[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/911#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/911","headline":"\u30df\u30cb\u30de\u30eb\u30a8\u30ea\u30a2-Wikipedia","name":"\u30df\u30cb\u30de\u30eb\u30a8\u30ea\u30a2-Wikipedia","description":"before-content-x4 \u4e00 \u6700\u5c0f\u9650\u306e\u30a8\u30ea\u30a2 \u90e8\u5c4b\u306e\u30a8\u30ea\u30a2\u304c\u6700\u5c0f\u9650\u306e\u30a8\u30ea\u30a2\u3067\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u3053\u306e\u3088\u3046\u306a\u5f62\u72b6\u306f\u3001\u5bfe\u5fdc\u3059\u308b\u30d5\u30ec\u30fc\u30e0\uff08\u98a8\u306e\u30ea\u30f3\u30b0\u306a\u3069\uff09\u3092\u4ecb\u3057\u3066\u5f35\u529b\u3092\u304b\u3051\u3089\u308c\u3066\u3044\u308b\u5834\u5408\u3001\u77f3\u9e78\u76ae\u3092\u5f15\u304d\u53d7\u3051\u307e\u3059\u3002 after-content-x4 \u6570\u5b66\u7684\u8a00\u8a9e\u3067\u306f\u3001\u6700\u5c0f\u9650\u306e\u9818\u57df\u304c\u9818\u57df\u306e\u30b3\u30f3\u30c6\u30f3\u30c4\u6a5f\u80fd\u306e\u91cd\u8981\u306a\u30dd\u30a4\u30f3\u30c8\u3067\u3059 a \uff08 \u30d0\u30c4 \uff09\uff09 = \u222b g(u)dn\u306e {displaystyle a\uff08mathbf {x}\uff09= int {sqrt {g\uff08u\uff09}}\u3001mathrm 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    (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4\u4e00 \u6700\u5c0f\u9650\u306e\u30a8\u30ea\u30a2 \u90e8\u5c4b\u306e\u30a8\u30ea\u30a2\u304c\u6700\u5c0f\u9650\u306e\u30a8\u30ea\u30a2\u3067\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u3053\u306e\u3088\u3046\u306a\u5f62\u72b6\u306f\u3001\u5bfe\u5fdc\u3059\u308b\u30d5\u30ec\u30fc\u30e0\uff08\u98a8\u306e\u30ea\u30f3\u30b0\u306a\u3069\uff09\u3092\u4ecb\u3057\u3066\u5f35\u529b\u3092\u304b\u3051\u3089\u308c\u3066\u3044\u308b\u5834\u5408\u3001\u77f3\u9e78\u76ae\u3092\u5f15\u304d\u53d7\u3051\u307e\u3059\u3002        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u6570\u5b66\u7684\u8a00\u8a9e\u3067\u306f\u3001\u6700\u5c0f\u9650\u306e\u9818\u57df\u304c\u9818\u57df\u306e\u30b3\u30f3\u30c6\u30f3\u30c4\u6a5f\u80fd\u306e\u91cd\u8981\u306a\u30dd\u30a4\u30f3\u30c8\u3067\u3059 a \uff08 \u30d0\u30c4 \uff09\uff09 = \u222b g(u)dn\u306e {displaystyle a\uff08mathbf {x}\uff09= int {sqrt {g\uff08u\uff09}}\u3001mathrm {d} ^{n} u} \u3002 \u3053\u308c\u304c\u30b5\u30a4\u30ba\u3067\u3059        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4g \uff08 \u306e \uff09\uff09 \uff1a=  \u2061 \uff08 g \u79c1 j \uff08 \u306e \uff09\uff09 \uff09\uff09 \u79c1 \u3001 j = \u521d\u3081 \u3001 … \u3001 n {displaystyle g\uff08u\uff09\uff1a= operaator {det}\uff08g_ {ij}\uff08u\uff09\uff09_ {i\u3001j = 1\u3001dotsc\u3001n}} \u3068 g \u79c1 j \uff08 \u306e \uff09\uff09 = \uff08 \u2202x\u2202ui\uff09\uff09 t \u2202x\u2202uj{displaystyle g_ {ij}\uff08u\uff09= left\uff08{tfrac {partial mathbf {x}} {partial u_ {i}}}\u53f3\uff09^{tfrac {partial mathbf {x}}}} {j}}}}}}}} \u305f\u3081\u306b \u79c1 \u3001 j = \u521d\u3081 \u3001 … \u3001 n {displaystyle i\u3001j = 1\u3001dotsc\u3001n} \u8aac\u660e\uff08Hesse-Matrix\u3092\u53c2\u7167\uff09\u3002\u6700\u5c0f\u9650\u306e\u9762\u7a4d\u306b\u306f\u5fc5\u305a\u3057\u3082\u6700\u5c0f\u9650\u306e\u9762\u7a4d\u304c\u3042\u308b\u308f\u3051\u3067\u306f\u306a\u304f\u3001\u9762\u7a4d\u30b3\u30f3\u30c6\u30f3\u30c4\u95a2\u6570\u306e\u9759\u6b62\u30dd\u30a4\u30f3\u30c8\u306b\u3059\u304e\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002 2\u3064\u306e\u90e8\u5c4b\u306e\u5bf8\u6cd5\u306b\u304a\u3051\u308b\u30a8\u30ea\u30a2\u30b3\u30f3\u30c6\u30f3\u30c4\u95a2\u6570\u306e\u6700\u521d\u306e\u5909\u52d5\u306e\u6d88\u5931\u306f\u3001\u8003\u616e\u3055\u308c\u305f\u30de\u30cb\u30db\u30fc\u30eb\u30c9\u304c\u5341\u5206\u306b\u898f\u5247\u7684\u3067\u3042\u308b\u5834\u5408\u3001\u4e2d\u592e\u306e\u66f2\u7387h\u3092\u6d88\u6ec5\u3055\u305b\u308b\u306e\u3068\u540c\u7b49\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 19\u4e16\u7d00\u4ee5\u6765\u3001\u6700\u5c0f\u9650\u306e\u5206\u91ce\u306f\u6570\u5b66\u7684\u7814\u7a76\u306b\u7126\u70b9\u3092\u5f53\u3066\u3066\u3044\u307e\u3059\u3002\u30d9\u30eb\u30ae\u30fc\u306e\u7269\u7406\u5b66\u8005\u30b8\u30e7\u30bb\u30d5\u30fb\u30d7\u30e9\u30c8\u30fc\u306b\u3088\u308b\u5b9f\u9a13\u306f\u3001\u3053\u308c\u306b\u5927\u304d\u306a\u8ca2\u732e\u3067\u3057\u305f\u3002        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4 \u6700\u5c0f\u9650\u306e\u9818\u57df\u306e\u4f8b\u3068\u3057\u3066\u306eenneper\u30a8\u30ea\u30a2 Table of Contents2\u3064\u306e\u5909\u6570\u306e\u5b58\u5728\u7406\u8ad6\u306b\u95a2\u3059\u308b\u4f8b\u5916 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3\u306e\u554f\u984c\u3068\u3057\u3066\u306e\u5b9a\u5f0f\u5316 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] 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F. Scherk\u306e\u6700\u5c0f\u9762\u7a4d [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30c0\u30b9\u30fb\u30ab\u30c6\u30ce\u30a4\u30c9 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u3089\u305b\u3093\u30a8\u30ea\u30a2 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30d8\u30f3\u30cd\u30d0\u30fc\u30b0\u30a8\u30ea\u30a2 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] MaximumPrinzip [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30a4\u30e9\u30b9\u30c8\u306e\u5747\u4e00\u5316 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30ec\u30ec\u30e9\u30f3\u5206\u6790\u30c1\u30e3\u30fc\u30c8 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] 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xu\u3001 xv\uff09\uff09 )d \u306e d \u306e {displaystyle a\uff08mathbf {x}\uff09= int {big\uff08} | mathbf {x} _ {u}\u500d\u306emathbf {x} | +2\uff08q\uff08mathbf {x}\uff09\u3001mathbf {x}\u3001mathbf {x} _ {v {v {v {v {{d} {d} {v {v {d} } \u3002 \u3053\u306e\u95a2\u6570\u306e\u5fc5\u8981\u306a\u6700\u5c0f\u6761\u4ef6\u3068\u3057\u3066\u306e\u30aa\u30a4\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306f\u3001Franz Rellich\u306b\u5f93\u3063\u3066\u540d\u524d\u304c\u4ed8\u3051\u3089\u308c\u305fH\u8868\u9762\u30b7\u30b9\u30c6\u30e0\u3067\u3059 d \u30d0\u30c4 = 2 h xu\u00d7 xv\u3001 xu2 –  xv2= 0 = xuxv{displaystyle delta mathbf {x} = 2hmathbf {x} _ {u} times mathbf {x} _ {v}\u3001qquad mathbf {x} _ {u}^{2} -mathbf {x}} _ {v}^{2}^{2} _ {x} _ {2} = 0} {x} {2} = 0 = 