[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki13\/2022\/03\/30\/%e3%83%a9%e3%83%97%e3%83%a9%e3%82%b9%e4%bd%9c%e7%94%a8%e7%b4%a0-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki13\/2022\/03\/30\/%e3%83%a9%e3%83%97%e3%83%a9%e3%82%b9%e4%bd%9c%e7%94%a8%e7%b4%a0-wikipedia\/","headline":"\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20 – Wikipedia","name":"\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20 – Wikipedia","description":"\u6570\u5b66\u306b\u304a\u3051\u308b\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\uff08\u30e9\u30d7\u30e9\u30b9\u3055\u3088\u3046\u305d\u3001\u82f1: Laplace operator\uff09\u3042\u308b\u3044\u306f\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\uff08\u82f1: Laplacian)\u306f\u3001\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u4e0a\u306e\u51fd\u6570\u306e\u52fe\u914d\u306e\u767a\u6563\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u308b\u5fae\u5206\u4f5c\u7528\u7d20\u3067\u3042\u308b\u3002\u8a18\u53f7\u3067\u306f \u2207\u00b7\u2207, \u22072, \u3042\u308b\u3044\u306f \u2206 \u3067\u8868\u3055\u308c\u308b\u306e\u304c\u666e\u901a\u3067\u3042\u308b\u3002\u51fd\u6570 f \u306e\u70b9 p \u306b\u304a\u3051\u308b\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3 \u2206f(p) \u306f\uff08\u6b21\u5143\u306b\u4f9d\u5b58\u3059\u308b\u5b9a\u6570\u306e\u9055\u3044\u3092\u9664\u3044\u3066\uff09\u70b9 p \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u7403\u9762\u3092\u534a\u5f84\u304c\u5897\u5927\u3059\u308b\u3088\u3046\u306b\u52d5\u304b\u3059\u3068\u304d\u306e f(p) \u304b\u3089\u5f97\u3089\u308c\u308b\u5e73\u5747\u5024\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u306f\u3001\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306f\u5404\u72ec\u7acb\u5909\u6570\u306b\u95a2\u3059\u308b\u51fd\u6570\u306e\u4e8c\u968e\uff08\u975e\u6df7\u5408\uff09\u504f\u5c0e\u51fd\u6570\u306e\u548c\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u3001\u307e\u305f\u307b\u304b\u306b\u5186\u7b52\u5ea7\u6a19\u7cfb\u3084\u7403\u5ea7\u6a19\u7cfb\u306a\u3069\u306e\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u3082\u6709\u7528\u306a\u8868\u793a\u3092\u6301\u3064\u3002 \u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306e\u540d\u79f0\u306f\u3001\u5929\u4f53\u529b\u5b66\u306e\u7814\u7a76\u306b\u540c\u4f5c\u7528\u7d20\u3092\u6700\u521d\u306b\u7528\u3044\u305f\u30d5\u30e9\u30f3\u30b9\u4eba\u6570\u5b66\u8005\u306e\u30d4\u30a8\u30fc\u30eb\uff1d\u30b7\u30e2\u30f3\u30fb\u30c9\u30fb\u30e9\u30d7\u30e9\u30b9","datePublished":"2022-03-30","dateModified":"2022-03-30","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki13\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki13\/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\/5ab5e4338f893605ed09b430b6ffd36f5cfd2bbd","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5ab5e4338f893605ed09b430b6ffd36f5cfd2bbd","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki13\/2022\/03\/30\/%e3%83%a9%e3%83%97%e3%83%a9%e3%82%b9%e4%bd%9c%e7%94%a8%e7%b4%a0-wikipedia\/","about":["Wiki"],"wordCount":9996,"articleBody":"\u6570\u5b66\u306b\u304a\u3051\u308b\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\uff08\u30e9\u30d7\u30e9\u30b9\u3055\u3088\u3046\u305d\u3001\u82f1: Laplace operator\uff09\u3042\u308b\u3044\u306f\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\uff08\u82f1: Laplacian)\u306f\u3001\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u4e0a\u306e\u51fd\u6570\u306e\u52fe\u914d\u306e\u767a\u6563\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u308b\u5fae\u5206\u4f5c\u7528\u7d20\u3067\u3042\u308b\u3002\u8a18\u53f7\u3067\u306f \u2207\u00b7\u2207, \u22072, \u3042\u308b\u3044\u306f \u2206 \u3067\u8868\u3055\u308c\u308b\u306e\u304c\u666e\u901a\u3067\u3042\u308b\u3002\u51fd\u6570 f \u306e\u70b9 p \u306b\u304a\u3051\u308b\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3 \u2206f(p) \u306f\uff08\u6b21\u5143\u306b\u4f9d\u5b58\u3059\u308b\u5b9a\u6570\u306e\u9055\u3044\u3092\u9664\u3044\u3066\uff09\u70b9 p \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u7403\u9762\u3092\u534a\u5f84\u304c\u5897\u5927\u3059\u308b\u3088\u3046\u306b\u52d5\u304b\u3059\u3068\u304d\u306e