[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki6\/archives\/6324#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki6\/archives\/6324","headline":"\u52fe\u914d (\u30d9\u30af\u30c8\u30eb\u89e3\u6790) – Wikipedia","name":"\u52fe\u914d (\u30d9\u30af\u30c8\u30eb\u89e3\u6790) – Wikipedia","description":"\u4e8c\u3064\u306e\u56f3\u3067\u3001\u767d\u3068\u9ed2\u3067\u8868\u3055\u308c\u308b\u30b9\u30ab\u30e9\u30fc\u5834\u306f\u9ed2\u306e\u65b9\u304c\u5024\u304c\u9ad8\u304f\u3001\u5bfe\u5fdc\u3059\u308b\u52fe\u914d\u306f\u9752\u77e2\u5370\u3067\u8868\u3055\u308c\u3066\u3044\u308b\u3002 \u30d9\u30af\u30c8\u30eb\u89e3\u6790\u306b\u304a\u3051\u308b\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d\uff08\u3053\u3046\u3070\u3044\u3001\u82f1: gradient; \u30b0\u30e9\u30c7\u30a3\u30a8\u30f3\u30c8\uff09\u306f\u3001\u5404\u70b9\u306b\u304a\u3044\u3066\u305d\u306e\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u5909\u5316\u7387\u304c\u6700\u5927\u3068\u306a\u308b\u65b9\u5411\u3078\u306e\u5909\u5316\u7387\u306e\u5024\u3092\u5927\u304d\u3055\u306b\u3082\u3064\u30d9\u30af\u30c8\u30eb\u3092\u5bfe\u5fdc\u3055\u305b\u308b\u30d9\u30af\u30c8\u30eb\u5834\u3067\u3042\u308b\u3002\u7c21\u5358\u306b\u8a00\u3048\u3070\u3001\u4efb\u610f\u306e\u91cf\u306e\u7a7a\u9593\u306b\u304a\u3051\u308b\u5909\u4f4d\u3092\u3001\u50be\u304d\u3068\u3057\u3066\u8868\u73fe\uff08\u4f8b\u3048\u3070\u56f3\u793a\uff09\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304c\u3001\u305d\u3053\u3067\u52fe\u914d\u306f\u3053\u306e\u50be\u304d\u306e\u5411\u304d\u3084\u50be\u304d\u306e\u304d\u3064\u3055\u3092\u8868\u3057\u3066\u3044\u308b\u3002 \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u4e0a\u306e\u95a2\u6570\u306e\u52fe\u914d\u3092\u3001\u5225\u306a\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u5024\u3092\u6301\u3064\u5199\u50cf\u306b\u5bfe\u3057\u3066\u4e00\u822c\u5316\u3057\u305f\u3082\u306e\u306f\u3001\u30e4\u30b3\u30d3\u884c\u5217\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u3055\u3089\u306b\u4e00\u822c\u5316\u3057\u3066\u3001\u30d0\u30ca\u30c3\u30cf\u7a7a\u9593\u304b\u3089\u5225\u306e\u30d0\u30ca\u30c3\u30cf\u7a7a\u9593\u3078\u306e\u5199\u50cf\u306e\u52fe\u914d\u3092\u30d5\u30ec\u30b7\u30a7\u5fae\u5206\u3092\u901a\u3058\u3066\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002 2 \u5909\u6570\u95a2\u6570 f(x,y)=xe\u2212x2\u2212y2{displaystyle f(x,y)=xe^{-x^{2}-y^{2}}} \u306e\u52fe\u914d\u3092\u64ec\u8272\u63cf\u753b\u3055\u308c\u305f\u95a2\u6570\u306e\u4e0a\u306e\u9752\u77e2\u5370\u3068\u3057\u3066\u63cf\u753b\u3057\u305f\u3082\u306e \u4e00\u3064\u306e\u90e8\u5c4b\u3092\u3001\u305d\u306e\u90e8\u5c4b\u306e\u6e29\u5ea6\u3092\u4e0e\u3048\u308b\u30b9\u30ab\u30e9\u30fc\u5834 T \u3068\u8003\u3048\u308c\u3070\u3001\u5404\u70b9 (x, y, z) \u306b\u304a\u3051\u308b\u6e29\u5ea6\u3092 T(x, y, z)","datePublished":"2019-11-28","dateModified":"2019-11-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki6\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki6\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?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\/0\/0f\/Gradient2.svg\/300px-Gradient2.