{"id":10248,"date":"2025-05-26T17:19:44","date_gmt":"2025-05-26T08:19:44","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=10248"},"modified":"2025-06-03T12:21:36","modified_gmt":"2025-06-03T03:21:36","slug":"2%e6%ac%a1%e5%85%83%e3%81%ae%e3%83%a9%e3%83%97%e3%83%a9%e3%82%b7%e3%82%a2%e3%83%b3%e3%82%92%e6%a5%b5%e5%ba%a7%e6%a8%99%e3%81%a7%e8%a1%a8%e3%81%99","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\/%e5%90%88%e6%88%90%e9%96%a2%e6%95%b0%e3%81%ae%e5%81%8f%e5%be%ae%e5%88%86%e6%b3%95\/2%e6%ac%a1%e5%85%83%e3%81%ae%e3%83%a9%e3%83%97%e3%83%a9%e3%82%b7%e3%82%a2%e3%83%b3%e3%82%92%e6%a5%b5%e5%ba%a7%e6%a8%99%e3%81%a7%e8%a1%a8%e3%81%99\/","title":{"rendered":"2\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3092\u6975\u5ea7\u6a19\u3067\u8868\u3059"},"content":{"rendered":"

\u5408\u6210\u95a2\u6570\u306e\u504f\u5fae\u5206\u306e\u5fdc\u7528\u3068\u3057\u3066\uff0c\uff08\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3067\u5b9a\u7fa9\u3055\u308c\u305f\uff092\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3 $\\nabla^2 = \\dfrac{\\partial^2}{\\partial x^2} + \\dfrac{\\partial^2}{\\partial y^2}$ \u3092\u6975\u5ea7\u6a19 $r, \\phi$ \u3092\u4f7f\u3063\u3066\u8868\u3057\u3066\u307f\u308b\u3002<\/p>\n

<\/p>\n

2\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3<\/h3>\n

2\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y$ \u306e2\u968e\u504f\u5fae\u5206\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\uff1a<\/p>\n

$$\\nabla^2 \\equiv \\dfrac{\\partial^2}{\\partial x^2} + \\dfrac{\\partial^2}{\\partial y^2}$$<\/p>\n

2\u6b21\u5143\u6975\u5ea7\u6a19<\/h3>\n

2\u6b21\u5143\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 \\(x, y\\) \u304b\u3089\u6975\u5ea7\u6a19 \\(r, \\phi\\) \u3078\u306e\u5ea7\u6a19\u5909\u63db\uff08\u3064\u307e\u308a\u5143\u306e\u5ea7\u6a19 \\(x, y\\) \u3092\u4f7f\u3063\u3066\u65b0\u3057\u3044\u5ea7\u6a19 \\(r, \\phi\\) \u3092\u8868\u3059\u5f0f\uff09\u306f<\/p>\n

\\begin{eqnarray}
\nr &=& \\sqrt{x^2 + y^2} \\\\
\n\\phi &=& \\tan^{-1} \\frac{y}{x}
\n\\end{eqnarray}<\/p>\n

\u305d\u306e\u9006\u5909\u63db\uff08\u3064\u307e\u308a\u65b0\u3057\u3044\u5ea7\u6a19 \\(r, \\phi\\) \u3092\u4f7f\u3063\u3066\u5143\u306e\u5ea7\u6a19 \\(x, y\\) \u3092\u8868\u3059\u5f0f\uff09\u306f\uff0c<\/p>\n

\\begin{eqnarray}
\nx &=&\u00a0 r \\cos\\phi\\\\
\ny &=&\u00a0 r \\sin\\phi
\n\\end{eqnarray}<\/p>\n

1\u968e\u504f\u5fae\u5206\u3092\u6975\u5ea7\u6a19\u3067\u66f8\u304d\u76f4\u3059<\/h3>\n

\u3059\u3067\u306b\u300c\u4f8b\u984c\uff1a2\u6b21\u5143\u6975\u5ea7\u6a19<\/a>\u300d\u3067\u4ee5\u4e0b\u306e\u504f\u5c0e\u95a2\u6570\u3092\u8a08\u7b97\u3057\u3066\u3044\u308b\u3002\u3059\u3050\u306b\u4f7f\u3046\u306e\u3067\u3053\u3053\u306b\u307e\u3068\u3081\u3066\u66f8\u304d\u51fa\u3057\u3066\u304a\u304f\u3002<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial r}{\\partial x} &=& \\cos\\phi \\\\
\n\\frac{\\partial r}{\\partial y} &=& \\sin\\phi \\\\
\n\\frac{\\partial \\phi}{\\partial x} &=& -\\frac{\\sin\\phi}{r} \\\\
\n\\frac{\\partial \\phi}{\\partial y} &=& \\frac{\\cos\\phi}{r}
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\n\\therefore\\ \\ \\frac{\\partial }{\\partial x} &=& \\frac{\\partial r}{\\partial x}\\frac{\\partial}{\\partial r} + \\frac{\\partial \\phi}{\\partial x}\\frac{\\partial}{\\partial \\phi} \\\\
\n&=& \\cos\\phi \\frac{\\partial}{\\partial r} -\\frac{\\sin\\phi}{r}\\frac{\\partial}{\\partial \\phi} \\\\
\n\\therefore\\ \\ \\frac{\\partial }{\\partial y} &=& \\frac{\\partial r}{\\partial y}\\frac{\\partial}{\\partial r} + \\frac{\\partial \\phi}{\\partial y}\\frac{\\partial}{\\partial \\phi} \\\\
\n&=& \\sin\\phi \\frac{\\partial}{\\partial r} +\\frac{\\cos\\phi}{r}\\frac{\\partial}{\\partial \\phi}
\n\\end{eqnarray}<\/p>\n

