[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/14964#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/14964","headline":"Multi -love Notation -Wikipedia\u3001\u7121\u6599\u767e\u79d1\u4e8b\u5178","name":"Multi -love Notation -Wikipedia\u3001\u7121\u6599\u767e\u79d1\u4e8b\u5178","description":"\u30de\u30eb\u30c1\u30c6\u30a4\u30e9\u30fc\u8868\u8a18 – \u6570\u5b66\u7684\u8868\u8a18\u6307\u6a19\uff08\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\uff09\u306e\u6982\u5ff5\u3092\u30a4\u30f3\u30b8\u30b1\u30fc\u30bf\u30fc\u30d9\u30af\u30c8\u30eb\u306b\u4e00\u822c\u5316\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u591a\u304f\u306e\u5909\u6570\u306e\u5206\u6790\u306e\u5f0f\u3001\u90e8\u5206\u5fae\u5206\u65b9\u7a0b\u5f0f\u3001\u304a\u3088\u3073\u5206\u5e03\u7406\u8ad6\u3092\u7c21\u7d20\u5316\u3057\u307e\u3059\u3002 \u30de\u30eb\u30c1\u30de\u30f3 n {displaystyle n} – \u30df\u30de\u30ea\u30fc\u306f\u30d9\u30af\u30c8\u30eb\u3067\u3059 a = \uff08 a 1\u3001 a 2\u3001 … \u3001 a n\uff09\uff09","datePublished":"2020-10-20","dateModified":"2020-10-20","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/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\/a601995d55609f2d9f5e233e36fbe9ea26011b3b","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a601995d55609f2d9f5e233e36fbe9ea26011b3b","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/14964","wordCount":8872,"articleBody":"\u30de\u30eb\u30c1\u30c6\u30a4\u30e9\u30fc\u8868\u8a18  – \u6570\u5b66\u7684\u8868\u8a18\u6307\u6a19\uff08\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\uff09\u306e\u6982\u5ff5\u3092\u30a4\u30f3\u30b8\u30b1\u30fc\u30bf\u30fc\u30d9\u30af\u30c8\u30eb\u306b\u4e00\u822c\u5316\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u591a\u304f\u306e\u5909\u6570\u306e\u5206\u6790\u306e\u5f0f\u3001\u90e8\u5206\u5fae\u5206\u65b9\u7a0b\u5f0f\u3001\u304a\u3088\u3073\u5206\u5e03\u7406\u8ad6\u3092\u7c21\u7d20\u5316\u3057\u307e\u3059\u3002   \u30de\u30eb\u30c1\u30de\u30f3  n {displaystyle n}  – \u30df\u30de\u30ea\u30fc\u306f\u30d9\u30af\u30c8\u30eb\u3067\u3059 a = \uff08 a 1\u3001 a 2\u3001 … \u3001 a n\uff09\uff09 {displaystyle alpha =\uff08alpha _ {1}\u3001alpha _ {2}\u3001dots\u3001alpha _ {n}\uff09} \u975e\u9670\u6027\u6574\u6570\u3002\u30de\u30eb\u30c1\u30da\u30c3\u30c8\u7528 a \u3001 b \u2208 n 0 n {displaystyle alpha\u3001athbb\u306e\u30d9\u30fc\u30bf{n} _ {0}^{n}} \u3068 \u30d0\u30c4 = \uff08 \u30d0\u30c4 \u521d\u3081 \u3001 \u30d0\u30c4 2 \u3001 … \u3001 \u30d0\u30c4 n \uff09\uff09 \u2208 r n {displaystyle x =\uff08x_ {1}\u3001x_ {2}\u3001dots\u3001x_ {n}\uff09in mathbb {r} ^{n}} \u5b9a\u7fa9\u6e08\u307f\uff1a \u5408\u8a08\u3068\u9055\u3044\uff08\u5ea7\u6a19\u5f8c\uff09\u3001\u03b1\u00b1\u03b2=(\u03b11\u00b1\u03b21,\u03b12\u00b1\u03b22,\u2026,\u03b1n\u00b1\u03b2n);{displaystyle alpha pm beta =\uff08alpha _ {1} pm beta _ {1}\u3001alpha _ {2} pm beta _ {2}\u3001dots ,, alpha _ {n} pm beta _ {n}\uff09;} \u90e8\u5206\u9806\u5e8f\u3001\u03b1\u2a7d\u03b2\u27fa\u03b1i\u2a7d\u03b2i\u2200i\u2208{1,\u2026,n};{displaystyle alpha leqslant beta iff alpha _ {i} leqslant beta _ {i} qquad forall _ {iin {1\u3001dots\u3001n}};};} \u5ea7\u6a19\u306e\u5408\u8a08\uff08\u7d76\u5bfe\u5024\uff09\u3001|\u03b1|=\u03b11+\u03b12+\u2026+\u03b1n;{displaystyle | alpha | = alpha _ {1}+alpha _ {2}+ldots+alpha _ {n};} \u5f37\u304f\u3001\u03b1!=\u03b11!\u22c5\u03b12!\u2026\u03b1n!;{displaystyle alpha\uff01= alpha _ {1}\uff01cdot alpha _ {2}\uff01ldots alpha _ {n} !;} \u30cb\u30e5\u30fc\u30c8\u30f3\u30b7\u30f3\u30dc\u30eb\u3001(\u03b1\u03b2)=(\u03b11\u03b21)(\u03b12\u03b22)\u2026(\u03b1n\u03b2n);{displaystyle {alpha seoce beta} = {alpha _ {1} beta _ {1}} {alpha _ {2} choces beta _ {2}} ldots {alpha _ {n} choice beta _ {n}};};};}; \u529b\u3001x\u03b1=x1\u03b11x2\u03b12\u2026xn\u03b1n;{displaystyle x^{alpha} = x_ {1}^{alpha _ {1}} x_ {2}^{alpha _ {2}} ldots x_ {n}^{alpha _ {n}};};} \u3088\u308a\u9ad8\u3044\u884c\u306e\u30d1\u30fc\u30c8\u5c55\u958b\u8a98\u5c0e\u4f53\u3001\u2202\u03b1=\u22021\u03b11\u22022\u03b12\u2026\u2202n\u03b1n,{displaystyle partial^{alpha} = partial _ {1}^{alpha _ {1}} partial _ {2}^{alpha _ {2}} ldots partial _ {n}^{alpha _ {n}}\u3001}}} \u3069\u3053 \u2202i\u03b1i:=\u2202\u03b1i\u2202xi\u03b1i.{displaystyle partial _ {i}^{alpha _ {i}}\uff1a= {tfrac {partial^{alpha _ {i}}}} {i}^{alpha _ {i}}}}}}}} \u30de\u30eb\u30c1\u30c6\u30a4\u30e9\u30fc\u8868\u8a18\u306b\u3088\u308a\u3001\u591a\u304f\u306e\u5909\u6570\u306e\u5206\u6790\u306b\u304a\u3044\u3066\u3001\u591a\u304f\u306e\u57fa\u672c\u5206\u6790\u30d1\u30bf\u30fc\u30f3\u3092\u5bfe\u5fdc\u3059\u308b\u30b1\u30fc\u30b9\u306b\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u306f\u3044\u304f\u3064\u304b\u306e\u4f8b\u3092\u793a\u3057\u307e\u3059\u3002 \u591a\u9805\u5f0f\u306e\u5b9a\u7406 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] (\u2211i=1n\u00a0xi)k= \u2211 |\u03b1|=k k!\u03b1!