{"id":2321,"date":"2024-06-19T15:00:59","date_gmt":"2024-06-19T06:00:59","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2321"},"modified":"2025-01-29T11:33:43","modified_gmt":"2025-01-29T02:33:43","slug":"%e3%83%86%e3%82%a4%e3%83%a9%e3%83%bc%e5%b1%95%e9%96%8b%ef%bc%882%e5%a4%89%e6%95%b0%ef%bc%89","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%81%8f%e5%be%ae%e5%88%86%ef%bc%9a%e5%a4%9a%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86\/%e3%83%86%e3%82%a4%e3%83%a9%e3%83%bc%e5%b1%95%e9%96%8b%ef%bc%882%e5%a4%89%e6%95%b0%ef%bc%89\/","title":{"rendered":"\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\uff082\u5909\u6570\uff09"},"content":{"rendered":"

2\u5909\u6570\u95a2\u6570 \\(f(x, y)\\) \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f\uff0c$h, \\ k$ \u3092\uff08\u5c0f\u3055\u3044\uff09\u5b9a\u6570\u3068\u3057\u3066\uff0c
\n\\begin{eqnarray}
\nf(x+h, y+k) = f(x,y) &\\,+& \\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right) f(x,y) \\\\
\n&+& \\frac{1}{2!} \\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right)^2 f(x,y) + \\cdots \\\\
\n&+& \\frac{1}{n!} \\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right)^n f(x,y) + \\cdots
\n\\end{eqnarray}
\n<\/p>\n

1\u5909\u6570\u95a2\u6570 \\(f(x)\\) \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h3>\n

1\u5909\u6570\u95a2\u6570 \\(f(x)\\) \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u306e\u3060\u3063\u305f\u3002$h$ \u3092\uff08\u5c0f\u3055\u3044\uff09\u5b9a\u6570\u3068\u3057\u3066\uff0c<\/p>\n

\\begin{eqnarray}
\nf({\\color{green}{x + h}}) &=& f(x) + f'(x) \\,h + \\frac{1}{2!} f^{”}(x) \\,h^2 + \\cdots + \\frac{1}{n!} f^{({n})}(x) \\,h^n + \\cdots\\\\
\n&=& f(x) + h \\frac{d}{dx} f(x) + \\frac{1}{2!} \\left(h\\frac{d}{dx}\\right)^2 f(x) + \\cdots + \\frac{1}{n!} \\left(h\\frac{d}{dx}\\right)^n f(x) + \\cdots
\n\\end{eqnarray}<\/p>\n

\u8868\u8a18\u306b\u95a2\u3059\u308b\u88dc\u8db3<\/a><\/h4>\n

\u6388\u696d\u306e\u300c\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/a>\u300d\u306e\u30da\u30fc\u30b8\u3067 $f({\\color{blue}{a + x}})$ \u306b\u3064\u3044\u3066\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3068\u3057\u3066<\/p>\n

\\begin{eqnarray}
\nf({\\color{blue}{a + x}}) &=& f(a) + f'(a)\\, x + \\frac{f^{\u201d}(a)}{2!}\\, x^2 + \\cdots +
\n\\frac{f^{({n})}(a)}{n!}\\, x^n + \\cdots
\n\\end{eqnarray}<\/p>\n

\u3068\u66f8\u3044\u305f\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u3053\u3053\u3067\u306f $f({\\color{green}{x + h}})$ \u3068\u66f8\u304d\u76f4\u3057\u305f\u7406\u7531\u306f\u4ee5\u4e0b\u306e\u3068\u304a\u308a\u3002$f({\\color{blue}{a + x}})$ \u306e\u307e\u307e\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3060\u3068\uff0c2\u5909\u6570\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3092\u898b\u3059\u3048\u3066\uff0c\u4e0d\u7528\u610f\u306b<\/p>\n

\\begin{eqnarray}
\nf(a + x)
\n&=& f(a) + x \\frac{d}{dx} f(a) + \\frac{1}{2!} \\left(x\\frac{d}{dx}\\right)^2 f(a) + \\cdots + \\frac{1}{n!} \\left(x\\frac{d}{dx}\\right)^n f(a) + \\cdots
\n\\end{eqnarray}<\/p>\n

