{"id":2344,"date":"2022-02-26T10:03:57","date_gmt":"2022-02-26T01:03:57","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2344"},"modified":"2024-07-10T11:45:24","modified_gmt":"2024-07-10T02:45:24","slug":"%e5%8f%82%e8%80%83%ef%bc%9a%e6%a5%b5%e5%ba%a7%e6%a8%99%e3%81%ab%e3%82%88%e3%82%8b2%e9%87%8d%e7%a9%8d%e5%88%86%e3%81%a8%e3%83%a4%e3%82%b3%e3%83%93%e3%82%a2%e3%83%b3","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%a4%9a%e9%87%8d%e7%a9%8d%e5%88%86%ef%bc%9a%e5%a4%9a%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e5%88%86\/%e5%86%86%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%82%922%e9%87%8d%e7%a9%8d%e5%88%86%e3%81%a7%e6%b1%82%e3%82%81%e3%82%8b\/%e5%8f%82%e8%80%83%ef%bc%9a%e6%a5%b5%e5%ba%a7%e6%a8%99%e3%81%ab%e3%82%88%e3%82%8b2%e9%87%8d%e7%a9%8d%e5%88%86%e3%81%a8%e3%83%a4%e3%82%b3%e3%83%93%e3%82%a2%e3%83%b3\/","title":{"rendered":"\u53c2\u8003\uff1a\u6975\u5ea7\u6a19\u306b\u3088\u308b2\u91cd\u7a4d\u5206\u3068\u30e4\u30b3\u30d3\u30a2\u30f3"},"content":{"rendered":"

<\/p>\n

\u6975\u5ea7\u6a19\u306b\u3088\u308b2\u91cd\u7a4d\u5206\u306e\u3068\u304d\uff0c\u306a\u305c\u5fae\u5c0f\u9762\u7a4d\u8981\u7d20\u304c \\(dx\\, dy \\rightarrow rdr\\,d\\theta\\) \u3068\u306a\u308b\u306e\u304b\u3002<\/p>\n

\n

2\u6b21\u5143\u6975\u5ea7\u6a19\u3068\u30e4\u30b3\u30d3\u30a2\u30f3<\/h3>\n

\u5e73\u884c\u56db\u8fba\u5f62\u306e\u5bfe\u89d2\u7dda\u30d9\u30af\u30c8\u30eb $\\overrightarrow{{PQ}}$ \u304c2\u3064\u306e\u7dda\u5f62\u72ec\u7acb\u306a 2 \u3064\u306e\u30d9\u30af\u30c8\u30eb\u3092\u4f7f\u3063\u3066<\/p>\n

$$\\overrightarrow{{PQ}} =\u00a0 {\\color{blue}{\\boldsymbol{a}}} + {\\color{red}{\\boldsymbol{b}}}$$<\/p>\n

\u3068\u8868\u3055\u308c\u308b\u3068\u304d\uff0c\u3053\u308c\u3089 2 \u3064\u306e\u30d9\u30af\u30c8\u30eb\u304b\u3089\u3064\u304f\u3089\u308c\u308b\u5e73\u884c\u56db\u8fba\u5f62\u306e\u9762\u7a4d $S$ \u306f<\/p>\n

$$S = |{\\color{blue}{\\boldsymbol{a}}} \\times {\\color{red}{\\boldsymbol{b}}}| = |{\\color{blue}{\\boldsymbol{a}}}|\\,|{\\color{red}{\\boldsymbol{b}}}|\\, \\sin{\\color{green}{\\theta}}$$<\/p>\n

\u3068\u306a\u308b\u3002\u3053\u3053\u3067 ${\\color{blue}{\\boldsymbol{a}}} \\times {\\color{red}{\\boldsymbol{b}}}$ \u306f\u30d9\u30af\u30c8\u30eb ${\\color{blue}{\\boldsymbol{a}}} $ \u3068\u30d9\u30af\u30c8\u30eb ${\\color{red}{\\boldsymbol{b}}}$ \u3068\u306e\u5916\u7a4d<\/strong><\/span>\u3067\u3042\u308a\uff0c${\\color{green}{\\theta}}$ \u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3002<\/p>\n

