{"id":8985,"date":"2024-06-27T13:21:08","date_gmt":"2024-06-27T04:21:08","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=8985"},"modified":"2024-06-29T08:59:54","modified_gmt":"2024-06-28T23:59:54","slug":"%e3%83%a9%e3%82%b0%e3%83%a9%e3%83%b3%e3%82%b8%e3%83%a5%e3%81%ae%e6%9c%aa%e5%ae%9a%e4%b9%97%e6%95%b0%e6%b3%95%ef%bc%882%e6%ac%a1%e5%85%83%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%a9%e3%82%b0%e3%83%a9%e3%83%b3%e3%82%b8%e3%83%a5%e3%81%ae%e6%9c%aa%e5%ae%9a%e4%b9%97%e6%95%b0%e6%b3%95%ef%bc%882%e6%ac%a1%e5%85%83%ef%bc%89\/","title":{"rendered":"\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\uff082\u6b21\u5143\uff09"},"content":{"rendered":"

$f(x, y) = 0$ \u306e\u6761\u4ef6\u306e\u3082\u3068\u3067\uff0c\u95a2\u6570 $g(x, y)$ \u304c\u6975\u5024\u3092\u3068\u308b\u70b9 $(x, y) = (a, b)$ \u3067\u306f\uff0c<\/p>\n

$$F(x, y, \\lambda) \\equiv g(x, y) + \\lambda\\, f(x, y)$$<\/p>\n

\u306b\u3064\u3044\u3066<\/p>\n

$$\\frac{\\partial F}{\\partial x} = \\frac{\\partial F}{\\partial y} = \\frac{\\partial F}{\\partial \\lambda} = 0$$<\/p>\n

\u3092\u6e80\u305f\u3059\u3088\u3046\u306a\u5b9a\u6570 $\\lambda$ \u304c\u5b58\u5728\u3059\u308b\u3002\uff08\u4ee5\u4e0b\u306e\u8a3c\u660e\u3067\u308f\u304b\u308b\u3088\u3046\u306b\uff0c$\\displaystyle \\frac{\\partial f}{\\partial x}$ \u3068 $\\displaystyle \\frac{\\partial f}{\\partial y}$ \u304c\u540c\u6642\u306b\u30bc\u30ed\u306b\u306a\u308b\u3053\u3068\u306f\u306a\u3044\uff0c\u3068\u3044\u3046\u6761\u4ef6\u304c\u3042\u308b\u3002\uff09<\/p>\n

\u8a3c\u660e\uff1a\u30d1\u30fc\u30c8 1<\/h3>\n

\u307e\u305a\uff0c\u70b9 $(x, y) = (a, b)$ \u3067 $\\displaystyle \\frac{\\partial f}{\\partial y} \\neq 0$ \u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\uff0c$f(x, y) = 0$ \u3092\u6e80\u305f\u3059\u9670\u95a2\u6570\u3092 $y = y(x)$ \u3068\u3057\u3066\uff0c\u9670\u95a2\u6570\u5b9a\u7406\u3088\u308a<\/p>\n

$$\\frac{dy}{dx} = – \\frac{\\ \\ \\frac{\\partial f}{\\partial x}\\ \\ }{\\frac{\\partial f}{\\partial y}}$$<\/p>\n

\u3053\u306e\u9670\u95a2\u6570 $y = y(x)$ \u3092\u4f7f\u3046\u3068\uff0c$g(x, y) = g(x, y(x))$ \u3068\u306a\u308a\u95a2\u6570 $g$ \u306f $x$ \u306e\u307f\u306e1\u5909\u6570\u95a2\u6570\u3068\u306a\u308b\u3002\u3057\u305f\u304c\u3063\u3066 $g(x, y)$ \u304c\u6975\u5024\u3092\u3068\u308b\u70b9\u3067\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{dg}{dx} &=& \\frac{\\partial g}{\\partial x} + \\frac{\\partial g}{\\partial y} \\frac{d y}{d x} \\\\
\n&=& \\frac{\\partial g}{\\partial x} + \\frac{\\partial g}{\\partial y}\\cdot\\left( -\\frac{\\ \\ \\frac{\\partial f}{\\partial x}\\ \\ }{\\frac{\\partial f}{\\partial y}} \\right) \\\\
\n&=& \\frac{\\partial g}{\\partial x} + \\left(-\\frac{\\ \\ \\frac{\\partial g}{\\partial y}\\ \\ }{\\frac{\\partial f}{\\partial y}} \\right)\\frac{\\partial f}{\\partial x} = 0 \\ \\ \\mbox{at} \\ \\ (x, y) = (a, b)\\\\
\n\\mbox{\u3053\u3053\u3067} \\ \\ \\lambda &\\equiv& -\\frac{\\ \\ \\frac{\\partial g}{\\partial y}\\ \\ }{\\frac{\\partial f}{\\partial y}} \\ \\ \\mbox{\u3068\u304a\u304f\u3068} \\\\
\n\\frac{\\partial g}{\\partial x} + \\lambda\\,\\frac{\\partial f}{\\partial x} &=& 0 \\ \\ \\mbox{at} \\ \\ (x, y) = (a, b) \\\\
\n\\therefore\\ \\ \\frac{\\partial F}{\\partial x} &=& 0 \\ \\ \\mbox{at} \\ \\ (x, y) = (a, b)
\n\\end{eqnarray}<\/p>\n