f(p) \u304b\u3089\u5f97\u3089\u308c\u308b\u5e73\u5747\u5024\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u306f\u3001\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306f\u5404\u72ec\u7acb\u5909\u6570\u306b\u95a2\u3059\u308b\u51fd\u6570\u306e\u4e8c\u968e\uff08\u975e\u6df7\u5408\uff09\u504f\u5c0e\u51fd\u6570\u306e\u548c\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u3001\u307e\u305f\u307b\u304b\u306b\u5186\u7b52\u5ea7\u6a19\u7cfb\u3084\u7403\u5ea7\u6a19\u7cfb\u306a\u3069\u306e\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u3082\u6709\u7528\u306a\u8868\u793a\u3092\u6301\u3064\u3002  \u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306e\u540d\u79f0\u306f\u3001\u5929\u4f53\u529b\u5b66\u306e\u7814\u7a76\u306b\u540c\u4f5c\u7528\u7d20\u3092\u6700\u521d\u306b\u7528\u3044\u305f\u30d5\u30e9\u30f3\u30b9\u4eba\u6570\u5b66\u8005\u306e\u30d4\u30a8\u30fc\u30eb\uff1d\u30b7\u30e2\u30f3\u30fb\u30c9\u30fb\u30e9\u30d7\u30e9\u30b9 (1749\u20131827) \u306b\u56e0\u3093\u3067\u3044\u308b\u3002\u540c\u4f5c\u7528\u7d20\u306f\u4e0e\u3048\u3089\u308c\u305f\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306b\u9069\u7528\u3059\u308b\u3068\u8cea\u91cf\u5bc6\u5ea6\u306e\u5b9a\u6570\u500d\u3092\u4e0e\u3048\u308b\u3002\u73fe\u5728\u3067\u306f\u30e9\u30d7\u30e9\u30b9\u65b9\u7a0b\u5f0f\u3068\u547c\u3070\u308c\u308b\u65b9\u7a0b\u5f0f \u2206f = 0 \u306e\u89e3\u306f\u8abf\u548c\u51fd\u6570\u3068\u547c\u3070\u308c\u3001\u81ea\u7531\u7a7a\u9593\u306b\u304a\u3044\u3066\u53ef\u80fd\u306a\u91cd\u529b\u5834\u3092\u8868\u73fe\u3059\u308b\u3082\u306e\u3067\u3042\u308b\u3002\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u304a\u3044\u3066\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306f\u96fb\u6c17\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3001\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3001\u71b1\u3084\u6d41\u4f53\u306e\u62e1\u6563\u65b9\u7a0b\u5f0f\u3001\u6ce2\u306e\u4f1d\u642c\u3001\u91cf\u5b50\u529b\u5b66\u3068\u3044\u3063\u305f\u3001\u591a\u304f\u306e\u7269\u7406\u73fe\u8c61\u3092\u8a18\u8ff0\u3059\u308b\u306e\u306b\u73fe\u308c\u308b\u3002\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306f\u3001\u51fd\u6570\u306e\u52fe\u914d\u30d5\u30ed\u30fc\u306e\u6d41\u675f\u5bc6\u5ea6\u3092\u8868\u3059\u3002\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306fn \u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u4e0a\u306e\u51fd\u6570 f \u306e\u52fe\u914d \u2207f \u306e\u767a\u6563 \u2207\u00b7 \u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u4e8c\u968e\u306e\u5fae\u5206\u4f5c\u7528\u7d20\u3067\u3042\u308b\u3002\u3064\u307e\u308a\u3001f \u304c\u4e8c\u56de\u5fae\u5206\u53ef\u80fd\u5b9f\u6570\u5024\u51fd\u6570\u306a\u3089\u3070 f \u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306f  \u0394f\u2261\u22072f:=\u2207\u22c5\u2207f{displaystyle Delta fequiv nabla ^{2}f:=nabla cdot nabla f}(1)\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u305f\u3060\u3057\u3001\u3042\u3068\u306e\u8a18\u6cd5\u306f\u5f62\u5f0f\u7684\u306b \u2207=(\u2202\/\u2202x1,\u2026,\u2202\/\u2202xn){displaystyle nabla =(partial \/partial x_{1},dotsc ,partial \/partial x_{n})} \u3068\u66f8\u3044\u305f\u3082\u306e\u3067\u3042\u308b\u3002\u3042\u308b\u3044\u306f\u540c\u3058\u3053\u3068\u3060\u304c\u3001f \u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306f\u76f4\u4ea4\u5ea7\u6a19\u7cfb xi \u306b\u304a\u3051\u308b\u975e\u6df7\u5408\u4e8c\u968e\u504f\u5c0e\u51fd\u6570\u306e\u5168\u3066\u306b\u308f\u305f\u308b\u548c  \u0394f=\u2211i=1n\u22022f\u2202xi2{displaystyle Delta f=sum _{i=1}^{n}{frac {partial ^{2}f}{partial x_{i}^{2}}}}(2)\u3068\u3057\u3066\u3082\u66f8\u3051\u308b\u3002\u4e8c\u968e\u306e\u5fae\u5206\u4f5c\u7528\u7d20\u3068\u3057\u3066\u3001\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306fCk \u7d1a\u51fd\u6570\u3092 Ck\u2009\u2212\u20092 \u7d1a\u306e\u51fd\u6570\u3078\u5199\u3059 (k\u00a0\u2265\u00a02)\u3002\u3064\u307e\u308a\u3001\u5f0f 1 (\u3042\u308b\u3044\u306f\u540c\u5024\u306a 2) \u306f\u4f5c\u7528\u7d20 \u2206: Ck(Rn) \u2192 Ck\u2009\u2212\u20092(Rn) \u3092\u5b9a\u3081\u308b\u3002\u3042\u308b\u3044\u306f\u3088\u308a\u4e00\u822c\u306b\u4efb\u610f\u306e\u958b\u96c6\u5408 \u03a9 \u306b\u5bfe\u3057\u3066\u4f5c\u7528\u7d20 \u2206: Ck(\u03a9) \u2192 Ck\u2009\u2212\u20092(\u03a9) \u3092\u5b9a\u3081\u308b\u3002Table of Contents\u6570\u5b66\u7684\u7279\u5fb4\u3065\u3051[\u7de8\u96c6]\u52d5\u6a5f\u4ed8\u3051[\u7de8\u96c6]\u62e1\u6563[\u7de8\u96c6]\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306b\u4ed8\u968f\u3059\u308b\u5bc6\u5ea6[\u7de8\u96c6]\u30a8\u30cd\u30eb\u30ae\u30fc\u6700\u5c0f\u5316[\u7de8\u96c6]\u5404\u7a2e\u5ea7\u6a19\u8868\u793a[\u7de8\u96c6]\u4e8c\u6b21\u5143[\u7de8\u96c6]\u4e09\u6b21\u5143[\u7de8\u96c6]\u4e00\u822c\u6b21\u5143[\u7de8\u96c6]\u30b9\u30da\u30af\u30c8\u30eb\u8ad6[\u7de8\u96c6]\u30e9\u30d7\u30e9\u30b9\uff1d\u30d9\u30eb\u30c8\u30e9\u30df\u4f5c\u7528\u7d20[\u7de8\u96c6]\u30c0\u30e9\u30f3\u30d9\u30fc\u30eb\u4f5c\u7528\u7d20[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]\u6570\u5b66\u7684\u7279\u5fb4\u3065\u3051[\u7de8\u96c6]\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306f\u3001\u5408\u540c\u5909\u63db\u3068\u53ef\u63db\u3067\u3042\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u4efb\u610f\u306eC\u221e\u7d1a\u95a2\u6570\u03c6\u00a0: Rn  \u2192 R\u3068\u4efb\u610f\u306e\u5408\u540c\u5909\u63dbT\u306b\u5bfe\u3057\u3001\u0394(\u03c6(T(x)))=T(\u0394(\u03c6(x))){displaystyle Delta (varphi (T(x)))=T(Delta (varphi (x)))}\u304c\u6210\u7acb\u3059\u308b[1]\u3002\u3057\u304b\u3082\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306f\u3001\u4e0a\u8a18\u306e\u6027\u8cea\u3092\u6e80\u305f\u3059\u975e\u81ea\u660e\u306a\u5fae\u5206\u6f14\u7b97\u5b50\u3067\u6700\u3082\u7c21\u5358\u306a\u3082\u306e\u3068\u3057\u3066\u7279\u5fb4\u3065\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u308c\u3092\u8aac\u660e\u3059\u308b\u70ba\u3001\u8a18\u53f7\u3092\u5c0e\u5165\u3059\u308b\u3002R\u3092\u5b9f\u6570\u306e\u96c6\u5408\u3068\u3057\u3001n\u500b\u306e\u5b9f\u6570\u304b\u3089\u306a\u308b\u7d44\u306e\u96c6\u5408\u3092Rn\u3068\u3059\u308b\u3002x=(x1,\u2026,xn)\u2208Rn\u3068n\u500b\u306e\u975e\u8ca0\u6574\u6570\u306e\u7d44\u03b1=(\u03b11,\u2026,\u03b1n)\u306b\u5bfe\u3057\u3001\u2202\u2202x\u03b1:=\u2202n\u2202x1\u03b11\u22ef\u2202xn\u03b1n,{displaystyle {partial  over partial x^{alpha }}:={partial ^{n} over partial x_{1}{}^{alpha _{1}}cdots partial x_{n}{}^{alpha _{n}}},}|\u03b1|:=\u03b11+\u22ef+\u03b1n{displaystyle |alpha |:=alpha _{1}+cdots +alpha _{n}}\u3068\u8868\u8a18\u3059\u308b\u3002\u5fae\u5206\u6f14\u7b97\u5b50D:=\u2211\u03b1:|\u03b1|\u2264ka\u03b1\u2202\u2202x\u03b1{displaystyle D:=sum _{alpha :|alpha |leq k}a_{alpha }{partial  over partial x^{alpha }}}\u304c\u4efb\u610f\u306eC\u221e\u7d1a\u95a2\u6570\u03c6\u00a0: Rn  \u2192 R\u3068\u5411\u304d\u3092\u4fdd\u3064\u4efb\u610f\u306e\u5408\u540c\u5909\u63dbT\u306b\u5bfe\u3057\u3001D(\u03c6(T(x)))=T(D(\u03c6(x))){displaystyle D(varphi (T(x)))=T(D(varphi (x)))}\u304c\u6210\u7acb\u3057\u3066\u3044\u305f\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u3001\u5b9f\u6570\u4fc2\u6570\u306e1\u5909\u6570\u591a\u9805\u5f0fp(X)=\u03a3m umXm\u304c\u5b58\u5728\u3057\u3001D=p(\u0394)=\u2211mum\u0394m{displaystyle D=p(Delta )=sum _{m}u_{m}Delta ^{m}}\u304c\u6210\u7acb\u3059\u308b[1]\u3002\u3088\u3063\u3066\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306f\u3001\u5408\u540c\u5909\u63db\u306b\u5bfe\u3057\u3066\u4e0d\u5909\u306a\u5fae\u5206\u6f14\u7b97\u5b50\u306e\u4e2d\u3067\u3001\u81ea\u660e\u306a\u3082\u306e\uff08\uff1d\u6052\u7b49\u7684\u306b0\u3092\u5bfe\u5fdc\u3055\u305b\u308b\u5fae\u5206\u6f14\u7b97\u5b50\uff09\u3092\u9664\u3051\u3070\u6700\u3082\u7c21\u5358\u306a\u3082\u306e\u3067\u3042\u308b\u3002\u52d5\u6a5f\u4ed8\u3051[\u7de8\u96c6]\u62e1\u6563[\u7de8\u96c6]\u62e1\u6563\u306e\u7269\u7406\u7406\u8ad6\u306b\u304a\u3044\u3066\u3001\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306f\uff08\u30e9\u30d7\u30e9\u30b9\u65b9\u7a0b\u5f0f\u3092\u901a\u3058\u3066\uff09\u5e73\u8861\u306e\u6570\u5b66\u7684\u8a18\u8ff0\u306b\u81ea\u7136\u306b\u73fe\u308c\u308b[2]\u3002\u5177\u4f53\u7684\u306b\u3001u \u304c\u5316\u5b66\u6fc3\u5ea6\u306e\u3088\u3046\u306a\u9069\u5f53\u306a\u91cf\u306e\u5e73\u8861\u5bc6\u5ea6\u3067\u3042\u308b\u3068\u304d\u3001u \u306e\u6ed1\u3089\u304b\u306a\u5883\u754c\u3092\u6301\u3064\u9818\u57df V \u3092\u901a\u308b\u6d41\u675f\u304c\u3001V \u306b\u6d41\u5165\u3082\u6f0f\u51fa\u3082\u7121\u3044\u3068\u3059\u308c\u3070\u30010 \u3067\u3042\u308b\u304b\u3089\u222b\u2202V\u2207u\u22c5ndS=0{displaystyle int _{partial V}nabla ucdot {boldsymbol {n}},dS=0}\u3068\u66f8\u3051\u308b\u3002\u305f\u3060\u3057\u3001n{textstyle {boldsymbol {n}}}\u306f\u9818\u57df V \u306e\u5883\u754c\u306b\u5bfe\u3057\u3066\u5916\u5074\u3092\u5411\u304f\u5358\u4f4d\u6cd5\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3002\u767a\u6563\u5b9a\u7406\u306b\u3088\u308a\u222bVdiv\u2061\u2207udV=\u222b\u2202V\u2207u\u22c5ndS=0{displaystyle int _{V}operatorname {div} nabla u,dV=int _{partial V}nabla ucdot {boldsymbol {n}},dS=0}\u306f\u9818\u57df V \u304c\u6ed1\u3089\u304b\u306a\u5883\u754c\u3092\u6301\u3064\u9650\u308a\u306b\u304a\u3044\u3066\u6210\u308a\u7acb\u3064\u304b\u3089\u3001\u3053\u308c\u306b\u3088\u308adiv\u2061\u2207u=\u0394u=0{displaystyle operatorname {div} nabla u=Delta u=0}\u304c\u5c0e\u304b\u308c\u308b\u3002\u65b9\u7a0b\u5f0f\u306e\u5de6\u8fba\u306f\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u3067\u3042\u308b\u3002\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u305d\u308c\u81ea\u8eab\u306f\u62e1\u6563\u65b9\u7a0b\u5f0f\u306b\u3088\u3063\u3066\u8a18\u8ff0\u3055\u308c\u308b\u3088\u3046\u306a\u3001\u5316\u5b66\u6fc3\u5ea6\u306e\u6d41\u5165\u3084\u6f0f\u51fa\u3092\u8868\u3059\u70b9\u3092\u542b\u3080\u975e\u5e73\u8861\u62e1\u6563\u306b\u5bfe\u3059\u308b\u7269\u7406\u7684\u89e3\u91c8\u3092\u6301\u3064\u3002\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306b\u4ed8\u968f\u3059\u308b\u5bc6\u5ea6[\u7de8\u96c6]\u03c6 \u304c\u96fb\u8377\u5206\u5e03 q \u306b\u4ed8\u968f\u3057\u305f\u96fb\u4f4d\u3092\u8a18\u8ff0\u3059\u308b\u3082\u306e\u3068\u3059\u308b\u3068\u3001\u96fb\u8377\u5206\u5e03\u81ea\u8eab\u306f \u03c6 \u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3068\u3057\u3066q=\u0394\u03c6{displaystyle q=Delta varphi }(1)\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u3053\u308c\u306f\u30ac\u30a6\u30b9\u306e\u6cd5\u5247\u306e\u5e30\u7d50\u3067\u3042\u308b\u3002\u5b9f\u969b\u3001V \u304c\u4efb\u610f\u306e\u6ed1\u3089\u304b\u306a\u9818\u57df\u306a\u3089\u3070\u3001\u96fb\u5834 E{textstyle {boldsymbol {E}}} \u306e\u96fb\u675f\u306b\u95a2\u3059\u308b\u30ac\u30a6\u30b9\u306e\u6cd5\u5247\u306b\u3088\u308a\u3001\uff08\u5358\u4f4d\u5f53\u305f\u308a\u306e\uff09\u96fb\u8377\u306f\u222b\u2202VE\u22c5ndS=\u222b\u2202V\u2207\u03c6\u22c5ndS=\u222bVqdV{displaystyle int _{partial V}{boldsymbol {E}}cdot {boldsymbol {n}},dS=int _{partial V}nabla varphi cdot {boldsymbol {n}},dS=int _{V}q,dV}\u306b\u306a\u308b\u3002\u305f\u3060\u3057\u3001\u6700\u521d\u306e\u7b49\u53f7\u306f\u9759\u96fb\u5834\u306f\u9759\u96fb\u4f4d\u306e\u52fe\u914d\u306b\u7b49\u3057\u3044\u3068\u3044\u3046\u4e8b\u5b9f\u3092\u7528\u3044\u305f\u3002\u767a\u6563\u5b9a\u7406\u306b\u3088\u308a\u3001\u222bV\u0394\u03c6dV=\u222bVqdV{displaystyle int _{V}Delta varphi ,dV=int _{V}q,dV}\u304c\u6210\u308a\u7acb\u3061\u3001\u3053\u308c\u306f\u4efb\u610f\u306e\u9818\u57df V \u306b\u5bfe\u3057\u3066\u6210\u308a\u7acb\u3064\u3053\u3068\u304b\u3089 (1) \u3092\u5f97\u308b\u3002\u540c\u3058\u8aac\u660e\u306b\u3088\u3063\u3066\u3001\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u304c\u8cea\u91cf\u5206\u5e03\u3068\u306a\u308b\u3053\u3068\u304c\u5c0e\u304b\u308c\u308b\u3002\u96fb\u8377\u3084\u8cea\u91cf\u306e\u5206\u5e03\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u3066\u305d\u308c\u3089\u306b\u4ed8\u968f\u3059\u308b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306f\u672a\u77e5\u3068\u3044\u3046\u3053\u3068\u306f\u3088\u304f\u3042\u308b\u3053\u3068\u3067\u3042\u308b\u3002\u9069\u5f53\u306a\u5883\u754c\u6761\u4ef6\u306e\u4e0b\u3067\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u51fd\u6570\u3092\u6c42\u3081\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u3001\u30dd\u30ef\u30bd\u30f3\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u306b\u540c\u3058\u3067\u3042\u308b\u3002\u30a8\u30cd\u30eb\u30ae\u30fc\u6700\u5c0f\u5316[\u7de8\u96c6]\u7269\u7406\u5b66\u306b\u304a\u3044\u3066\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u304c\u73fe\u308c\u308b\u5225\u306a\u7406\u7531\u306f\u3001\u9818\u57df