\u3068\u66f8\u3044\u3066\u307f\u305f\u304f\u306a\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u3053\u308c\u306f2\u3064\u306e\u610f\u5473\u3067\u4e0d\u5177\u5408\u304c\u3042\u308a\uff0c\u9593\u9055\u3044\u3002<\/p>\n

\u307e\u305a $\\displaystyle \\frac{d}{dx} f(a)$ \u3060\u3068\uff0c\u5148\u306b $x=a$ \u3092 $f(x)$ \u306b\u4ee3\u5165\u3057\u305f $f(a)$ \u306f\u5b9a\u6570\u3067\u3042\u308a\uff0c$x$\u00a0 \u3067\u5fae\u5206\u3057\u305f\u3089\u30bc\u30ed\u3060\u308d\uff1f\u3068\u8aa4\u89e3\u3055\u308c\u308b\u3053\u3068\u3002<\/p>\n

\u307e\u305f\uff0c\u305f\u3068\u3048\u30702\u968e\u5fae\u5206\u306e\u9805\u304c<\/p>\n

$$\\displaystyle \\left(x\\frac{d}{dx}\\right)^2 f(x) = x \\frac{d}{dx} \\left( x \\frac{df}{dx} \\right) = x \\frac{df}{dx} + x^2 \\frac{d^2f}{dx^2} \\ \\mbox{?}$$<\/p>\n

\u3068\u8aa4\u89e3\u3055\u308c\u308b\u6050\u308c\u3082\u5341\u5206\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c$h$ \u3092\u5b9a\u6570\u3068\u3057\u3066<\/p>\n

\\begin{eqnarray}
\nf({\\color{green}{x + h}})
\n&=& f(x) + h \\frac{d}{dx} f(x) + \\frac{1}{2!} \\left(h\\frac{d}{dx}\\right)^2 f(x) + \\cdots + \\frac{1}{n!} \\left(h\\frac{d}{dx}\\right)^n f(x) + \\cdots
\n\\end{eqnarray}<\/p>\n

\u3068\u66f8\u3044\u3066\u304a\u3051\u3070\uff0c\u4e0a\u8a18\u306e\u3088\u3046\u306a\u8aa4\u89e3\u304c\u751f\u3058\u308b\u9699\u306f\u306a\u304f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b2\u5909\u6570\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306b\u3082\u540c\u69d8\u306a\u8868\u8a18\u304c\u4f7f\u3048\u308b\u3002<\/p>\n

2\u5909\u6570\u95a2\u6570 \\(f(x, y)\\) \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h3>\n

2\u5909\u6570\u95a2\u6570 \\(f(x, y)\\) \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3082\u540c\u69d8\u3002$h, \\ k$ \u3092\uff08\u5c0f\u3055\u3044\uff09\u5909\u6570\u3068\u3057\u3066\uff0c
\n\\begin{eqnarray}
\nf(x+h, y+k) = f(x,y) &\\,+& \\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right) f(x,y) \\\\
\n&+& \\frac{1}{2!} \\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right)^2 f(x,y) + \\cdots \\\\
\n&+& \\frac{1}{n!} \\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right)^n f(x,y) + \\cdots
\n\\end{eqnarray}<\/p>\n

\u5404\u9805\u3092\u5177\u4f53\u7684\u306b\u8a08\u7b97\u3059\u308b\u3068…<\/p>\n

\\begin{eqnarray}
\n\\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right) f(x,y) &=& h \\frac{\\partial f}{\\partial x} + k \\frac{\\partial f}{\\partial y}\\\\
\n\\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right)^2 f(x,y) &=&
\n\\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right) \\left( h \\frac{\\partial f}{\\partial x} + k \\frac{\\partial f}{\\partial y}\\right)\\\\
\n&=& h^2 \\frac{\\partial^2 f}{\\partial x^2} + hk \\frac{\\partial^2 f}{\\partial x \\partial y} + kh \\frac{\\partial^2 f}{\\partial y \\partial x} + k^2 \\frac{\\partial^2 f}{\\partial y^2} \\\\
\n&=& h^2 \\frac{\\partial^2 f}{\\partial x^2} + 2 hk \\frac{\\partial^2 f}{\\partial x \\partial y}+ k^2 \\frac{\\partial^2 f}{\\partial y^2}\\\\
\n\\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right)^3 f(x,y) &=& \\dots
\n\\end{eqnarray}<\/p>\n