\"\"<\/p>\n

\u8fd1\u63a5\u3057\u305f 2 \u70b9 \\(P(x, y), \\ Q(x + dx, y + dy)\\) \u3092\u8003\u3048\u308b\u30022\u70b9\u3092\u7d50\u3076\u30d9\u30af\u30c8\u30eb\u306f
$$ \\overrightarrow{{PQ}} = dx \\,\\boldsymbol{i} + dy\\, \\boldsymbol{j}$$
\u3053\u3053\u3067\uff0c\\(\\boldsymbol{i}, \\ \\boldsymbol{j}\\) \u306f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u7cfb\u306e\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u3002<\/p>\n

\u4e00\u822c\u306b\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304b\u3089\u4f5c\u3089\u308c\u308b\u5e73\u884c\u56db\u8fba\u5f62\u306e\u9762\u7a4d\u306f\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u3092\u4f7f\u3063\u3066\u66f8\u3051\u308b\u304b\u3089\uff0c\u3053\u306e \\(\\overrightarrow{{PQ}}\\) \u3092\u5bfe\u89d2\u7dda\u3068\u3059\u308b\u5fae\u5c0f\u5e73\u884c\u56db\u8fba\u5f62\u306e\u9762\u7a4d \\(S\\) \u306f<\/p>\n

$$ S = |dx\\, \\boldsymbol{i} \\times dy\\, \\boldsymbol{j}| = dx\\,dy\\,|\\boldsymbol{k}| = dx\\,dy$$<\/p>\n

 <\/p>\n

\u5ea7\u6a19\u5909\u63db
\n\\begin{eqnarray}x = x(r, \\theta) = r\\cos\\theta, \\quad y = y(r, \\theta)=r\\sin\\theta
\n\\end{eqnarray}
\n\u306b\u3088\u3063\u3066\uff0c
\n\\begin{eqnarray}\\overrightarrow{PQ} &=& dx \\,\\boldsymbol{i} + dy\\, \\boldsymbol{j}\\\\
\n&=& \\left(\\frac{\\partial x}{\\partial r} dr + \\frac{\\partial x}{\\partial \\theta} d\\theta\\right) \\boldsymbol{i} +\\left(\\frac{\\partial y}{\\partial r} dr + \\frac{\\partial y}{\\partial \\theta} d\\theta\\right) \\boldsymbol{j}\\\\
\n&=&dr \\left(\\frac{\\partial x}{\\partial r} \\boldsymbol{i} + \\frac{\\partial y}{\\partial r}\\boldsymbol{j}\\right) + d\\theta \\left(\\frac{\\partial x}{\\partial \\theta} \\boldsymbol{i} + \\frac{\\partial y}{\\partial \\theta}\\boldsymbol{j}\\right) \\\\
\n&\\equiv& dr \\,\\boldsymbol{e}_r + d\\theta\\, \\boldsymbol{e}_{\\theta}
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\\(\\overrightarrow{PQ}\\) \u3092\u5bfe\u89d2\u7dda\u3068\u3059\u308b\u5fae\u5c0f\u5e73\u884c\u56db\u8fba\u5f62\u306e\u9762\u7a4d \\(dS\\) \u3092\u6975\u5ea7\u6a19\u3067\u8868\u3059\u3068\uff0c
\n\\begin{eqnarray} dS &=& |dr\\,\\boldsymbol{e}_r \\times d\\theta\\,\\boldsymbol{e}_{\\theta} |\\\\
\n&=& dr\\,d\\theta \\left| \\left(\\frac{\\partial x}{\\partial r} \\boldsymbol{i} + \\frac{\\partial y}{\\partial r}\\boldsymbol{j}\\right)\\times \\left(\\frac{\\partial x}{\\partial \\theta} \\boldsymbol{i} + \\frac{\\partial y}{\\partial \\theta}\\boldsymbol{j}\\right)\\right|\\\\
\n&=& dr\\,d\\theta\\left(\\frac{\\partial x}{\\partial r} \\frac{\\partial y}{\\partial \\theta} – \\frac{\\partial y}{\\partial r}\\frac{\\partial x}{\\partial \\theta}\\right) |\\boldsymbol{i} \\times \\boldsymbol{j}|\\\\
\n&=& dr\\,d\\theta\\left(\\frac{\\partial x}{\\partial r} \\frac{\\partial y}{\\partial \\theta} – \\frac{\\partial y}{\\partial r}\\frac{\\partial x}{\\partial \\theta}\\right) |\\boldsymbol{k}|\\\\
\n&\\equiv& dr\\,d\\theta \\frac{\\partial(x,y)}{\\partial(r,\\theta)}
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c$$ \\frac{\\partial(x,y)}{\\partial(r,\\theta)} \\equiv
\n\\begin{vmatrix}
\n\\frac{\\partial x}{\\partial r} & \\frac{\\partial x}{\\partial \\theta}\\\\
\n\\frac{\\partial y}{\\partial r} & \\frac{\\partial y}{\\partial \\theta}\\\\
\n\\end{vmatrix}
\n= \\frac{\\partial x}{\\partial r} \\frac{\\partial y}{\\partial \\theta} – \\frac{\\partial y}{\\partial r}\\frac{\\partial x}{\\partial \\theta}$$ \u3092\u30e4\u30b3\u30d3\u30a2\u30f3<\/b><\/span>\u3068\u3044\u3046\u3002<\/p>\n

