{"id":10300,"date":"2025-05-30T15:44:12","date_gmt":"2025-05-30T06:44:12","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=10300"},"modified":"2025-06-03T12:20:27","modified_gmt":"2025-06-03T03:20:27","slug":"3%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\/3%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":"3\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\uff093\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3 $\\nabla^2 = \\dfrac{\\partial^2}{\\partial x^2} + \\dfrac{\\partial^2}{\\partial y^2}+ \\dfrac{\\partial^2}{\\partial z^2}$ \u3092\u6975\u5ea7\u6a19 $r, \\theta, \\phi$ \u3092\u4f7f\u3063\u3066\u8868\u3057\u3066\u307f\u308b\u3002\u7df4\u7fd2\u554f\u984c\u306b\u3057\u3088\u3046\u304b\u3068\u601d\u3063\u305f\u304c\u5358\u306b\u8a08\u7b97\u91cf\u304c\u591a\u304f\u306a\u308b\u3060\u3051\u306a\u306e\u3067\uff0c\u5099\u5fd8\u9332\u3068\u3057\u3066\u3002<\/p>\n

<\/p>\n

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

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

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

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

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

\\begin{eqnarray}
\nr &=& \\sqrt{x^2 + y^2 + z^2} \\\\
\n\\theta &=& \\tan^{-1} \\frac{\\sqrt{x^2 + y^2}}{z}\\\\
\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, \\theta, \\phi\\) \u3092\u4f7f\u3063\u3066\u5143\u306e\u5ea7\u6a19 \\(x, y, z\\) \u3092\u8868\u3059\u5f0f\uff09\u306f\uff0c<\/p>\n

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

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

\u6975\u5ea7\u6a19 \\(r, \\theta, \\phi\\) \u3092 \\(x, y, z\\) \u30671\u968e\u504f\u5fae\u5206\u3057\u305f\u7d50\u679c\u3092\u6975\u5ea7\u6a19\u3067\u8868\u3059\u3002<\/p>\n

\\(r(x, y, z) \\) \u306e\u504f\u5fae\u5206\u306f\uff0c$u \\equiv x^2 + y^2 + z^2$ \u3068\u304a\u3044\u3066\u5408\u6210\u95a2\u6570\u306e\u504f\u5fae\u5206\u306e\u8981\u9818\u3067…<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial r}{\\partial x} &=& \\frac{d}{du} \\sqrt{u} \\cdot \\frac{\\partial u}{\\partial x} \\\\
\n&=& \\frac{1}{2} \\frac{1}{\\sqrt{u}} \\cdot 2 x \\\\
\n&=&\\frac{x}{r} = \\sin\\theta \\cos\\phi \\\\
\n\\frac{\\partial r}{\\partial y} &=& \\frac{y}{r} = \\sin\\theta \\sin\\phi \\\\
\n\\frac{\\partial r}{\\partial z} &=& \\frac{z}{r} = \\cos\\theta
\n\\end{eqnarray}<\/p>\n

\\(\\theta(x, y, z) \\) \u306e\u504f\u5fae\u5206\u306f\uff0c$u \\equiv \\dfrac{\\sqrt{x^2+y^2}}{z}$ \u3068\u304a\u3044\u3066\u5408\u6210\u95a2\u6570\u306e\u504f\u5fae\u5206\u306e\u8981\u9818\u3067…<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial \\theta}{\\partial x} &=& \\frac{d}{du} \\tan^{-1} {u} \\cdot \\frac{\\partial u}{\\partial x} \\\\
\n&=& \\frac{1}{1+u^2} \\frac{1}{z}\\cdot \\frac{x}{\\sqrt{x^2+y^2}} \\\\
\n&=&\\frac{z}{r^2}\u00a0 \\cdot \\frac{r \\sin\\theta \\cos\\phi}{r \\sin\\theta}\\\\
\n&=& \\frac{\\cos\\theta \\cos\\phi}{r} \\\\
\n\\frac{\\partial \\theta}{\\partial y} &=& \\frac{\\cos\\theta \\sin\\phi}{r} \\\\
\n\\frac{\\partial \\theta}{\\partial z} &=& -\\frac{\\sin\\theta}{r}
\n\\end{eqnarray}<\/p>\n

