u7b2c\u4e00\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7[\u7de8\u96c6]\u7b2c\u4e00\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f\u57fa\u672c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u304b\u3089[jk,a]=12(\u2202gja\u2202xk+\u2202gka\u2202xj\u2212\u2202gjk\u2202xa){displaystyle left[j;k,aright]={frac {1}{2}}left({frac {partial g_{ja}}{partial x^{k}}}+{frac {partial g_{ka}}{partial x^{j}}}-{frac {partial g_{jk}}{partial x^{a}}}right)}  \u3068\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u7b2c\u4e8c\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7[\u7de8\u96c6]\u7b2c\u4e8c\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f\u540c\u3058\u304f\u57fa\u672c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u307e\u305f\u306f\u7b2c\u4e00\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u304b\u3089  {ijk}=12gia(\u2202gja\u2202xk+\u2202gka\u2202xj\u2212\u2202gjk\u2202xa)=gia[jk,a]{displaystyle left{{{i} atop {j;k}}right}={frac {1}{2}}g^{ia}left({frac {partial g_{ja}}{partial x^{k}}}+{frac {partial g_{ka}}{partial x^{j}}}-{frac {partial g_{jk}}{partial x^{a}}}right)=g^{ia}left[j;k,aright]}\u3068\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u5171\u5909\u5fae\u5206\u306b\u3088\u308b\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u5c0e\u51fa[\u7de8\u96c6]\u7b2c\u4e8c\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u304c\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u306a\u3044\u4ee3\u308f\u308a\u306b\u3001\u63a5\u7d9a\u306e\u8a18\u53f7 \u0393kij \u3068\u3068\u3082\u306b\u5171\u5909\u5fae\u5206\u304c\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b\u5834\u5408[\u6ce8\u91c8 8]\u3001\u63a5\u7d9a\u306e\u8a18\u53f7\u3068\u3057\u3066\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b[\u6ce8\u91c8 9]\u3002\u4e8c\u968e\u5171\u5909\u30c6\u30f3\u30bd\u30eb Sij \u306e\u5171\u5909\u5fae\u5206\u306f\u5b9a\u7fa9\u3088\u308a\u3001\u2207jSik=\u2202Sik\u2202xj\u2212\u0393jiaSak\u2212\u0393jkaSia{displaystyle nabla _{j}S_{ik}={frac {partial S_{ik}}{partial x^{j}}}-Gamma _{ji}^{a}S_{ak}-Gamma _{jk}^{a}S_{ia}}\uff08\uff12\u968e\u5171\u5909\u30c6\u30f3\u30bd\u30eb\u306e\u5171\u5909\u5fae\u5206\uff09\u3067\u3042\u308b\u3002\u307e\u305f\u3001\u4e8c\u968e\u5171\u5909\u30c6\u30f3\u30bd\u30eb\u3067\u3042\u308b\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53 M \u306e\u57fa\u672c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb gik \u306e\u5171\u5909\u5fae\u5206\u306b\u3064\u3044\u3066\u30ea\u30c3\u30c1\u306e\u88dc\u5b9a\u7406  \u2207jgik=0,\u2207jgik=0{displaystyle nabla _{j}g_{ik}=0,;;nabla _{j}g^{ik}=0}\uff08\u30ea\u30c3\u30c1\u306e\u88dc\u5b9a\u7406\uff09\u304c\u4e00\u822c\u306e\u63a5\u7d9a\u306e\u8a18\u53f7 \u0393kij \u304b\u3089\u5b9a\u7fa9\u3055\u308c\u308b\u5171\u5909\u5fae\u5206\u306b\u3064\u3044\u3066\u3082\u305d\u306e\u307e\u307e\u6210\u308a\u7acb\u3064\u3082\u306e\u3068\u3055\u308c\u3066\u3044\u308b\u3068\u3059\u308b\u3068\u3001\u2207jgik=\u2202gik\u2202xj\u2212\u0393jiagak\u2212\u0393jkagia=0{displaystyle nabla _{j}g_{ik}={frac {partial g_{ik}}{partial x^{j}}}-Gamma _{ji}^{a}g_{ak}-Gamma _{jk}^{a}g_{ia}=0}\u3067\u3042\u308a\u3001\u6dfb\u5b57\u3092\u4e26\u3079\u66ff\u3048\u3001\u88dc\u3046\u3053\u3068\u306b\u3088\u308a\u3001\u4e0a\u5f0f\u3092\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u95a2\u6570\u3068\u3057\u3066\u63a5\u7d9a\u306e\u8a18\u53f7\u306b\u3064\u3044\u3066\u967d\u306b\u89e3\u3044\u3066\u0393kjl=12glm(\u2202gmk\u2202xj+\u2202gmj\u2202xk\u2212\u2202gkj\u2202xm)={lkj}{displaystyle Gamma _{kj}^{l}={frac {1}{2}}g^{lm}left({frac {partial g_{mk}}{partial x^{j}}}+{frac {partial g_{mj}}{partial x^{k}}}-{frac {partial g_{kj}}{partial x^{m}}}right)=left{{{l} atop {k;j}}right}}\u3068\u3001\u63a5\u7d9a\u306e\u8a18\u53f7\u3068\u3057\u3066\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3092\u5c0e\u51fa\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b[\u6ce8\u91c8 10]\u3002\u7b2c\u4e8c\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f\u4e0b\u306e\u6dfb\u5b57\u306b\u3064\u3044\u3066\u5bfe\u79f0[\u7de8\u96c6]\u5b9a\u7fa9\u304b\u3089\u660e\u3089\u304b\u306b{ijk}={ikj}{displaystyle left{{{i} atop {j;k}}right}=left{{{i} atop {k;j}}right}}\u304c\u6210\u308a\u7acb\u3064\u3002\u7b2c\u4e8c\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f\u30c6\u30f3\u30bd\u30eb\u3067\u306f\u306a\u3044[\u7de8\u96c6]\u7b2c\u4e8c\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306b\u3064\u3044\u3066\u3001\u5ea7\u6a19\u7cfb xh \u304b\u3089\u5ea7\u6a19\u7cfb uh (h = 1, 2, …, n) \u3078\u306e\u5909\u6570\u5909\u63db\u3092\u884c\u3046\u3068\u2202xi\u2202ua{abc\u00af}=\u2202xj\u2202ub\u2202xk\u2202uc{ijk}+\u22022xi\u2202ub\u2202uc{displaystyle {frac {partial x^{i}}{partial u^{a}}}left{{overline {{a} atop {b;c}}}right}={frac {partial x^{j}}{partial u^{b}}}{frac {partial x^{k}}{partial u^{c}}}left{{{i} atop {j;k}}right}+{frac {partial ^{2}x^{i}}{partial u^{b}partial u^{c}}}}\u3053\u3053\u3067\u3001\u4e0a\u7dda\u306f u-\u5ea7\u6a19\u7cfb\u306b\u95a2\u3059\u308b\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3067\u3042\u308b\u3053\u3068\u3092\u8868\u3059\u3002\u3068\u306a\u308b\u3002\u3053\u306e\u5f0f\u304b\u3089\u7b2c\u4e8c\u7a2e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u3067\u306f\u306a\u3044\u3053\u3068\u304c\u5224\u308b\u3002\u6e2c\u5730\u5ea7\u6a19\u7cfb\u306e\u3042\u308b\u4e00\u70b9\u306b\u304a\u3044\u3066\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f 0[\u7de8\u96c6]\u66f2\u9762\u4e0a\u306e\u3059\u3079\u3066\u306e\u70b9\u3067\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u304c 0 \u3068\u306a\u308b\u3088\u3046\u306a\u5ea7\u6a19\u7cfb\u304c\u5b58\u5728\u3059\u308b\u306a\u3089\u3070\u3001\u305d\u306e\u66f2\u9762\u306f\u4f38\u7e2e\u3059\u308b\u3053\u3068\u306a\u304f\u5e73\u9762\u4e0a\u306b\u5c55\u958b\u53ef\u80fd\u306a\u3082\u306e\u3060\u3051\u3067\u3042\u308a\u3001\u305d\u308c\u4ee5\u5916\u306e\u5834\u5408\u306b\u306f\u3001\u66f2\u9762\u4e0a\u306e\u3059\u3079\u3066\u306e\u70b9\u3067 {ijk}=0{displaystyle left{{{i} atop {j;k}}right}=0} \u3068\u306a\u308b\u3088\u3046\u306a\u5ea7\u6a19\u7cfb\u306f\u4e00\u822c\u306b\u5b58\u5728\u3057\u306a\u3044[3]\u3002\u305f\u3060\u3057\u3001\u66f2\u9762\u4e0a\u306e\u3042\u308b\u7279\u5b9a\u306e\u4e00\u70b9 xi0 \u3067\u306a\u3089\u3070 {ijk}0=0{displaystyle