\u307e\u305f\uff0c$\\displaystyle \\frac{\\partial F}{\\partial y} $ \u306b\u3064\u3044\u3066\u306f\uff0c$\\displaystyle \\lambda = -\\frac{\\ \\ \\frac{\\partial g}{\\partial y}\\ \\ }{\\frac{\\partial f}{\\partial y}}$ \u3092\u4f7f\u3046\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial F}{\\partial y} &=& \\frac{\\partial g}{\\partial y} + \\lambda\\, \\frac{\\partial f}{\\partial y} \\\\
\n&=& \\frac{\\partial g}{\\partial y} + \\left(-\\frac{\\ \\ \\frac{\\partial g}{\\partial y}\\ \\ }{\\frac{\\partial f}{\\partial y}} \\right) \\frac{\\partial f}{\\partial y} \\\\
\n&=& \\frac{\\partial g}{\\partial y}\u00a0 -\\frac{\\partial g}{\\partial y} \\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

\u6700\u5f8c\u306b $\\displaystyle \\frac{\\partial F}{\\partial \\lambda} $ \u306b\u3064\u3044\u3066\u306f\uff0c\u305d\u3082\u305d\u3082\u306e\u62d8\u675f\u6761\u4ef6 $f(x, y) = 0$ \u3088\u308a<\/p>\n

$$\\frac{\\partial F}{\\partial \\lambda} = f(x, y) = 0$$<\/p>\n

\u3068\u306a\u308a\uff0c<\/p>\n

$$\\frac{\\partial F}{\\partial x} = \\frac{\\partial F}{\\partial y} = \\frac{\\partial F}{\\partial \\lambda} = 0$$<\/p>\n

\u3092\u6e80\u305f\u3059\u3088\u3046\u306a\u5b9a\u6570 $\\lambda$ \u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f\u3002<\/p>\n

\n

\u8a3c\u660e\uff1a\u30d1\u30fc\u30c8 2<\/h3>\n

\u70b9 $(x, y) = (a, b)$ \u3067 $\\displaystyle \\frac{\\partial f}{\\partial y} = 0$ \u306e\u5834\u5408\u306f\uff0c$\\displaystyle \\frac{\\partial f}{\\partial x} \\neq 0$ \u3067\u3042\u308b\u304b\u3089\uff0c$f(x, y) = 0$ \u3092\u6e80\u305f\u3059\u9670\u95a2\u6570\u3092 $x = x(y)$ \u3068\u3057\u3066\u3084\u308c\u3070\u540c\u69d8\u306b\u8a3c\u660e\u3067\u304d\u308b\u3002<\/p>\n

\u307e\u305a\uff0c\u70b9 $(x, y) = (a, b)$ \u3067 $\\displaystyle \\frac{\\partial f}{\\partial x} \\neq 0$ \u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\uff0c$f(x, y) = 0$ \u3092\u6e80\u305f\u3059\u9670\u95a2\u6570\u3092 $x = x(y)$ \u3068\u3057\u3066\uff0c\u9670\u95a2\u6570\u5b9a\u7406\u3088\u308a<\/p>\n

$$\\frac{dx}{dy} = – \\frac{\\ \\ \\frac{\\partial f}{\\partial y}\\ \\ }{\\frac{\\partial f}{\\partial x}}$$<\/p>\n