0 = 0 = x} _ {v}} \u3002 \u3053\u3053\u306f h = div Q {displaystyle h = operatorname {div}\u3001q} \u4e2d\u592e\u306e\u66f2\u7387\u3002 \u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u30b1\u30fc\u30b9 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u3053\u306e\u6a5f\u80fd\u306e\u305f\u3081\u306b\u3001\u5c40\u6240\u7684\u306a\u6700\u5c0f\u5024\u306e\u5b58\u5728\u306e\u554f\u984c\u306f\u3001\u6709\u9650\u9577\u306e\u6240\u5b9a\u306e\u5b89\u5b9a\u3057\u305f\u5468\u8fba\u66f2\u7dda\u3067\u751f\u3058\u307e\u3059\u3002\u3053\u306e\u30bf\u30b9\u30af\u306f\u3001\u6587\u732e\u3067\u3082\u8aac\u660e\u3055\u308c\u3066\u3044\u307e\u3059 \u30d7\u30e9\u30c3\u30c8\u30d5\u30a9\u30fc\u30e0\u306e\u554f\u984c\u3002 \u4e2d\u7a0b\u5ea6\u306e\u66f2\u7387\u306b\u5bfe\u3059\u308b\u5c0f\u3055\u3055\u306e\u6761\u4ef6\u3092\u4eee\u5b9a\u3059\u308b\u3068\u3001\u6700\u5c0f\u9762\u7a4d\u304c\u767a\u751f\u3057\u305f\u5834\u5408\u306b\u5e38\u306b\u6e80\u305f\u3055\u308c\u3066\u3044\u308b\u305f\u3081\u3001\u3053\u306e\u8cea\u554f\u306f\u7a4d\u6975\u7684\u306b\u7b54\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u81ea\u5206\u3092\u7d0d\u5f97\u3055\u305b\u308b\u305f\u3081\u306b\u3001\u3042\u306a\u305f\u306f\u540c\u6642\u306b\u6700\u5c0f\u5316\u3057\u307e\u3059 a {displaystyle a} \u30a8\u30cd\u30eb\u30ae\u30fc\u6a5f\u80fd \u3068 \uff08 \u30d0\u30c4 \uff09\uff09 = \u222c (12| \u2207 \u30d0\u30c4 |2+ 2 \uff08 Q \uff08 \u30d0\u30c4 \uff09\uff09 \u3001 xu\u3001 xv\uff09\uff09 )d \u306e d \u306e {displaystyle e\uff08mathbf {x}\uff09= int {big {1} {2} {2}} | nabla mathbf {x} |^{x} +2\uff08q\uff08x {x {x {x {x {x {x}\uff09\u3001mathbf {x} {d} {d} v} \u3044\u308f\u3086\u308b\u9ad8\u901f\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u306e\u5c0e\u5165\u4e2d\u3002 1884\u5e74\u3001\u30d8\u30eb\u30de\u30f3\u30fb\u30a2\u30de\u30f3\u30c0\u30b9\u30fb\u30b7\u30e5\u30ef\u30eb\u30c4\u306f\u5211\u3092\u898b\u305b\u305f \u6027\u5225\u30bc\u30ed\uff08\u3064\u307e\u308a\u3001\u7a74\u306e\u306a\u3044\uff09\u304b\u3089\u3001\u7740\u5b9f\u306b\u533a\u5225\u3055\u308c\u3001\u9589\u3058\u305f\u3001\u9589\u3058\u305f\u65b9\u5411\u306e\u3042\u308b\u8868\u9762\u306e\u91cf\u306e\u91cf\u3067\u306f\u3001\u30dc\u30fc\u30eb\u306f\u8868\u9762\u3067\u3042\u308a\u3001\u4e0e\u3048\u3089\u308c\u305f\u8868\u9762\u306e\u6700\u5927\u306e\u7a7a\u9593\u3092\u5909\u5316\u3055\u305b\u307e\u3059\u3002 \u5206\u5c90\u70b9 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u3069\u306e\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u3092\u7f6e\u304d\u307e\u3059 | \u30d0\u30c4 \u306e \u00d7 \u30d0\u30c4 \u306e | = 0 {displaystyle | mathbf {x} _ {u}\u500d\u306eMathbf {x} _ {v} | = 0} \u6e80\u305f\u3055\u308c\u305f\u3001\u547c\u3070\u308c\u307e\u3059 \u30d6\u30e9\u30f3\u30c1\u30dd\u30a4\u30f3\u30c8\u3002 \u3053\u308c\u3089\u306e\u30dd\u30a4\u30f3\u30c8\u3067\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u5316\u304c\u7279\u7570\u306b\u306a\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\u305f\u3081\u3001\u5206\u5c90\u70b9\u306f\u8208\u5473\u6df1\u3044\u3082\u306e\u3067\u3059\u3002\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u304c\u3082\u306f\u3084\u9818\u57df\u3067\u306f\u306a\u3044\u53ef\u80fd\u6027\u306f\u3055\u3089\u306b\u60aa\u5316\u3057\u307e\u3059\u304c\u3001\u66f2\u7dda\u306e\u307f\u3067\u3059\u3002 \u73fe\u5728\u3001Carleman\u3068Vekua\u306b\u3088\u308b\u4f5c\u54c1\u306b\u5927\u304d\u304f\u89e6\u767a\u3055\u308c\u305f\u6a5f\u80fd\u7684\u306a\u8003\u616e\u4e8b\u9805\u306f\u3001\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u304c\u6700\u7d42\u7684\u306b\u591a\u304f\u306e\u305d\u306e\u3088\u3046\u306a\u5206\u5c90\u3092\u6301\u3064\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u3092\u63d0\u4f9b\u3057\u3066\u3044\u307e\u3059\u3002\u6b8b\u5ff5\u306a\u304c\u3089\u3001\u4e0a\u8a18\u306e\u65b9\u6cd5\u3067\u306f\u3001\u305d\u306e\u3088\u3046\u306a\u5206\u5c90\u70b9\u306f\u5148\u9a13\u7684\u306b\u9664\u5916\u3055\u308c\u3066\u3044\u307e\u305b\u3093\u3002\u305d\u308c\u306f\u30ac\u30ea\u30d0\u30fc\u30fb\u30a2\u30eb\u30c8\u30fb\u30aa\u30c3\u30b5\u30fc\u30de\u30f3\u306e\u7cbe\u5de7\u306a\u5211\u3068\u306e\u307f\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u679d\u306e\u306a\u3044H\u8868\u9762\u306e\u30af\u30e9\u30b9\u3067\u30d7\u30e9\u30c3\u30c8\u30d5\u30a9\u30fc\u30e0\u306e\u554f\u984c\u3092\u89e3\u6c7a\u3057\u305f\u3044\u3068\u3044\u3046\u9858\u671b\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u308c\u306f\u307e\u3060\u672a\u89e3\u6c7a\u306e\u8cea\u554f\u3067\u3059\u3002 \u975e\u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u30b1\u30fc\u30b9\u3001\u6700\u5c0f\u9762\u7a4d\u65b9\u7a0b\u5f0f [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u305f\u3060\u3057\u3001\u4e0a\u8a18\u306e\u65b9\u6cd5\u306f\u5b9a\u6570\u306b\u306e\u307f\u3064\u306a\u304c\u308a\u307e\u3059 h {displaystyle h} \u6210\u529f\u3078\u3002\u5e73\u5747\u66f2\u7387\u3082\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u306b\u4f9d\u5b58\u3057\u3066\u3044\u308b\u5834\u5408\u3067\u3082\u3001\u30b0\u30e9\u30d5\u304c\u767a\u751f\u3057\u305f\u5834\u5408\u3067\u3082\u4f55\u304b\u3092\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 \u30d0\u30c4 {displaystyle mathbf {x}} \u30b0\u30e9\u30d5\u3001\u5f7c\u306fa\u3068\u3057\u3066\u66f8\u304d\u307e\u3059 \u30d0\u30c4 \uff08 \u306e \u3001 \u306e \uff09\uff09 = \uff08 \u306e \u3001 \u306e \u3001 z \uff08 \u306e \u3001 \u306e \uff09\uff09 \uff09\uff09 t {displaystyle mathbf {x}\uff08u\u3001v\uff09=\uff08u\u3001v\u3001zeta\uff08u\u3001v\uff09{t}}}}} \u3068\u95a2\u6570 z {displaystyle zeta} \u6e80\u305f\u3055\u308c\u305f \u51e6\u65b9\u3055\u308c\u305f\u4e2d\u7a0b\u5ea6\u306e\u66f2\u7387\u306e\u975e\u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u65b9\u7a0b\u5f0f  \uff08 \u521d\u3081 + z v2\uff09\uff09 z uu –  2 z uz vz uv+ \uff08 \u521d\u3081 + z u2\uff09\uff09 z vv= h \uff08 \u306e \u3001 \u306e \u3001 z \uff08 \u306e \u3001 \u306e \uff09\uff09 \uff09\uff09 {displaystyle\uff081+zeta _ {v} {2}\uff09Zeta _ {uu} -2zeta _ {u} Zeta _} Zeta _ {uv}+\uff081+Zeta _ {2}\uff09Zeta _ {VV} = H = H \u3002 \u9818\u57df\u304c\u6700\u5c0f\u306e\u5834\u5408\u3001h = 