\u6975\u5ea7\u6a19\u3078\u306e\u5909\u63db\u306e\u5834\u5408\u306f\u30e4\u30b3\u30d3\u30a2\u30f3\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u3066
\n$$\\frac{\\partial(x,y)}{\\partial(r,\\theta)} = \\cos\\theta\\cdot r\\,\\sin\\theta – \\sin\\theta\\,\\cdot (-r\\,\\sin\\theta) = r$$
\n\u6700\u7d42\u7684\u306b
\n$$ dS = dx\\,dy = \\frac{\\partial(x,y)}{\\partial(r,\\theta)} dr\\,d\\theta = r\\, dr\\,d\\theta$$ \u3068\u306a\u308b\u3002<\/p>\n

\u4e00\u822c\u306e2\u6b21\u5143\u306e\u5ea7\u6a19\u5909\u63db $x = x(u,v), \\ y = y(u, v)$ \u306b\u5bfe\u3057\u3066\u3082
\n$$ dS = dx\\,dy = \\frac{\\partial(x,y)}{\\partial(u,v)} du\\,dv$$<\/p>\n

\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u30e4\u30b3\u30d3\u30a2\u30f3<\/b><\/span>\u3068\u3044\u3046\u3082\u306e\u304c\u306a\u305c\u73fe\u308c\u308b\u306e\u304b\u3092\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u304c\u9762\u7a4d\u306b\u306a\u308b\u306e\u3060\u3068\u3044\u3046\u3053\u3068\u304b\u3089\u7406\u89e3\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u305f\u3002<\/p>\n

3\u6b21\u5143\u306e\u5ea7\u6a19\u5909\u63db\u3068\u30e4\u30b3\u30d3\u30a2\u30f3<\/h3>\n

\u307e\u305f\uff0c3\u6b21\u5143\u306e\u5ea7\u6a19\u5909\u63db $x = x(u,v,w), \\ y = y(u, v,w), \\ z = z(u,v,w)$ \u306b\u3064\u3044\u3066\u3082\uff0c3\u3064\u306e\u7dda\u5f62\u72ec\u7acb\u306a\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\ \\boldsymbol{b}, \\boldsymbol{c}$ \u304b\u3089\u4f5c\u3089\u308c\u308b\u7acb\u4f53\uff08\u5e73\u884c\u516d\u9762\u4f53\uff09\u306e\u4f53\u7a4d $V$ \u304c<\/p>\n

$$ V = \\boldsymbol{a}\\cdot(\\boldsymbol{b}\\times\\boldsymbol{c})$$<\/p>\n

\u3067\u3042\u308b\u3053\u3068\u3092\u4f7f\u3046\u3068\uff0c\u5fae\u5c0f\u4f53\u7a4d\u8981\u7d20 $dV$ \u304c<\/p>\n

$$ dV = dx\\,dy\\,dz = \\frac{\\partial(x,y,z)}{\\partial(u,v,w)} du\\,dv\\,dw$$<\/p>\n