\\(\\phi(x, y, z) \\) \u306e\u504f\u5fae\u5206\u306f\uff0c$u \\equiv \\dfrac{y}{x}$ \u3068\u304a\u3044\u3066\u5408\u6210\u95a2\u6570\u306e\u504f\u5fae\u5206\u306e\u8981\u9818\u3067…<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial \\phi}{\\partial x} &=& \\frac{d}{du} \\tan^{-1} {u} \\cdot \\frac{\\partial u}{\\partial x} \\\\
\n&=& \\frac{1}{1+u^2} \\cdot \\left( -\\frac{y}{x^2} \\right)\\\\
\n&=&-\\frac{y}{x^2 + y^2}\\\\
\n&=& -\\frac{\\sin\\phi}{r \\sin\\theta} \\\\
\n\\frac{\\partial \\phi}{\\partial y} &=& \\frac{\\cos\\phi}{r \\sin\\theta} \\\\
\n\\frac{\\partial \\phi}{\\partial z} &=& 0
\n\\end{eqnarray}<\/p>\n

\u5f93\u3063\u3066\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial }{\\partial x} &=&
\n\\frac{\\partial r}{\\partial x}\\frac{\\partial}{\\partial r}
\n+\\frac{\\partial\\theta}{\\partial x} \\frac{\\partial}{\\partial \\theta}
\n+\\frac{\\partial \\phi}{\\partial x}\\frac{\\partial}{\\partial \\phi} \\\\
\n&=& \\sin\\theta \\cos\\phi \\frac{\\partial}{\\partial r}
\n+\\frac{\\cos\\theta \\cos\\phi}{r} \\frac{\\partial}{\\partial \\theta}
\n-\\frac{\\sin\\phi}{r \\sin\\theta} \\frac{\\partial}{\\partial \\phi} \\\\
\n\\frac{\\partial }{\\partial y} &=&
\n\\sin\\theta \\sin\\phi \\frac{\\partial}{\\partial r}
\n+\\frac{\\cos\\theta \\sin\\phi}{r} \\frac{\\partial}{\\partial \\theta}
\n+\\frac{\\cos\\phi}{r \\sin\\theta} \\frac{\\partial}{\\partial \\phi} \\\\
\n\\frac{\\partial }{\\partial z} &=&
\n\\cos\\theta\u00a0 \\frac{\\partial}{\\partial r}
\n-\\frac{\\sin\\theta}{r} \\frac{\\partial}{\\partial \\theta}
\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(\\sin\\theta \\cos\\phi \\frac{\\partial}{\\partial r}
\n+\\frac{\\cos\\theta \\cos\\phi}{r} \\frac{\\partial}{\\partial \\theta}
\n-\\frac{\\sin\\phi}{r \\sin\\theta} \\frac{\\partial}{\\partial \\phi} \\right)^2 \\\\
\n&=& \\left(\\sin\\theta \\cos\\phi \\frac{\\partial}{\\partial r} \\right)
\n\\left(\\sin\\theta \\cos\\phi \\frac{\\partial}{\\partial r} \\right) \\\\
\n&& +\\left(\\sin\\theta \\cos\\phi \\frac{\\partial}{\\partial r} \\right)
\n\\left(\\frac{\\cos\\theta \\cos\\phi}{r} \\frac{\\partial}{\\partial \\theta} \\right) \\\\
\n&& +\\left(\\sin\\theta \\cos\\phi \\frac{\\partial}{\\partial r} \\right)
\n\\left( -\\frac{\\sin\\phi}{r \\sin\\theta} \\frac{\\partial}{\\partial \\phi}\\right) \\\\
\n%
\n&& +\\left(\\frac{\\cos\\theta \\cos\\phi}{r} \\frac{\\partial}{\\partial \\theta} \\right)
\n\\left(\\sin\\theta \\cos\\phi \\frac{\\partial}{\\partial r} \\right) \\\\
\n&& + \\left(\\frac{\\cos\\theta \\cos\\phi}{r} \\frac{\\partial}{\\partial \\theta} \\right)
\n\\left(\\frac{\\cos\\theta \\cos\\phi}{r} \\frac{\\partial}{\\partial \\theta} \\right) \\\\
\n&& + \\left(\\frac{\\cos\\theta \\cos\\phi}{r} \\frac{\\partial}{\\partial \\theta} \\right)
\n\\left( -\\frac{\\sin\\phi}{r \\sin\\theta} \\frac{\\partial}{\\partial \\phi}\\right) \\\\
\n%