U \u306b\u304a\u3051\u308b\u65b9\u7a0b\u5f0f \u2206f = 0 \u306e\u89e3\u306f\u30c7\u30a3\u30ea\u30af\u30ec\u30a8\u30cd\u30eb\u30ae\u30fc\u6c4e\u51fd\u6570\u3092\u505c\u7559\u3055\u305b\u308b\u51fd\u6570E(f):=12\u222bU\u2016\u2207f\u20162dx{displaystyle E(f):={frac {1}{2}}int _{U}Vert nabla fVert ^{2},dx}\u3068\u306a\u308b\u3053\u3068\u3067\u3042\u308b\u3002\u3053\u308c\u3092\u898b\u308b\u305f\u3081\u306b f: U \u2192 R \u306f\u51fd\u6570\u3067\u3001\u51fd\u6570 u: U \u2192 R \u306f U \u306e\u5883\u754c\u4e0a\u3067\u6d88\u3048\u3066\u3044\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3002\u3053\u306e\u3068\u304ddd\u03b5|\u03b5=0E(f+\u03b5u)=\u222bU\u2207f\u22c5\u2207udx=\u2212\u222bUu\u0394fdx{displaystyle {frac {d}{dvarepsilon }}{Bigg |}_{varepsilon =0}E(f+varepsilon u)=int _{U}nabla fcdot nabla u,dx=-int _{U}uDelta f,dx}\u304c\u6210\u308a\u7acb\u3064\uff08\u305f\u3060\u3057\u3001\u6700\u5f8c\u306e\u7b49\u53f7\u306f\u30b0\u30ea\u30fc\u30f3\u306e\u7b2c\u4e00\u6052\u7b49\u5f0f\uff08\u82f1\u8a9e\u7248\uff09\u3092\u7528\u3044\u305f\uff09\u3002\u3053\u306e\u8a08\u7b97\u306b\u3088\u308a\u3001\u2206f = 0 \u306a\u3089\u3070 E \u306f f \u306e\u5468\u308a\u3067\u505c\u7559\u3059\u308b\u3002\u9006\u306b E \u304c f \u306e\u5468\u308a\u3067\u505c\u7559\u3059\u308b\u306a\u3089\u3070\u5909\u5206\u6cd5\u306e\u57fa\u672c\u88dc\u984c\uff08\u82f1\u8a9e\u7248\uff09 \u306b\u3088\u308a \u2206f = 0 \u3067\u3042\u308b\u3002\u5404\u7a2e\u5ea7\u6a19\u8868\u793a[\u7de8\u96c6]\u4e8c\u6b21\u5143[\u7de8\u96c6]\u4e8c\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306f x, y \u3092 xy-\u5e73\u9762\u4e0a\u306e\u6a19\u6e96\u76f4\u4ea4\u5ea7\u6a19\u3068\u3057\u3066\u0394f:=\u22022f\u2202x2+\u22022f\u2202y2{displaystyle Delta f:={frac {partial ^{2}f}{partial x^{2}}}+{frac {partial ^{2}f}{partial y^{2}}}}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u6975\u5ea7\u6a19\u0394f=1r\u2202\u2202r(r\u2202f\u2202r)+1r2\u22022f\u2202\u03b82=1r\u2202f\u2202r+\u22022f\u2202r2+1r2\u22022f\u2202\u03b82.{displaystyle {begin{aligned}Delta f&={1 over r}{partial  over partial r}left(r{partial f over partial r}right)+{1 over r^{2}}{partial ^{2}f over partial theta ^{2}}\\&={1 over r}{partial f over partial r}+{partial ^{2}f over partial r^{2}}+{1 over r^{2}}{partial ^{2}f over partial theta ^{2}}.end{aligned}}}\u4e09\u6b21\u5143[\u7de8\u96c6]\u4e09\u6b21\u5143\u3067\u306f\u69d8\u3005\u306a\u5ea7\u6a19\u7cfb\u304c\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3092\u8a18\u8ff0\u3059\u308b\u305f\u3081\u306b\u5e83\u304f\u7528\u3044\u3089\u308c\u308b\u3002\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u0394f=\u22022f\u2202x2+\u22022f\u2202y2+\u22022f\u2202z2.{displaystyle Delta f={frac {partial ^{2}f}{partial x^{2}}}+{frac {partial ^{2}f}{partial y^{2}}}+{frac {partial ^{2}f}{partial z^{2}}}.}\u5186\u7b52\u5ea7\u6a19\u7cfb\u0394f=1\u03c1\u2202\u2202\u03c1(\u03c1\u2202f\u2202\u03c1)+1\u03c12\u22022f\u2202\u03c62+\u22022f\u2202z2.{displaystyle Delta f={1 over rho }{partial  over partial rho }left(rho {partial f over partial rho }right)+{1 over rho ^{2}}{partial ^{2}f over partial varphi ^{2}}+{partial ^{2}f over partial z^{2}}.}\u7403\u9762\u5ea7\u6a19\u7cfb\u0394f=1r2\u2202\u2202r(r2\u2202f\u2202r)+1r2sin\u2061\u03b8\u2202\u2202\u03b8(sin\u2061\u03b8\u2202f\u2202\u03b8)+1r2sin2\u2061\u03b8\u22022f\u2202\u03c62.{displaystyle Delta f={1 over r^{2}}{partial  over partial r}left(r^{2}{partial f over partial r}right)+{1 over r^{2}sin theta }{partial  over partial theta }left(sin theta {partial f over partial theta }right)+{1 over r^{2}sin ^{2}theta }{partial ^{2}f over partial varphi ^{2}}.