\u8868\u8a18\u306b\u95a2\u3059\u308b\u88dc\u8db3<\/h4>\n

\u4e16\u306b\u3042\u307e\u305f\u3042\u308b\u6559\u79d1\u66f8\u306e\u4e2d\u306b\u306f\uff0c2\u5909\u6570\u306e\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3068\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u516c\u5f0f\u3092\u63b2\u3052\u3066\u3044\u308b\u672c\u304c\u3042\u3063\u305f\u3002<\/p>\n

\\begin{eqnarray}
\nf(x, y) = f(0, 0) \\ &+& \\left( x \\frac{\\partial}{\\partial x} + y \\frac{\\partial}{\\partial y} \\right) \\ f(0, 0) \\\\
\n&+& \\frac{1}{2!} \\left( x \\frac{\\partial}{\\partial x} + y \\frac{\\partial}{\\partial y} \\right)^2\u00a0 f(0, 0) + \\cdots\\\\
\n&+& \\frac{1}{n!} \\left( x \\frac{\\partial}{\\partial x} + y \\frac{\\partial}{\\partial y} \\right)^n\u00a0 f(0, 0) + \\cdots\\\\
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c<\/p>\n

$$\\left( x \\frac{\\partial}{\\partial x} + y \\frac{\\partial}{\\partial y} \\right)^2\u00a0 f(0, 0)
\n= x^2 f_{xx}(0, 0) + 2 x y f_{xy}(0, 0) + y^2 f_{yy}(0, 0)$$<\/p>\n

\u306a\u3069\u3068\u88dc\u8db3\u8aac\u660e\u3057\u3066\u3044\u308b\u306e\u3067\u610f\u56f3\u306f\u308f\u304b\u308b\u304c\uff0c1\u5909\u6570\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e\u8868\u8a18\u306b\u95a2\u3059\u308b\u88dc\u8db3\u306b\u66f8\u3044\u305f\u3088\u3046\u306b<\/a>\uff0c\u8457\u3057\u304f\u8aa4\u89e3\u3092\u751f\u3058\u304b\u306d\u306a\u3044\u8868\u8a18\u306a\u306e\u3067\uff0c\u3053\u306e\u3088\u3046\u306a\u8868\u8a18\u306f\u907f\u3051\u305f\u65b9\u304c\u3088\u308d\u3057\u3044\u304b\u3068\u3002<\/p>\n

\u307e\u305f\uff0c$\\displaystyle \\frac{\\partial f}{\\partial x} = f_x, \\ \\frac{\\partial^2 f}{\\partial x \\partial y} = f_{x y}$ \u306a\u3069\u3068\u3044\u3046\u8868\u8a18\u3082\uff0c$f_x$ \u306f\u30d9\u30af\u30c8\u30eb $\\boldsymbol{f}$ \u306e $x$ \u6210\u5206\u3067\u3059\u304b\uff1f\u3068\u304b\uff0c$f_{xy}$ \u306f\u30c6\u30f3\u30bd\u30eb\u306e $xy$ \u6210\u5206\u3067\u3059\u304b\uff1f\u306a\u3069\u3068\u8aa4\u89e3\u3055\u308c\u308b\u3068\u9762\u5012\u306a\u306e\u3067\uff0c\u3053\u306e\u624b\u306e\u7701\u7565\u8a18\u6cd5\u306f\u907f\u3051\u305f\u65b9\u304c\u3088\u308d\u3057\u3044\u306e\u3067\u306f\u306a\u3044\u3067\u3057\u3087\u3046\u304b\uff0c\u3068\u3044\u3046\u306e\u304c\u500b\u4eba\u306e\u611f\u60f3\u3002\u79c1\u7684\u306b\u306f\uff0c\u76f8\u5bfe\u8ad6\u5c4b\u304c\u901a\u5e38\u4f7f\u3046<\/p>\n

$$\\frac{\\partial f}{\\partial x} = f_{, x}, \\quad \\frac{\\partial^2 f}{\\partial x \\partial y} = f_{, x y}$$<\/p>\n