left{{{i} atop {j;k}}right}_{0}=0} \u3068\u306a\u308b\u3088\u3046\u306a\u5ea7\u6a19\u7cfb\u3092\u3068\u308b\u3053\u3068\u304c\u3067\u304d\u308b[\u6ce8\u91c8 11]\u3002\u3053\u3053\u3067\u3001xi\u2212x0i=ui\u221212{ijk}0ujuk{displaystyle x^{i}-x_{0}^{i}=u^{i}-{frac {1}{2}}left{{{i} atop {j;k}}right}_{0}u^{j}u^{k}}  \u305f\u3060\u3057\u3001u0i=0{displaystyle u_{0}^{i}=0}\u306a\u308b\u5ea7\u6a19\u5909\u63db\u3092\u884c\u3046\u3002\u3053\u306e\u3068\u304d\u3001uh \u3067\u504f\u5fae\u5206\u3092\u884c\u3046\u3068\u2202xi\u2202uh=\u03b4hi\u221212(\u2202{ijk}0\u2202uhujuk+{ijk}0\u03b4hjuk+{ijk}0uj\u03b4hk)=\u03b4hi\u221212\u2202{ijk}0\u2202uhujuk\u2212{ihk}0uk{displaystyle {frac {partial x^{i}}{partial u^{h}}}=delta _{h}^{i}-{frac {1}{2}}left({frac {partial left{{{i} atop {j;k}}right}_{0}}{partial u^{h}}}u^{j}u^{k}+left{{{i} atop {j;k}}right}_{0}delta _{h}^{j}u^{k}+left{{{i} atop {j;k}}right}_{0}u^{j}delta _{h}^{k}right)=delta _{h}^{i}-{frac {1}{2}}{frac {partial left{{{i} atop {j;k}}right}_{0}}{partial u^{h}}}u^{j}u^{k}-left{{{i} atop {h;k}}right}_{0}u^{k}}\u3068\u306a\u308a\u3001\u3055\u3089\u306b ul \u3067\u504f\u5fae\u5206\u3092\u884c\u3046\u3068\u22022xi\u2202ul\u2202uh=\u221212(\u22022{ijk}0\u2202ul\u2202uhujuk+\u2202{ijk}0\u2202uh\u03b4ljuk+\u2202{ijk}0\u2202uhuj\u03b4lk)\u2212\u2202{ihk}0\u2202uluk\u2212{ihk}0\u03b4lk{displaystyle {frac {partial ^{2}x^{i}}{partial u^{l}partial u^{h}}}=-{frac {1}{2}}left({frac {partial ^{2}left{{{i} atop {j;k}}right}_{0}}{partial u^{l}partial u^{h}}}u^{j}u^{k}+{frac {partial left{{{i} atop {j;k}}right}_{0}}{partial u^{h}}}delta _{l}^{j}u^{k}+{frac {partial left{{{i} atop {j;k}}right}_{0}}{partial u^{h}}}u^{j}delta _{l}^{k}right)-{frac {partial left{{{i} atop {h;k}}right}_{0}}{partial u^{l}}}u^{k}-left{{{i} atop {h;k}}right}_{0}delta _{l}^{k}}\u3068\u306a\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001xi = xi0 \u306e\u3068\u304d ui = 0 \u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001(\u2202xi\u2202uh)0=\u03b4hi,(\u22022xi\u2202ul\u2202uh)0=\u2212{ihl}0{displaystyle left({frac {partial x^{i}}{partial u^{h}}}right)_{0}=delta _{h}^{i},;;;left({frac {partial ^{2}x^{i}}{partial u^{l}partial u^{h}}}right)_{0}=-left{{{i} atop {hl}}right}_{0}}\u3092\u5f97\u308b\u3002\u3088\u3063\u3066\u3001\u3042\u308b\u4e00\u70b9 xi0 \u306b\u304a\u3051\u308b\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u5909\u6570\u5909\u63db\u5f0f\u304c(\u2202xi\u2202uk)0{khl\u00af}0=(\u2202xj\u2202uh)0(\u2202xk\u2202ul)0{ijk}0+(\u22022xi\u2202uh\u2202ul)0=\u03b4hj\u03b4lk{ijk}0\u2212{ihl}0=0{displaystyle left({frac {partial x^{i}}{partial u^{k}}}right)_{0}left{{overline {{k} atop {h;l}}}right}_{0}=left({frac {partial x^{j}}{partial u^{h}}}right)_{0}left({frac {partial x^{k}}{partial u^{l}}}right)_{0}left{{{i} atop {j;k}}right}_{0}+left({frac {partial ^{2}x^{i}}{partial u^{h}partial u^{l}}}right)_{0}=delta _{h}^{j}delta _{l}^{k}left{{{i} atop {j;k}}right}_{0}-left{{{i} atop {h;l}}right}_{0}=0}\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001\u03b4ki{khl\u00af}0={ihl\u00af}0=0{displaystyle delta _{k}^{i}left{{overline {{k} atop {h;l}}}right}_{0}=left{{overline {{i} atop {h;l}}}right}_{0}=0}\u3059\u306a\u308f\u3061\u3001\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f\u3042\u308b\u4e00\u70b9 xi0 \u306b\u304a\u3044\u3066\u306f\u5168\u3066\uff10\u3068\u306a\u308b\u3053\u3068\u304c\u5c0e\u304b\u308c\u308b\u3002\u3053\u306e\u3088\u3046\u306a\u5ea7\u6a19\u7cfb\u3092\u3001\u70b9 u0 = 0 \u3092\u539f\u70b9\u3068\u3059\u308b\u6e2c\u5730\u5ea7\u6a19\u7cfb\uff08geodetic coordinate system\uff09\u3068\u547c\u3076\u3002\u306a\u304a\u3001\u6e2c\u5730\u5ea7\u6a19\u306e\u539f\u70b9\u306b\u304a\u3044\u3066\u306f\u3001\u30c6\u30f3\u30bd\u30eb\u306e\u5171\u5909\u5fae\u5206\u3068\u901a\u5e38\u306e\u5fae\u5206\u304c\u4e00\u81f4\u3059\u308b[4]\u3002\u6e2c\u5730\u5ea7\u6a19\u7cfb\u306e\u539f\u70b9\u306b\u304a\u3044\u3066\u5171\u5909\u5fae\u5206\u306f\u901a\u5e38\u306e\u5fae\u5206\u3068\u4e00\u81f4\u3059\u308b[\u7de8\u96c6]\u4e00\u968e\u5171\u5909\u30c6\u30f3\u30bd\u30eb\u3092 wi \u3068\u3059\u308b\u3068\u304d\u3001\u305d\u306e\u5171\u5909\u5fae\u5206\u306f\u2207jwi=\u2202wi\u2202xj\u2212{aji}wa{displaystyle nabla _{j}w_{i}={frac {partial w_{i}}{partial x^{j}}}-left{{{a} atop {j;i}}right}w_{a}}\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u5ea7\u6a19\u7cfb (xh) \u3092\u6e2c\u5730\u5ea7\u6a19\u7cfb (uh) \u3078\u5ea7\u6a19\u5909\u63db\u3059\u308b\u3068\u3001\u305d\u306e\u539f\u70b9\u306b\u304a\u3044\u3066 {ihl\u00af}0=0{displaystyle left{{overline {{i} atop {h;l}}}right}_{0}=0} \u3068\u306a\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001wi \u306e\u5171\u5909\u5fae\u5206\u306f ui = 0 \u306b\u304a\u3044\u3066\u3001(\u2207jwi)0=(\u2202wi\u2202uj)0{displaystyle left(nabla _{j}w_{i}right)_{0}=left({frac {partial w_{i}}{partial u^{j}}}right)_{0}}\u3068\u3001\u5171\u5909\u5fae\u5206\u3068\u901a\u5e38\u306e\u5fae\u5206\u304c\u4e00\u81f4\u3059\u308b\u3002\u57fa\u672c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u884c\u5217\u5f0f\u306b\u3088\u308b\u8868\u793a[\u7de8\u96c6]n \u6b21\u5143\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u306e\u57fa\u672c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb gij \u306f n \u00d7 n\u306e\u6b63\u65b9\u884c\u5217\u3067\u3042\u308b\u3068\u898b\u306a\u305b\u308b\u3053\u3068\u304b\u3089\u305d\u306e\u884c\u5217\u5f0f gg=det(gij)=|g11g12\u22efg21g22\u22ef\u22ee\u22ee\u22f1|{displaystyle g=det(g_{ij})=left|{begin{array}{ccc}g_{11}&g_{12}&cdots g_{21}&g_{22}&cdots vdots &vdots &ddots end{array}}right|}\u3092\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u3053\u3067\u3001gij \u306e\u4f59\u56e0\u5b50\u884c\u5217\u3092 Gij \u3068\u3057\u3001g \u3092 xk \u3067\u504f\u5fae\u5206\u3059\u308b\u3068\u2202g\u2202xk=\u2202g\u2202gij\u2202gij\u2202xk=Gij\u2202gij\u2202xk{displaystyle {frac {partial g}{partial x^{k}}}={frac {partial g}{partial g_{ij}}}{frac {partial g_{ij}}{partial x^{k}}}=G_{ij}{frac {partial g_{ij}}{partial x^{k}}}}\u3068\u306a\u308b[\u6ce8\u91c8 12]\u3002\u3055\u3089\u306b\u4f59\u56e0\u5b50\u884c\u5217\u3092\u884c\u5217\u5f0f\u3067\u5272\u3063\u305f\u3082\u306e\u306f\u9006\u884c\u5217\u3068\u306a\u308b\u304c\u3001\u305d\u308c\u306f\u53cd\u5909\u7248\u306e\u57fa\u672c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306b\u4ed6\u306a\u3089\u306a\u3044\u3002\u3064\u307e\u308a\u3001Gij = ggij\u3002\u3088\u3063\u3066\u2202g\u2202xk=ggij\u2202gij\u2202xk=ggij(gia{ajk}+gaj{aik})=g(\u03b4aj{ajk}+\u03b4ai{aik})=g({aak}+{aak})=2g{aak}{displaystyle {begin{aligned}{frac {partial g}{partial x^{k}}}&=gg^{ij}{frac {partial g_{ij}}{partial x^{k}}}=gg^{ij}left(g_{ia}left{{{a} atop {j;k}}right}+g_{aj}left{{{a} atop {i;k}}right}right)&=gleft(delta _{a}^{j}left{{{a} atop {j;k}}right}+delta _{a}^{i}left{{{a} atop {i;k}}right}right)=gleft(left{{{a} atop {a;k}}right}+left{{{a} atop {a;k}}right}right)&=2gleft{{{a} atop {a;k}}right}end{aligned}}}\u3088\u3063\u3066{aak}={aka}=121g\u2202g\u2202xk=\u2202log\u2061g\u2202xk=1g\u2202g\u2202xk{displaystyle left{{{a} atop {a;k}}right}=left{{{a} atop {k;a}}right}={frac {1}{2}}{frac {1}{g}}{frac {partial g}{partial x^{k}}}={frac {partial log {sqrt {g}}}{partial x^{k}}}={frac {1}{sqrt {g}}}{frac {partial {sqrt {g}}}{partial x^{k}}}}\u304c\u5f97\u3089\u308c\u308b\u3002\u4e00\u822c\u76f8\u5bfe\u8ad6\u3078\u306e\u5fdc\u7528[\u7de8\u96c6]\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306f\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u306e\u4e00\u822c\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066\u983b\u7e41\u306b\u7528\u3044\u3089\u308c\u308b\u3002\u4e00\u822c\u76f8\u5bfe\u8ad6\u306f\u6642\u7a7a\u3092\u3001\u30ec\u30f4\u30a3-\u30c1\u30f4\u30a3\u30bf\u63a5\u7d9a\u3092\u5099\u3048\u305f\u3001\u6e7e\u66f2\u3057\u305f 4-\u6b21\u5143\u30ed\u30fc\u30ec\u30f3\u30c4\u591a\u69d8\u4f53\u306b\u3088\u3063\u3066\u8868\u73fe\u3059\u308b\u3002\uff08\u7269\u4f53\u306e\u5b58\u5728\u306b\u3088\u3063\u3066\u6642\u7a7a\u306e\u5f62\u72b6\u3092\u6c7a\u5b9a\u3059\u308b\u3068\u3044\u3046\uff09\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u306e\u5834\u306e\u65b9\u7a0b\u5f0f\u306f\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u3092\u542b\u307f\u3001\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u672c\u8cea\u7684\u3067\u3042\u308b\u3002\u4e00\u65e6\u5f62\u72b6\u304c\u6c7a\u5b9a\u3055\u308c\u305f\u306a\u3089\u3070\u3001\u7c92\u5b50\u3068\u5149\u7dda\u306e\u8ecc\u8de1\u306f\uff08\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u304c\u967d\u306b\u73fe\u308c\u308b\uff09\u6e2c\u5730\u7684\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u306b\u3088\u3063\u3066\u8a08\u7b97\u3067\u304d\u308b\u3002\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u6ce8\u91c8[\u7de8\u96c6]^ 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\u4ee5\u5f8c\u3001xi=x0i{displaystyle x^{i}=x_{0}^{i}} \u306b\u304a\u3051\u308b\u5024\u3092\u793a\u3057\u3066\u3044\u308b\u3082\u306e\u306b\u3064\u3044\u3066\u306f\u53f3\u4e0b\u306b 0 \u3092\u3064\u3051\u308b\u3002^ \u4f59\u56e0\u5b50\u884c\u5217\u3068\u884c\u5217\u5f0f\u306e\u9593\u306b\u2202g\u2202gij=Gij{displaystyle {frac {partial g}{partial g_{ij}}}=G_{ij}}\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u305f\u3081[5]\u3002\u51fa\u5178[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki7\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/3851#breadcrumbitem","name":"\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7 – Wikipedia"}}]}]