\u3068\u306a\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u308b\u3002<\/p>\n

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

$$d\\boldsymbol{r} = dx \\,\\boldsymbol{i} + dy\\,\\boldsymbol{j} + dz\\,\\boldsymbol{k}$$<\/p>\n

3\u672c\u306e\uff08\u7dda\u5f62\u72ec\u7acb\u306a\uff09\u30d9\u30af\u30c8\u30eb $dx\\, \\boldsymbol{i}, \\ dy\\, \\boldsymbol{j}, \\ dz\\,\\boldsymbol{k}$ \u304b\u3089\u3064\u304f\u3089\u308c\u308b\u5fae\u5c0f\u306a\u5e73\u884c6\u9762\u4f53\u306e\u4f53\u7a4d $dV$ \u306f<\/p>\n

\\begin{eqnarray}
\ndV &=& \\left(dx\\, \\boldsymbol{i} \\right) \\cdot \\left( \\left(dy\\, \\boldsymbol{j} \\right)\\times\\left(dz\\,\\boldsymbol{k} \\right)\\right) \\\\
\n&=& dx\\,dy\\, dz\\, \\boldsymbol{i} \\cdot(\\boldsymbol{j}\\times\\boldsymbol{k}) \\\\
\n&=& dx\\,dy\\, dz\\, \\boldsymbol{i} \\cdot \\boldsymbol{i} \\\\
\n&=& dx\\,dy\\, dz
\n\\end{eqnarray}<\/p>\n

\u4e00\u65b9\uff0c\u5ea7\u6a19\u5909\u63db<\/p>\n

\\begin{eqnarray}
\nx &=& x(u, v, w) \\\\
\ny &=& y(u, v, w) \\\\
\nz &=& z(u, v, w) \\\\
\n\\end{eqnarray}<\/p>\n

\u306b\u3088\u308a\uff0c\u5168\u5fae\u5206\u306f<\/p>\n

\\begin{eqnarray}
\ndx &=& \\frac{\\partial x}{\\partial u} du + \\frac{\\partial x}{\\partial v} dv +\\frac{\\partial x}{\\partial w} dw \\\\
\ndy &=& \\frac{\\partial y}{\\partial u} du + \\frac{\\partial y}{\\partial v} dv +\\frac{\\partial y}{\\partial w} dw \\\\
\ndz &=& \\frac{\\partial z}{\\partial u} du + \\frac{\\partial z}{\\partial v} dv +\\frac{\\partial z}{\\partial w} dw \\\\
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308b\u304b\u3089\uff0c\u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

\\begin{eqnarray}
\nd \\boldsymbol{r}&=& dx \\,\\boldsymbol{i} + dy\\,\\boldsymbol{j} + dz\\,\\boldsymbol{k} \\\\
\n&=& \\ \\ \\ \\left(\\frac{\\partial x}{\\partial u} du + \\frac{\\partial x}{\\partial v} dv +\\frac{\\partial x}{\\partial w} dw \\right)\\boldsymbol{i} \\\\
\n&& + \\left(\\frac{\\partial y}{\\partial u} du + \\frac{\\partial y}{\\partial v} dv +\\frac{\\partial y}{\\partial w} dw \\right) \\boldsymbol{j} \\\\
\n&& + \\left(\\frac{\\partial z}{\\partial u} du + \\frac{\\partial z}{\\partial v} dv +\\frac{\\partial z}{\\partial w} dw \\right) \\boldsymbol{k} \\\\
\n&=& du\\, \\boldsymbol{e}_u + dv\\,\\boldsymbol{e}_v + dw\\, \\boldsymbol{e}_w
\n\\end{eqnarray}<\/p>\n