\u30d0\u30c4 \u03b1{displaystyle left\uff08sum _ {i = 1}^{n} \u301cx_ {i} right\uff09^{k} = sum _ {| alpha | = k}\u301c{frac {k\uff01} {alpha\uff01}}\u3001x^{alpha}}} \u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u30d1\u30bf\u30fc\u30f3 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6ed1\u3089\u304b\u306a\u6a5f\u80fd\u306e\u5834\u5408 f {displaystyle f} \u79c1 g {displaystyle g}  \u2202 \u03b1\uff08 f g \uff09\uff09 = \u2211 \u03bd\u2a7d\u03b1(\u03b1\u03bd)\u2202 \u03bdf \u2202 \u03b1\u2212\u03bdg \u3002 {displaystyle partial ^{alpha}\uff08fg\uff09= sum _ {nu leqslant alpha} {alpha choice nu} partial ^{nu} f\u3001partial ^{alpha -nu} g\u3002} \u4e00\u9023\u306e\u30c6\u30a4\u30e9\u30fc [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u5206\u6790\u6a5f\u80fd\u306e\u5834\u5408 f {displaystyle f} o n {displaystyle n} \u5909\u6570\u306f\u3067\u3059 f \uff08 \u30d0\u30c4 + h \uff09\uff09 = \u2211 \u03b1\u2208N0n\u2202\u03b1f(x)\u03b1!h\u03b1\u3002 {displaystyle f\uff08x+h\uff09= sum _ {alpha in mathbb {n} _ {0}^{n}} {{frac {partial^{alpha} f\uff08x\uff09} {alpha\uff01}} h^{alpha}}}}}} \u78ba\u304b\u306b\u3001\u5341\u5206\u306a\u6ed1\u3089\u304b\u306a\u6a5f\u80fd\u306e\u305f\u3081\u306b\u3082\u540c\u69d8\u3067\u3059 \u30c6\u30a4\u30e9\u30fc\u306e\u958b\u767a  f \uff08 \u30d0\u30c4 + h \uff09\uff09 = \u2211 |\u03b1|\u2a7dn \u2202\u03b1f(x)\u03b1!h\u03b1+ r n\uff08 \u30d0\u30c4 \u3001 h \uff09\uff09 \u3001 {displaystyle f\uff08x+h\uff09= sum _ {| alpha | leqslant n}\u301c{{frac {partial ^{alpha} f\uff08x\uff09} {alpha\uff01}} h ^{alpha}}+r_ {n}\uff08x\u3001h\uff09\u3001} \u6700\u5f8c\u306e\u5f0f\uff08\u6b8b\u308a\uff09\u306f\u3001\u30c6\u30a4\u30e9\u30fc\u30d1\u30bf\u30fc\u30f3\u306e\u7279\u5b9a\u306e\u30d0\u30fc\u30b8\u30e7\u30f3\u306b\u4f9d\u5b58\u3057\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u30ab\u30a6\u30c1\u30e3\u306e\u30d1\u30bf\u30fc\u30f3\uff08\u6b8b\u308a\u306e\u7a4d\u5206\uff09\u306b\u3064\u3044\u3066 r n\uff08 \u30d0\u30c4 \u3001 h \uff09\uff09 = \uff08 n + \u521d\u3081 \uff09\uff09 \u2211 |\u03b1|=n+1 h\u03b1\u03b1!\u222b 01 \uff08 \u521d\u3081  –  t )n\u2202\u03b1f \uff08 \u30d0\u30c4 + t h \uff09\uff09 d t \u3002 {displaystyle r_ {n}\uff08x\u3001h\uff09=\uff08n+1\uff09sum _ {| alpha | = n+1}\u301c{frac {h^{alpha}} {alpha\uff01}} \u4e00\u822c\u7684\u306a\u5f62\u306e\u90e8\u5206\u7684\u306a\u9055\u3044\u306e\u6f14\u7b97\u5b50 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u90e8\u5206\u7684\u306a\u9055\u3044\u306e\u30aa\u30da\u30ec\u30fc\u30bf\u30fc