\n&& +\\left( -\\frac{\\sin\\phi}{r \\sin\\theta} \\frac{\\partial}{\\partial \\phi}\\right)
\n\\left(\\sin\\theta \\cos\\phi \\frac{\\partial}{\\partial r} \\right) \\\\
\n&& + \\left( -\\frac{\\sin\\phi}{r \\sin\\theta} \\frac{\\partial}{\\partial \\phi}\\right)
\n\\left(\\frac{\\cos\\theta \\cos\\phi}{r} \\frac{\\partial}{\\partial \\theta} \\right) \\\\
\n&& + \\left( -\\frac{\\sin\\phi}{r \\sin\\theta} \\frac{\\partial}{\\partial \\phi}\\right)
\n\\left( -\\frac{\\sin\\phi}{r \\sin\\theta} \\frac{\\partial}{\\partial \\phi}\\right) \\\\
\n&=& \\sin^2\\theta \\cos^2\\phi \\frac{\\partial^2}{\\partial r^2}
\n+ \\frac{\\cos^2\\theta \\cos^2\\phi}{r^2} \\frac{\\partial^2}{\\partial \\theta^2}
\n+ \\frac{\\sin^2\\phi}{r^2 \\sin^2\\theta} \\frac{\\partial^2}{\\partial \\phi^2}\\\\
\n&& + \\frac{2}{r} \\sin\\theta\\cos\\theta \\cos^2\\phi \\frac{\\partial^2}{\\partial r \\partial\\theta} \\\\
\n&& -\\frac{2}{r} \\sin\\phi \\cos\\phi \\frac{\\partial^2}{\\partial r \\partial\\phi} \\\\
\n&& -\\frac{2}{r^2} \\frac{\\cos\\theta}{\\sin\\theta} \\sin\\phi \\cos\\phi \\frac{\\partial^2}{\\partial \\theta \\partial\\phi} \\\\
\n&& +\\frac{1}{r} \\left(\\cos^2\\theta \\cos^2\\phi + \\sin^2\\phi \\right) \\frac{\\partial}{\\partial r} \\\\
\n&& + \\frac{1}{r^2} \\left(\\frac{\\cos\\theta}{\\sin\\theta} \\sin^2\\phi -2 \\sin\\theta \\cos\\theta \\cos^2\\phi \\right)\\frac{\\partial}{\\partial \\theta} \\\\
\n&& +\\frac{2 \\sin\\phi \\cos\\phi}{r^2 \\sin^2\\theta} \\frac{\\partial}{\\partial \\phi} \\\\
\n%%
\n\\frac{\\partial^2}{\\partial y^2} &=&\\sin^2\\theta \\sin^2\\phi \\frac{\\partial^2}{\\partial r^2}
\n+ \\frac{\\cos^2\\theta \\sin^2\\phi}{r^2} \\frac{\\partial^2}{\\partial \\theta^2}
\n+ \\frac{\\cos^2\\phi}{r^2 \\sin^2\\theta} \\frac{\\partial^2}{\\partial \\phi^2}\\\\
\n&& + \\frac{2}{r} \\sin\\theta\\cos\\theta \\sin^2\\phi \\frac{\\partial^2}{\\partial r \\partial\\theta} \\\\
\n&& +\\frac{2}{r} \\sin\\phi \\cos\\phi \\frac{\\partial^2}{\\partial r \\partial\\phi} \\\\
\n&& +\\frac{2}{r^2} \\frac{\\cos\\theta}{\\sin\\theta} \\sin\\phi \\cos\\phi \\frac{\\partial^2}{\\partial \\theta \\partial\\phi} \\\\
\n&& +\\frac{1}{r} \\left(\\cos^2\\theta \\sin^2\\phi + \\cos^2\\phi \\right) \\frac{\\partial}{\\partial r} \\\\
\n&& + \\frac{1}{r^2} \\left(\\frac{\\cos\\theta}{\\sin\\theta} \\cos^2\\phi -2 \\sin\\theta \\cos\\theta \\sin^2\\phi \\right)\\frac{\\partial}{\\partial \\theta} \\\\
\n&& -\\frac{2 \\sin\\phi \\cos\\phi}{r^2 \\sin^2\\theta} \\frac{\\partial}{\\partial \\phi} \\\\
\n%%
\n\\frac{\\partial^2}{\\partial z^2}
\n&=&
\n\\cos^2\\theta \\frac{\\partial^2}{\\partial r^2}
\n+ \\frac{1}{r^2} \\sin^2\\theta \\frac{\\partial^2}{\\partial \\theta^2} \\\\
\n&& -\\frac{2}{r} \\sin\\theta \\cos\\theta \\frac{\\partial^2}{\\partial r \\partial\\theta} \\\\
\n&& + \\frac{1}{r} \\sin^2\\theta \\frac{\\partial}{\\partial r} + \\frac{2}{r^2} \\sin\\theta \\cos\\theta \\frac{\\partial}{\\partial\\theta}
\n\\end{eqnarray}<\/p>\n