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

\\begin{eqnarray}
\n\\boldsymbol{e}_u&\\equiv& \\frac{\\partial x}{\\partial u} \\boldsymbol{i} + \\frac{\\partial y}{\\partial u} \\boldsymbol{j} + \\frac{\\partial z}{\\partial u} \\boldsymbol{k} \\\\
\n\\boldsymbol{e}_v&\\equiv& \\frac{\\partial x}{\\partial v} \\boldsymbol{i} + \\frac{\\partial y}{\\partial v} \\boldsymbol{j} + \\frac{\\partial z}{\\partial v} \\boldsymbol{k} \\\\
\n\\boldsymbol{e}_w&\\equiv& \\frac{\\partial x}{\\partial w} \\boldsymbol{i} + \\frac{\\partial y}{\\partial w} \\boldsymbol{j} + \\frac{\\partial z}{\\partial w} \\boldsymbol{k}
\n\\end{eqnarray}<\/p>\n

\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u4f53\u7a4d\u8981\u7d20 $dV$ \u306f3\u672c\u306e\u30d9\u30af\u30c8\u30eb $du\\, \\boldsymbol{e}_u,\\\u00a0 dv\\,\\boldsymbol{e}_v,\u00a0 \\,\u00a0 dw\\, \\boldsymbol{e}_w$ \u304b\u3089\u3064\u304f\u3089\u308c\u308b\u5e73\u884c6\u9762\u4f53\u306e\u4f53\u7a4d\u3068\u3057\u3066\u3082\u8868\u3055\u308c\u308b\u304b\u3089<\/p>\n

\\begin{eqnarray}
\ndV = dx\\,dy\\,dz = du\\,dv\\,dw \\, \\boldsymbol{e}_u \\cdot(\\boldsymbol{e}_v \\times\\boldsymbol{e}_w)
\n\\end{eqnarray}<\/p>\n

\u30d9\u30af\u30c8\u30eb\u306e\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306f\uff0c3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u3092\u4e26\u3079\u305f $3\\times 3$ \u884c\u5217\u306e\uff08\u8ee2\u7f6e\u884c\u5217\u3067\u3082\u53ef\uff09\u306e\u884c\u5217\u5f0f\u3068\u7b49\u3057\u3044\u3053\u3068\u306f\u3053\u3053<\/strong><\/span><\/a>\u3067\u308f\u304b\u3063\u3066\u3044\u308b\u306e\u3067<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{e}_u \\cdot(\\boldsymbol{e}_v \\times\\boldsymbol{e}_w)
\n&=& \\det \\begin{pmatrix}
\n\\frac{\\partial x}{\\partial u} & \\frac{\\partial y}{\\partial u} & \\frac{\\partial z}{\\partial u} \\\\
\n\\frac{\\partial x}{\\partial v} & \\frac{\\partial y}{\\partial v} & \\frac{\\partial z}{\\partial v} \\\\
\n\\frac{\\partial x}{\\partial w} & \\frac{\\partial y}{\\partial w} & \\frac{\\partial z}{\\partial w}
\n\\end{pmatrix} \\\\
\n&=&
\n\\det \\begin{pmatrix}
\n\\frac{\\partial x}{\\partial u} & \\frac{\\partial x}{\\partial v} & \\frac{\\partial x}{\\partial w} \\\\
\n\\frac{\\partial y}{\\partial u} & \\frac{\\partial y}{\\partial v} & \\frac{\\partial y}{\\partial w} \\\\
\n\\frac{\\partial z}{\\partial u} & \\frac{\\partial z}{\\partial v} & \\frac{\\partial z}{\\partial w}
\n\\end{pmatrix} \\\\
\n&=& \\frac{\\partial(x,y,z)}{\\partial(u,v,z)}
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c<\/p>\n

\\begin{eqnarray}
\ndV = dx\\,dy\\,dz = \\frac{\\partial(x,y,z)}{\\partial(u,v,z)} du\\,dv\\,dw
\n\\end{eqnarray}<\/p>\n