\u3053\u306e\u9670\u95a2\u6570 $y = y(x)$ \u3092\u4f7f\u3046\u3068\uff0c$g(x, y) = g(x(y),\u00a0 y)$ \u3068\u306a\u308a\u95a2\u6570 $g$ \u306f $y$ \u306e\u307f\u306e1\u5909\u6570\u95a2\u6570\u3068\u306a\u308b\u3002\u3057\u305f\u304c\u3063\u3066 $g(x, y)$ \u304c\u6975\u5024\u3092\u3068\u308b\u70b9\u3067\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{dg}{dy} &=& \\frac{\\partial g}{\\partial x} \\frac{d x}{d y}+ \\frac{\\partial g}{\\partial y} \\\\
\n&=& \\frac{\\partial g}{\\partial y} + \\frac{\\partial g}{\\partial x}\\cdot\\left( -\\frac{\\ \\ \\frac{\\partial f}{\\partial y}\\ \\ }{\\frac{\\partial f}{\\partial x}} \\right) \\\\
\n&=& \\frac{\\partial g}{\\partial y} + \\left(-\\frac{\\ \\ \\frac{\\partial g}{\\partial x}\\ \\ }{\\frac{\\partial f}{\\partial x}} \\right)\\frac{\\partial f}{\\partial y} = 0 \\ \\ \\mbox{at} \\ \\ (x, y) = (a, b)\\\\
\n\\mbox{\u3053\u3053\u3067} \\ \\ \\lambda &\\equiv& -\\frac{\\ \\ \\frac{\\partial g}{\\partial x}\\ \\ }{\\frac{\\partial f}{\\partial x}} \\ \\ \\mbox{\u3068\u304a\u304f\u3068} \\\\
\n\\frac{\\partial g}{\\partial y} + \\lambda\\,\\frac{\\partial f}{\\partial y} &=& 0 \\ \\ \\mbox{at} \\ \\ (x, y) = (a, b) \\\\
\n\\therefore\\ \\ \\frac{\\partial F}{\\partial y} &=& 0 \\ \\ \\mbox{at} \\ \\ (x, y) = (a, b)
\n\\end{eqnarray}<\/p>\n

\u307e\u305f\uff0c$\\displaystyle \\frac{\\partial F}{\\partial x} $ \u306b\u3064\u3044\u3066\u306f\uff0c$\\displaystyle \\lambda = -\\frac{\\ \\ \\frac{\\partial g}{\\partial x}\\ \\ }{\\frac{\\partial f}{\\partial x}}$ \u3092\u4f7f\u3046\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial F}{\\partial x} &=& \\frac{\\partial g}{\\partial x} + \\lambda\\, \\frac{\\partial f}{\\partial x} \\\\
\n&=& \\frac{\\partial g}{\\partial x} + \\left(-\\frac{\\ \\ \\frac{\\partial g}{\\partial x}\\ \\ }{\\frac{\\partial f}{\\partial x}} \\right) \\frac{\\partial f}{\\partial x} \\\\
\n&=& \\frac{\\partial g}{\\partial x}\u00a0 -\\frac{\\partial g}{\\partial x} \\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

\u6700\u5f8c\u306b $\\displaystyle \\frac{\\partial F}{\\partial \\lambda} $ \u306b\u3064\u3044\u3066\u306f\uff0c\u305d\u3082\u305d\u3082\u306e\u62d8\u675f\u6761\u4ef6 $f(x, y) = 0$ \u3088\u308a<\/p>\n

$$\\frac{\\partial F}{\\partial \\lambda} = f(x, y) = 0$$<\/p>\n

\u3068\u306a\u308a\uff0c<\/p>\n

$$\\frac{\\partial F}{\\partial x} = \\frac{\\partial F}{\\partial y} = \\frac{\\partial F}{\\partial \\lambda} = 0$$<\/p>\n

\u3092\u6e80\u305f\u3059\u3088\u3046\u306a\u5b9a\u6570 $\\lambda$ \u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

$f(x, y) = 0$ \u306e\u6761\u4ef6\u306e\u3082\u3068\u3067\uff0c\u95a2\u6570 $g(x, y)$ \u304c\u6975\u5024\u3092\u3068\u308b\u70b9 $(x, y) = (a, b)$ \u3067\u306f\uff0c<\/p>\n

$$F(x, y, \\lambda) \\equiv g(x, y) + \\lambda\\, f(x, y)$$<\/p>\n

\u306b\u3064\u3044\u3066<\/p>\n

$$\\frac{\\partial F}{\\partial x} = \\frac{\\partial F}{\\partial y} = \\frac{\\partial F}{\\partial \\lambda} = 0$$<\/p>\n

\u3092\u6e80\u305f\u3059\u3088\u3046\u306a\u5b9a\u6570 $\\lambda$ \u304c\u5b58\u5728\u3059\u308b\u3002\uff08\u4ee5\u4e0b\u306e\u8a3c\u660e\u3067\u308f\u304b\u308b\u3088\u3046\u306b\uff0c$\\displaystyle \\frac{\\partial f}{\\partial x}$ \u3068 $\\displaystyle \\frac{\\partial f}{\\partial y}$ \u304c\u540c\u6642\u306b\u30bc\u30ed\u306b\u306a\u308b\u3053\u3068\u306f\u306a\u3044\uff0c\u3068\u3044\u3046\u6761\u4ef6\u304c\u3042\u308b\u3002\uff09<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2226,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-8985","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\/8985","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=8985"}],"version-history":[{"count":21,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/8985\/revisions"}],"predecessor-version":[{"id":9040,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/8985\/revisions\/9040"}],"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=8985"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}