{"id":10272,"date":"2025-05-27T13:11:24","date_gmt":"2025-05-27T04:11:24","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=10272"},"modified":"2025-05-30T16:07:58","modified_gmt":"2025-05-30T07:07:58","slug":"2%e6%ac%a1%e5%85%83%e3%81%ae%e7%b7%9a%e7%b4%a0%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%85%a8%e5%be%ae%e5%88%86\/2%e6%ac%a1%e5%85%83%e3%81%ae%e7%b7%9a%e7%b4%a0%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\u7dda\u7d20\u3092\u6975\u5ea7\u6a19\u3067\u8868\u3059"},"content":{"rendered":"

\u5168\u5fae\u5206\u306e\u4f8b\u3068\u3057\u3066\uff0c\u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u306e\u4e8c\u4e57\u3067\u3042\u308b\u300c\u7dda\u7d20<\/strong><\/span>\u300d\u3092\u6975\u5ea7\u6a19\u3067\u8868\u3057\u3066\u307f\u308b\u3002<\/p>\n

<\/p>\n

\u5e73\u9762\u4e0a\u306e\u8fd1\u63a52\u70b9<\/h3>\n

\u5e73\u9762\u4e0a\u306e\u8fd1\u63a5\u3057\u305f2\u70b9 $P(x, y)$ \u3068 $Q(\\tilde{x}, \\tilde{y})$ \u3092\u8003\u3048\u308b\u30022\u70b9\u306f\u6975\u3081\u3066\u8fd1\u3044\u306e\u3067\uff0c\u70b9 $Q$ \u306e\u6210\u5206\u306f\u5fae\u5c0f\u5909\u4f4d $dx, dy$ \u3092\u4f7f\u3063\u3066<\/p>\n

\\begin{eqnarray}
\n\\tilde{x} &=& x + dx \\\\
\n\\tilde{y} &=& y + dy
\n\\end{eqnarray}<\/p>\n

\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

\u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb<\/h3>\n

2\u70b9\u3092\u7d50\u3076\u30d9\u30af\u30c8\u30eb $\\overrightarrow{PQ}$ \u3092\u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb $d\\vec{x}$ \u3068\u5b9a\u7fa9\u3059\u308b\u3068\uff0c\u305d\u306e\u6210\u5206\u306f<\/p>\n

$$d\\vec{x} = (dx, dy)$$<\/p>\n

\u7dda\u7d20<\/h3>\n

\u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u306e\u4e8c\u4e57\u3092\u300c\u7dda\u7d20<\/strong><\/span>\u300d\u3068\u547c\u3076\u3053\u3068\u306b\u3059\u308b\u3002\u3053\u306e\u547c\u3073\u65b9\u306f\uff0c\u306e\u3061\u306b\u4e00\u822c\u76f8\u5bfe\u8ad6\u3067\u4f7f\u308f\u308c\u308b\u6570\u5b66\u7684\u9053\u5177\u7acb\u3066\u306e\u4e00\u3064\u3068\u3057\u3066\u73fe\u308c\u308b\uff08\u3053\u306e\u3042\u305f\u308a<\/strong><\/span><\/a>\u3092\u53c2\u7167\uff09\u3002<\/p>\n

$$d\\ell^2 \\equiv d\\vec{x}\\cdot d\\vec{x} = dx^2 + dy^2$$<\/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<\/h3>\n

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

\u5168\u5fae\u5206<\/h3>\n

\\begin{eqnarray}
\ndx &=& \\frac{\\partial x}{\\partial r} dr + \\frac{\\partial x}{\\partial \\phi}\u00a0 d\\phi \\\\
\n&=& \\cos\\phi\\, dr -r \\sin\\phi\\, d\\phi \\\\
\ndy &=& \\frac{\\partial y}{\\partial r} dr + \\frac{\\partial y}{\\partial \\phi}\u00a0 d\\phi \\\\
\n&=& \\sin\\phi\\, dr + r \\cos\\phi\\, d\\phi
\n\\end{eqnarray}<\/p>\n