\u5186\u7b52\u5ea7\u6a19\u3068\u30e4\u30b3\u30d3\u30a2\u30f3<\/h4>\n

\\begin{eqnarray}
\nx &=& \\rho \\cos\\phi \\\\
\ny &=& \\rho \\sin\\phi \\\\
\nz &=& z
\n\\end{eqnarray}<\/p>\n

$$\\frac{\\partial(x,y,z)}{\\partial(\\rho, \\phi, z)} =\\frac{\\partial(x,y)}{\\partial(\\rho, \\phi)} = \\rho$$<\/p>\n

\u3057\u305f\u304c\u3063\u3066<\/p>\n

$$dV = \\frac{\\partial(x,y,z)}{\\partial(\\rho, \\phi, z)} d\\rho\\,d\\phi\\, dz = \\rho\\, d\\rho\\,d\\phi\\, dz$$<\/p>\n

\u6975\u5ea7\u6a19\u3068\u30e4\u30b3\u30d3\u30a2\u30f3<\/h4>\n

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

$$\\frac{\\partial(x,y,z)}{\\partial(r, \\theta, \\phi)} = r^2 \\sin\\theta$$<\/p>\n

\u3057\u305f\u304c\u3063\u3066<\/p>\n

$$dV = \\frac{\\partial(x,y,z)}{\\partial(r, \\theta, \\phi)} dr\\, d\\theta\\, d\\phi = r^2 dr\\, \\sin\\theta\\, d\\theta\\, d\\phi$$<\/p>\n