2\u968e\u504f\u5fae\u5206\u3092\u6975\u5ea7\u6a19\u3067\u66f8\u304d\u76f4\u3059<\/h3>\n

\\begin{eqnarray}
\n\\frac{\\partial^2}{\\partial x^2} &=& \\left( \\cos\\phi \\frac{\\partial}{\\partial r} -\\frac{\\sin\\phi}{r}\\frac{\\partial}{\\partial \\phi}\\right)\\left( \\cos\\phi \\frac{\\partial}{\\partial r} -\\frac{\\sin\\phi}{r}\\frac{\\partial}{\\partial \\phi}\\right) \\\\
\n&=& \\left( \\cos\\phi \\frac{\\partial}{\\partial r}\\right)\\left( \\cos\\phi \\frac{\\partial}{\\partial r}\\right) \\\\
\n&& -\\left( \\cos\\phi \\frac{\\partial}{\\partial r}\\right)\\left(\\frac{\\sin\\phi}{r}\\frac{\\partial}{\\partial \\phi}\\right) \\\\
\n&&-\\left(\\frac{\\sin\\phi}{r}\\frac{\\partial}{\\partial \\phi}\\right)\\left( \\cos\\phi \\frac{\\partial}{\\partial r}\\right)\\\\
\n&&+\\left(\\frac{\\sin\\phi}{r}\\frac{\\partial}{\\partial \\phi}\\right)\\left(\\frac{\\sin\\phi}{r}\\frac{\\partial}{\\partial \\phi}\\right) \\\\
\n&=& \\cos^2\\phi \\frac{\\partial^2}{\\partial r^2} \\\\
\n&&+\\cos\\phi \\frac{\\sin\\phi }{r^2} \\frac{\\partial }{\\partial \\phi} – \\cos\\phi \\frac{\\sin\\phi}{r}\\frac{\\partial^2}{\\partial r \\partial\\phi}\\\\
\n&&+\\frac{\\sin\\phi}{r} \\sin\\phi \\frac{\\partial}{\\partial r} -\\frac{\\sin\\phi}{r}\\cos\\phi \\frac{\\partial^2}{\\partial r\\partial \\phi}\\\\
\n&&+\\frac{\\sin\\phi}{r} \\frac{\\cos\\phi}{r} \\frac{\\partial}{\\partial\\phi} + \\frac{\\sin^2\\phi}{r^2} \\frac{\\partial^2}{\\partial \\phi^2} \\\\
\n&=& \\cos^2\\phi \\frac{\\partial^2}{\\partial r^2} + \\frac{\\sin^2\\phi}{r} \\frac{\\partial}{\\partial r} -\\frac{2\\sin\\phi\\cos\\phi}{r} \\frac{\\partial^2}{\\partial r \\partial \\phi} \\\\
\n&&+\\frac{\\sin^2}{r^2} \\frac{\\partial^2}{\\partial \\phi^2} + \\frac{2\\sin\\phi \\cos\\phi}{r^2} \\frac{\\partial}{\\partial\\phi}
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial^2}{\\partial y^2} &=& \\left( \\sin\\phi \\frac{\\partial}{\\partial r} +\\frac{\\cos\\phi}{r}\\frac{\\partial}{\\partial \\phi}\\right)\\left( \\sin\\phi \\frac{\\partial}{\\partial r} +\\frac{\\cos\\phi}{r}\\frac{\\partial}{\\partial \\phi}\\right) \\\\
\n&=& \\cdots \\\\
\n&=& \\sin^2\\phi \\frac{\\partial^2}{\\partial r^2} + \\frac{\\cos^2\\phi}{r} \\frac{\\partial}{\\partial r} +\\frac{2\\sin\\phi\\cos\\phi}{r} \\frac{\\partial^2}{\\partial r \\partial \\phi} \\\\
\n&&+\\frac{\\cos^2}{r^2} \\frac{\\partial^2}{\\partial \\phi^2} -\\frac{2\\sin\\phi \\cos\\phi}{r^2} \\frac{\\partial}{\\partial\\phi}
\n\\end{eqnarray}<\/p>\n