\u306e\u3088\u3046\u306b\u30ab\u30f3\u30de $(,)$ \u306e\u3042\u3068\u306b\u504f\u5fae\u5206\u3059\u308b\u5909\u6570\u3092\u66f8\u304f\u8868\u8a18\u6cd5\u304c\u3082\u3063\u3068\u5e02\u6c11\u6a29\u3092\u5f97\u3066\u5e83\u307e\u308b\u3053\u3068\u3092\u5e0c\u671b\u3059\u308b\u3002<\/p>\n

\u3060\u3044\u305f\u3044\u3053\u306e\u8868\u8a18\u6cd5\u3067\u306f\uff0c$f_{xx}$ \u306f\u30d9\u30af\u30c8\u30eb $\\boldsymbol{f}$ \u306e $x$ \u6210\u5206\u306e\u504f\u5fae\u5206 $\\displaystyle \\frac{\\partial f_x}{\\partial x}$ \u306a\u306e\u304b\uff0c\u30b9\u30ab\u30e9\u30fc\u95a2\u6570 $f$ \u306e2\u968e\u504f\u5fae\u5206 $\\displaystyle \\frac{\\partial^2 f}{\\partial x^2}$ \u306a\u306e\u304b\uff0c\u533a\u5225\u3064\u304b\u306a\u3044\u3067\u3057\u3087\u3002<\/p>\n

\u76f8\u5bfe\u8ad6\u5c4b\u306e\u8a18\u6cd5\u3067\u306f $f_{x, x}$ \u3068 $f_{, x x}$ \u306f\u3061\u3083\u3093\u3068\u533a\u5225\u3064\u304f\u3093\u3060\u304b\u3089\u3002<\/p>\n

\u307e\u305f\u3053\u306e\u4eba\uff0c\u4ed6\u4eba\u306b\u3044\u3061\u3083\u3082\u3093\u3064\u3051\u3066… \u3068\u3044\u3046\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u30d9\u30af\u30c8\u30eb\u306e $x$ \u6210\u5206\u306e $x$ \u504f\u5fae\u5206\u306a\u3093\u304b\u306f\uff0c\u96fb\u78c1\u6c17\u5b66\u3067\u3059\u3050\u306b\u3042\u3089\u308f\u308c\u308b\u306e\u3067\uff0c\u3069\u306e\u96fb\u78c1\u6c17\u5b66\u306e\u6559\u79d1\u66f8\u3067\u3082 $x$ \u504f\u5fae\u5206\u3092 $f_x$ \u306a\u3069\u3068\u306f\u66f8\u304b\u306a\u3044\u3093\u3067\u3059\u3088\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

2\u5909\u6570\u95a2\u6570 \\(f(x, y)\\) \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f\uff0c$h, \\ k$ \u3092\uff08\u5c0f\u3055\u3044\uff09\u5b9a\u6570\u3068\u3057\u3066\uff0c \\begin{eqnarray} f(x+h, y+k) = f(x,y) &\\,+& \\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right) f(x,y) \\\\ &+& \\frac{1}{2!} \\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right)^2 f(x,y) + \\cdots \\\\ &+& \\frac{1}{n!} \\left(h\\frac{\\partial}{\\partial x} + k\\frac{\\partial}{\\partial y}\\right)^n f(x,y) + \\cdots \\end{eqnarray}<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2226,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2321","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2321","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=2321"}],"version-history":[{"count":17,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2321\/revisions"}],"predecessor-version":[{"id":10138,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2321\/revisions\/10138"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2226"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2321"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}