0\u3068\u65b9\u7a0b\u5f0f\u306f\u6700\u5c0f\u9762\u7a4d\u65b9\u7a0b\u5f0f\u3068\u547c\u3070\u308c\u307e\u3059\u3002 [\u521d\u3081] \u6df1\u3044\u751f\u8a08\u306e\u7d50\u679c\u306f\u3001\u3053\u306e\u90e8\u5206\u7684\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u30c7\u30a3\u30ea\u30af\u30ec\u306e\u554f\u984c\u306e\u6eb6\u89e3\u5ea6\u3092\u63d0\u4f9b\u3057\u3001\u5c0f\u3055\u3055\u6761\u4ef6\u3084\u305d\u306e\u4ed6\u306e\u6280\u8853\u7684\u8981\u4ef6\u3092\u4eee\u5b9a\u3057\u307e\u3059\u3002\u4e00\u610f\u6027\u306f\u30012\u3064\u306e\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u9593\u306e\u9055\u3044\u306e\u6700\u5927\u539f\u7406\u306b\u3088\u3063\u3066\u3082\u660e\u3089\u304b\u306b\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u306e\u305f\u3081\u306b\u30b0\u30e9\u30d5\u3082\u3042\u308a\u307e\u3059 | xu\u00d7 xv| = 1+\u2207\u03b6(u,v)2\u2265 \u521d\u3081 {displaystyle | mathbf {x} _ {u}\u500d\u306emathbf {x} _ {v} | = {sqrt {1+nabla zeta\uff08u\u3001v\uff09^{2}}} geq 1}} \u5e38\u306b\u30d6\u30e9\u30f3\u30c1\u30d5\u30ea\u30fc\u3002 \u3053\u3053\u3067\u306f\u3001\u3055\u307e\u3056\u307e\u306a\u6700\u5c0f\u9818\u57df\u306e\u3044\u304f\u3064\u304b\u306e\u4f8b\u304c\u30013\u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u793a\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u305d\u308c\u3089\u306e\u3044\u304f\u3064\u304b\u306f\u3001\u81ea\u5df1\u30ab\u30c3\u30c8\u306a\u3057\u30673\u6b21\u5143\u7a7a\u9593\u306b\u57cb\u3081\u8fbc\u3080\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u6700\u521d\u306e\u4f8b\u304c\u793a\u3059\u3088\u3046\u306b\u3001\u4ed6\u306e\u4eba\u306f\u5b9a\u7fa9\u9818\u57df\u306e\u7aef\u307e\u3067\u7d99\u7d9a\u7684\u306b\u7d99\u7d9a\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002 H. F. Scherk\u306e\u6700\u5c0f\u9762\u7a4d [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ]  \u30d1\u30e9\u30e1\u30fc\u30bf\u30fcC = 1\u3078\u306eScherk\u306e\u6700\u5c0f\u9818\u57df\u306e\u63a5\u7d9a\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8 \u30cf\u30a4\u30f3\u30ea\u30c3\u30d2\u30d5\u30a7\u30eb\u30c7\u30a3\u30ca\u30f3\u30c9\u30b7\u30a7\u30eb\u30af\u306e\u6700\u5c0f\u9762\u7a4d\uff081835\uff09\uff1a\u975e\u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u6700\u5c0f\u9818\u57df\u65b9\u7a0b\u5f0f\u306e\u3059\u3079\u3066\u306e\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u3092\u63a2\u3057\u3066\u3044\u307e\u3059 \u3068 = z \uff08 \u306e \u3001 \u306e \uff09\uff09 = f \uff08 \u306e \uff09\uff09 + g \uff08 \u306e \uff09\uff09 {displaystyle z = zeta {big\uff08} u\u3001v {big\uff09} = f\uff08u\uff09+g\uff08v\uff09} \u66f8\u3044\u305f\u3053\u3068\u3068\u6761\u4ef6 z \uff08 0 \u3001 0 \uff09\uff09 = 0 {displaystyle zeta {big\uff08} 0,0 {big\uff09} = 0} \u3001 \u2207 z \uff08 0 \u3001 0 \uff09\uff09 = 0 {displaystyle nabla zeta\uff080.0\uff09= 0} \u5341\u5206\u3067\u3059\u3002\u6700\u521d\u306b\u3053\u306e\u69cb\u9020\u3092\u6700\u5c0f\u9818\u57df\u65b9\u7a0b\u5f0f\u306b\u633f\u5165\u3057\u3001\u4fdd\u5b58\u3057\u307e\u3059\u3002 0 = \uff08 \u521d\u3081 + z v2\uff09\uff09 z uu –  2 z uz vz uv+ \uff08 \u521d\u3081 + z u2\uff09\uff09 z vv{displaystyle 0 ,, =\uff081+Zeta _ {v} {2}\uff09Zeta _ {uu} -2Zeta _ {u} Zeta _} Zeta _ {UV}+\uff081+Zeta _ {U} {2}\uff09+\uff081+Zeta _ {U} {2}\uff09  =(1+(g\u2032(v))2)f\u2033(u)+(1+(f\u2032(u))2)g\u2033(v){displaystyle = {big\uff08} 1+\uff08g ‘\uff08v\uff09\uff09^{2} {big\uff09} f’ ‘\uff08u\uff09+\uff081+\uff08f’\uff08u\uff09\uff09^{2}\uff09g ”\uff08v\uff09} \u540c\u7b49\u306e\u5909\u66f4\u3092\u63d0\u4f9b\u3057\u307e\u3059  –  f\u2033(u)1+(f\u2032(u))2= c = g\u2033(v)1+(g\u2032(v))2{displaystyle- {frac {f ”\uff08u\uff09} {1+\uff08u\uff09\uff09^{2}}} = c = {frac {g ”\uff08v\uff09} {1+\uff08g ‘\uff08v\uff09\uff09^{2}}}}}}}}}}}}}}}}}} \u3068\u3068\u3082\u306b c \u2208 r {displaystyle cin mathbb {r}} \u3002\u901a\u5e38\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u7406\u8ad6\u306b\u3088\u308c\u3070\u3001\u521d\u671f\u5024\u306e\u554f\u984c\u306b\u306f\u6b63\u78ba\u306b1\u3064\u306e\u89e3\u304c\u3042\u308a\u307e\u3059 f \u300c (\u306e )+ c \uff08 \u521d\u3081 + \uff08 f ‘ \uff08 \u306e \uff09\uff09 \uff09\uff09 2\uff09\uff09 = 0 {displaystyle f ” {big\uff08} u {big\uff09}+c\uff081+\uff08f ‘\uff08u\uff09\uff09^{2}\uff09= 0} \u30c7\u30fc\u30bf\u306b f \uff08 0 \uff09\uff09 = 0 \u3001 f ‘ \uff08 0 \uff09\uff09 = 0 {displaystyle f\uff080\uff09= 0 ,,, f ‘\uff080\uff09= 0} \u3068 g \u300c (\u306e ) –  c \uff08 \u521d\u3081 + \uff08 g ‘ \uff08 \u306e \uff09\uff09 \uff09\uff09 2\uff09\uff09 = 0 {displaystyle g ” {big\uff08} v {big\uff09}  –  c\uff081+\uff08g ‘\uff08v\uff09\uff09^{2}\uff09= 0} \u30c7\u30fc\u30bf\u306b g \uff08 0 \uff09\uff09 = 0 \u3001 g ‘ \uff08 0 \uff09\uff09 = 0\u3002 {displaystyle g\uff080\uff09= 0 ,,, g ‘\uff080\uff09= 0\u3002} \u3053\u308c\u3089\u306e\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u306f\u305d\u3046\u3067\u3059 f \uff08 \u306e \uff09\uff09 = 1c\u30ed\u30b0 \u2061 cos \u2061 \uff08 c \u306e \uff09\uff09 {displaystyle f\uff08u\uff09= {frac {1} {c}} log cos\uff08cu\uff09}} \u3068 g \uff08 \u306e \uff09\uff09 =  –  1c\u30ed\u30b0 \u2061 cos \u2061 \uff08 c \u306e \uff09\uff09 \u3002 {displaystyle g\uff08v\uff09=  –  {frac {1} {c}} log cos\uff08cv\uff09\u3002}  \u30d1\u30e9\u30e1\u30fc\u30bf\u30fcC = 0.2\u306eScherkschen\u6700\u5c0f\u9818\u57df\u306e\u3044\u304f\u3064\u304b\u306e\u30b3\u30f3\u30c6\u30ad\u30b9\u30c8\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8 \u3053\u3053\u3067\u79c1\u305f\u3061\u306f\u307e\u3060\u6700\u521d\u306e\u5024\u306b\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u3044\u307e\u305b\u3093 f \uff08 0 \uff09\uff09 = d {displaystyle f {big\uff08} 0\uff09= d} \u3068 g \uff08 0 \uff09\uff09 =  –  d {displaystyle g {big\uff08} 0\uff09= -d} \u3068\u3068\u3082\u306b d \u2208 r {displaystyle din mathbb {r}} \u7570\u306a\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u307e\u3059\u3002\u305f\u3060\u3057\u3001\u69cb\u9020\u7684\u6761\u4ef6\u3068\u6a5f\u80fd\u81ea\u4f53\u304c\u901a\u5e38\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u306f\u767a\u751f\u3057\u306a\u3044\u3068\u3044\u3046\u4e8b\u5b9f\u306e\u305f\u3081\u3001\u767a\u751f\u3059\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u307e\u3059 