}\u4e00\u822c\u306e\u66f2\u7dda\u5ea7\u6a19\u7cfb\uff08\u82f1\u8a9e\u7248\uff09 (\u03be1,\u03be2,\u03be3){displaystyle (xi ^{1},xi ^{2},xi ^{3})}\u22072=\u2207\u03bem\u22c5\u2207\u03ben\u22022\u2202\u03bem\u2202\u03ben+\u22072\u03bem\u2202\u2202\u03bem,{displaystyle nabla ^{2}=nabla xi ^{m}cdot nabla xi ^{n}{partial ^{2} over partial xi ^{m}partial xi ^{n}}+nabla ^{2}xi ^{m}{partial  over partial xi ^{m}},}\u3053\u3053\u3067\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u306e\u548c\u306e\u898f\u7d04\u3092\u7528\u3044\u305f\u3002\u4e00\u822c\u6b21\u5143[\u7de8\u96c6]N \u6b21\u5143\u7403\u5ea7\u6a19\u7cfb\u306b\u304a\u3044\u3066\u3001r \u3092\u6b63\u306e\u5b9f\u6570\u3092\u3068\u308b\u534a\u5f84\u3001\u03b8 \u306f\u5358\u4f4d\u7403\u9762 SN\u22121 \u306e\u5143\u3068\u3057\u3066\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\u8868\u793a x = r\u03b8 \u2208 RN \u3092\u3059\u308c\u3070\u0394f=\u22022f\u2202r2+N\u22121r\u2202f\u2202r+1r2\u0394SN\u22121f{displaystyle Delta f={frac {partial ^{2}f}{partial r^{2}}}+{frac {N-1}{r}}{frac {partial f}{partial r}}+{frac {1}{r^{2}}}Delta _{S^{N-1}}f}\u3068\u66f8\u3051\u308b\u3002\u305f\u3060\u3057\u3001\u2206SN\u22121 \u306f\u7403\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3068\u3082\u547c\u3070\u308c\u308b (N\u22121)-\u6b21\u5143\u7403\u9762\u4e0a\u306e\u30e9\u30d7\u30e9\u30b9\uff1d\u30d9\u30eb\u30c8\u30e9\u30df\u4f5c\u7528\u7d20\u3067\u3042\u308b\u3002\u4e8c\u3064\u306e\u7403\u5bfe\u79f0\u5fae\u5206\u9805\u306f1rN\u22121\u2202\u2202r(rN\u22121\u2202f\u2202r){displaystyle {frac {1}{r^{N-1}}}{frac {partial }{partial r}}{Bigl (}r^{N-1}{frac {partial f}{partial r}}{Bigr )}}\u3068\u66f8\u3044\u3066\u3082\u540c\u3058\u3053\u3068\u3067\u3042\u308b\u3002\u4e00\u3064\u306e\u5e30\u7d50\u3068\u3057\u3066\u3001SN\u22121 \u2282 RN \u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u51fd\u6570\u306e\u7403\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306f RN\u2216{0}{textstyle mathbf {R} ^{N}backslash {0}} \u3078\u5ef6\u9577\u3057\u305f\u51fd\u6570\u306e\u901a\u5e38\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3068\u3057\u3066\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001\u305d\u308c\u306f\u534a\u76f4\u7dda\u306b\u6cbf\u3063\u3066\u5b9a\u6570\uff08\u3064\u307e\u308a\u3001\u6589\u96f6\u6b21\u306e\u6589\u6b21\u51fd\u6570\uff09\u306b\u306a\u308b\u3002\u30b9\u30da\u30af\u30c8\u30eb\u8ad6[\u7de8\u96c6]\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306e\u30b9\u30da\u30af\u30c8\u30eb\u306f\u3001\u5bfe\u5fdc\u3059\u308b\u56fa\u6709\u51fd\u6570 f \u304c\u2212\u0394f=\u03bbf{displaystyle -Delta f=lambda f}\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u3067\u304d\u308b\u56fa\u6709\u5024 \u2212\u03bb \u306e\u5168\u3066\u304b\u3089\u306a\u308b[\u8981\u691c\u8a3c \u2013 \u30ce\u30fc\u30c8]\u3002\u4e0a\u306e\u5f0f\u306f\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u65b9\u7a0b\u5f0f\u3068\u547c\u3070\u308c\u308b\u3082\u306e\u3067\u3042\u308b\u3002 \u03a9 \u3092 Rn \u306e\u6709\u754c\u9818\u57df\u3068\u3059\u308c\u3070\u3001\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306e\u56fa\u6709\u51fd\u6570\u5168\u4f53\u306f\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593 L2(\u03a9) \u306e\u6b63\u898f\u76f4\u4ea4\u57fa\u5e95\u3092\u6210\u3059\u3002\u3053\u306e\u7d50\u679c\u306f\u672c\u8cea\u7684\u306b\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u81ea\u5df1\u968f\u4f34\u4f5c\u7528\u7d20\u306b\u95a2\u3059\u308b\u30b9\u30da\u30af\u30c8\u30eb\u5b9a\u7406\u3092\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306e\u9006\u4f5c\u7528\u7d20\uff08\u3053\u308c\u306f\u30dd\u30ef\u30f3\u30ab\u30ec\u4e0d\u7b49\u5f0f\u304a\u3088\u3073\u30b3\u30f3\u30c9\u30e9\u30b3\u30d5\u57cb\u8535\u5b9a\u7406\uff08\u82f1\u8a9e\u7248\uff09\u306b\u3088\u3063\u3066\u30b3\u30f3\u30d1\u30af\u30c8\uff09\u306b\u9069\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u5f93\u3046[3]\u3002\u56fa\u6709\u51fd\u6570\u304c\u7121\u9650\u56de\u5fae\u5206\u53ef\u80fd\u51fd\u6570\u3067\u3042\u308b\u3053\u3068\u3082\u793a\u305b\u308b[4]\u3002\u3053\u306e\u7d50\u679c\u306f\u3088\u308a\u4e00\u822c\u306b\u3001\u4efb\u610f\u306e\u5883\u754c\u4ed8\u304d\u30b3\u30f3\u30d1\u30af\u30c8\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u4e0a\u306e\u30e9\u30d7\u30e9\u30b9\uff1d\u30d9\u30eb\u30c8\u30e9\u30e0\u4f5c\u7528\u7d20\u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3061\u3001\u307e\u305f\u5b9f\u969b\u306b\u6709\u754c\u9818\u57df\u4e0a\u6ed1\u3089\u304b\u306a\u4fc2\u6570\u3092\u6301\u3064\u4efb\u610f\u306e\u6955\u5186\u578b\u4f5c\u7528\u7d20\u306b\u5bfe\u3059\u308b\u30c7\u30a3\u30ea\u30af\u30ec\u56fa\u6709\u5024\u554f\u984c\u306b\u3064\u3044\u3066\u3082\u6b63\u3057\u3044\u3002\u03a9 \u304c\u8d85\u7403\u9762\u3067\u3042\u308b\u3068\u304d\u306e\u3001\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306e\u56fa\u6709\u51fd\u6570\u306f\u7403\u9762\u8abf\u548c\u51fd\u6570\u3068\u547c\u3070\u308c\u308b\u3002\u30e9\u30d7\u30e9\u30b9\uff1d\u30d9\u30eb\u30c8\u30e9\u30df\u4f5c\u7528\u7d20[\u7de8\u96c6]\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306e\u6982\u5ff5\u306f\u3001\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u30e9\u30d7\u30e9\u30b9\uff1d\u30d9\u30eb\u30c8\u30e9\u30df\u4f5c\u7528\u7d20\uff08\u82f1\u8a9e\u7248\uff09\u3068\u547c\u3070\u308c\u308b\u6955\u5186\u578b\u4f5c\u7528\u7d20\u306b\u4e00\u822c\u5316\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u540c\u69d8\u306b\u30c0\u30e9\u30f3\u30d9\u30fc\u30eb\u4f5c\u7528\u7d20\u306f\u64ec\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u4e0a\u306e\u53cc\u66f2\u578b\u4f5c\u7528\u7d20\u306b\u4e00\u822c\u5316\u3055\u308c\u308b\u3002\u30e9\u30d7\u30e9\u30b9\uff1d\u30d9\u30eb\u30c8\u30e9\u30df\u4f5c\u7528\u7d20\u3092\u51fd\u6570\u306b\u9069\u7528\u3059\u308c\u3070\u3001\u305d\u306e\u51fd\u6570\u306e\u30d8\u30c3\u30bb\u884c\u5217\u306e\u30c8\u30ec\u30fc\u30b9\u0394f=tr\u2061(H(f)){displaystyle Delta f=operatorname {tr} (H(f))}\u304c\u5f97\u3089\u308c\u308b\u3002\u305f\u3060\u3057\u3001\u30c8\u30ec\u30fc\u30b9\u306f\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u9006\u306b\u95a2\u3057\u3066\u53d6\u308b\u3082\u306e\u3068\u3059\u308b\u3002\u30e9\u30d7\u30e9\u30b9\uff1d\u30d9\u30eb\u30c8\u30e9\u30df\u4f5c\u7528\u7d20\u3092\u540c\u69d8\u306e\u5f0f\u3067\u30c6\u30f3\u30bd\u30eb\u5834\u306b\u4f5c\u7528\u3059\u308b\u4f5c\u7528\u7d20\uff08\u3053\u308c\u3082\u307e\u305f\u30e9\u30d7\u30e9\u30b9\uff1d\u30d9\u30eb\u30c8\u30e9\u30df\u4f5c\u7528\u7d20\u3068\u547c\u3070\u308c\u308b\uff09\u306b\u4e00\u822c\u5316\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306e\u5225\u306a\u4e00\u822c\u5316\u3068\u3057\u3066\u3001\u64ec\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u5916\u5fae\u5206\u3092\u7528\u3044\u305f\u300c\u5e7e\u4f55\u5b66\u8005\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u300d\u3068\u547c\u3070\u308c\u308b\u0394f=d\u2217df{displaystyle Delta f=d^{*}df}\u3092\u8003\u3048\u308b\u3053\u3068\u3082\u3067\u304d\u308b\u3002\u3053\u3053\u3067 d\u2217\u306f\u4f59\u5fae\u5206\u3067\u3001\u30db\u30c3\u30b8\u53cc\u5bfe\u3092\u4f7f\u3063\u3066\u66f8\u304f\u3053\u3068\u3082\u3067\u304d\u308b\u3002\u3053\u308c\u304c\u4e0a\u306b\u8ff0\u3079\u305f\u300c\u89e3\u6790\u5b66\u8005\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u300d\u3068\u306f\u7570\u306a\u308b\u3082\u306e\u3067\u3042\u308b\u3053\u3068\u306b\u306f\u6ce8\u610f\u3059\u3079\u304d\u3067\u3042\u308b\u3002\u305d\u306e\u3053\u3068\u306f\u5927\u57df\u89e3\u6790\u5b66\u306e\u8ad6\u6587\u3092\u8aad\u3080\u3068\u304d\u306b\u306f\u5e38\u306b\u6c17\u3092\u4ed8\u3051\u306d\u3070\u306a\u3089\u306a\u3044\u3002\u3088\u308a\u4e00\u822c\u306b\u3001\u5fae\u5206\u5f62\u5f0f\u306b\u5bfe\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u300c\u30db\u30c3\u30b8\u300d\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3 \u03b1 \u306f\u0394\u03b1=d\u2217d\u03b1+dd\u2217\u03b1{displaystyle Delta alpha =d^{*}dalpha +dd^{*}alpha }\u3068\u66f8\u3051\u308b\u3002\u3053\u308c\u306f\u307e\u305f\u30e9\u30d7\u30e9\u30b9\uff1d\u30c9\u30e9\u30fc\u30e0\u4f5c\u7528\u7d20\uff08\u82f1\u8a9e\u7248\uff09\u3068\u3082\u547c\u3070\u308c\u3001\u30f4\u30a1\u30a4\u30c4\u30a7\u30f3\u30d9\u30c3\u30af\u4e0d\u7b49\u5f0f\uff08\u82f1\u8a9e\u7248\uff09\u306b\u3088\u3063\u3066\u30e9\u30d7\u30e9\u30b9\uff1d\u30d9\u30eb\u30c8\u30e9\u30df\u4f5c\u7528\u7d20\u3068\u95a2\u4fc2\u3059\u308b\u3002\u30c0\u30e9\u30f3\u30d9\u30fc\u30eb\u4f5c\u7528\u7d20[\u7de8\u96c6]\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3092\u9069\u5f53\u306a\u4ed5\u65b9\u306b\u3088\u3063\u3066\u975e\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u4e00\u822c\u5316\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001\u305d\u308c\u306b\u306f\u6955\u5186\u578b\u3001\u53cc\u66f2\u578b\u3001\u8d85\u53cc\u66f2\u578b\uff08\u82f1\u8a9e\u7248\uff09\u306a\u3069\u304c\u53ef\u80fd\u3067\u3042\u308b\u3002\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u7a7a\u9593\u306b\u304a\u3051\u308b\u30e9\u30d7\u30e9\u30b9\uff1d\u30d9\u30eb\u30c8\u30e9\u30df\u4f5c\u7528\u7d20\u306f\u30c0\u30e9\u30f3\u30d9\u30fc\u30eb\u4f5c\u7528\u7d20\u25fb=1c2\u22022\u2202t2\u2212\u22022\u2202x2\u2212\u22022\u2202y2\u2212\u22022\u2202z2{displaystyle