n {displaystyle n} -y\u6ce8\u6587 n {displaystyle n} \u5909\u6570\u306f\u6b63\u5f0f\u306b\u66f8\u304b\u308c\u3066\u3044\u307e\u3059 p \uff08 \u2202 \uff09\uff09 = \u2211 |\u03b1|\u2a7dN a\u03b1\uff08 \u30d0\u30c4 \uff09\uff09 \u2202\u03b1\u3002 {displaystyle P\uff08partial\uff09= sum _ {| alpha | leqslant n}\u301c{a_ {alpha}\uff08x\uff09partial ^{alpha}}\u3002}} \u90e8\u54c1\u3054\u3068\u306e\u7d71\u5408 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u9650\u3089\u308c\u305f\u30d5\u30a3\u30fc\u30eb\u30c9\u306b\u30b3\u30f3\u30d1\u30af\u30c8\u5a92\u4f53\u3092\u4f7f\u7528\u3057\u305f\u6ed1\u3089\u304b\u306a\u6a5f\u80fd\u306e\u5834\u5408 \u304a\u304a \u2282 r n {displaystyle omega subset mathbb {r} ^{n}} \u306f \u222b \u03a9 \u306e \uff08 \u2202\u03b1\u306e \uff09\uff09 d \u30d0\u30c4 = \uff08  –  \u521d\u3081 \uff09\uff09 |\u03b1|\u222b \u03a9 \uff08 \u2202\u03b1\u306e \uff09\uff09 \u306e d \u30d0\u30c4 \u3002 {displaystyle int limits _ {omega}\u301c{u\uff08partial ^{alpha} v\uff09}\u3001dx =\uff08 –  1\uff09 ^{| alpha |} int limits _ {omega}\u301c{\uff08partial ^{alpha} u\uff09v\u3001dx}\u3002 \u3053\u306e\u30d1\u30bf\u30fc\u30f3\u306f\u3001\u5206\u5e03\u3068\u5f31\u3044\u5c0e\u95a2\u6570\u3092\u5b9a\u7fa9\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002 \u3082\u3057\u3082 a \u3001 b \u2208 n 0 n {displaystyle alpha\u3001athbb\u306e\u30d9\u30fc\u30bf{n} _ {0}^{n}} \u5f7c\u3089\u306f\u30de\u30eb\u30c1\u30d7\u30ec\u30a4\u30e4\u30fc\u3067\u3059\u3001a \u30d0\u30c4 = \uff08 \u30d0\u30c4 \u521d\u3081 \u3001 … \u3001 \u30d0\u30c4 n \uff09\uff09 \u3001 {displaystyle x =\uff08x_ {1}\u3001dots\u3001x_ {n}\uff09\u3001} \u306b \u2202 \u03b1\u30d0\u30c4 \u03b2= {\u03b2!(\u03b2\u2212\u03b1)!x\u03b2\u2212\u03b1,gdy\u03b1\u2a7d\u03b2,0w p.p.{displaystyle partial^{alpha} x^{beta} = {begin {cases} {frac {beta\uff01} {\uff08beta -alpha\uff09\uff01}} x^{beta -alpha}\u3001\uff06{hbox {gdy}}} }} \u8a3c\u62e0 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u8a3c\u62e0\u306f\u3001\u901a\u5e38\u306e\u30c7\u30ea\u30d0\u30c6\u30a3\u30d6\u306e\u6a29\u529b\u306e\u30eb\u30fc\u30eb\u3092\u3082\u305f\u3089\u3057\u307e\u3059\u3002\u3082\u3057\u3082 a \u3001 b \u2208 { 0 \u3001 \u521d\u3081 \u3001 2 \u3001 … } \u3001 {displaystyle alpha\u3001{0,1,2\u3001dots}\u306e\u30d9\u30fc\u30bf\u7248\u3001} \u305d\u308c\u304b\u3089 d\u03b1dx\u03b1\u30d0\u30c4 \u03b2= {\u03b2!