d = 0 {displaystyled = 0} \u8981\u6c42\u3002\u3060\u304b\u3089\u79c1\u305f\u3061\u306f\u53d7\u3051\u53d6\u308a\u307e\u3059\uff1a \u3068 = z (\u306e \u3001 \u306e )= f \uff08 \u306e \uff09\uff09 + g \uff08 \u306e \uff09\uff09 = 1c\u30ed\u30b0 \u2061 cos\u2061(cu)cos\u2061(cv){displaystyle z = zeta {big\uff08} u\u3001v {big\uff09} = f\uff08u\uff09+g\uff08v\uff09= {frac {1} {c}} log {frac {cos\uff08cu\uff09} {cos\uff08cv\uff09}}}}} \u6b63\u65b9\u5f62\u306e\u3053\u306e\u6700\u5c0f\u9818\u57df\u306b\u6c17\u3065\u304d\u307e\u3059 \u304a\u304a \uff1a= { (u,v)\u2208R2:|u+k\u03c0c|c|Zmitk\u2261l(mod2)} {displaystyle omega\uff1a= left {\uff08u\u3001v\uff09in mathbb {r} ^{2}\uff1a\u3001{big |} u+k {frac {pi} {c}} {big |}  0\u3092\u6e80\u305f\u3057\u3066\u3044\u307e\u3059 x2+y2= c \u30b3\u30b9 \u2061 zc\u3002 {displaystyle {sqrt {x^{2}+y^{2}}} = c\u3001operatorname {cosh} {frac {z} {c}}\u3002}} \u3053\u308c\u306f\u3001\u5b9f\u9a13\u7684\u306b\u767a\u898b\u3055\u308c\u305f\u8840\u5c0f\u677f\u306e\u6700\u521d\u306e\u89e3\u6c7a\u7b56\u306e1\u3064\u3067\u3057\u305f\u3002\u30a8\u30c3\u30b8\u30c7\u30fc\u30bf\u306f2\u3064\u306e\u5186\u5f62\u30ea\u30f3\u30b0\u3067\u3001\u30b3\u30fc\u30f3\u307e\u305f\u306f\u30b7\u30ea\u30f3\u30c0\u30fc\u306e\u4e0a\u9650\u3068\u4e0b\u306e\u30a8\u30c3\u30b8\u66f2\u7dda\u3092\u5f62\u6210\u3057\u307e\u3057\u305f\u3002 \u6700\u5c0f\u9650\u306e\u9818\u57df\u3068\u3057\u3066\u306e\u30ab\u30c6\u30ce\u30a4\u30c9\u306f\u30011740\u5e74\u9803\u306e\u30ec\u30aa\u30f3\u30cf\u30eb\u30c8\u30aa\u30a4\u30e9\u30fc\u304b\u3089\u6765\u3066\u3044\u307e\u3059\u3002 \u3089\u305b\u3093\u30a8\u30ea\u30a2 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ]  parameter c = 1\u304b\u3089helikoid \u30b9\u30d1\u30a4\u30e9\u30eb\u30a8\u30ea\u30a2\u307e\u305f\u306f\u30d8\u30ea\u30b3\u30a4\u30c9\u306f\u3001\u30ab\u30c6\u30ce\u30a4\u30c9\u3068\u5bc6\u63a5\u306b\u95a2\u9023\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u9818\u57df\u306f\u3001\u4e0d\u5b89\u5b9a\u3067\u3042\u308b\u304c\u7b49\u5c3a\u6027\u306e\u5909\u5f62\u306e\u305f\u3081\u306b\u30ab\u30c6\u30ce\u30a4\u30c9\u304b\u3089\u51fa\u73fe\u3057\u307e\u3059\u3002\u30d1\u30e9\u30e1\u30fc\u30bf\u30fcc> 0\u306b\u3001\u6b21\u306e\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3057\u3066\u3044\u307e\u3059\u3002 \u30d0\u30c4 (\u306e \u3001 \u306e \uff09\uff09 = \u306e cos \u2061 \u306e {displaystyle x {big\uff08} u\u3001v\uff09= ucos v} \u3068 (\u306e \u3001 \u306e \uff09\uff09 = \u306e \u7f6a \u2061 \u306e {displaystyle y {big\uff08} u\u3001v\uff09= usin v} \u3068 (\u306e \u3001 \u306e \uff09\uff09 = c \u306e {displaystyle z {big\uff08} u\u3001v\uff09= cv} \u3053\u306e\u6700\u5c0f\u9762\u7a4d\u306f\u30013\u6b21\u5143\u7a7a\u9593\u306b\u3082\u57cb\u3081\u8fbc\u307e\u308c\u3066\u3044\u307e\u3059\u3002 \u6700\u5c0f\u9650\u306e\u30a8\u30ea\u30a2\u3068\u3057\u3066\u306e\u30b9\u30d1\u30a4\u30e9\u30eb\u30a8\u30ea\u30a2\u306f\u3001Jean-Baptiste Meusnier de La Place\uff081776\uff09\u304b\u3089\u6765\u3066\u3044\u307e\u3059\u3002\u30d8\u30ea\u30af\u30b9\u306b\u306f\u30c8\u30dd\u30ed\u30b8\u30ab\u30eb\u306a\u6027\u5225\u304c\u3042\u308a\u307e\u3059\u3002\u30c7\u30a4\u30d3\u30c3\u30c9\u30a2\u30ec\u30f3\u30db\u30d5\u30de\u30f3\u3068\u540c\u50da\u306f\u30011990\u5e74\u4ee3\u306b\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u30fc\u63f4\u52a9\u3067\u5efa\u8a2d\u3055\u308c\u307e\u3057\u305f\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u7121\u9650\u3067arbitrary\u610f\u7684\u306a\u30c8\u30dd\u30ed\u30b8\u30fc\u306e\u6027\u5225\u3092\u5099\u3048\u305f\u6700\u5c0f\u9650\u306e\u9818\u57df\u3001\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u30fc\u88dc\u52a9\u5177\u306f\u7121\u9650\u306e\u6027\u5225\u3068\u6027\u5225\u306e\u307f\u306e\u305f\u3081\u306e\u3082\u306e\u3067\u3057\u305f\uff08Michael Wolf\u3001Hoffman\u3001Matthias Weber 2009\uff09\u3002 0\u306e\u5834\u5408\u3001\u30d8\u30ea\u30b7\u30c9\uff08\u7279\u5225\u306a\u30b1\u30fc\u30b9\u3068\u3057\u3066\u3082\u30ab\u30c6\u30ce\u30a4\u30c9\u3068\u3057\u3066\u3082\u542b\u307e\u308c\u307e\u3059\uff09\u3068\u30ec\u30d9\u30eb\u306f\u3001\u552f\u4e00\u306e\u5b8c\u5168\u306a\u30df\u30f3\u30c8\u53ef\u80fd\u306a\u6700\u5c0f\u9818\u57df\u3067\u3059\uff08William Meeks\u3001Harold William Rosenberg 2005\uff09\u3002 \u30d8\u30f3\u30cd\u30d0\u30fc\u30b0\u30a8\u30ea\u30a2 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ]  Henneberg Minimal\u30a8\u30ea\u30a2 \u30d8\u30f3\u30cd\u30d0\u30fc\u30b0\u30a8\u30ea\u30a2\u306f\u30013\u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u6d78\u308b\u3053\u3068\u306e\u30a4\u30e1\u30fc\u30b8\u3067\u3042\u308b\u6700\u5c0f\u9650\u306e\u9818\u57df\u306e\u4f8b\u3067\u3059\u304c\u30013\u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u57cb\u3081\u8fbc\u3080\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u3042\u306a\u305f\u306e\u6c7a\u5b9a\u65b9\u7a0b\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 [2] \u30d0\u30c4 \uff08 \u306e \u3001 \u306e \uff09\uff09 = 2 \u751f\u307e\u308c\u308b \u2061 \u306e cos \u2061 \u306e  –  23\u751f\u307e\u308c\u308b \u2061 \uff08 3 \u306e \uff09\uff09 cos \u2061 \uff08 3 \u306e \uff09\uff09 {displaystyle x\uff08u\u3001v\uff09= 2sinh ucos v- {frac {2} {3}} sinh\uff083u\uff09cos\uff083v\uff09} \u3068 \uff08 \u306e \u3001 \u306e \uff09\uff09 = 2 \u751f\u307e\u308c\u308b \u2061 \u306e \u7f6a \u2061 \u306e + 23\u751f\u307e\u308c\u308b \u2061 \uff08 3 \u306e \uff09\uff09 \u7f6a \u2061 \uff08 3 \u306e \uff09\uff09 {displaystyle y\uff08u\u3001v\uff09= 2sinh usin v+{frac {2} {3}} biological\uff083u\uff09sin\uff083v\uff09} \u3068 \uff08 \u306e \u3001 \u306e \uff09\uff09 = 2 \u30b3\u30b9 \u2061 \uff08 2 \u306e \uff09\uff09 cos \u2061 \uff08 2 \u306e \uff09\uff09 {displaystyle z\uff08u\u3001v\uff09= 2cosh\uff082u\uff09cos\uff082v\uff09} \u3055\u3089\u306b\u3001\u3053\u306e\u9818\u57df\u3092\u65b9\u5411\u3065\u3051\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\uff1a\u3042\u306a\u305f\u306f\u3053\u306e\u9818\u57df\u306e\u3069\u3061\u3089\u306e\u5074\u9762\u3092\u9bae\u660e\u306b\u8a71\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059 \u305d\u306e\u4e0a \u305d\u3057\u3066\u3069\u3061\u3089 \u4e0b \u306f\u3002 \u5f7c\u5973\u306f\u30011875\u5e74\u306b\u5f7c\u306e\u8ad6\u6587\u3067\u5f7c\u5973\u3092\u7d39\u4ecb\u3057\u305fLife Right\u306b\u3088\u308b\u3068Henneberg\u3068\u540d\u4ed8\u3051\u3089\u308c\u307e\u3057\u305f\u3002 \u90e8\u5c4b\u306e\u5bf8\u6cd5\u304c\u9ad8\u3044\u5834\u5408\u3001\u8840\u5c0f\u677f\u306e\u554f\u984c\u3078\u306e\u30a2\u30af\u30bb\u30b9\u306f\u56f0\u96e3\u3067\u3059\u3002\u3053\u3053\u3067\u306f\u3001\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u3092\u30b0\u30e9\u30d5\u3068\u3057\u3066\u7406\u89e3\u3059\u308b\u6a5f\u4f1a\u3057\u304b\u3042\u308a\u307e\u305b\u3093\u3002\u30b0\u30e9\u30d5\u306e\u6700\u5c0f\u9762\u7a4d\u65b9\u7a0b\u5f0f\u304c\u8a18\u8ff0\u3055\u308c\u3066\u3044\u307e\u3059 div \u2061 \u2207\u03b61+\u2207\u03b62= 0 {displaystyle operatorname {div} {frac {nabla zeta} {sqrt {1+nabla zeta ^{2}}}} = 0} \u3002 \u6955\u5186\u5f62\u306e\u9650\u754c\u4fa1\u5024\u306e\u554f\u984c\u306e\u30bd\u30fc\u30d9\u30f3\u30b7\u30fc\u304c\u5f31\u3044\u3068\u3044\u3046\u7406\u8ad6\u306e\u305f\u3081\u3001\u3053\u306e\u72b6\u6cc1\u3067\u306e\u89e3\u6c7a\u7b56\u306e\u5b58\u5728\u3092\u4fdd\u8a3c\u3067\u304d\u307e\u3059\u3002\u305d\u306e\u5f8c\u306e\u5b9a\u671f\u7684\u306a\u8003\u616e\u4e8b\u9805\u306f\u3001\u53e4\u5178\u7684\u306a\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u3092\u63d0\u4f9b\u3057\u307e\u3059\u3002 2\u3064\u306e\u90e8\u5c4b\u306e\u5bf8\u6cd5\u3068\u540c\u69d8\u306b\u30012\u3064\u306e\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u9593\u306e\u9055\u3044\u306e\u6700\u5927\u539f\u7406\u304b\u3089\u4e00\u610f\u6027\u3092\u53d6\u5f97\u3067\u304d\u307e\u3059\u3002 \u6700\u5c0f\u9650\u306e\u9818\u57df\u306b\u5341\u5206\u306a\u65b9\u7a0b\u5f0f\u306e\u6bd4\u8f03\u7684\u5358\u7d14\u306a\u69cb\u9020\u306b\u3088\u308a\u3001Homorph\u3084\u8abf\u548c\u306e\u3068\u308c\u305f\u95a2\u6570\u3067\u7279\u306b\u77e5\u3089\u308c\u3066\u3044\u308b\u3044\u304f\u3064\u304b\u306e\u65e2\u77e5\u306e\u30b9\u30c6\u30fc\u30c8\u30e1\u30f3\u30c8\u3092\u30012\u3064\u306e\u5909\u6570\u306e\u6700\u5c0f\u9818\u57df\u306b\u9001\u4fe1\u3067\u304d\u307e\u3059\u3002 MaximumPrinzip [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6700\u5c0f\u9650\u306e\u9818\u57df\u306e\u5834\u5408 \u30d0\u30c4 {displaystyle mathbf {x}} \u4e0d\u5e73\u7b49\u304c\u9069\u7528\u3055\u308c\u307e\u3059 \u3059\u3059\u308b w\u2208B| \u30d0\u30c4 \uff08 \u306e \uff09\uff09 | \u2264 \u3059\u3059\u308b w\u2208\u2202B| \u30d0\u30c4 \uff08 \u306e \uff09\uff09 | {displaystyle sup _ {win b} | mathbf {x}\uff08w\uff09| leq sup _ {win partial b} | mathbf {x}\uff08w\uff09|} \u3002 \u6700\u5c0f\u9762\u7a4d\u306f\u3001\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u30a8\u30ea\u30a2\u306e\u7aef\u306b\u6700\u5927\u5024\u3092\u5e2f\u3073\u307e\u3059\u3002 \u30a4\u30e9\u30b9\u30c8\u306e\u5747\u4e00\u5316 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Geodesy\u3067\u306f\u3001\u305d\u306e\u3088\u3046\u306a\u7b49\u6e29\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u3092\u5c0e\u5165\u3067\u304d\u307e\u3059\u3002\u3053\u308c\u3092\u884c\u3046\u30a4\u30e9\u30b9\u30c8\u306f\u547c\u3070\u308c\u307e\u3059 \u30a4\u30e9\u30b9\u30c8\u306e\u5747\u4e00\u5316\u3002 \u6700\u5c0f\u9650\u306e\u9818\u57df\u306e\u30a4\u30e9\u30b9\u30c8\u306e\u5747\u4e00\u5316\u306f\u3001\u8abf\u548c\u306e\u3068\u308c\u305f\u95a2\u6570\u3067\u3059\u3002 \u30ec\u30ec\u30e9\u30f3\u5206\u6790\u30c1\u30e3\u30fc\u30c8 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u305d\u308c\u3089\u304c\u7b49\u6e29\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u306b\u3042\u308b\u9650\u308a\u3001\u6700\u5c0f\u9650\u306e\u9818\u57df\u306f\u3001\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u9818\u57df\u5185\u306e\u5b9f\u969b\u306e\u5206\u6790\u6a5f\u80fd\u3067\u3059\u3002\u3053\u308c\u306f\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u30d7\u30ec\u30bc\u30f3\u30c6\u30fc\u30b7\u30e7\u30f3\u3092\u3001\u3053\u306e\u30dd\u30a4\u30f3\u30c8\u306e\u9818\u57df\u306e\u9818\u57df\u306e\u4efb\u610f\u306e\u30dd\u30a4\u30f3\u30c8\u306e\u53ce\u675f\u30b7\u30ea\u30fc\u30ba\u306e\u5897\u5f37\u306b\u767a\u5c55\u3055\u305b\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u305d\u308c\u306f\u3042\u306a\u305f\u304c\u597d\u304d\u306a\u3088\u3046\u306b\u983b\u7e41\u306b\u533a\u5225\u3055\u308c\u307e\u3059\u3002\u5468\u8fba\u66f2\u7dda\u30821\u30dd\u30a4\u30f3\u30c8\u306e\u30dd\u30a4\u30f3\u30c8\u306b\u3042\u308b\u5834\u5408\u3001\u3053\u306e\u30dd\u30a4\u30f3\u30c8\u306e\u9818\u57df\u306e\u6700\u5c0f\u9762\u7a4d\u306f\u3001\u30a8\u30c3\u30b8\u3092\u8d85\u3048\u3066\u5206\u6790\u7684\u306a\u65b9\u6cd5\u3067\u7d99\u7d9a\u3067\u304d\u307e\u3059\u3002 \u30d0\u30fc\u30f3\u30b9\u30bf\u30a4\u30f3\u3068\u30ea\u30a6\u30f4\u30a3\u30eb\u306e\u5211 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6700\u5c0f\u9650\u306e\u30a8\u30ea\u30a2\u306b\u95a2\u3059\u308b\u30bb\u30eb\u30b2\u30a4\u30d0\u30fc\u30f3\u30b9\u30bf\u30a4\u30f3\u306e\u5224\u6c7a\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 \u5168\u4f53\u3068\u3057\u3066 R2{displaystyle mathbb {r} ^{2}} \u8aac\u660e\u3055\u308c\u305f\u89e3\u6c7a\u7b56 z (\u306e \u3001 \u306e ){displaystyle zeta {big\uff08} u\u3001v {big\uff09}} \u5f0f\u3092\u6e80\u305f\u3059\u306b\u306f\u3001\u975e\u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u6700\u5c0f\u9762\u7a4d\u65b9\u7a0b\u5f0f\u304c\u5fc5\u8981\u3067\u3059 \u03b6(u,v)=au+bv+c{displaystyle zeta {big\uff08} u\u3001v {big\uff09} = au+bv+c} \u4e00\u5b9a\u3067 a \u3001 b \u3001 c \u2208 R{displaystyle a\u3001b\u3001cin mathbb {r}} \u3002 \u9ad8\u6b21\u5143\u3078\u306e\u4e00\u822c\u5316\u306e\u554f\u984c\u306f\u3001\u30a2\u30f3\u30d0\u30fc\u306e\u554f\u984c\u3068\u3057\u3066\u77e5\u3089\u308c\u3066\u304a\u308a\u3001\u30a8\u30f3\u30cb\u30aa\u30fb\u30c7\u30fb\u30b8\u30e7\u30eb\u30b8\u3001\u30a8\u30f3\u30ea\u30b3\u30fb\u30dc\u30f3\u30d3\u30a8\u30ea\u306a\u3069\u306b\u3088\u3063\u3066\u89e3\u6c7a\u3055\u308c\u307e\u3057\u305f\u3002 \u3053\u306e\u6587\u304b\u3089\u3001\u6700\u5c0f\u9650\u306e\u9818\u57df\u306e\u30ea\u30a6\u30d3\u30eb\u304b\u3089\u306e\u6587\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 \u5168\u4f53\u3068\u3057\u3066 R2{displaystyle mathbb {r} ^{2}} \u9650\u3089\u308c\u305f\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u3092\u5ba3\u8a00\u3057\u307e\u3057\u305f z (\u306e \u3001 \u306e ){displaystyle zeta {big\uff08} u\u3001v {big\uff09}} \u975e\u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u6700\u5c0f\u9762\u7a4d\u65b9\u7a0b\u5f0f\u304c\u5fc5\u8981\u3067\u3059 \u03b6(u,v)=const{displaystyle zeta {big\uff08} u\u3001v {big\uff09} = operatorname {const}} \u3002 \u3053\u308c\u306f\u3001\u6a5f\u80fd\u7406\u8ad6\u306e\u30ea\u30a6\u30d3\u30eb\u6587\u306e\u985e\u4f3c\u7269\u3067\u3059\u3002 \u6700\u5c0f\u9762\u7a4d\u306e\u9762\u7a4d [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6700\u5c0f\u9762\u7a4d\u306e\u9762\u7a4d \u30d0\u30c4 {displaystyle mathbf {x}} \u30e6\u30cb\u30c3\u30c8\u304c\u6b63\u5e38\u3067 n {displaystyle mathbf {n}} \u30d5\u30a9\u30fc\u30e0\u306b\u66f8\u3044\u3066\u3044\u307e\u3059 a = \u222e \uff08 n \u3001 \u30d0\u30c4 \u3001 d \u30d0\u30c4 \uff09\uff09 {displaystyle a = oint\uff08mathbf {n}\u3001mathbf {x}\u3001mathrm {d} mathbf {x}\uff09} \u3002 \u30a8\u30c3\u30b8\u66f2\u7dda\u306f\u5358\u306b\u9589\u3058\u3066\u7740\u5b9f\u306b\u533a\u5225\u3055\u308c\u308b\u3068\u60f3\u5b9a\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002 \u6700\u5c0f\u9650\u306e\u9818\u57df\u3092\u3088\u308a\u3088\u304f\u7406\u89e3\u3059\u308b\u305f\u3081\u306b\u306f\u3001\u5f7c\u3089\u304c\u51fa\u4f1a\u3046\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u307f\u3092\u898b\u308b\u3060\u3051\u3067\u306f\u5341\u5206\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u304c\u3001\u6c7a\u5b9a\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002 \u8907\u96d1\u306a\u8868\u73fe [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u7b49\u6e29\u30d1\u30e9\u30e1\u30fc\u30bf\u30fcu\u3068v\u306e\u5c0e\u5165\u306b\u3088\u308a\u3001\u6700\u521d\u306bH = 0\u306eH\u8868\u9762\u30b7\u30b9\u30c6\u30e0\u3092\u53d7\u3051\u53d6\u308a\u307e\u3059\u3002 d \u30d0\u30c4 = 0 \u3001 xu2 –  xv2= 0 = xuxv{displaystyle delta mathbf {x} = 0\u3001qquad mathbf {x} _ {u}^{2} -mathbf {x} _ {v}^{2} = 0 = mathbf {x} _ {u} mathbf {x} _ {v}} \u3053\u308c\u306b\u3088\u308a\u3001\u30d5\u30a9\u30fc\u30e0\u306e2\u6b21\u65b9\u7a0b\u5f0f\u304c\u8a18\u8f09\u3055\u308c\u3066\u3044\u307e\u3059 \u22022x\u2202z1\u2202z2= 0 {displaystyle {frac {partial^{2} mathbf {x}} {partial z^{1} partial z^{2}} = 0}} \u8907\u96d1\u306a\u5909\u6570\u3067 \u3068 \u521d\u3081 = \u306e + \u79c1 \u306e {displaystyle z^{1} = u+iv} \u3068 \u3068 2 = \u306e  –  \u79c1 \u306e {displaystyle z^{2} = u-iv} \u305d\u3057\u3066\u3001\u30d7\u30ec\u30bc\u30f3\u30c6\u30fc\u30b7\u30e7\u30f3\u3092\u53d6\u5f97\u3057\u307e\u3059 2 \u30d0\u30c4 = \u3068 \uff08 \u3068 1\uff09\uff09 + \u3068 \uff08 \u3068 2\uff09\uff09 {displaystyle 2mathbf {x} = mathbf {y}\uff08z^{1}\uff09+mathbf {z}\uff08z^{2}\uff09qquad} \u3068 \uff08 y‘ \uff09\uff09 2= 0 = \uff08 z‘ \uff09\uff09 2\u3001 y‘ z‘ \u2260 0 {displaystyle qquad\uff08mathbf {y} ‘\uff09^{2} = 0 =\uff08mathbf {z}’\uff09^{2}\u3001qquad mathbf {y} ‘mathbf {z}’ neq 0} \u3002 \u8907\u96d1\u306a\u66f2\u7dda\u3092\u547c\u3073\u51fa\u3057\u307e\u3059 \u3068 \uff08 \u3068 \uff09\uff09 {displaystyle mathbf {y}\uff08z\uff09} \u6761\u4ef6 \uff08 \u3068 ‘ \uff09\uff09 2 = 0 {displaystyle\uff08mathbf {y} ‘\uff09^{2} = 0} \u3068 \u3068 ‘ \u2260 0 {displaystyle mathbf {\u304a\u3088\u3073 ‘neo 0} \u5341\u5206\u30011\u3064 \u7b49\u65b9\u6027\u30ab\u30fc\u30d6\u3002 \u307e\u305f\u3001\u30a8\u30ea\u30a2\u3092\u547c\u3073\u51fa\u3057\u307e\u3059 \u30d0\u30c4 \uff08 \u306e \u3001 \u306e \uff09\uff09 {displaystyle mathbf {x}\uff08u\u3001v\uff09} \u305d\u308c\u306f\u5f62\u3067 \u30d0\u30c4 \uff08 \u306e \u3001 \u306e \uff09\uff09 = \u3068 \uff08 \u306e \uff09\uff09 + \u3068 \uff08 \u306e \uff09\uff09 {displaystyle mathbf {x}\uff08u\u3001v\uff09= mathbf {y}\uff08u\uff09+mathbf {z}\uff08v\uff09} \u66f8\u3044\u3066\u307f\u307e\u3057\u3087\u3046 \u30b9\u30e9\u30a4\u30c9\u9762\u3002  \u6700\u5c0f\u9818\u57df\u306e\u4e00\u822c\u5316\u3055\u308c\u305f\u5b9a\u7fa9\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 \u6700\u5c0f\u9650\u306e\u9818\u57df\u3068\u306f\u3001\u751f\u6210\u7b49\u65b9\u6027\u66f2\u7dda\u304c\u3042\u308b\u6ed1\u308a\u9762\u3067\u3059\u3002 \u305d\u306e\u5f8c\u3001\u771f\u306e\u6700\u5c0f\u9650\u306e\u9818\u57df\u304c\u6761\u4ef6\u3092\u6e80\u305f\u3057\u307e\u3059 z1\u00af= \u3068 2{displaystyle {overline {z^{1}}} = z^{2} qquad} \u3068 \u3068 \uff08 z1\u00af\uff09\uff09 = y(z1)\u00af{displaystyle qquad mathbf {z}\uff08{overline {z^{1}}}}\uff09= {overline {mathbf {y}\uff08z_ {1}\uff09}}}}} \u3002 \u7d71\u5408 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] KarlWeierstra\u00df\u3068Alfred Enneper\u306b\u3061\u306a\u3093\u3067\u540d\u4ed8\u3051\u3089\u308c\u305f\u8868\u73fe\u5f0f\u306f\u3001\u5fae\u5206\u5f62\u72b6\u3068\u6a5f\u80fd\u306e\u7406\u8ad6\u3068\u306e\u95a2\u4fc2\u3092\u63d0\u4f9b\u3057\u307e\u3059\u3002 WeierStrass\u306f\u73fe\u5728\u3001\u6a5f\u80fd\u7406\u8ad6\u306e\u51fa\u73fe\u306b\u5927\u304d\u306a\u5f71\u97ff\u3092\u4e0e\u3048\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u516c\u5f0f\u306f\u3001\u3053\u306e\u6bd4\u8f03\u7684\u65b0\u3057\u3044\u6570\u5b66\u306e\u679d\u304c\u771f\u5263\u306b\u53d7\u3051\u6b62\u3081\u3089\u308c\u3001\u975e\u5e38\u306b\u6210\u529f\u3057\u305f\u7406\u7531\u306e1\u3064\u3067\u3057\u305f\u3002\u5f7c\u306f\u3001\u3059\u3079\u3066\u306e\u975e\u30b3\u30f3\u30c6\u30f3\u30c4\u30df\u30cb\u30de\u30eb\u30a8\u30ea\u30a2\u3067\u3042\u308b\u3053\u3068\u3092\u767a\u898b\u3057\u307e\u3057\u305f \u30d0\u30c4 \uff08 \u306e \uff09\uff09 = \uff08 \u30d0\u30c4 \uff08 \u306e \uff09\uff09 \u3001 \u3068 \uff08 \u306e \uff09\uff09 \u3001 \u3068 \uff08 \u306e \uff09\uff09 \uff09\uff09 {displaystyle mathbf {x}\uff08w\uff09= {big\uff08} x\uff08w\uff09\u3001y\uff08w\uff09\u3001z\uff08w\uff09{big\uff09}} 2\u3064\u306e\u30db\u30ed\u30e2\u30eb\u30d5\u30a3\u30c3\u30af\u95a2\u6570G\u304a\u3088\u3073H\u3068\u306e\u7a4d\u5206\u3068\u3057\u3066\u3002\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8\u306b\u3082\u540c\u3058\u3053\u3068\u304c\u5f53\u3066\u306f\u307e\u308a\u307e\u3059\u3002 \u30d0\u30c4 \uff08 \u306e \uff09\uff09 = c 1+ \u518d \u2061 \u222b w0wh(\u03b6)(1\u2212g(\u03b6)2)2d z {displaystyle