\u6975\u5ea7\u6a19\u3067\u8868\u3057\u305f\u7dda\u7d20<\/h3>\n

\\begin{eqnarray}
\n\\therefore\\ \\ d\\ell^2 &=& dx^2 + dy^2 \\\\
\n&=& \\left( \\cos\\phi\\, dr -r \\sin\\phi\\, d\\phi\\right)^2 + \\left( \\sin\\phi\\, dr + r \\cos\\phi\\, d\\phi\\right)^2\\\\
\n&=& dr^2 + r^2 d\\phi^2
\n\\end{eqnarray}<\/p>\n

\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3067\u306f\uff0c\u7dda\u7d20\u306f\u5fae\u5c0f\u5909\u4f4d\u306e\u4e8c\u4e57\u548c $d\\ell^2 = dx^2 + dy^2$ \u3067\u3042\u308b\u304c\uff0c\u6975\u5ea7\u6a19\uff08\u304a\u3088\u3073\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u4ee5\u5916\u306e\u4e00\u822c\u5ea7\u6a19\uff09\u3067\u306f\u305d\u3046\u306f\u306a\u3089\u306a\u3044\u3002<\/p>\n

\u8fd1\u63a52\u70b9\u306e\u5ea7\u6a19\u3092\u6975\u5ea7\u6a19\u3067\u66f8\u304f\u3068 $P(r, \\phi), Q(r+dr, \\phi + d\\phi)$ \u3067\u3042\u308a\u6975\u5ea7\u6a19\u3067\u66f8\u3044\u305f\u5fae\u5c0f\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u3082<\/p>\n

$$\\overrightarrow{PQ} \\equiv d\\vec{x} = (dr, d\\phi)$$<\/p>\n

\u3067\u3042\u308b\u304b\u3089\u3068\u3044\u3063\u3066\uff0c\u7dda\u7d20\u306f $d\\ell^2 = dr^2 + {\\color{red}{1}}\\cdot d\\phi^2$ \u3068\u306f\u306a\u3089\u306a\u3044\uff01<\/strong><\/span>$d\\phi^2$ \u306e\u524d\u306f $ {\\color{red}{1}}$ \u3067\u306f\u306a\u304f\uff0c${\\color{blue}{r^2}}$ \u304c\u3064\u3044\u3066 $d\\ell^2 = dr^2 + {\\color{blue}{r^2}} \\,d\\phi^2$ \u3068\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\uff01<\/strong><\/span>
\n<\/strong><\/span><\/p>\n

\u53c2\u8003\uff1a2\u6b21\u5143\u6975\u5ea7\u6a19\u306b\u304a\u3051\u308b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206<\/h3>\n

\u3053\u306e\u3053\u3068\u306f\uff0c\u306e\u3061\u306b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u3092\u4f7f\u3063\u3066\u8aac\u660e\u3055\u308c\u308b\u3053\u3068\u306b\u306a\u308b\uff08\u3053\u306e\u3078\u3093<\/strong><\/span><\/a>\u3068\u304b\u3053\u306e\u3078\u3093<\/strong><\/span><\/a>\u3092\u53c2\u7167\uff09\u3002\u3053\u3053\u3067\u306f\u8aac\u660e\u3092\u7701\u304f\u306e\u3067\u306a\u3093\u306e\u3053\u3068\u304b\u308f\u304b\u3089\u306a\u3044\u3068\u306f\u601d\u3046\u304c\uff0c\u5c06\u6765\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u3092\u5b66\u3093\u3060\u3042\u3068\u3067\u3053\u306e\u30da\u30fc\u30b8\u3092\u898b\u76f4\u3057\u3066\u307f\u308b\u3068\uff0c2\u6b21\u5143\u6975\u5ea7\u6a19\u306b\u304a\u3051\u308b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u304c<\/p>\n

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

\u3067\u3042\u308a\uff0c\u7dda\u7d20\u304c<\/p>\n

$$d\\ell^2 = g_{rr} \\,dr^2 + g_{\\phi\\phi} \\,d\\phi^2$$<\/p>\n

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

\u5168\u5fae\u5206\u306e\u4f8b\u3068\u3057\u3066\uff0c\u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u306e\u4e8c\u4e57\u3067\u3042\u308b\u300c\u7dda\u7d20\u300d\u3092\u6975\u5ea7\u6a19\u3067\u8868\u3057\u3066\u307f\u308b\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2316,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-10272","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\/10272","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=10272"}],"version-history":[{"count":14,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/10272\/revisions"}],"predecessor-version":[{"id":10337,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/10272\/revisions\/10337"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2316"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=10272"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}