[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/6368#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/6368","headline":"\u63a5\u7d9a\uff08\u5dee\u52d5\u5e7e\u4f55\u5b66\uff09 – \u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2","name":"\u63a5\u7d9a\uff08\u5dee\u52d5\u5e7e\u4f55\u5b66\uff09 – \u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2","description":"before-content-x4 \u5fae\u5206\u5f62\u72b6\u306e\u6570\u5b66\u7684\u30b5\u30d6\u30a8\u30ea\u30a2\u3067 \u7e4b\u304c\u308a \u52d5\u304d\u306e\u904e\u7a0b\u3067\u65b9\u5411\u5909\u5316\u3092\u5b9a\u91cf\u5316\u3057\u3001\u7570\u306a\u308b\u30dd\u30a4\u30f3\u30c8\u306e\u65b9\u5411\u3092\u95a2\u9023\u4ed8\u3051\u308b\u305f\u3081\u306e\u63f4\u52a9\u3002 \u3053\u306e\u8a18\u4e8b\u306f\u3001\u57fa\u672c\u7684\u306b\u3001\u5dee\u5225\u5316\u3055\u308c\u305f\u591a\u69d8\u6027\u307e\u305f\u306f\u30d9\u30af\u30c8\u30eb\u306e\u675f\u306b\u95a2\u3059\u308b\u63a5\u7d9a\u3092\u6271\u3044\u307e\u3059\u3002\u30c6\u30f3\u30bd\u30eb\u30d3\u30e5\u30fc\u30f3\u30c7\u30eb\u306e\u512a\u308c\u305f\u3064\u306a\u304c\u308a\u3001\u30d9\u30af\u30c8\u30eb\u306e\u7279\u5225\u306a\u675f\u304c\u547c\u3070\u308c\u307e\u3059 Kovariant\u6d3e\u751f \u3002\u4e00\u822c\u306b\u3001\u30a2\u30ca\u30ed\u30b0\u5b9a\u7fa9\u30d7\u30ed\u30d1\u30c6\u30a3\u3092\u5099\u3048\u305f\u4e3b\u8981\u306a\u30d0\u30f3\u30c9\u30eb\u306b\u3082\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002 \u5fae\u5206\u30b8\u30aa\u30e1\u30c8\u30ea\u3067\u306f\u3001\u66f2\u7dda\u3001\u7279\u306b\u30b8\u30aa\u30fc\u30c7\u306e\u66f2\u7387\u306b\u8208\u5473\u304c\u3042\u308a\u307e\u3059\u3002\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u90e8\u5c4b\u3067\u306f\u3001\u66f2\u7387\u306f\u5358\u306b2\u756a\u76ee\u306e\u6d3e\u751f\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002 2\u756a\u76ee\u306e\u5fae\u5206\u306f\u3001\u5dee\u5225\u5316\u53ef\u80fd\u306a\u591a\u69d8\u6027\u3067\u76f4\u63a5\u5f62\u6210\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u306f c {displaystyle\u30ac\u30f3\u30de} \u66f2\u7dda\u306a\u306e\u3067\u3001\u3053\u306e\u66f2\u7dda\u306e2\u756a\u76ee\u306e\u5c0e\u51fa\u306e\u305f\u3081\u306b\u30d9\u30af\u30c8\u30eb\u3068\u306e\u9055\u3044\u306e\u5546\u3092\u884c\u3046\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059 c ‘ \uff08 t \uff09\uff09 {displaystyle\u30ac\u30f3\u30de ‘\uff08t\uff09} \u3068","datePublished":"2019-04-13","dateModified":"2019-04-13","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?