2\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3092\u6975\u5ea7\u6a19\u3067<\/h3>\n

\\begin{eqnarray}
\n\\therefore\\ \\ \\nabla^2 &=& \\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2} \\\\
\n&=& \\frac{\\partial^2}{\\partial r^2} +\\frac{1}{r} \\frac{\\partial}{\\partial r} + \\frac{1}{r^2} \\frac{\\partial^2}{\\partial \\phi^2}\\\\
\n&=& \\frac{1}{r} \\frac{\\partial}{\\partial r} \\left(r \\frac{\\partial}{\\partial r} \\right)+ \\frac{1}{r^2} \\frac{\\partial^2}{\\partial \\phi^2}
\n\\end{eqnarray}<\/p>\n

\u53c2\u8003\uff1a\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u3092\u4f7f\u3063\u3066\u8868\u3059<\/h3>\n

\u5c06\u6765\uff0c\u300c\u30d9\u30af\u30c8\u30eb\u306e\u767a\u6563\u3084\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3092\u5171\u5909\u5fae\u5206\u3067\u7406\u89e3\u3059\u308b<\/a><\/strong><\/span>\u300d\u306e\u30da\u30fc\u30b8\u3067\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u3084\u5171\u5909\u5fae\u5206\u3092\u5b66\u3093\u3060\u3042\u3068\u3067\u3053\u306e\u30da\u30fc\u30b8\u3092\u898b\u304b\u3048\u3057\u3066\u307f\u308b\u3068\uff0c2\u6b21\u5143\u6975\u5ea7\u6a19\u306b\u304a\u3051\u308b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb $g_{ij}$ \u306e\u6210\u5206\uff08\u3053\u306e\u3078\u3093<\/strong><\/span><\/a>\u3067\u3084\u3063\u3066\u305f\uff09\u304c<\/p>\n

$$
\ng_{ij} = \\left(\\begin{array}{cc}
\ng_{rr} & g_{r\\phi} \\\\
\ng_{\\phi r} & g_{\\phi\\phi}\\end{array}\\right)
\n=\u00a0 \\left(\\begin{array}{cc}
\n1 & 0 \\\\
\n0 & r^2\\end{array}\\right)
\n$$<\/p>\n

\u3067\u3042\u308a\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u3092\u884c\u5217\u3068\u307f\u306a\u3057\u305f\u3068\u304d\u306e\u884c\u5217\u5f0f $g$ \u306f<\/p>\n

$$g \\equiv \\det (g_{ij}) = r^2$$<\/p>\n

\u307e\u305f\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u9006\u884c\u5217 $g^{ij}$ \u306f<\/p>\n

$$
\ng^{ij} = \\left(\\begin{array}{cc}
\ng^{rr} & g^{r\\phi} \\\\
\ng^{\\phi r} & g^{\\phi\\phi}\\end{array}\\right)
\n=\u00a0 \\left(\\begin{array}{cc}
\n1 & 0 \\\\
\n0 & r^{-2}\\end{array}\\right)
\n$$<\/p>\n

\u3053\u308c\u3089\u3092\u4f7f\u3046\u30682\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u304c<\/p>\n

\\begin{eqnarray}
\n\\nabla^2 &=& \\frac{1}{\\sqrt{g}} \\partial_r \\left( \\sqrt{g} \\ r^{rr}\\\u00a0 \\partial_r\\right)
\n+ \\frac{1}{\\sqrt{g}} \\partial_{\\phi} \\left( \\sqrt{g} \\ r^{{\\phi} {\\phi} } \\ \\partial_{\\phi} \\right)\\\\
\n&=& \\frac{1}{r} \\partial_r \\left( r \\ \\partial_r\\right)
\n+ \\frac{1}{r}\\partial_{\\phi} \\left( r \\ r^{-2 } \\ \\partial_{\\phi} \\right)
\n\\end{eqnarray}<\/p>\n

\u3068\u66f8\u3051\u3066\u3044\u308b\u306e\u3060\u306a\u3041\u3068\u3044\u3046\u3053\u3068\u304c\u308f\u304b\u308b\u3088\u3046\u306b\u306a\u308b\u3068\u601d\u3046\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5408\u6210\u95a2\u6570\u306e\u504f\u5fae\u5206\u306e\u5fdc\u7528\u3068\u3057\u3066\uff0c\uff08\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3067\u5b9a\u7fa9\u3055\u308c\u305f\uff092\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3 $\\nabla^2 = \\dfrac{\\partial^2}{\\partial x^2} + \\dfrac{\\partial^2}{\\partial y^2}$ \u3092\u6975\u5ea7\u6a19 $r, \\phi$ \u3092\u4f7f\u3063\u3066\u8868\u3057\u3066\u307f\u308b\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2325,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-10248","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\/10248","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=10248"}],"version-history":[{"count":29,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/10248\/revisions"}],"predecessor-version":[{"id":10402,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/10248\/revisions\/10402"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2325"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=10248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}