x\uff08w\uff09= c_ {1}+operatorname {re} int _ {w_ {0}}^{w} {frac {h\uff08zeta\uff09left\uff081-g\uff08zeta\uff09^{2} right\uff09} {2}} mathrm {d} Zeta} \u3001 \u3068 \uff08 \u306e \uff09\uff09 = c 2+ \u518d \u2061 \u222b w0wh(\u03b6)(1+g(\u03b6)2)2d z {displaystyle y\uff08w\uff09= c_ {2}+operatorname {re} int _ {w_ {0}}^{w} {frac {h\uff08zeta\uff09\u5de6\uff081+g\uff08zeta\uff09^{2}\u53f3\uff09} {2}} mathrm {d} Zeta} \u3001 \u3068 \uff08 \u306e \uff09\uff09 = c 3+ \u518d \u2061 \u222b w0wh \uff08 z \uff09\uff09 g \uff08 z \uff09\uff09 d z {displaystyle z\uff08w\uff09= c_ {3}+operatorname {re} int _ {w_ {0}}^{w} h\uff08zeta\uff09g\uff08zeta\uff09mathrm {d} zeta} \u3053\u306e\u30c7\u30a3\u30b9\u30d7\u30ec\u30a4\u30d5\u30a9\u30fc\u30df\u30e5\u30e9\u306b\u3088\u308a\u3001\u6700\u5c0f\u9650\u306e\u9818\u57df\u306e\u5199\u771f\u304c\u6700\u65b0\u306e\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u30fc\u4ee3\u6570\u30b7\u30b9\u30c6\u30e0\u306e\u52a9\u3051\u3092\u501f\u308a\u3066\u753b\u50cf\u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\u30e1\u30fc\u30d7\u30eb\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u3066\u3053\u308c\u3089\u306e\u5f0f\u3092\u4f7f\u7528\u3057\u3066\u3001\u6700\u5c0f\u9650\u306e\u9818\u57df\u306e\u753b\u50cf\u304c\u4f5c\u6210\u3055\u308c\u307e\u3057\u305f\u3002 \u7a4d\u5206 – \u30d5\u30ea\u30fc\u8868\u73fe [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u7b49\u65b9\u65b9\u7a0b\u5f0fh = 0\u3092\u7d71\u5408\u3057\u3066\u7b49\u65b9\u6027\u66f2\u7dda\u3092\u6c7a\u5b9a\u3059\u308b\u3060\u3051\u3067\u5341\u5206\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\u3001\u5b9f\u969b\u306e\u6700\u5c0f\u9818\u57df\u3067\u53d7\u3051\u53d6\u308a\u307e\u3059 \u30d0\u30c4 \uff08 \u306e \uff09\uff09 = \uff08 \u30d0\u30c4 \uff08 \u306e \uff09\uff09 \u3001 \u3068 \uff08 \u306e \uff09\uff09 \u3001 \u3068 \uff08 \u306e \uff09\uff09 \uff09\uff09 {displaystyle mathbf {x}\uff08w\uff09= {big\uff08} x\uff08w\uff09\u3001y\uff08w\uff09\u3001z\uff08w\uff09{big\uff09}} SO -CALLED INTECTLAL -FREE\u8868\u73fe \u30d0\u30c4 \uff08 \u306e \uff09\uff09 = \u518d \u2061 \uff08 \u00b1i(f(w)\u2212wf\u2032(w)\u22121\u2212w22f\u2033(w))\uff09\uff09 {displaystyle x\uff08w\uff09= operatorname {re} left\uff08pm ileft\uff08f\uff08w\uff09-wf ‘\uff08w\uff09 –  {frac {1-w^{2}} {2}} f’ ‘\uff08w\uff09\u53f3\uff09}} \u3068 \uff08 \u306e \uff09\uff09 = \u518d \u2061 \uff08 f(w)\u2212wf\u2032(w)+1+w22f\u2033(w)\uff09\uff09 {displaystyle y\uff08w\uff09= operatorname {re}\u5de6\uff08f\uff08w\uff09-wf ‘\uff08w\uff09+{frac {1+w^{2}} {2}} f’ ‘\uff08w\uff09\u53f3\uff09}} \u3068 \uff08 \u306e \uff09\uff09 = \u518d \u2061 \uff08 \u2212i(f\u2032(w)\u2212wf\u2033(w))\uff09\uff09 {displaystyle z\uff08w\uff09= operatorname {re} left\uff08-i {big\uff08} f ‘\uff08w\uff09-wf’ ‘\uff08w\uff09{big\uff09}\u53f3\uff09} \u30db\u30ed\u30e2\u30fc\u30d5\u30a3\u30c3\u30af\u6a5f\u80fd\u3092\u5099\u3048\u3066\u3044\u307e\u3059 f = f \uff08 \u306e \uff09\uff09 {displaystyle f = f\uff08w\uff09} \u524d\u63d0\u6761\u4ef6 f \u2034 \u2260 0 {displaystyle f ” ‘neq 0} \u4f1a\u308f\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u3002\u3053\u306e\u3088\u3046\u306b\u3001\u30ec\u30a4\u30e4\u30fc\u306f\u3053\u306e\u8868\u73fe\u3092\u907f\u3051\u307e\u3059\u3002\u4eca\u307e\u3067\u3001\u8907\u96d1\u306a\u5909\u6570\u306e\u610f\u5473 \u306e {displaystyle in} \u5b9f\u969b\u306e\u6700\u5c0f\u9818\u57df\u306e\u9577\u3044\u8acb\u6c42\u66f8\u3092\u660e\u78ba\u306b\u3059\u308b \u306e = n1+in2\u22121+n3{displaystyle w = {frac {n_ {1}+in_ {2}} {-1+n_ {3}}}}}} \u307e\u305f\u3002 1w= n1+in2\u22121+n3{displaystyle {frac {1} {w}} = {frac {n_ {1}+in_ {2}} {-1+n_ {3}}}}}}} \u3001 \u3053\u3053\u306f n = \uff08 n \u521d\u3081 \u3001 n 2 \u3001 n 3 \uff09\uff09 {displaystyle mathbf {n} =\uff08n_ {1}\u3001n_ {2}\u3001n_ {3}\uff09} \u6700\u5c0f\u9818\u57df\u306e\u30e6\u30cb\u30c3\u30c8\u901a\u5e38\u30d9\u30af\u30c8\u30eb\u3002\u8981\u7d04\u3057\u3066\u304f\u3060\u3055\u3044\uff1a \u8907\u96d1\u306a\u6570\u3092\u6307\u5b9a\u3059\u308b\u3053\u3068\u306b\u3088\u308a \u306e {displaystyle in} \u307e\u305f\u3002 \u521d\u3081 \/ \u306e {\u5c55\u793a1\/w} \u30e6\u30cb\u30c3\u30c8\u306f\u901a\u5e38\u306e\u30d9\u30af\u30c8\u30eb\u3067\u3059 n {displaystyle mathbf {n}} \u6700\u5c0f\u30a8\u30ea\u30a2\u3092\u8a2d\u5b9a\u3057\u307e\u3059\u3002\u9006\u306b\u3001\u30cf\u30f3\u30b0\u3057\u307e\u3059 \u306e {displaystyle in} \u307e\u305f\u3002 \u521d\u3081 \/ \u306e {\u5c55\u793a1\/w} \u304b\u3089\u306e\u307f n {displaystyle mathbf {n}} ab\u3002 \u3053\u306e\u30bb\u30af\u30b7\u30e7\u30f3\u306e\u58f0\u660e\u306f\u7279\u306b\u672c\u3067\u3059 \u57fa\u672c\u7684\u306a\u9451\u5225\u578b W. Blaschke\u3068K. Light White\u304c\u898b\u3064\u3051\u308b\u3053\u3068\u306f\u3001\u6587\u732e\u3082\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002 \u5f53\u521d\u3001\u3053\u308c\u3092\u6a5f\u80fd\u7684\u306b\u5c0e\u304d\u51fa\u3057\u3001\u7a4d\u6975\u7684\u306b\u6307\u5411\u3057\u305f\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u5909\u63db\u3068\u306e\u4e0d\u5909\u6027\u3092\u793a\u3057\u307e\u3059\u3002\u6700\u5f8c\u306b\u30011\u3064\u30682\u6b21\u5143\u306e\u30b1\u30fc\u30b9\u3092\u660e\u793a\u7684\u306b\u8a08\u7b97\u3057\u307e\u3059\u3002 \u6d3e\u751f\u3068\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u306e\u4e0d\u5909 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u79c1\u305f\u3061\u306e\u6700\u5c0f\u9650\u306e\u9818\u57df\u306f\u3001N\u6b21\u5143\u306e\u5b9f\u969b\u306e\u30d9\u30af\u30c8\u30eb\u7a7a\u9593\u306b\u304a\u3051\u308bM\u6b21\u5143\u306e\u591a\u69d8\u6027\u3068\u898b\u306a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u3053\u308c\u306f\u3001NASH\u304b\u3089\u306e\u57cb\u3081\u8fbc\u307f\u7387\u306e\u305f\u3081\u306b\u5e38\u306b\u53ef\u80fd\u3067\u3059\u3002\u6700\u521d\u306b\u8aac\u660e\u3057\u307e\u3059 \u30e1\u30fc\u30c8\u30eb\u30c6\u30f3\u30bd\u30eb  g ij\uff08 \u306e \uff09\uff09 \uff1a= \u2202x(u)\u2202ui\u2202x(u)\u2202uj{displaystyle g_ {ij}\uff08u\uff09\uff1a= {frac {partial mathbf {x}\uff08u\uff09} {partial u_ {i}}}}}}}\uff08u\uff09}\uff08u\uff09} {partial u_ {j}}}}}}}}} \u6c7a\u5b9a\u8981\u56e0\u3092\u4f7f\u7528 g \uff08 \u306e \uff09\uff09 \uff1a=  \u2061 \uff08 g ij\uff08 \u306e \uff09\uff09 \uff09\uff09 i,j=1,\u2026,m{displaystyle g\uff08u\uff09\uff1a= operaatorname {det}\uff08g_ {ij}\uff08u\uff09\uff09_ {i\u3001j = 1\u3001dotsc\u3001m}}} \u3002 \u79c1\u305f\u3061\u306f\u3001M\u6b21\u5143\u306e\u7a4d\u5206\u3068\u3057\u3066\u306em\u5bf8\u6cd5\u9818\u57df\u306e\u30b3\u30f3\u30c6\u30f3\u30c4\u304c\u4e0a\u306b\u3042\u308b\u3053\u3068\u3092\u899a\u3048\u3066\u3044\u307e\u3059 \u7279\u6027\u95a2\u6570 \u3053\u306e\u9818\u57df\u306f\u7d50\u679c\u3067\u3059\u3002\u7279\u5fb4\u7684\u306a\u95a2\u6570\u306f\u3001\u7fa4\u8846\u306e\u3069\u3053\u3067\u3082\u540c\u3058\u3082\u306e\u3067\u3042\u308a\u3001\u305d\u308c\u4ee5\u5916\u306e\u5834\u5408\u306f\u540c\u4e00\u306e\u30bc\u30ed\u3067\u3059\u3002\u3053\u308c\u306f\u3001\u8868\u9762\u8981\u7d20\u3092\u8868\u73fe\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002\u56fa\u5b9a\u70b9\u3067\u63a5\u7dda\u30d9\u30af\u30c8\u30eb\u3092\u8aac\u660e\u3057\u307e\u3059\u3002 \u2202x(u)\u2202ui{displaystyle {frac {partial mathbf {x}\uff08u\uff09} {partial u_ {i}}}}} \u305f\u3081\u306b \u79c1 \u3001 j = \u521d\u3081 \u3001 … \u3001 m {displaystyle i\u3001j = 1\u3001dotsc\u3001m} \u30d9\u30af\u30bf\u30fc\u3092\u9078\u629e\u3057\u307e\u3059 \u30d0\u30c4 m + \u521d\u3081 \u3001 … \u3001 \u30d0\u30c4 n {displaystyle xi _ {m+1}\u3001dotsc\u3001xi _ {n}} \u3001\u30b7\u30b9\u30c6\u30e0\u3067\u3059 \uff08 xu1\u3001 … \u3001 xum\u3001 \u30d0\u30c4 m+1\u3001 … \u3001 \u30d0\u30c4 n\uff09\uff09 {displaystyle\uff08mathbf {x} _ {u_ {1}}\u3001dotsc\u3001mathbf {x} _ {u_ {m}}\u3001xi _ {m+1}\u3001dotsc\u3001xi _ {n}\uff09} \u7a4d\u6975\u7684\u306b\u914d\u5411\u3055\u308c\u30012\u3064\u306e\u6761\u4ef6\u3067\u3059 \uff08 \u30d0\u30c4 ui\u3001 \u30d0\u30c4 j \uff09\uff09 = 0 {displaystyle\uff08mathbf {x} _ {u_ {i}}\u3001xi _ {j}\uff09= 0} \u3068 \uff08 \u30d0\u30c4 \u79c1 \u3001 \u30d0\u30c4 j \uff09\uff09 = d \u79c1 j {displaystyle {big\uff08} xi _ {i}\u3001xi _ {j}\uff09= delta _ {ij}}}} I\u3068J\u306e\u3059\u3079\u3066\u306e\u8ce2\u660e\u306a\u5024\u306e\u305f\u3081\u306b\u6e80\u305f\u3055\u308c\u305f\u3057\u305f\u304c\u3063\u3066\u3001\u8868\u9762\u8981\u7d20\u306f\u6b21\u306e\u3088\u3046\u306b\u66f8\u3044\u3066\u3044\u307e\u3059\u3002 d\u03c3(u)=det\u2061(xu1,\u2026,xum,\u03bem+1,\u2026,\u03ben)du1\u22efdum=det\u2061((xu1,\u2026,\u03ben)T(xu1,\u2026,\u03ben))du1\u22efdum=det\u2061(gij(u))i,j=1,\u2026,mdu1\u22efdum=g(u)du1\u22efdum{displaystyle {begin {aligned} mathrm {d} sigma\uff08u\uff09operatorame {det}\uff08mathbf {x {x {x}}}\u3001dotsc\u3001mathbf {xbf {x {mth}\u3001xi {m+ {m+ {m+ 1}\u3001dots _ {ndm } u_ {m} \\\uff06= {sqrt {operator {det}\uff08\uff08mathbf {x {x}}} _ {u_ {1}}\u3001dosc\u3001xi _ {n}\uff09\uff08mathbf {u_x} _ {u_ {u_ {{1}} {u_ {d oper {u_ {u_ {u_ {u_ {u_ {u_ {u_ {u_ {u_ {u_ {u_ {det}\uff08g_ {ij}\uff08u\uff09\uff09_ {i\u3001j = 1\u3001dotsc\u3001m}}}}} u_ {1} dotsm mathrm {d} u_ {m} \\\uff06= {sqrm {d} ugign }} \u6c7a\u5b9a\u8981\u56e02\u306e\u5834\u5408 m \u00d7 n {displaystyle mtimes n}  – \u30de\u30bf\u30e9\u30a4\u30ba\u30f3 m \u2264 n {displaystyle mleq n} \u8a72\u5f53\u3059\u308b\uff1a  \u2061 \uff08 a Tb \uff09\uff09 = \u2211 1\u2264i1n \u2061 a i1\u2026im \u2061 b i1\u2026im{displaystyle operatorname {det}\uff08a^{t} b\uff09= sum _ {1leq i_ {1} \u3001 \u79c1 m {displaystyle i_ {1}\u3001dotsc\u3001i_ {m}} \u69cb\u6210\u3055\u308c\u307e\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u8868\u9762\u8981\u7d20\u3092\u30d5\u30a9\u30fc\u30e0\u3067\u4f7f\u7528\u3067\u304d\u307e\u3059 d a \uff08 \u306e \uff09\uff09 = \u22111\u2264i1n(\u2202(xui1,\u2026,xuim)\u2202(u1,\u2026,um))2d \u306e 1\u22ef d \u306e m{displaystyle mathrm {d} sigma\uff08u\uff09= {sqrt {sum _ {1leq i_ {1} f 2= | xu\u00d7 xv|2{displaystyle eg-f^{2} = | mathbf {x} _ {u}\u500d\u306emathbf {x} _ {v} |^{2}} \u3053\u308c\u306f\u6a5f\u80fd\u7684\u306a\u9818\u57df\u306b\u9069\u7528\u3055\u308c\u307e\u3059\u3002 a \uff08 \u30d0\u30c4 \uff09\uff09 = \u222c | xu\u00d7 xv| d \u306e d \u306e {displaystyle a\uff08mathbf {x}\uff09= int | mathbf {x} _ {u} use mathbf {x} |\u3001mathrm {d} umathrm {d} v} \u30a6\u30a3\u30eb\u30d8\u30eb\u30e0\u30fb\u30d6\u30e9\u30c3\u30b7\u30e5\u30b1\u3001\u30ab\u30fc\u30c8\u30fb\u30e9\u30a4\u30c8\u30fb\u30db\u30ef\u30a4\u30c8\uff1a \u57fa\u672c\u7684\u306a\u9451\u5225\u578b \uff08=\u6570\u5b66\u79d1\u5b66\u306e\u57fa\u672c\u6559\u80b2\u3002\u7b2c1\u5dfb\uff09\u3002 Springer-verlag\u30011973\u5e74 \u30aa\u30f3\u30e9\u30a4\u30f3\u3002 \u30e8\u30cf\u30cd\u30b9C. C.\u30cb\u30c3\u30c1\u30a7\uff1a \u6700\u5c0f\u9650\u306e\u9818\u57df\u306b\u95a2\u3059\u308b\u8b1b\u7fa9 \uff08=\u6570\u5b66\u79d1\u5b66\u306e\u57fa\u672c\u7684\u306a\u6559\u80b2\u3002\u7b2c199\u5dfb\uff09\u3002 Springer-Verlag\u30011975\u5e74\u3002 \u30c7\u30d3\u30c3\u30c9\u30fb\u30ae\u30eb\u30d0\u30fc\u30b0\u3001\u30cb\u30fc\u30eb\u30fb\u30c8\u30eb\u30c7\u30a3\u30f3\u30b8\u30e3\u30fc\uff1a \u6955\u5186\u5f62\u306e\u90e8\u5206\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4e8c\u6b21\u65b9\u7a0b\u5f0f \uff08=\u6570\u5b66\u79d1\u5b66\u306e\u57fa\u672c\u6559\u80b2\u3002\u7b2c224\u5dfb\uff09\u3002 Springer-Verlag\u30011983\u5e74\u3002 Ulrich Dierkes\u3001Stefan Hildebrandt\u3001AlbrechtK\u00fcster\u3001Ortwin Wohlrab\uff1a \u6700\u5c0f\u9650\u306e\u8868\u9762 \uff08=\u6570\u5b66\u79d1\u5b66\u306e\u57fa\u672c\u6559\u80b2\u3002295\u5dfb\u3068296\u5dfb\uff09\u3002 Springer-Verlag\u30011992\u5e74\u30012\u5dfb\u3002 Stefan Hildebrandt\u3001Anthony Tromba\uff1a \u30dc\u30fc\u30eb\u3001\u30b5\u30fc\u30af\u30eb\u3001\u77f3\u9e78\u306e\u6ce1\u3001\u5e7e\u4f55\u5b66\u3068\u81ea\u7136\u306e\u6700\u9069\u306a\u5f62\u72b6\u3002 Birkh\u00e4user\u30011996\u5e74\u3002 \u30d5\u30ea\u30fc\u30c9\u30ea\u30c3\u30d2\u30fb\u30bd\u30fc\u30f4\u30a3\u30cb\u30fc\uff1a \u30b8\u30aa\u30e1\u30c8\u30ea\u3068\u7269\u7406\u5b66\u306e\u90e8\u5206\u5fae\u5206\u65b9\u7a0b\u5f0f\u3002 Springer-Verlag\u30012004 f\u3002\u30012\u5dfb\u3002 \u2191  \u6700\u5c0f\u9650\u306e\u8868\u9762 \u3001Mathworld  \u2191  Mathworld\u3001Henneberg\u6700\u5c0f\u8868\u9762       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/911#breadcrumbitem","name":"\u30df\u30cb\u30de\u30eb\u30a8\u30ea\u30a2-Wikipedia"}}]}]