square ={frac {1}{c^{2}}}{partial ^{2} over partial t^{2}}-{partial ^{2} over partial x^{2}}-{partial ^{2} over partial y^{2}}-{partial ^{2} over partial z^{2}}}\u3068\u306a\u308b\u3002\u3053\u308c\u306f\u8003\u3048\u308b\u7a7a\u9593\u4e0a\u306e\u7b49\u9577\u5909\u63db\u7fa4\u306e\u3082\u3068\u3067\u4e0d\u5909\u306a\u5fae\u5206\u4f5c\u7528\u7d20\u3067\u3042\u308b\u3068\u3044\u3046\u610f\u5473\u306b\u304a\u3044\u3066\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306e\u4e00\u822c\u5316\u3068\u306a\u308b\u3082\u306e\u3067\u3042\u308a\u3001\u6642\u9593\u4e0d\u5909\u51fd\u6570\u3078\u5236\u9650\u3059\u308b\u9650\u308a\u306b\u304a\u3044\u3066\u306f\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20\u306b\u5e30\u7740\u3055\u308c\u308b\u3002\u3053\u3053\u3067\u306f\u8a08\u91cf\u306e\u7b26\u53f7\u3092\u4f5c\u7528\u7d20\u306e\u7a7a\u9593\u6210\u5206\u306b\u95a2\u3057\u3066\u8ca0\u7b26\u53f7\u3092\u8a31\u3059\u3088\u3046\u306b\u3057\u3066\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\uff08\u9ad8\u30a8\u30cd\u30eb\u30ae\u30fc\u7c92\u5b50\u7269\u7406\u5b66\u3067\u306f\u3053\u3046\u4eee\u5b9a\u3059\u308b\u306e\u304c\u666e\u901a\uff09\u3002\u30c0\u30e9\u30f3\u30d9\u30fc\u30eb\u4f5c\u7528\u7d20\u306f\u6ce2\u52d5\u65b9\u7a0b\u5f0f\u306b\u73fe\u308c\u308b\u5fae\u5206\u4f5c\u7528\u7d20\u3067\u3042\u308b\u3068\u3044\u3046\u7406\u7531\u3067\u6ce2\u52d5\u4f5c\u7528\u7d20\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u3082\u3042\u308b\u3002\u3053\u308c\u306f\u307e\u305f\u30af\u30e9\u30a4\u30f3\uff1d\u30b4\u30eb\u30c9\u30f3\u65b9\u7a0b\u5f0f\uff08\u8cea\u91cf\u306e\u7121\u3044\u5834\u5408\u306b\u306f\u6ce2\u52d5\u65b9\u7a0b\u5f0f\u306b\u5e30\u7740\u3055\u308c\u308b\uff09\u306e\u6210\u5206\u3067\u3082\u3042\u308b\u3002\u8a08\u91cf\u306b\u304a\u3051\u308b\u4f59\u5206\u306a\u56e0\u5b50 c \u306f\u3001\u7269\u7406\u5b66\u306b\u304a\u3044\u3066\u7a7a\u9593\u3068\u6642\u9593\u3092\u7570\u306a\u308b\u5358\u4f4d\u3067\u6e2c\u3063\u3066\u3044\u308b\u5834\u5408\u306b\u5fc5\u8981\u3068\u306a\u308b\u3082\u306e\u3067\u3042\u308b\uff08\u4f8b\u3048\u3070\u540c\u69d8\u306e\u3053\u3068\u306f x-\u65b9\u5411\u3092\u30e1\u30fc\u30c8\u30eb\u3067 y-\u65b9\u5411\u3092\u30bb\u30f3\u30c1\u30e1\u30fc\u30c8\u30eb\u3067\u6e2c\u3063\u305f\u308a\u3059\u308b\u3088\u3046\u306a\u5834\u5408\u306b\u3082\u51fa\u3066\u304f\u308b\uff09\u3002\u5b9f\u969b\u3001\u7406\u8ad6\u7269\u7406\u5b66\u3067\u306f\u65b9\u7a0b\u5f0f\u3092\u7c21\u5358\u306b\u3059\u308b\u76ee\u7684\u3067\u3001\u81ea\u7136\u5358\u4f4d\u7cfb\u306a\u3069\u306e\u5358\u4f4d\u7cfb\u306e\u3082\u3068 c = 1 \u3068\u3057\u3066\u6271\u3046\u306e\u304c\u3075\u3064\u3046\u3067\u3042\u308b\u3002\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]Evans, L (1998), Partial Differential Equations, American Mathematical Society, ISBN\u00a0978-0-8218-0772-9\u00a0.Feynman, R, Leighton, R, and Sands, M (1970), \u201cChapter 12: Electrostatic Analogs\u201d, The Feynman Lectures on Physics, Volume 2, Addison-Wesley-Longman\u00a0.Gilbarg, D.; Trudinger, N. (2001), Elliptic partial differential equations of second order, Springer, ISBN\u00a0978-3-540-41160-4\u00a0.Schey, H. M. (1996), Div, grad, curl, and all that, W W Norton & Company, ISBN\u00a0978-0-393-96997-9\u00a0.\u91ce\u6751\u9686\u662d (2006\u5e74). \u201c\u6975\u5ea7\u6a19\u30fb\u56de\u8ee2\u7fa4\u30fbSL(2, R) (pdf)\u201d. 2017\u5e741\u67084\u65e5\u95b2\u89a7\u3002\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\/wiki13\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki13\/2022\/03\/30\/%e3%83%a9%e3%83%97%e3%83%a9%e3%82%b9%e4%bd%9c%e7%94%a8%e7%b4%a0-wikipedia\/#breadcrumbitem","name":"\u30e9\u30d7\u30e9\u30b9\u4f5c\u7528\u7d20 – Wikipedia"}}]}]