Maxima-Jupyter \u3067\u30e4\u30b3\u30d3\u30a2\u30f3\u306e\u8a08\u7b97<\/h3>\n

 <\/p>\n

\n
<\/div>\n
\n
\n

2\u6b21\u5143\u6975\u5ea7\u6a19\u3068\u30e4\u30b3\u30d3\u30a2\u30f3<\/h4>\n

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

\n
\n
In\u00a0[1]:<\/div>\n
\n
\n
\n
x<\/span>:<\/span> r<\/span> *<\/span> cos<\/span>(<\/span>phi<\/span>)<\/span>;\r\ny<\/span>:<\/span> r<\/span> *<\/span> sin<\/span>(<\/span>phi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}\\cos \\varphi\\,r\\]<\/div>\n<\/div>\n
\n
Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}\\sin \\varphi\\,r\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
Maxima \u306e jacobian()<\/code> \u306f\u300c\u95a2\u6570\u884c\u5217\u300d\u3067\u3042\u308a\uff0c\u300c\u95a2\u6570\u884c\u5217\u5f0f\u300d\u306b\u3059\u308b\u306b\u306f\u3055\u3089\u306b determinant()<\/code> \u3092\u3068\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[2]:<\/div>\n
\n
\n
\n
determinant<\/span>(<\/span>jacobian<\/span>([<\/span>x<\/span>,y<\/span>]<\/span>, [<\/span>r<\/span>,phi<\/span>]))<\/span>$\r\ntrigsimp<\/span>(<\/span>%<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{4}$}r\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
3\u6b21\u5143\u5186\u7b52\u5ea7\u6a19\u3068\u30e4\u30b3\u30d3\u30a2\u30f3<\/h4>\n
\\begin{eqnarray}
\nx &=& \\rho \\cos\\phi \\\\
\ny &=& \\rho \\sin\\phi \\\\
\nz &=& z
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[3]:<\/div>\n
\n
\n
\n
x<\/span>:<\/span> rho<\/span> *<\/span> cos<\/span>(<\/span>phi<\/span>)<\/span>;\r\ny<\/span>:<\/span> rho<\/span> *<\/span> sin<\/span>(<\/span>phi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}\\cos \\varphi\\,\\rho\\]<\/div>\n<\/div>\n
\n
Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}\\sin \\varphi\\,\\rho\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[4]:<\/div>\n
\n
\n
\n
determinant<\/span>(<\/span>jacobian<\/span>([<\/span>x<\/span>,y<\/span>,z<\/span>]<\/span>, [<\/span>rho<\/span>,phi<\/span>,z<\/span>]))<\/span>$\r\ntrigsimp<\/span>(<\/span>%<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{8}$}\\rho\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
3\u6b21\u5143\u6975\u5ea7\u6a19\u3068\u30e4\u30b3\u30d3\u30a2\u30f3<\/h4>\n
\\begin{eqnarray}
\nx &=& r\\sin\\theta \\cos\\phi \\\\
\ny &=& r\\sin\\theta \\sin\\phi \\\\
\nz &=& r \\cos\\theta
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[5]:<\/div>\n
\n
\n
\n
x<\/span>:<\/span> r<\/span> *<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span> *<\/span> cos<\/span>(<\/span>phi<\/span>)<\/span>;\r\ny<\/span>:<\/span> r<\/span> *<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span> *<\/span> sin<\/span>(<\/span>phi<\/span>)<\/span>;\r\nz<\/span>:<\/span> r<\/span> *<\/span> cos<\/span>(<\/span>theta<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{9}$}\\cos \\varphi\\,r\\,\\sin \\vartheta\\]<\/div>\n<\/div>\n
\n
Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{10}$}\\sin \\varphi\\,r\\,\\sin \\vartheta\\]<\/div>\n<\/div>\n
\n
Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}r\\,\\cos \\vartheta\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[6]:<\/div>\n
\n
\n
\n
determinant<\/span>(<\/span>jacobian<\/span>([<\/span>x<\/span>,y<\/span>,z<\/span>]<\/span>, [<\/span>r<\/span>,theta<\/span>,phi<\/span>]))<\/span>$\r\ntrigsimp<\/span>(<\/span>%<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{13}$}r^2\\,\\sin \\vartheta\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
SymPy \u3066\u3099\u30e4\u30b3\u30d2\u3099\u30a2\u30f3\u306e\u8a08\u7b97<\/h3>\n
\u30e2\u30b8\u30e5\u30fc\u30eb\u306e import<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[1]:<\/div>\n
\n
\n
\n
from<\/span> sympy.abc<\/span> import<\/span> *<\/span> \r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
2\u6b21\u5143\u6975\u5ea7\u6a19<\/h4>\n
\\begin{eqnarray}
\nx &=& r \\cos\\theta\\\\
\ny &=& r \\sin\\theta
\n\\end{eqnarray}<\/p>\n
\u30e4\u30b3\u30d3\u884c\u5217\uff08\u95a2\u6570\u884c\u5217\uff09<\/h5>\n
jacobian<\/code> \u306f\u95a2\u6570\u884c\u5217<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[2]:<\/div>\n
\n
\n
\n
xy<\/span> =<\/span> Matrix<\/span>([<\/span>r<\/span>*<\/span>cos<\/span>(<\/span>theta<\/span>),<\/span> r<\/span>*<\/span>sin<\/span>(<\/span>theta<\/span>)])<\/span>\r\nrtheta<\/span> =<\/span> Matrix<\/span>([<\/span>r<\/span>,<\/span> theta<\/span>])<\/span>\r\n\r\njaco<\/span> =<\/span> xy<\/span>.<\/span>jacobian<\/span>(<\/span>rtheta<\/span>)<\/span>\r\njaco<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[2]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\theta \\right)} & – r \\sin{\\left(\\theta \\right)}\\\\\\sin{\\left(\\theta \\right)} & r \\cos{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30e4\u30b3\u30d3\u30a2\u30f3\uff08\u95a2\u6570\u884c\u5217\u5f0f\uff09<\/h5>\n