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\/a223c880b0ce3da8f64ee33c4f0010beee400b1a","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a223c880b0ce3da8f64ee33c4f0010beee400b1a","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/6368","wordCount":13127,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4\u5fae\u5206\u5f62\u72b6\u306e\u6570\u5b66\u7684\u30b5\u30d6\u30a8\u30ea\u30a2\u3067 \u7e4b\u304c\u308a \u52d5\u304d\u306e\u904e\u7a0b\u3067\u65b9\u5411\u5909\u5316\u3092\u5b9a\u91cf\u5316\u3057\u3001\u7570\u306a\u308b\u30dd\u30a4\u30f3\u30c8\u306e\u65b9\u5411\u3092\u95a2\u9023\u4ed8\u3051\u308b\u305f\u3081\u306e\u63f4\u52a9\u3002 \u3053\u306e\u8a18\u4e8b\u306f\u3001\u57fa\u672c\u7684\u306b\u3001\u5dee\u5225\u5316\u3055\u308c\u305f\u591a\u69d8\u6027\u307e\u305f\u306f\u30d9\u30af\u30c8\u30eb\u306e\u675f\u306b\u95a2\u3059\u308b\u63a5\u7d9a\u3092\u6271\u3044\u307e\u3059\u3002\u30c6\u30f3\u30bd\u30eb\u30d3\u30e5\u30fc\u30f3\u30c7\u30eb\u306e\u512a\u308c\u305f\u3064\u306a\u304c\u308a\u3001\u30d9\u30af\u30c8\u30eb\u306e\u7279\u5225\u306a\u675f\u304c\u547c\u3070\u308c\u307e\u3059 Kovariant\u6d3e\u751f \u3002\u4e00\u822c\u306b\u3001\u30a2\u30ca\u30ed\u30b0\u5b9a\u7fa9\u30d7\u30ed\u30d1\u30c6\u30a3\u3092\u5099\u3048\u305f\u4e3b\u8981\u306a\u30d0\u30f3\u30c9\u30eb\u306b\u3082\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002 \u5fae\u5206\u30b8\u30aa\u30e1\u30c8\u30ea\u3067\u306f\u3001\u66f2\u7dda\u3001\u7279\u306b\u30b8\u30aa\u30fc\u30c7\u306e\u66f2\u7387\u306b\u8208\u5473\u304c\u3042\u308a\u307e\u3059\u3002\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u90e8\u5c4b\u3067\u306f\u3001\u66f2\u7387\u306f\u5358\u306b2\u756a\u76ee\u306e\u6d3e\u751f\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002 2\u756a\u76ee\u306e\u5fae\u5206\u306f\u3001\u5dee\u5225\u5316\u53ef\u80fd\u306a\u591a\u69d8\u6027\u3067\u76f4\u63a5\u5f62\u6210\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u306f c {displaystyle\u30ac\u30f3\u30de} \u66f2\u7dda\u306a\u306e\u3067\u3001\u3053\u306e\u66f2\u7dda\u306e2\u756a\u76ee\u306e\u5c0e\u51fa\u306e\u305f\u3081\u306b\u30d9\u30af\u30c8\u30eb\u3068\u306e\u9055\u3044\u306e\u5546\u3092\u884c\u3046\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059 c ‘ \uff08 t \uff09\uff09 {displaystyle\u30ac\u30f3\u30de ‘\uff08t\uff09} \u3068 c ‘ \uff08 t 0 \uff09\uff09 {displaystyle\u30ac\u30f3\u30de ‘\uff08t_ {0}\uff09} \u5f62\u72b6\u3002\u305f\u3060\u3057\u3001\u3053\u308c\u3089\u306e\u30d9\u30af\u30c8\u30eb\u306f\u7570\u306a\u308b\u30d9\u30af\u30c8\u30eb\u306b\u3042\u308b\u305f\u3081\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u9055\u3044\u3092\u5358\u7d14\u306b\u5f62\u6210\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u554f\u984c\u3092\u89e3\u6c7a\u3059\u308b\u305f\u3081\u306b\u3001\u63a5\u7d9a\u3068\u547c\u3070\u308c\u308b\u30a4\u30e9\u30b9\u30c8\u3092\u5b9a\u7fa9\u3057\u307e\u3057\u305f\u3002\u3053\u306e\u56f3\u306f\u3001\u95a2\u4fc2\u3059\u308b\u30d9\u30af\u30c8\u30eb\u30eb\u30fc\u30e0\u9593\u306e\u63a5\u7d9a\u3092\u63d0\u4f9b\u3059\u308b\u3053\u3068\u3092\u76ee\u7684\u3068\u3057\u3066\u3044\u308b\u305f\u3081\u3001\u3053\u306e\u540d\u524d\u3082\u4ed8\u3044\u3066\u3044\u307e\u3059\u3002 \u3053\u306e\u30bb\u30af\u30b7\u30e7\u30f3\u3067\u8aac\u660e\u3057\u307e\u3059 m {displaystyle m} \u30b9\u30e0\u30fc\u30ba\u306a\u591a\u69d8\u6027\u3001 t m {displaystyleTm} \u63a5\u7dda\u30d0\u30f3\u30c9\u30eb\u3068 pi \uff1a \u3068 \u2192 m {displaystyle pi colon eto m} \u30d9\u30af\u30c8\u30eb\u306e\u675f\u3002\u3068 c \uff08 \u3068 \uff09\uff09 {displaystyle\u30ac\u30f3\u30de\uff08e\uff09} \u30d9\u30af\u30c8\u30eb\u30d0\u30f3\u30c9\u30eb\u306e\u6ed1\u3089\u304b\u306a\u30ab\u30c3\u30c8\u306e\u91cf\u306f \u3068 {displaystyle e} \u66f8\u304d\u7559\u3081\u305f\u3002 Table of Contents\u7e4b\u304c\u308a [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u7dda\u5f62\u63a5\u7d9a [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u5b9f\u969b\u306e\u30b5\u30d6\u30de\u30cd\u30fc\u30b7\u30c6\u30a3\u306e\u63a5\u7d9a [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Tensb\u00fcndel\u306e\u63a5\u7d9a [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Riemann\u30e1\u30c8\u30ea\u30c3\u30af\u3068\u5bfe\u79f0\u6027\u3068\u306e\u4e92\u63db\u6027 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u7e4b\u304c\u308a [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u63a5\u7dda\u30d9\u30af\u30c8\u30eb\u306e\u65b9\u5411\u306b\u3042\u308b\u30d9\u30af\u30c8\u30eb\u5834\u306e\u65b9\u5411\u3092\u8a00\u3046\u3053\u3068\u3067\u3001\u5dee\u5225\u5316\u3055\u308c\u305f\u591a\u69d8\u6027\u306b\u63a5\u7d9a\u3092\u53d6\u5f97\u3057\u307e\u3059 m {displaystyle m}        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u30d9\u30af\u30c8\u30eb\u306e\u675f\u306e\u63a5\u7d9a\u306f\u30a4\u30e9\u30b9\u30c8\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059 \u2207:\u0393(TM)\u00d7\u0393(E)\u2192\u0393(E)(X,s)\u21a6\u2207Xs,{displaystyle {begin {aligned} nabla colon gamma\uff08tm\uff09\u500d\u30ac\u30f3\u30de\uff08e\uff09\uff06rightarrow\u30ac\u30f3\u30de\uff08e\uff09\\\uff08x\u3001s\uff09\uff06mapsto nabla _ {x} s\u3001end {aligned}}}} 1\u3064\u306e\u30d9\u30af\u30c8\u30eb\u30d5\u30a3\u30fc\u30eb\u30c9 \u30d0\u30c4 {displaystyle