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/0f\/Gradient2.svg\/300px-Gradient2.svg.png","height":"149","width":"300"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki6\/archives\/6324","about":["Wikipedia"],"wordCount":7820,"articleBody":"  \u4e8c\u3064\u306e\u56f3\u3067\u3001\u767d\u3068\u9ed2\u3067\u8868\u3055\u308c\u308b\u30b9\u30ab\u30e9\u30fc\u5834\u306f\u9ed2\u306e\u65b9\u304c\u5024\u304c\u9ad8\u304f\u3001\u5bfe\u5fdc\u3059\u308b\u52fe\u914d\u306f\u9752\u77e2\u5370\u3067\u8868\u3055\u308c\u3066\u3044\u308b\u3002\u30d9\u30af\u30c8\u30eb\u89e3\u6790\u306b\u304a\u3051\u308b\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d\uff08\u3053\u3046\u3070\u3044\u3001\u82f1: gradient; \u30b0\u30e9\u30c7\u30a3\u30a8\u30f3\u30c8\uff09\u306f\u3001\u5404\u70b9\u306b\u304a\u3044\u3066\u305d\u306e\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u5909\u5316\u7387\u304c\u6700\u5927\u3068\u306a\u308b\u65b9\u5411\u3078\u306e\u5909\u5316\u7387\u306e\u5024\u3092\u5927\u304d\u3055\u306b\u3082\u3064\u30d9\u30af\u30c8\u30eb\u3092\u5bfe\u5fdc\u3055\u305b\u308b\u30d9\u30af\u30c8\u30eb\u5834\u3067\u3042\u308b\u3002\u7c21\u5358\u306b\u8a00\u3048\u3070\u3001\u4efb\u610f\u306e\u91cf\u306e\u7a7a\u9593\u306b\u304a\u3051\u308b\u5909\u4f4d\u3092\u3001\u50be\u304d\u3068\u3057\u3066\u8868\u73fe\uff08\u4f8b\u3048\u3070\u56f3\u793a\uff09\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304c\u3001\u305d\u3053\u3067\u52fe\u914d\u306f\u3053\u306e\u50be\u304d\u306e\u5411\u304d\u3084\u50be\u304d\u306e\u304d\u3064\u3055\u3092\u8868\u3057\u3066\u3044\u308b\u3002  \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u4e0a\u306e\u95a2\u6570\u306e\u52fe\u914d\u3092\u3001\u5225\u306a\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u5024\u3092\u6301\u3064\u5199\u50cf\u306b\u5bfe\u3057\u3066\u4e00\u822c\u5316\u3057\u305f\u3082\u306e\u306f\u3001\u30e4\u30b3\u30d3\u884c\u5217\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u3055\u3089\u306b\u4e00\u822c\u5316\u3057\u3066\u3001\u30d0\u30ca\u30c3\u30cf\u7a7a\u9593\u304b\u3089\u5225\u306e\u30d0\u30ca\u30c3\u30cf\u7a7a\u9593\u3078\u306e\u5199\u50cf\u306e\u52fe\u914d\u3092\u30d5\u30ec\u30b7\u30a7\u5fae\u5206\u3092\u901a\u3058\u3066\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002    2 \u5909\u6570\u95a2\u6570 f(x,y)=xe\u2212x2\u2212y2{displaystyle f(x,y)=xe^{-x^{2}-y^{2}}} \u306e\u52fe\u914d\u3092\u64ec\u8272\u63cf\u753b\u3055\u308c\u305f\u95a2\u6570\u306e\u4e0a\u306e\u9752\u77e2\u5370\u3068\u3057\u3066\u63cf\u753b\u3057\u305f\u3082\u306e\u4e00\u3064\u306e\u90e8\u5c4b\u3092\u3001\u305d\u306e\u90e8\u5c4b\u306e\u6e29\u5ea6\u3092\u4e0e\u3048\u308b\u30b9\u30ab\u30e9\u30fc\u5834 T \u3068\u8003\u3048\u308c\u3070\u3001\u5404\u70b9 (x, y, z) \u306b\u304a\u3051\u308b\u6e29\u5ea6\u3092 T(x, y, z) \u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\uff08\u3053\u3053\u3067\u306f\u6e29\u5ea6\u306f\u6642\u9593\u5909\u5316\u3092\u8d77\u3053\u3055\u306a\u3044\u3082\u306e\u3068\u4eee\u5b9a\u3059\u308b\uff09\u3002\u90e8\u5c4b\u306e\u5404\u70b9\u306b\u304a\u3044\u3066\u3001T \u306e\u52fe\u914d\u306f\u6700\u3082\u65e9\u304f\u6e29\u5ea6\u304c\u4e0a\u6607\u3059\u308b\u65b9\u5411\u3092\u6307\u3057\u3001\u305d\u306e\u5927\u304d\u3055\u306f\u305d\u306e\u65b9\u5411\u3067\u3069\u308c\u307b\u3069\u65e9\u304f\u6e29\u5ea6\u304c\u4e0a\u6607\u3059\u308b\u304b\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u70b9 (x, y) \u306b\u304a\u3051\u308b\u6d77\u629c\u304c H(x, y) \u3067\u3042\u308b\u3088\u3046\u306a\u66f2\u9762\u3092\u8003\u3048\u308b\u3002\u3042\u308b\u70b9\u306b\u304a\u3051\u308b H \u306e\u52fe\u914d\u306f\u3001\u305d\u306e\u70b9\u306b\u304a\u3044\u3066\u3082\u3063\u3068\u3082\u50be\u304d\uff08\u7e26\u65ad\u52fe\u914d\uff09\u304c\u6025\u5cfb\u3067\u3042\u308b\u3088\u3046\u306a\u65b9\u5411\u3092\u6307\u3059\u30d9\u30af\u30c8\u30eb\u3067\u3001\u305d\u306e\u5927\u304d\u3055\u306f\u305d\u306e\u70b9\u3067\u306e\u3082\u3063\u3068\u3082\u6025\u5cfb\u306a\u50be\u304d\u306e\u5024\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u308b\u3002\u52fe\u914d\u304b\u3089\u306f\u3001\u5185\u7a4d\u3092\u53d6\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u6700\u3082\u5909\u5316\u306e\u5927\u304d\u3044\u65b9\u5411\u4ee5\u5916\u306e\u65b9\u5411\u3067\u3082\u3001\u305d\u306e\u30b9\u30ab\u30e9\u30fc\u5834\u304c\u3069\u308c\u307b\u3069\u5909\u5316\u3059\u308b\u304b\u3092\u77e5\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4e18\u9675\u306e\u3082\u3063\u3068\u3082\u6025\u5cfb\u306a\u50be\u304d\u304c 40% \u3068\u3059\u308b\u3068\u3001\u305d\u306e\u4e18\u9675\u3092\u771f\u3063\u76f4\u3050\u4e0a\u308b\u9053\u306e\u6700\u3082\u6025\u5cfb\u306a\u50be\u304d\u3082 40% \u3068\u306a\u308b\u304c\u3001\u4ee3\u308f\u308a\u306b\u9069\u5f53\u306a\u89d2\u5ea6\u3092\u3064\u3051\u3066\u4e18\u9675\u3092\u3050\u308b\u308a\u3068\u56de\u308b\u9053\u3092\u884c\u3051\u3070\u3001\u50be\u304d\u306f\u3082\u3063\u3068\u7de9\u3084\u304b\u306b\u306a\u308b\u306f\u305a\u3067\u3042\u308b\u3002\u4f8b\u3048\u3070\u3001\u9053\u3068\u771f\u3063\u76f4\u3050\u5742\u3092\u4e0a\u304c\u308b\u65b9\u5411\u3068\u306e\u9593\u306e\u89d2\u5ea6\u304c\u3001\u6c34\u5e73\u9762\u306b\u6295\u5f71\u3057\u3066 60\u00b0 \u306b\u306a\u3063\u3066\u3044\u308c\u3070\u3001\u305d\u306e\u9053\u306e\u6700\u3082\u6025\u5cfb\u306a\u50be\u304d\u306f 20%\uff0840% \u306b 60\u00b0 \u306e\u4f59\u5f26\u3092\u639b\u3051\u305f\u3082\u306e\uff09\u306b\u306a\u308b\u306f\u305a\u3067\u3042\u308b\u3002  \u3053\u306e\u8003\u5bdf\u3092\u6570\u5b66\u7684\u306b\u8ff0\u3079\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002\u4e18\u9675\u306e\u9ad8\u3055\u3092\u8868\u3059\u95a2\u6570 H \u304c\u5fae\u5206\u53ef\u80fd\u3067\u3042\u308b\u3082\u306e\u3068\u3059\u308c\u3070\u3001H \u306e\u52fe\u914d\u306b\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3068\u306e\u5185\u7a4d\u3092\u3068\u308c\u3070\u3001\u305d\u306e\u30d9\u30af\u30c8\u30eb\u306e\u65b9\u5411\u3078\u306e\u4e18\u9675\u306e\u50be\u304d\u304c\u5f97\u3089\u308c\u308b\u3002\u3082\u3046\u5c11\u3057\u5f62\u5f0f\u7684\u306b\u66f8\u304f\u3068\u3001H \u304c\u53ef\u5fae\u5206\u3067\u3042\u308b\u3068\u304d\u3001H \u306e\u52fe\u914d\u3068\u4e0e\u3048\u3089\u308c\u305f\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3068\u306e\u5185\u7a4d\u306f\u3001\u305d\u306e\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u306e\u65b9\u5411\u3078\u306e H \u306e\u65b9\u5411\u5fae\u5206\u306b\u7b49\u3057\u3044\u3002  \u95a2\u6570 f(x, y) = \u2212(cos2x + cos2y)2 \u306e\u52fe\u914d\u3092\u3001\u5e95\u9762\u306b\u5c04\u5f71\u3057\u305f\u30d9\u30af\u30c8\u30eb\u5834\u3068\u3057\u3066\u63cf\u3044\u305f\u3082\u306e\u30b9\u30ab\u30e9\u30fc\u95a2\u6570 f\u2009(x1, x2, x3, …, xn) \u306e\u52fe\u914d\uff08\u52fe\u914d\u30d9\u30af\u30c8\u30eb\u5834\uff09\u306f\u3001\u30d9\u30af\u30c8\u30eb\u5fae\u5206\u4f5c\u7528\u7d20 \u2207\uff08\u30ca\u30d6\u30e9\u8a18\u53f7\uff09\u3092\u7528\u3044\u3066\u3001\u2207 f \u3042\u308b\u3044\u306f \u2207\u2192 f \u3068\u66f8\u304b\u308c\u308b\u3002\u52fe\u914d\u3092 grad f \u3068\u66f8\u304f\u3053\u3068\u3082\u5e83\u304f\u884c\u308f\u308c\u3066\u3044\u308b\u3002f \u306e\u52fe\u914d \u2207 f \u3068\u306f\u3001\u5404\u70b9 x \u306b\u304a\u3044\u3066\u4efb\u610f\u306e\u7a7a\u9593\u30d9\u30af\u30c8\u30eb v \u3068\u306e\u30c9\u30c3\u30c8\u7a4d\u304c f \u306e v \u306b\u6cbf\u3046\u65b9\u5411\u5fae\u5206\u306b\u4e00\u81f4\u3059\u308b\u30d9\u30af\u30c8\u30eb\u5834\u3068\u3057\u3066\u4e00\u610f\u7684\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u5f0f\u3067\u66f8\u3051\u3070\u3001\u52fe\u914d\u306f(\u2207f(x))\u22c5v=Dvf(x){displaystyle (nabla f(x))cdot mathbf {v} =D_{mathbf {v} }f(x)}\u3067\u6c7a\u5b9a\u3055\u308c\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u3001\u52fe\u914d\u306f\u6210\u5206\u304c f \u306e\u504f\u5fae\u5206\u3067\u4e0e\u3048\u3089\u308c\u308b\u30d9\u30af\u30c8\u30eb\u5834\u2207f=\u2202f\u2202x1e1+\u22ef+\u2202f\u2202xnen{displaystyle nabla f={frac {partial f}{partial x_{1}}}mathbf {e} _{1}+cdots +{frac {partial f}{partial x_{n}}}mathbf {e} _{n}}\u3067\u3042\u308b\u3002\u305f\u3060\u3057\u3001ei \u306f\u3053\u306e\u5ea7\u6a19\u7cfb\u306e\u76ee\u5730\u3092\u63cf\u304f\u76f4\u4ea4\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3002\u95a2\u6570\u304c\u4f8b\u3048\u3070\u6642\u9593\u306e\u3088\u3046\u306a\u30d1\u30e9\u30e1\u30fc\u30bf\u306b\u3082\u4f9d\u5b58\u3059\u308b\u5834\u5408\u3001\u305d\u306e\u52fe\u914d\u3068\u306f\u5358\u306b\u7a7a\u9593\u6210\u5206\u306e\u5fae\u5206\u306e\u307f\u304b\u3089\u306a\u308b\u30d9\u30af\u30c8\u30eb\u3092\u6307\u3059\u3053\u3068\u3082\u591a\u3044\u3002\u4e09\u6b21\u5143\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u3053\u308c\u306f\u3001i, j, k \u3092\u57fa\u672c\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3068\u3057\u3066\u2202f\u2202xi+\u2202f\u2202yj+\u2202f\u2202zk{displaystyle {frac {partial f}{partial x}}mathbf {i} +{frac {partial f}{partial y}}mathbf {j} +{frac {partial f}{partial z}}mathbf {k} }\u3068\u66f8\u3051\u308b\u3002\u4f8b\u3048\u3070\u95a2\u6570 f (x, y, z) = 2x + 3y2 \u2212 sin(z) \u306e\u52fe\u914d\u306f \u2207f = 2i + 6yj \u2212 cos(z)k \u3068\u306a\u308b\u3002\u5fdc\u7528\u306b\u969b\u3057\u3066\u3001\u52fe\u914d\u3092\u305d\u306e\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u306b\u95a2\u3059\u308b\u6210\u5206\u306e\u6210\u3059\u884c\u30d9\u30af\u30c8\u30eb\u3082\u3057\u304f\u306f\u5217\u30d9\u30af\u30c8\u30eb\u3068\u3057\u3066\u8868\u793a\u3059\u308b\u3053\u3068\u3082\u3042\u308b\u3002Table of Contents\u52fe\u914d\u3068\u5168\u5fae\u5206\u306e\u95a2\u4fc2[\u7de8\u96c6] \u5199\u50cf\u306e\u7dda\u578b\u8fd1\u4f3c[\u7de8\u96c6]\u5168\u5fae\u5206[\u7de8\u96c6]\u5fae\u5206\u3068\u3057\u3066\u306e\u6027\u8cea[\u7de8\u96c6]\u66f4\u306a\u308b\u6027\u8cea\u3068\u5fdc\u7528[\u7de8\u96c6]\u7b49\u4f4d\u96c6\u5408[\u7de8\u96c6]\u4fdd\u5b58\u30d9\u30af\u30c8\u30eb\u5834\u3068\u52fe\u914d\u5b9a\u7406[\u7de8\u96c6]\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53[\u7de8\u96c6]\u5186\u7b52\u5ea7\u6a19\u7cfb\u304a\u3088\u3073\u7403\u9762\u5ea7\u6a19\u7cfb\u3067\u306e\u8868\u793a[\u7de8\u96c6]\u30d9\u30af\u30c8\u30eb\u5024\u95a2\u6570\u306e\u52fe\u914d[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]\u52fe\u914d\u3068\u5168\u5fae\u5206\u306e\u95a2\u4fc2[\u7de8\u96c6] \u5199\u50cf\u306e\u7dda\u578b\u8fd1\u4f3c[\u7de8\u96c6]\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593 Rn \u304b\u3089 R \u3078\u306e\u95a2\u6570 f \u306e\u3001\u4efb\u610f\u306e\u70b9 x0 \u2208 Rn \u306b\u304a\u3051\u308b\u52fe\u914d\u306f\u3001x0 \u306b\u304a\u3051\u308b f \u306e\u6700\u9069\u7dda\u578b\u8fd1\u4f3c\u3092\u7279\u5fb4\u3065\u3051\u308b\u3082\u306e\u3067\u3042\u308b\u3002\u5373\u3061\u3001\u7dda\u578b\u8fd1\u4f3c\u5f0f\u306f x0 \u306b\u307b\u3069\u8fd1\u3044 x \u306b\u5bfe\u3057\u3066f(x)\u2248f(x0)+(\u2207f)x0\u22c5(x\u2212x0){displaystyle f(x)approx f(x_{0})+(nabla f)_{x_{0}}cdot (x-x_{0})}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u3053\u3053\u3067 (\u2207 f\u2009)x0 \u306f x0 \u306b\u304a\u3051\u308b f \u306e\u52fe\u914d\u3067\u3042\u308a\u3001\u4e2d\u9ed2\u306f Rn \u306b\u304a\u3051\u308b\u30c9\u30c3\u30c8\u7a4d\u3067\u3042\u308b\u3002\u3053\u306e\u5f0f\u306f f \u306e x0 \u306b\u304a\u3051\u308b\u591a\u5909\u6570\u30c6\u30a4\u30e9\u30fc\u7d1a\u6570\u5c55\u958b\u306e\u6700\u521d\u306e 2 \u9805\u3092\u3068\u3063\u305f\u3082\u306e\u3068\u540c\u5024\u3067\u3042\u308b\u3002\u5168\u5fae\u5206[\u7de8\u96c6]\u95a2\u6570 f: Rn \u2192 R \u306e\u70b9 x \u2208 Rn \u306b\u304a\u3051\u308b\u6700\u9069\u7dda\u578b\u8fd1\u4f3c\u306f\u3001Rn \u304b\u3089 R \u3078\u306e\u7dda\u578b\u6c4e\u95a2\u6570\u3067\u3042\u308a\u3001x \u306b\u304a\u3051\u308b f \u306e\u5fae\u5206\u4fc2\u6570\u3042\u308b\u3044\u306f\u5168\u5fae\u5206\u4fc2\u6570 dfx, Df(x) \u3068\u547c\u3070\u308c\u308b\u3002\u5f93\u3063\u3066\u52fe\u914d\u306f\u5168\u5fae\u5206\u4fc2\u6570\u3068\u306e\u9593\u306b(\u2207f)x\u22c5v=dfx(v)(v\u2208Rn){displaystyle (nabla f)_{x}cdot v=df_{x}(v)quad (vin mathbb {R} ^{n})}\u306a\u308b\u95a2\u4fc2\u3067\u7d50\u3070\u308c\u3066\u3044\u308b\u3002x \u3092 dfx \u3078\u5199\u3059\u95a2\u6570 df \u306f f \u306e\u5168\u5fae\u5206\u307e\u305f\u306f\u5168\u5c0e\u95a2\u6570\u3068\u547c\u3070\u308c\u3001\u3053\u308c\u3092\u4e00\u6b21\u5fae\u5206\u5f62\u5f0f\u3068\u89e3\u91c8\u3057\u3066 f \u306e\u5916\u5fae\u5206\u3068\u898b\u505a\u3059\u3053\u3068\u3082\u3067\u304d\u308b\u3002Rn \u3092\uff08\u9577\u3055 n \u3067\u6210\u5206\u304c\u5b9f\u6570\u5024\u306e\uff09\u5217\u30d9\u30af\u30c8\u30eb\u5168\u4f53\u306e\u6210\u3059\u7a7a\u9593\u3068\u898b\u308b\u3068\u304d\u3001\u5168\u5fae\u5206 df \u3092\u884c\u30d9\u30af\u30c8\u30ebdf=(\u2202f\u2202x1,\u2026,\u2202f\u2202xn){displaystyle df=left({frac {partial f}{partial x_{1}}},dots ,{frac {partial f}{partial x_{n}}}right)}\u3068\u898b\u505a\u3057\u3066\u3001dfx(v) \u3092\u884c\u5217\u306e\u7a4d\u3067\u4e0e\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u306e\u3068\u304d\u3001\u52fe\u914d\u306f\u5217\u30d9\u30af\u30c8\u30eb\u2207f=t(df){displaystyle nabla f={}^{t}(df)}\u306b\u5bfe\u5fdc\u3059\u308b\u3002\u5fae\u5206\u3068\u3057\u3066\u306e\u6027\u8cea[\u7de8\u96c6]U \u3092 Rn \u306e\u958b\u96c6\u5408\u3068\u3057\u3001\u95a2\u6570 f\u00a0: U \u2192 R \u304c\u30d5\u30ec\u30b7\u30a7\u5fae\u5206\u53ef\u80fd\u3068\u3059\u308b\u3068\u3001f \u306e\u5168\u5fae\u5206\u306f f \u306e\u30d5\u30ec\u30b7\u30a7\u5c0e\u95a2\u6570\u3067\u3042\u308a\u3001\u5f93\u3063\u3066 \u2207f \u306f U \u304b\u3089\u7a7a\u9593 R \u3078\u306e\u5199\u50cf\u3067limh\u21920\u2016f(x+h)\u2212f(x)\u2212\u2207f(x)\u22c5h\u2016\u2016h\u2016=0{displaystyle lim _{hto 0}{frac {|f(x+h)-f(x)-nabla f(x)cdot h|}{|h|}}=0}\u3092\u6e80\u305f\u3059\u3082\u306e\u3067\u3042\u308b\uff08\u4e2d\u9ed2\u306f\u30c9\u30c3\u30c8\u7a4d\uff09\u3002\u3053\u306e\u5e30\u7d50\u3068\u3057\u3066\u3001\u52fe\u914d\u304c\u901a\u5e38\u306e\u5fae\u5206\u304c\u6301\u3064\u5fae\u5206\u6cd5\u5247\u3092\u6e80\u8db3\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u7dda\u578b\u6027\u4e8c\u3064\u306e\u5b9f\u6570\u5024\u95a2\u6570 f, g \u304c\u70b9 a \u2208 Rn \u306b\u304a\u3044\u3066\u5fae\u5206\u53ef\u80fd\u3067\u3001\u03b1, \u03b2 \u304c\u5b9f\u5b9a\u6570\u3067\u3042\u308b\u3068\u304d\u3001\u7dda\u578b\u7d50\u5408 \u03b1f + \u03b2g \u306f a \u306b\u304a\u3044\u3066\u5fae\u5206\u53ef\u80fd\u3067\u3042\u308a\u3001\u3055\u3089\u306b \u2207(\u03b1f + \u03b2g)(a) = \u03b1\u2207f(a) + \u03b2\u2207g(a) \u3092\u6e80\u305f\u3059\u3068\u3044\u3046\u610f\u5473\u3067\u3001\u52fe\u914d\u306f\u7dda\u578b\u3067\u3042\u308b\u3002\u7a4d\u306e\u5fae\u5206\u6cd5\u5247f \u3068 g \u304c\u5b9f\u6570\u5024\u95a2\u6570\u3067\u70b9 a \u2208 Rn \u306b\u304a\u3044\u3066\u5fae\u5206\u53ef\u80fd\u306a\u3089\u3070\u3001\u305d\u308c\u3089\u306e\u7a4d (fg)(x) = f(x)g(x) \u306f a \u306b\u304a\u3044\u3066\u5fae\u5206\u53ef\u80fd\u3067\u3001\u2207(fg)(a) = f(a)\u2207g(a) + g(a)\u2207f(a) \u306a\u308b\u7a4d\u306e\u6cd5\u5247\u3092\u6e80\u305f\u3059\u3002\u9023\u9396\u5f8bRn \u306e\u90e8\u5206\u96c6\u5408 A \u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u5b9f\u6570\u5024\u95a2\u6570 f\u00a0: A \u2192 R \u304c\u70b9 a \u306b\u304a\u3044\u3066\u5fae\u5206\u53ef\u80fd\u3068\u3059\u308b\u3002\u52fe\u914d\u306b\u95a2\u3059\u308b\u9023\u9396\u5f8b\u306b\u306f 2 \u3064\u306e\u5f62\u304c\u5b58\u5728\u3059\u308b\u30021 \u3064\u76ee\u306f\u3001\u95a2\u6570 g \u3092\u66f2\u7dda\u306e\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3001\u5373\u3061 R \u306e\u90e8\u5206\u96c6\u5408 I \u304b\u3089 Rn \u3078\u306e\u95a2\u6570 g\u00a0: I \u2192 Rn \u3068\u3059\u308b\u3068\u304d\u3001g \u304c g(c) = a \u306a\u308b I \u306e\u70b9 c \u3067\u5fae\u5206\u53ef\u80fd\u306a\u3089\u3070\u3001(f\u25cbg)‘(c) = \u2207f(a) \u00b7 g‘(c) \u304c\u6210\u7acb\u3059\u308b\u3068\u3044\u3046\u3082\u306e\u3002\u305f\u3060\u3057 \u25cb \u306f\u5199\u50cf\u306e\u5408\u6210\u3067\u3042\u308b\u3002\u3088\u308a\u4e00\u822c\u306b\u3001I \u2282 Rk \u3067\u3042\u308b\u5834\u5408\u306b\u3082 \u2207(f\u25cbg)(c) = t(Dg(c))(\u2207f(a)) \u304c\u6210\u7acb\u3059\u308b\u3002\u305f\u3060\u3057 t(Dg) \u306f\u8ee2\u7f6e\u95a2\u6570\u884c\u5217\u3067\u3042\u308b\u3002\u4e8c\u3064\u76ee\u306e\u9023\u9396\u5f8b\u306f\u3001R \u306e\u90e8\u5206\u96c6\u5408 I \u4e0a\u306e\u5b9f\u6570\u5024\u95a2\u6570 h: I \u2192 R \u304c f(a) \u2208 I \u306a\u308b\u70b9\u306b\u304a\u3044\u3066\u5fae\u5206\u53ef\u80fd\u306a\u3089\u3070 \u2207(h\u25cbf)(a) = h‘(f(a))\u2207f(a) \u3068\u3044\u3046\u3082\u306e\u3067\u3042\u308b\u3002\u66f4\u306a\u308b\u6027\u8cea\u3068\u5fdc\u7528[\u7de8\u96c6]\u7b49\u4f4d\u96c6\u5408[\u7de8\u96c6]f \u304c\u53ef\u5fae\u5206\u3067\u3042\u308b\u3068\u304d\u3001\u70b9 x \u306b\u304a\u3051\u308b\u52fe\u914d\u3068\u30d9\u30af\u30c8\u30eb v \u3068\u306e\u30c9\u30c3\u30c8\u7a4d (\u2207f)x \u22c5 v \u306f x \u306b\u304a\u3051\u308b f \u306e v \u65b9\u5411\u3078\u306e\u65b9\u5411\u5fae\u5206\u3092\u4e0e\u3048\u308b\u3002\u5f93\u3063\u3066\u3053\u306e\u5834\u5408\u3001f \u306e\u52fe\u914d\u306f f \u306e\u3059\u3079\u3066\u306e\u7b49\u4f4d\u96c6\u5408\u3068\u76f4\u4ea4\u3059\u308b\u3002\u4f8b\u3048\u3070\u3001\u4e09\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u7b49\u4f4d\u9762\u306f F(x, y, z) = c \u306a\u308b\u5f62\u306e\u65b9\u7a0b\u5f0f\u3067\u5b9a\u7fa9\u3055\u308c\u3001\u305d\u3057\u3066 F \u306e\u52fe\u914d\u306f\u3053\u306e\u9762\u306e\u6cd5\u7dda\u65cf\u3068\u306a\u308b\u3002\u3088\u308a\u4e00\u822c\u306b\u3001\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u306b\u57cb\u3081\u8fbc\u307e\u308c\u305f\u4efb\u610f\u306e\u8d85\u66f2\u9762\u306f  F(P) = 0\uff08\u305f\u3060\u3057 dF \u306f\u81f3\u308b\u6240\u96f6\u3067\u306a\u3044\uff09\u306e\u5f62\u306e\u65b9\u7a0b\u5f0f\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u3066\u3001F \u306e\u52fe\u914d\u306f\u3053\u306e\u8d85\u66f2\u9762\u306e\u6cd5\u7dda\u65cf\u306b\u306a\u308b\u3002\u4e00\u70b9 P \u306b\u304a\u3044\u3066\u95a2\u6570 f \u3092\u8003\u3048\u308b\u3068\u304d\u3001\u3053\u306e\u70b9 P \u3092\u901a\u308b\u66f2\u9762\u3092\u63cf\u304d\u3001\u3053\u306e\u66f2\u9762\u4e0a\u306e\u5404\u70b9\u3067\u95a2\u6570\u304c\u540c\u3058\u5024\u3092\u53d6\u308b\u3082\u306e\u3068\u3059\u308c\u3070\u3001\u3053\u306e\u66f2\u9762\u306f\u300c\u7b49\u4f4d\u9762\u300d\u3068\u547c\u3070\u308c\u308b\u3002\u4fdd\u5b58\u30d9\u30af\u30c8\u30eb\u5834\u3068\u52fe\u914d\u5b9a\u7406[\u7de8\u96c6]\u95a2\u6570\u306e\u52fe\u914d\u3092\u52fe\u914d\u5834\u3068\u547c\u3076\u3002\u9023\u7d9a\u52fe\u914d\u5834\u306f\u5e38\u306b\u4fdd\u5b58\u5834\u3067\u3001\u4efb\u610f\u306e\u7a4d\u5206\u8def\u306b\u6cbf\u3063\u305f\u7dda\u7a4d\u5206\u306f\u7a4d\u5206\u8def\u306e\u7aef\u70b9\u306b\u306e\u307f\u4f9d\u5b58\u3057\u3066\u6c7a\u307e\u308a\u3001\u305d\u306e\u5024\u306f\u52fe\u914d\u5b9a\u7406\uff08\u7dda\u7a4d\u5206\u306b\u5bfe\u3059\u308b\u5fae\u5206\u7a4d\u5206\u5b66\u306e\u57fa\u672c\u5b9a\u7406\uff09\u3067\u6c42\u3081\u3089\u308c\u308b\u3002\u9006\u306b\u9023\u7d9a\u4fdd\u5b58\u30d9\u30af\u30c8\u30eb\u5834\u306f\u5fc5\u305a\u3042\u308b\u95a2\u6570\u306e\u52fe\u914d\u5834\u3068\u3057\u3066\u5f97\u3089\u308c\u308b\u3002\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53[\u7de8\u96c6]\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53 (M, g) \u4e0a\u306e\u4efb\u610f\u306e\u6ed1\u3089\u304b\u306a\u95a2\u6570 f \u306b\u5bfe\u3057\u3001f \u306e\u52fe\u914d \u2207f \u3068\u306f\u3001\u4efb\u610f\u306e\u30d9\u30af\u30c8\u30eb\u5834 X \u306b\u3064\u3044\u3066g(\u2207f,X)=\u2202Xf,i.e.,gx((\u2207f)x,Xx)=(\u2202Xf)(x){displaystyle g(nabla f,X)=partial _{X}f,quad {text{i.e.,}}quad g_{x}((nabla f)_{x},X_{x})=(partial _{X}f)(x)}\u3092\u6e80\u305f\u3059\u30d9\u30af\u30c8\u30eb\u5834\u3092\u8a00\u3046\u3002\u305f\u3060\u3057 gx( , ) \u306f\u8a08\u91cf g \u306e\u5b9a\u3081\u308b x \u306b\u304a\u3051\u308b\u63a5\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3067\u3001\u2202Xf\uff08X(f) \u3068\u3082\u66f8\u304f\uff09\u306f\u5404\u70b9 x \u2208 M \u306b\u304a\u3044\u3066 X \u65b9\u5411\u3078\u306e f \u306e\u65b9\u5411\u5fae\u5206\u306e x \u306b\u304a\u3051\u308b\u5024\u3092\u3068\u308b\u95a2\u6570\u3067\u3042\u308b\u3002\u8a00\u3044\u63db\u3048\u308c\u3070\u3001\u5ea7\u6a19\u30c1\u30e3\u30fc\u30c8 \u03c6 \u306b\u304a\u3044\u3066 M \u306e\u958b\u96c6\u5408\u304b\u3089 Rn \u306e\u958b\u96c6\u5408\u3078\u306e\u5199\u50cf (\u2202Xf)(x) \u306f\u2211j=1nXj(\u03c6(x))\u2202\u2202xj(f\u2218\u03c6\u22121)|\u03c6(x){displaystyle sum _{j=1}^{n}X^{j}(varphi (x)){frac {partial }{partial x_{j}}}(fcirc varphi ^{-1}){Big |}_{varphi (x)}}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u3053\u3053\u306b Xj \u306f\u3001\u3053\u306e\u5ea7\u6a19\u30c1\u30e3\u30fc\u30c8\u306b\u304a\u3051\u308b X \u306e\u7b2c j \u6210\u5206\u3092\u8868\u3059\u3002\u6545\u306b\u3053\u306e\u52fe\u914d\u306e\u5c40\u6240\u5f62\u306f\u2207f=gik\u2202f\u2202xk\u2202\u2202xi{displaystyle nabla f=g^{ik}{frac {partial f}{partial x^{k}}}{frac {partial }{partial x^{i}}}}\u3068\u306a\u308b\u3002M = Rn \u306e\u5834\u5408\u3092\u4e00\u822c\u5316\u3057\u3066\u3001\u95a2\u6570\u306e\u52fe\u914d\u3068\u5916\u5fae\u5206\u3068\u3092(\u2202Xf)(x)=dfx(Xx){displaystyle (partial _{X}f)(x)=df_{x}(X_{x})}\u306b\u3088\u3063\u3066\u95a2\u4fc2\u3065\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3088\u308a\u7d30\u304b\u304f\u8a00\u3048\u3070\u3001\u52fe\u914d\u30d9\u30af\u30c8\u30eb\u5834 \u2207f \u306f\u5fae\u5206\u4e00\u6b21\u5f62\u5f0f df \u3068 g \u306e\u5b9a\u3081\u308b\u4e0a\u3052\u540c\u578b\uff08\u82f1\u8a9e\u7248\uff09\uff08\u30b7\u30e3\u30fc\u30d7\uff09\u266f=\u266fg:T\u2217M\u2192TM{displaystyle sharp =sharp ^{g}colon T^{*}Mto TM}\u3092\u7528\u3044\u3066\u5bfe\u5fdc\u4ed8\u3051\u3089\u308c\u308b\u3002Rn \u4e0a\u306e\u95a2\u6570\u306e\u52fe\u914d\u3068\u5916\u5fae\u5206\u3068\u306e\u9593\u306e\u95a2\u4fc2\u306f\u3001\u3053\u306e\u8a08\u91cf\u304c\u30c9\u30c3\u30c8\u7a4d\u306e\u4e0e\u3048\u308b\u5e73\u5766\u8a08\u91cf\u3067\u3042\u308b\u7279\u5225\u306e\u5834\u5408\u3067\u3042\u308b\u3002\u5186\u7b52\u5ea7\u6a19\u7cfb\u304a\u3088\u3073\u7403\u9762\u5ea7\u6a19\u7cfb\u3067\u306e\u8868\u793a[\u7de8\u96c6]\u5186\u7b52\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u52fe\u914d\u306f\u2207f(\u03c1,\u03d5,z)=\u2202f\u2202\u03c1e\u03c1+1\u03c1\u2202f\u2202\u03d5e\u03d5+\u2202f\u2202zez{displaystyle nabla f(rho ,phi ,z)={frac {partial f}{partial rho }}mathbf {e} _{rho }+{frac {1}{rho }}{frac {partial f}{partial phi }}mathbf {e} _{phi }+{frac {partial f}{partial z}}mathbf {e} _{z}}\u3067\u4e0e\u3048\u3089\u308c\u308b(Schey 1992, pp.\u00a0139\u2013142)\u3002\u3053\u3053\u3067 \u03d5 \u306f\u65b9\u4f4d\u89d2\u3001z \u306f\u8ef8\u65b9\u5411\u306e\u5ea7\u6a19\u304a\u3088\u3073 e\u03c1, e\u03c6, ez \u306f\u5404\u5ea7\u6a19\u8ef8\u65b9\u5411\u306b\u6cbf\u3063\u305f\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3002\u7403\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u306f\u2207f(r,\u03b8,\u03d5)=\u2202f\u2202rer+1r\u2202f\u2202\u03b8e\u03b8+1rsin\u2061\u03b8\u2202f\u2202\u03d5e\u03d5{displaystyle nabla f(r,theta ,phi )={frac {partial f}{partial r}}mathbf {e} _{r}+{frac {1}{r}}{frac {partial f}{partial theta }}mathbf {e} _{theta }+{frac {1}{rsin theta }}{frac {partial f}{partial phi }}mathbf {e} _{phi }}\u3068\u306a\u308b(Schey 1992, pp.\u00a0139\u2013142)\u3002\u3053\u3053\u306b \u03d5 \u306f\u65b9\u4f4d\u89d2\u3067 \u03b8 \u306f\u5929\u9802\u89d2\u3067\u3042\u308b\u3002\u30d9\u30af\u30c8\u30eb\u5024\u95a2\u6570\u306e\u52fe\u914d[\u7de8\u96c6]\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u3001\u30d9\u30af\u30c8\u30eb f = (f1, f2, f3) \u306e\u52fe\u914d\u306f\u2207f=\u2202fi\u2202xjeiej{displaystyle nabla mathbf {f} ={frac {partial {{f}_{i}}}{partial {{x}_{j}}}}{{mathbf {e} }_{i}}{{mathbf {e} }_{j}}}\u3042\u308b\u3044\u306f\u95a2\u6570\u884c\u5217\u2202(f1,f2,f3)\u2202(x1,x2,x3){displaystyle {frac {partial ({{f}_{1}},{{f}_{2}},{{f}_{3}})}{partial ({{x}_{1}},{{x}_{2}},{{x}_{3}})}}}\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u66f2\u9762\u5ea7\u6a19\u7cfb\u306b\u304a\u3051\u308b\u52fe\u914d\u306b\u306f\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u304c\u73fe\u308c\u308b\u3002\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]Korn, Theresa M.; Korn, Granino Arthur (2000), Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, New York: Dover Publications, pp.\u00a0157\u2013160, ISBN\u00a00-486-41147-8, OCLC\u00a043864234\u00a0.Schey, H.M. (1992), Div, Grad, Curl, and All That (2nd ed.), W.W. Norton, ISBN\u00a00-393-96251-2, OCLC\u00a025048561\u00a0.Dubrovin, B.A.; A.T. Fomenko, S.P. Novikov (1991), Modern Geometry–Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Graduate Texts in Mathematics) (2nd ed.), Springer, pp.\u00a014\u201317, ISBN\u00a0978-0-387-97663-1\u00a0\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki6\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki6\/archives\/6324#breadcrumbitem","name":"\u52fe\u914d (\u30d9\u30af\u30c8\u30eb\u89e3\u6790) – Wikipedia"}}]}]