(\u03b2\u2212\u03b1)!x\u03b2\u2212\u03b1,gdy\u03b1\u2a7d\u03b2,0w p.p.\uff08 \u521d\u3081 \uff09\uff09 {displaystyle {frac {d^{alpha}} {dx^{alpha}} x^{beta} = {begin {case} {frac} {beta\uff01} {\uff08beta -alpha\uff09\uff01}}} x^{beta} 0\uff06{hbox {w p.p\u3002}} end {cases}} qquad\uff081\uff09} \u305d\u308c\u3092\u4eee\u5b9a\u3057\u307e\u3057\u3087\u3046 a = \uff08 a \u521d\u3081 \u3001 … \u3001 a n \uff09\uff09 \u3001 {displaystyle alpha =\uff08alpha _ {1}\u3001dots\u3001alpha _ {n}\uff09\u3001}  b = \uff08 b \u521d\u3081 \u3001 … \u3001 b n \uff09\uff09 \u3001 {displaystyle beta =\uff08beta _ {1}\u3001dots\u3001beta _ {n}\uff09\u3001}  \u30d0\u30c4 = \uff08 \u30d0\u30c4 \u521d\u3081 \u3001 … \u3001 \u30d0\u30c4 n \uff09\uff09 \u3002 {displaystyle x =\uff08x_ {1}\u3001dots\u3001x_ {n}\uff09\u3002} \u305d\u308c\u304b\u3089 \u2202\u03b1x\u03b2=\u2202|\u03b1|\u2202x1\u03b11\u2026\u2202xn\u03b1nx1\u03b21\u2026xn\u03b2n=\u2202\u03b11\u2202x1\u03b11x1\u03b21\u2026\u2202\u03b1n\u2202xn\u03b1nxn\u03b2n.{displaystyle {begin {aligned} partial^{alpha} x^{beta}\uff06= {frac {partial^{| alpha |}}^{alpha _ {1}} ldots partial x_ {n}^{{n} {{{{{{{{{{{{n} {n} {{{n} {n} {{n} {n} {n} {n} {n} {n}\uff09 beta _ {1}} ldots x_ {n}^{beta _ {n}} \\ [1ex]\uff06= {frac {partial^{alpha _ {1}}}} {partial x_ {1}^{alpha _ {1}}}}}}}}}}}}}}}}}}}}}}}}} {frac {partial^{alpha _ {n}}} {partial x_ {n}^{alpha _ {n}}}} x_ {n}^{beta _ {n}}\u3002 \u3059\u3079\u3066\u306e\u4eba\u306e\u305f\u3081\u306b \u79c1 \u2208 { \u521d\u3081 \u3001 … \u3001 n } \u3001 {displaystyle iin {1\u3001dots\u3001n}\u3001} \u95a2\u6570 \u30d0\u30c4 \u79c1 \u03b2i{displaystyle x_ {i}^{beta _ {i}}} \u4f9d\u5b58\u3057\u307e\u3059 \u30d0\u30c4 \u79c1 \u3002 {displaystyle x_ {i}\u3002} \u3057\u305f\u304c\u3063\u3066\u3001\u4e0a\u8a18\u306e\u30d1\u30bf\u30fc\u30f3\u3067\u306f\u3001\u5404\u90e8\u5206\u7684\u306a\u5206\u5316 \u2202\u2202xi{displaystyle {tfrac {partial} {partial x_ {i}}}}} \u9069\u5207\u306a\u5fae\u5206\u306b\u7e2e\u5c0f\u3055\u308c\u307e\u3059 ddxi\u3002 {displaystyle {tfrac {d} {dx_ {i}}}}} \u3057\u305f\u304c\u3063\u3066\u3001\u5f0f\uff081\uff09\u306f\u305d\u308c\u3092\u793a\u3057\u3066\u3044\u307e\u3059 \u2202 a \u30d0\u30c4 b {displaystyle partial ^{alpha} x ^{beta}} \u5834\u5408\u306f\u6d88\u3048\u307e\u3059 "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/14964#breadcrumbitem","name":"Multi -love Notation -Wikipedia\u3001\u7121\u6599\u767e\u79d1\u4e8b\u5178"}}]}]