x} \u306e\u4e0a        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4m {displaystyle m} \u305d\u3057\u3066\u30ab\u30c3\u30c8 s {displaystyleS} \u30d9\u30af\u30c8\u30eb\u30d0\u30f3\u30c9\u30eb \u3068 {displaystyle e} \u5225\u306e\u30ab\u30c3\u30c8\u30a4\u30f3 \u3068 {displaystyle e} \u6b21\u306e\u6761\u4ef6\u304c\u6e80\u305f\u3055\u308c\u308b\u3088\u3046\u306b\u5272\u308a\u5f53\u3066\u3089\u308c\u307e\u3059\u3002        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u2207 Xs {displaystyle nabla _ {x} s} \u5165\u3063\u3066\u3044\u307e\u3059 \u30d0\u30c4 \u2208 c \uff08 t m \uff09\uff09 {DisplayStyle Please Gamma\uff08TM\uff09} \u7dda\u5f62 c \u221e\uff08 m \uff09\uff09 {displaystyle c^{infty}\uff08m\uff09} \u3001\u305d\u308c\u306f\u610f\u5473\u3057\u307e\u3059 \u2207fX1+gX2s=f\u22c5\u2207X1s+g\u22c5\u2207X2s{displaystyle nabla _ {fx_ {1}+gx_ {2}} s = fcdot nabla _ {x_ {1}} s+gcdot nabla _ {x_ {2}} s}} \u305f\u3081\u306b f \u3001 g \u2208 c \u221e\uff08 m \uff09\uff09 {displaystyle F\u3001gin c^{infty}\uff08m\uff09} \u3068 \u30d0\u30c4 1\u3001 \u30d0\u30c4 2\u2208 c \uff08 t m \uff09\uff09 \u3002 {displaystyle x_ {1}\u3001x_ {2} in gamma\uff08tm\uff09\u3002} \u2207 Xs {displaystyle nabla _ {x} s} \u306f r {displaystyle mathbb {r}} -Linear in s \u3001 {displaystyle s\u3001} \u305d\u308c\u306f\u305d\u308c\u304c\u9069\u7528\u3055\u308c\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059 \u2207X(\u03bb1s1+\u03bb2s2)=\u03bb1\u22c5\u2207Xs1+\u03bb2\u22c5\u2207Xs2{displaystyle nabla _ {x}}\uff08lambda _ {1}} s__lamba _ {2} s} s\uff09= lambada _ {1} cdot nabala _ {x} s} s} s} s} s} s} s} s} s} s} s} s} s} } cdot nabala\u3002 \u305f\u3081\u306b l 1\u3001 l 2\u2208 r {displaystyle lambda _ {1}\u3001lambda _ {2} in mathbb {r}} \u3002 \u3055\u3089\u306b\u3001\u88fd\u54c1\u30eb\u30fc\u30eb\u307e\u305f\u306f\u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u30eb\u30fc\u30eb\u304c\u9069\u7528\u3055\u308c\u307e\u3059 \u2207X(fs)=Xf\u22c5s+f\u22c5\u2207Xs{displaystyle nabla _ {x}\uff08fs\uff09= xfcdot s+fcdot nabla _ {x} s} \u3059\u3079\u3066\u306e\u95a2\u6570\u306b\u5bfe\u3057\u3066 f \u2208 c \u221e\uff08 m \uff09\uff09 {displaystyle fin c^{infty}\uff08m\uff09} \u3002 \u3053\u3053\u3067\u8aac\u660e\u3057\u307e\u3059 \u30d0\u30c4 f {displaystyle xf} \u95a2\u6570\u306e\u65b9\u5411\u306e\u65b9\u5411 f {displaystyle f} \u65b9\u5411 \u30d0\u30c4 {displaystyle x} \uff08\u3057\u305f\u304c\u3063\u3066\u3001\u63a5\u7dda\u30d9\u30af\u30c8\u30eb\u306f\u5c0e\u51fa\u3068\u3057\u3066\u7406\u89e3\u3055\u308c\u307e\u3059\uff09\u3002\u5225\u306e\u30b9\u30da\u30eb \u30d0\u30c4 f {displaystyle xf} \u306f d f \uff08 \u30d0\u30c4 \uff09\uff09 {displaystyle mathrm {d} f\uff08x\uff09} \u3002 \u3042\u308b\u3044\u306f\u3001\u63a5\u7d9a\u3092\u30a4\u30e9\u30b9\u30c8\u3068\u3057\u3066\u4f5c\u6210\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059 \u2207 \uff1a c \uff08 \u3068 \uff09\uff09 \u2192 c \uff08 t \u2217m \u2297 \u3068 \uff09\uff09 {displaystyle nabla\u30b3\u30ed\u30f3\u30ac\u30f3\u30de\uff08e\uff09\u304b\u3089\u7bc4\u56f2\uff08t^{*} motimese\uff09} \u540c\u3058\u30d7\u30ed\u30d1\u30c6\u30a3\u3067\u5b9a\u7fa9\u3057\u307e\u3059\u3002 \u7dda\u5f62\u63a5\u7d9a [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u7dda\u5f62\u307e\u305f\u306f\u30a2\u30d5\u30a3\u30ca\u30fc\u306e\u30b3\u30f3\u30c6\u30ad\u30b9\u30c8 m {displaystyle m} \u63a5\u7d9a\u3067\u3059 t m {displaystyleTm} \u3002\u305d\u308c\u306f\u305d\u308c\u304c\u30a4\u30e9\u30b9\u30c8\u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059 \u2207 \uff1a c \uff08 t m \uff09\uff09 \u00d7 c \uff08 t m \uff09\uff09 \u2192 c \uff08 t m \uff09\uff09 \u3001 {displaystyle nabla\u30b3\u30ed\u30f3\u30ac\u30f3\u30de\uff08TM\uff09\u30bf\u30a4\u30e0\u30ac\u30f3\u30de\uff08TM\uff09\u304b\u3089\u30ac\u30f3\u30de\uff08TM\uff09\u3001} \u4e0a\u8a18\u306e\u30bb\u30af\u30b7\u30e7\u30f3\u306e3\u3064\u306e\u5b9a\u7fa9\u30d7\u30ed\u30d1\u30c6\u30a3\u3092\u6e80\u305f\u3057\u3066\u3044\u307e\u3059\u3002 \u81ea\u7136\u306a\u65b9\u6cd5\u3067\u4ed6\u306e\u30d9\u30af\u30c8\u30eb\u306e\u30d0\u30f3\u30c9\u30eb\u306b\u63a5\u7d9a\u3092\u8a98\u5c0e\u3059\u308b\u3055\u307e\u3056\u307e\u306a\u65b9\u6cd5\u304c\u3042\u308a\u307e\u3059\u3002 \u5b9f\u969b\u306e\u30b5\u30d6\u30de\u30cd\u30fc\u30b7\u30c6\u30a3\u306e\u63a5\u7d9a [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u591a\u5206 \u2202 \u521d\u3081 \u3001 … \u3001 \u2202 n {displaystyle partial _ {1}\u3001ldots\u3001partial _ {n}} \u306e\u6a19\u6e96\u30d9\u30fc\u30b9 r n {displaystyle mathbb {r} ^{n}} \u3001\u305d\u308c\u304b\u3089\u4e0a\u306b\u306a\u308a\u307e\u3059 r n {displaystyle mathbb {r} ^{n}} \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u6587\u8108 \u2207 \u30d0\u30c4 Rn{displaystyle nabla _ {x} ^{mathbb {r} ^{n}}}}} \u7d42\u3048\u305f \u2207 XRn\u3068 \uff1a= \u2211 i,j\uff08 \u30d0\u30c4 i\u2202 i\u3068 j\uff09\uff09 \u2202 j{displaystyle textStyle nabla _ {x}^{mathbb {r}^{n}} y\uff1a= sum _ {i\u3001j}\uff08x^{i} partial _ {i} y^{j}\uff09partial _ {j}}}} \u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059 \u30d0\u30c4 = \u2211 i\u30d0\u30c4 i\u2202 i{displaystyle textStyle x = sum _ {i} x^{i} partial _ {i}} \u3068 \u3068 = \u2211 j\u3068 j\u2202 j{displaystyle textStyle y = sum _ {j} y^{j} bias _ {j}} \u30d9\u30af\u30c8\u30eb\u30d5\u30a3\u30fc\u30eb\u30c9\u306e\u8868\u73fe \u30d0\u30c4 \u3001 \u3068 {displaystyle x\u3001y} \u6a19\u6e96\u30d9\u30fc\u30b9\u3067\u3059\u3002\u306f m {displaystyle m} \u306e\u30b5\u30d6\u30de\u30cd\u30fc\u30b7\u30c6\u30a3 r n {displaystyle mathbb {r} ^{n}} \u3001\u3060\u304b\u3089\u3042\u306a\u305f\u306f\u8d77\u304d\u307e\u3059 m {displaystyle m} \u306e\u4e00\u3064 r n {displaystyle mathbb {r} ^{n}} \u8a98\u5c0e\u63a5\u7d9a\u3002\u3053\u308c\u306f\u901a\u308a\u3067\u3059 \u2207 XM\u3068 \uff1a= pi \uff08 \u2207 XRn\u3068 \uff09\uff09 {displaystyle nabla _ {x}^{m} y\uff1a= pi\uff08nabla _ {x}^{mathbb {r}^{n}} y\uff09}} \u305d\u3046\u3067\u3059\u3002\u8aac\u660e\u3055\u308c\u305f pi \uff1a t p r n \u2192 t p m {displaystyle pi colon t_ {p} mathbb {r} ^{n}\u304b\u3089t_ {p} m}\u304b\u3089t_ {p} m} \u76f4\u4ea4\u6295\u5f71\u3002 Tensb\u00fcndel\u306e\u63a5\u7d9a [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u591a\u5206 \u2207 {displaystyle nabla} \u591a\u69d8\u6027\u306e\u7dda\u5f62\u63a5\u7d9a m {displaystyle m} \u3002 Tensb\u00fcndel t l k m {displaystyle t_ {l}^{k} m} \u660e\u78ba\u306a\u3064\u306a\u304c\u308a\u306b\u306a\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059 c \uff08 t m \uff09\uff09 \u00d7 c \uff08 t l k m \uff09\uff09 \u2192 c \uff08 t l k m \uff09\uff09 {displaystyle\u30ac\u30f3\u30de\uff08TM\uff09\u30bf\u30a4\u30e0\u30ac\u30f3\u30de\uff08T_ {l}^{k} m\uff09\u304b\u3089\u30ac\u30f3\u30de\uff08T_ {l}^{k} m\uff09} \u3067\u8a98\u5c0e\u3057\u307e\u3059 \u2207 {displaystyle nabla} \u6ce8\u76ee\u3055\u308c\u3001\u6b21\u306e\u7279\u6027\u304c\u6e80\u305f\u3055\u308c\u3066\u3044\u307e\u3059\u3002 \u306e\u4e0a t m \u2245 t 01m {displaystyle tmcong t_ {0}^{1} m} \u6b63\u3057\u3044 \u2207 {displaystyle nabla} \u6307\u5b9a\u3055\u308c\u305f\u63a5\u7d9a\u3067\u3002 \u306e\u4e0a t 0m {displaystylet^{0} m} \u306f \u2207 {displaystyle nabla} \u95a2\u6570\u306e\u65b9\u5411\u306e\u901a\u5e38\u306e\u65b9\u5411\uff1a \u2207 Xf = \u30d0\u30c4 f \u3002 {displaystyle nabla _ {x} f = xf\u3002} \u305f\u3081\u306b \u2207 {displaystyle nabla} \u6b21\u306e\u88fd\u54c1\u30eb\u30fc\u30eb\u304c\u9069\u7528\u3055\u308c\u307e\u3059 \u2207 X\uff08 f \u2297 g \uff09\uff09 = \uff08 \u2207 Xf \uff09\uff09 \u2297 g + f \u2297 \uff08 \u2207 Xg \uff09\uff09 \u3002 {displaysyle nabla _ {x}\uff08fotimes g\uff09=\uff08nabla _ {x} f\uff09otimes g+photos\uff08nabla _ {x} g\uff09\u3002}} \u95a2\u4fc2 \u2207 {displaystyle nabla} \u30c6\u30f3\u30bd\u30eb\u306e\u82e5\u8fd4\u308a\u306b\u901a\u3046 tr {displaystyle operatorname {tr}} \u3001\u305d\u308c\u306f\u610f\u5473\u3057\u307e\u3059 \u2207 X\uff08 tr \u2061 f \uff09\uff09 = tr \u2061 \uff08 \u2207 Xf \uff09\uff09 \u3002 {displaystyle nabla _ {x}\uff08operatorname {tr} f\uff09= operatorname {tr}\uff08nabla _ {x} f\uff09\u3002} \u3053\u306e\u63a5\u7d9a t l k m {displaystyle t_ {l}^{k} m} \u30b3\u30d0\u30ea\u30a2\u30f3\u30c8\u6d3e\u751f\u3068\u3082\u547c\u3070\u308c\u307e\u3059\u3002 Riemann\u30e1\u30c8\u30ea\u30c3\u30af\u3068\u5bfe\u79f0\u6027\u3068\u306e\u4e92\u63db\u6027 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u591a\u5206 \uff08 m \u3001 g \uff09\uff09 {displaystyle\uff08m\u3001g\uff09} Riemann\u307e\u305f\u306fPseudo-Riemannsche\u306e\u591a\u69d8\u6027\u3002\u63a5\u7d9a \u2207 {displaystyle nabla} \u30e1\u30c8\u30ea\u30c3\u30af\u3068\u4e92\u63db\u6027\u304c\u3042\u308b\u3068\u547c\u3070\u308c\u307e\u3059 g {displaystyle g} \u3053\u306e\u591a\u69d8\u6027\u3001if \u30d0\u30c4 \uff08 g \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \uff09\uff09 = g \uff08 \u2207 X\u3068 \u3001 \u3068 \uff09\uff09 + g \uff08 \u3068 \u3001 \u2207 X\u3068 \uff09\uff09 {displaystyle x\uff08g\uff08y\u3001z\uff09\uff09= g\uff08nabla _ {x} y\u3001z\uff09+g\uff08y\u3001nabla _ {x} z\uff09} \u9069\u7528\u53ef\u80fd\u3067\u3059\u3002\u30bb\u30af\u30b7\u30e7\u30f3\u304b\u30893\u756a\u76ee\u306e\u30d7\u30ed\u30d1\u30c6\u30a3\u304c\u3042\u308a\u307e\u3059 Tensb\u00fcndel\u306e\u63a5\u7d9a \u65b9\u7a0b\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059 \uff08 \u2207 Xg \uff09\uff09 \uff08 \u3068 \u3001 \u3068 \uff09\uff09 = \u30d0\u30c4 \uff08 g \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \uff09\uff09  –  g \uff08 \u2207 X\u3068 \u3001 \u3068 \uff09\uff09  –  g \uff08 \u3068 \u3001 \u2207 X\u3068 \uff09\uff09 {displaystyle\uff08nabla _ {x} g\uff09\uff08y\u3001z\uff09= x\uff08y\u3001z\uff09\uff09 –  g\uff08nabla _ {x} y\u3001z\uff09-g\uff08y\u3001nabla _ {x} z}} \u3057\u305f\u304c\u3063\u3066\u3001\u4e92\u63db\u6027\u6761\u4ef6\u306f\u540c\u7b49\u3067\u3059 \uff08 \u2207 Xg \uff09\uff09 \uff08 \u3068 \u3001 \u3068 \uff09\uff09 = 0\u3002 {displaystyle\uff08nabla _ {x} g\uff09\uff08y\u3001z\uff09= 0.