$$\\frac{\\partial (x, y)}{\\partial (r, \\theta)} \\equiv
\n\\begin{vmatrix}
\n\\frac{\\partial x}{\\partial r} & \\frac{\\partial x}{\\partial \\theta}\\\\
\n\\frac{\\partial y}{\\partial r} & \\frac{\\partial y}{\\partial \\theta}\\\\
\n\\end{vmatrix}
\n= \\frac{\\partial x}{\\partial r} \\frac{\\partial y}{\\partial \\theta} \u2013 \\frac{\\partial y}{\\partial r}\\frac{\\partial x}{\\partial \\theta}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[3]:<\/div>\n
\n
\n
\n
det<\/span>(<\/span>jaco<\/span>)<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[3]:<\/div>\n
$\\displaystyle r$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
3\u6b21\u5143\u5186\u7b52\u5ea7\u6a19<\/h4>\n
\\begin{eqnarray}
\nx &=& \\rho \\cos\\phi \\\\
\ny &=& \\rho \\sin\\phi \\\\
\nz &=& z
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30e4\u30b3\u30d3\u884c\u5217\uff08\u95a2\u6570\u884c\u5217\uff09<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[4]:<\/div>\n
\n
\n
\n
xyz<\/span> =<\/span> Matrix<\/span>([<\/span>rho<\/span>*<\/span>cos<\/span>(<\/span>phi<\/span>),<\/span> rho<\/span>*<\/span>sin<\/span>(<\/span>phi<\/span>),<\/span> z<\/span>])<\/span>\r\nrhophiz<\/span> =<\/span> Matrix<\/span>([<\/span>rho<\/span>,<\/span> phi<\/span>,<\/span> z<\/span>])<\/span>\r\n\r\njaco<\/span> =<\/span> xyz<\/span>.<\/span>jacobian<\/span>(<\/span>rhophiz<\/span>)<\/span>\r\njaco<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[4]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\phi \\right)} & – \\rho \\sin{\\left(\\phi \\right)} & 0\\\\\\sin{\\left(\\phi \\right)} & \\rho \\cos{\\left(\\phi \\right)} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30e4\u30b3\u30d3\u30a2\u30f3\uff08\u95a2\u6570\u884c\u5217\u5f0f\uff09<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[5]:<\/div>\n
\n
\n
\n
det<\/span>(<\/span>jaco<\/span>)<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[5]:<\/div>\n
$\\displaystyle \\rho$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
3\u6b21\u5143\u6975\u5ea7\u6a19<\/h4>\n
\\begin{eqnarray}
\nx &=& r\\sin\\theta \\cos\\phi \\\\
\ny &=& r\\sin\\theta \\sin\\phi \\\\
\nz &=& r \\cos\\theta
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30e4\u30b3\u30d3\u884c\u5217\uff08\u95a2\u6570\u884c\u5217\uff09<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[6]:<\/div>\n
\n
\n
\n
xyz<\/span> =<\/span> Matrix<\/span>([<\/span>r<\/span>*<\/span>sin<\/span>(<\/span>theta<\/span>)<\/span>*<\/span>cos<\/span>(<\/span>phi<\/span>),<\/span> r<\/span>*<\/span>sin<\/span>(<\/span>theta<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>phi<\/span>),<\/span> r<\/span>*<\/span>cos<\/span>(<\/span>theta<\/span>)])<\/span>\r\nrthetaphi<\/span> =<\/span> Matrix<\/span>([<\/span>r<\/span>,<\/span> theta<\/span>,<\/span> phi<\/span>])<\/span>\r\n\r\njaco<\/span> =<\/span> xyz<\/span>.<\/span>jacobian<\/span>(<\/span>rthetaphi<\/span>)<\/span>\r\njaco<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[6]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}\\sin{\\left(\\theta \\right)} \\cos{\\left(\\phi \\right)} & r \\cos{\\left(\\phi \\right)} \\cos{\\left(\\theta \\right)} & – r \\sin{\\left(\\phi \\right)} \\sin{\\left(\\theta \\right)}\\\\\\sin{\\left(\\phi \\right)} \\sin{\\left(\\theta \\right)} & r \\sin{\\left(\\phi \\right)} \\cos{\\left(\\theta \\right)} & r \\sin{\\left(\\theta \\right)} \\cos{\\left(\\phi \\right)}\\\\\\cos{\\left(\\theta \\right)} & – r \\sin{\\left(\\theta \\right)} & 0\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30e4\u30b3\u30d3\u30a2\u30f3\uff08\u95a2\u6570\u884c\u5217\u5f0f\uff09<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[7]:<\/div>\n
\n
\n
\n
det<\/span>(<\/span>jaco<\/span>)<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[7]:<\/div>\n
$\\displaystyle r^{2} \\sin{\\left(\\theta \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2338,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2344","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\/2344","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=2344"}],"version-history":[{"count":26,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2344\/revisions"}],"predecessor-version":[{"id":9194,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2344\/revisions\/9194"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2338"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2344"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}