 <\/p>\n

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

\u6700\u7d42\u7684\u306b\uff0c<\/p>\n

\\begin{eqnarray}
\n\\nabla^2 &=& \\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}+\\frac{\\partial^2}{\\partial z^2} \\\\
\n&=& \\frac{\\partial^2}{\\partial r^2} +\\frac{2}{r} \\frac{\\partial}{\\partial r}
\n+\\frac{1}{r^2} \\frac{\\partial^2}{\\partial\\theta^2} + \\frac{1}{r^2} \\frac{\\cos\\theta}{\\sin\\theta}\\frac{\\partial}{\\partial \\theta}
\n+ \\frac{1}{r^2 \\sin^2\\theta} \\frac{\\partial^2}{\\partial \\phi^2} \\\\
\n&=& \\frac{1}{r^2} \\frac{\\partial}{\\partial r} \\left( r^2 \\frac{\\partial}{\\partial r}\\right)
\n+\\frac{1}{r^2\\sin\\theta} \\frac{\\partial}{\\partial\\theta} \\left(\\sin\\theta\\frac{\\partial}{\\partial\\theta} \\right)
\n+ \\frac{1}{r^2 \\sin^2\\theta} \\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\uff0c3\u6b21\u5143\u6975\u5ea7\u6a19\u306b\u304a\u3051\u308b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb $g_{ij}$ \u306e\u6210\u5206\u304c<\/p>\n

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

$$g \\equiv \\det (g_{ij}) = r^4 \\sin^2\\theta, \\ \\therefore\\ \\sqrt{g} = r^2 \\sin\\theta$$<\/p>\n

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

$$
\ng^{ij} = \\left(\\begin{array}{ccc}
\ng^{rr} & g^{r\\theta} & g^{r\\phi} \\\\
\ng^{\\theta r} & g^{\\theta\\theta} & g^{\\theta\\phi} \\\\
\ng^{\\phi r} & g^{\\phi\\theta} & g^{\\phi\\phi}
\n\\end{array}\\right)
\n=\u00a0 \\left(\\begin{array}{ccc}
\n1 & 0& 0 \\\\
\n0 & \\dfrac{1}{r}& 0 \\\\
\n0 & 0 & \\dfrac{1}{r^{2} \\sin^{2}\\theta}
\n\\end{array}\\right)$$<\/p>\n

\u3053\u308c\u3089\u3092\u4f7f\u3046\u30683\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_{\\theta} \\left( \\sqrt{g} \\ r^{{\\theta}{\\theta}}\\\u00a0 \\partial_{\\theta}\\right)
\n+ \\frac{1}{\\sqrt{g}} \\partial_{\\phi} \\left( \\sqrt{g} \\ r^{{\\phi} {\\phi} } \\ \\partial_{\\phi} \\right)\\\\
\n&=& \\frac{1}{r^2 \\sin\\theta} \\partial_r \\left( r^2 \\sin\\theta \\ \\partial_r\\right)
\n+ \\frac{1}{r^2 \\sin\\theta} \\partial_{\\theta} \\left( \\frac{r^2 \\sin\\theta}{ \\ r^{2}}\\ \\partial_{\\theta}\\right)
\n+ \\frac{1}{r^2 \\sin\\theta}\\partial_{\\phi} \\left( \\frac{r^2 \\sin\\theta}{r^{2 } \\sin^{2}\\theta}\\ \\partial_{\\phi} \\right)\\\\
\n&=& \\frac{1}{r^2 } \\partial_r \\left( r^2 \\ \\partial_r\\right)
\n+ \\frac{1}{r^2 \\sin\\theta} \\partial_{\\theta} \\left( \\sin\\theta \\ \\partial_{\\theta}\\right)
\n+ \\frac{1}{r^2 \\sin^2\\theta}\\partial_{\\phi} \\left( \\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\uff093\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3 $\\nabla^2 = \\dfrac{\\partial^2}{\\partial x^2} + \\dfrac{\\partial^2}{\\partial y^2}+ \\dfrac{\\partial^2}{\\partial z^2}$ \u3092\u6975\u5ea7\u6a19 $r, \\theta, \\phi$ \u3092\u4f7f\u3063\u3066\u8868\u3057\u3066\u307f\u308b\u3002\u7df4\u7fd2\u554f\u984c\u306b\u3057\u3088\u3046\u304b\u3068\u601d\u3063\u305f\u304c\u5358\u306b\u8a08\u7b97\u91cf\u304c\u591a\u304f\u306a\u308b\u3060\u3051\u306a\u306e\u3067\uff0c\u5099\u5fd8\u9332\u3068\u3057\u3066\u3002<\/p>

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