} \u63a5\u7d9a\u306f\u3001\u306d\u3058\u308c\u30ac\u30f3\u30c8\u30ec\u30c3\u30c8\u304c\u6d88\u3048\u308b\u3068\u304d\u3001\u3064\u307e\u308a\u3001\u305d\u308c\u304c\u9069\u7528\u3055\u308c\u308b\u3068\u304d\u306b\u5bfe\u79f0\u6027\u307e\u305f\u306f\u306d\u3058\u308c\u306e\u306a\u3044\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059 \u2207 X\u3068  –  \u2207 Y\u30d0\u30c4 = [ \u30d0\u30c4 \u3001 \u3068 ] \u3002 {displaystyle nabla _ {x} y-nabla _ {y} x = [x\u3001y]\u3002} \u3082\u3061\u308d\u3093\u3001\u3053\u308c\u3089\u306e2\u3064\u306e\u30d7\u30ed\u30d1\u30c6\u30a3\u306f\u3001\u5b9f\u969b\u306e\u30a2\u30f3\u30c0\u30fc\u30de\u30f3\u30ca\u30a4\u30c8\u3067\u8a98\u5c0e\u3055\u308c\u305f\u63a5\u7d9a\u306b\u3088\u3063\u3066\u3059\u3067\u306b\u6e80\u305f\u3055\u308c\u3066\u3044\u308b\u305f\u3081\u3067\u3059\u3002\u3053\u308c\u3089\u306e2\u3064\u306e\u30d7\u30ed\u30d1\u30c6\u30a3\u3092\u6e80\u305f\u3059\uff08\u8981\u7d04\uff09\u591a\u69d8\u6027\u306e\u63a5\u7d9a\u304c\u660e\u78ba\u306b\u6c7a\u5b9a\u3055\u308c\u307e\u3059\u3002\u3053\u306e\u58f0\u660e\u306f\u3001Riemann\u306e\u5e7e\u4f55\u5b66\u306e\u4e3b\u8981\u6761\u9805\u3068\u547c\u3070\u308c\u3001\u660e\u78ba\u306b\u6c7a\u5b9a\u3055\u308c\u305f\u63a5\u7d9a\u306fLevi-Civita\u307e\u305f\u306fRiemann\u306e\u63a5\u7d9a\u3068\u547c\u3070\u308c\u307e\u3059\u3002 Riemann\u30e1\u30c8\u30ea\u30c3\u30af\u3068\u4e92\u63db\u6027\u306e\u3042\u308b\u63a5\u7d9a\u306f\u3001\u30e1\u30c8\u30ea\u30c3\u30af\u30b3\u30f3\u30c6\u30ad\u30b9\u30c8\u3068\u547c\u3070\u308c\u307e\u3059\u3002\u30ea\u30fc\u30de\u30f3\u306e\u591a\u69d8\u6027\u306f\u3001\u4e00\u822c\u306b\u3044\u304f\u3064\u304b\u306e\u7570\u306a\u308b\u30e1\u30c8\u30ea\u30c3\u30af\u95a2\u4fc2\u3092\u6301\u3064\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 \uff08 \u2207 X\u3068 1\uff09\uff09 \uff08 p \uff09\uff09 = \uff08 \u2207 X\u3068 2\uff09\uff09 \uff08 p \uff09\uff09 \u3002 {displaystyle\uff08nabla _ {x} y_ {1}\uff09\uff08p\uff09=\uff08nabla _ {x} y_ {2}\uff09\uff08p\uff09\u3002} \u3088\u308a\u4e00\u822c\u7684\u3067\u3059 \u3068 \u521d\u3081 {displaystyle y_ {1}} \u3068 \u3068 2 {displaystyle y_ {2}} \u74b0\u5883\u5168\u4f53\u3067\u540c\u3058\u3067\u3055\u3048\u3042\u308a\u307e\u305b\u3093\u3002\u3088\u308a\u6b63\u78ba\u306b\u306f\u3001\u6ed1\u3089\u304b\u306a\u66f2\u7dda\u304c\u3042\u308b\u5834\u5408 c \uff1a \uff08  –  \u03f5 \u3001 \u03f5 \uff09\uff09 \u2282 r \u2192 m {displaystyle\u30ac\u30f3\u30de\uff1a\uff08  – \u30a8\u30d7\u30b7\u30ed\u30f3\u3001\u30a8\u30d7\u30b7\u30ed\u30f3\uff09\u30b5\u30d6\u30bb\u30c3\u30c8\u6570\u5b66{r}\u304b\u3089m} \u4e0e\u3048\u308b\uff08\u9069\u5207\u306a\u3082\u306e\u306e\u305f\u3081\u306b "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/6368#breadcrumbitem","name":"\u63a5\u7d9a\uff08\u5dee\u52d5\u5e7e\u4f55\u5b66\uff09 – \u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2"}}]}]