{"id":2615,"date":"2022-03-26T14:38:11","date_gmt":"2022-03-26T05:38:11","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2615"},"modified":"2025-07-15T17:24:11","modified_gmt":"2025-07-15T08:24:11","slug":"%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e5%a0%b4%e3%81%ae%e7%a9%8d%e5%88%86","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e5%a0%b4%e3%81%ae%e7%a9%8d%e5%88%86\/","title":{"rendered":"\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7a4d\u5206"},"content":{"rendered":"<p><!--more--><\/p>\n<p>\u6700\u5f8c\u306b\uff0c\u5fae\u5206\u5f62\u3067\u66f8\u304b\u308c\u3066\u3044\u308b\u30de\u30af\u30b9\u30a6\u30a7\u30eb\u65b9\u7a0b\u5f0f\u3092\u7a4d\u5206\u5f62\u3067\u7406\u89e3\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\uff0c\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7a4d\u5206\u516c\u5f0f\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n<h3>1\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206<\/h3>\n<p>1\u5909\u6570\u95a2\u6570 $f(x)$ \u306e\u7a4d\u5206\u306f<\/p>\n<p>$$\\int_a^b f(x) \\,dx$$<\/p>\n<p>\u3068\u66f8\u304f\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n<h4>\u7dda\u7a4d\u5206<\/h4>\n<p>$x$ \u8ef8\u306b\u305d\u3063\u305f\u30b9\u30ab\u30e9\u30fc\u95a2\u6570 $\\psi(x, y, z)$ \u306e\u7a4d\u5206\u3082<\/p>\n<p>$$\\int_a^b \\psi(x, y, z) \\,dx$$<\/p>\n<p>\u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u308b\u3002\u3053\u308c\u3092\u4e00\u822c\u5316\u3057\u3066\uff0c\u30d1\u30e9\u30e1\u30fc\u30bf $v$ \u3067\u30d1\u30e9\u30e1\u30c8\u30e9\u30a4\u30ba\u3055\u308c\u308b\u66f2\u7dda $C: \\, \\boldsymbol{r}(v)$ \u306b\u6cbf\u3063\u305f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7dda\u7a4d\u5206<\/strong><\/span>\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3059\u3002<\/p>\n<p>$$\\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} \\psi\\bigl(\\boldsymbol{r}\\left(v\\right)\\bigr)\\, dv$$<\/p>\n<p>\u3053\u3053\u3067\uff0c$\\boldsymbol{r}_1 = \\boldsymbol{r}(v_1), \\ \\boldsymbol{r}_2=\\boldsymbol{r}(v_2)$ \u3067\u3042\u308b\u3002<\/p>\n<p>\u7279\u306b\uff0c\u3042\u308b\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}$ \u306e\uff0c\u3053\u306e\u66f2\u7dda\u306b\u6cbf\u3063\u305f\u7dda\u7a4d\u5206\u306e\u5834\u5408\u306f\uff0c\u3053\u306e\u66f2\u7dda\u306e\u63a5\u30d9\u30af\u30c8\u30eb $\\boldsymbol{t}$ \u306f<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\boldsymbol{t} &amp;\\equiv&amp; \\frac{d\\boldsymbol{r}}{dv}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3067\u3042\u308b\u304b\u3089\uff0c$\\psi \\Rightarrow \\boldsymbol{a}\\cdot \\boldsymbol{t}$ \u3068\u3057\u3066\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} \\left(\\boldsymbol{a}\\cdot \\boldsymbol{t}\\right) \\, dv<br \/>\n&amp;=&amp; \\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} \\left(\\boldsymbol{a}\\cdot\\frac{d\\boldsymbol{r}}{dv}\\right)\\, dv \\\\<br \/>\n&amp;\\equiv&amp; \\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} \\boldsymbol{a}\\cdot d\\boldsymbol{r}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u66f8\u304f\u3002\u7279\u306b\uff0c\u9589\u66f2\u7dda $C$ \u306b\u6cbf\u3063\u305f1\u5468\u7dda\u7a4d\u5206\u306f\uff08$\\boldsymbol{r} = \\boldsymbol{r}_1$ \u304b\u3089\u540c\u3058 $\\boldsymbol{r} = \\boldsymbol{r}_1$ \u307e\u3067\u306a\u306e\u3067\u3042\u308b\u304c\uff09<\/p>\n<p>$$\\oint_C \\boldsymbol{a}\\cdot d\\boldsymbol{r}$$<\/p>\n<p>\u3068\u66f8\u304f\u3002<\/p>\n<h3>\u591a\u91cd\u7a4d\u5206\uff1a\u591a\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206<\/h3>\n<p>\u96fb\u78c1\u6c17\u5b66\u3067\u306f1\u5909\u6570\u95a2\u6570\u3060\u3051\u3067\u306a\u304f\uff0c\u4e00\u822c\u306b\u7a7a\u9593\u5ea7\u6a19 $x, y, z$ \u304a\u3088\u3073\u6642\u9593\u5ea7\u6a19 $y$ \u306e4\u3064\u306e\u5909\u6570\u306b\u4f9d\u5b58\u3059\u308b\u95a2\u6570\uff08\u591a\u5909\u6570\u95a2\u6570\uff0c\u30b9\u30ab\u30e9\u30fc\u5834\u30fb\u30d9\u30af\u30c8\u30eb\u5834\uff09\u306e\u7a4d\u5206\u304c\u51fa\u3066\u304f\u308b\u3002\u3053\u306e\u3088\u3046\u306a\u591a\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206\u306e\u3053\u3068\u3092<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u591a\u91cd\u7a4d\u5206<\/strong><\/span>\u3068\u3044\u3046\u3002<\/p>\n<h4>2\u91cd\u7a4d\u5206\u30fb\u9762\u7a4d\u5206<\/h4>\n<p>2\u5909\u6570\u95a2\u6570 $f(x, y)$ \u306e\u9818\u57df $D$ \u3067\u306e\u7a4d\u5206<\/p>\n<p>$$\\iint_D f(x, y) \\\u00a0 dx\\, dy$$<\/p>\n<p>\u306f\uff0c\u591a\u91cd\u7a4d\u5206\u306e\u3046\u3061\u3067\u7279\u306b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>2\u91cd\u7a4d\u5206<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u3002<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u9762\u7a4d\u8981\u7d20<\/strong><\/span> $dx\\, dy$ \u306f\u4f55\u3082 $xy$ \u5e73\u9762\u4e0a\u306b\u9650\u3089\u306a\u3044\u306e\u3067\u4e00\u822c\u306b $ dx\\, dy \\ \\rightarrow \\ dS$ \u3068\u66f8\u3044\u3066<\/p>\n<p>$$\\iint_D f(x, y) \\\u00a0 dS$$<\/p>\n<p>\u3092\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u9762\u7a4d\u5206<\/strong><\/span>\u300d\u3068\u3044\u3046\u3002\uff08\u300c\u9762\u7a4d\u5206\u300d\u3068\u3059\u3079\u304d\u304b\u300c\u9762\u7a4d\u7a4d\u5206\u300d\u3068\u3059\u3079\u304d\u306f\u60a9\u3080\u3068\u3053\u308d\u3002Wikipedia \u3067\u3082\u5fae\u5999\u306b\u3076\u308c\u3066\u3044\u308b\u3068\u601d\u308f\u308c\uff0c2\u91cd\u7a4d\u5206\u306f\u300c<a href=\"https:\/\/ja.wikipedia.org\/wiki\/%E9%9D%A2%E7%A9%8D%E5%88%86\" target=\"_blank\" rel=\"noopener\"><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u9762\u7a4d\u5206<\/strong><\/span><\/a>\u300d\u306a\u306e\u306b3\u91cd\u7a4d\u5206\u306f\u300c<a href=\"https:\/\/ja.wikipedia.org\/wiki\/%E4%BD%93%E7%A9%8D%E7%A9%8D%E5%88%86\" target=\"_blank\" rel=\"noopener\"><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4f53\u7a4d\u7a4d\u5206<\/strong><\/span><\/a>\u300d\u3068\u3057\u3066\u3044\u308b\u3002\uff09<\/p>\n<p>\u5177\u4f53\u7684\u306a\u8a08\u7b97\u306f\uff0c\u305f\u3068\u3048\u3070\u9818\u57df $D$ \u304c<br \/>\n$$D: x_1 \\leq x \\leq x_2, \\ y_1 \\leq y \\leq y_2$$<br \/>\n\u3067\u3042\u308c\u3070<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\iint_D f(x, y) \\ dS &amp;=&amp;\\iint_D f(x, y) \\\u00a0 dx\\, dy \\\\<br \/>\n&amp;=&amp; \\int_{y_1}^{y_2} \\left\\{\\int_{x_1}^{x_2} f(x, y) \\,dx \\right\\} dy<br \/>\n\\end{eqnarray}<\/p>\n<p>\u306e\u3088\u3046\u306b1\u5909\u6570\u305a\u3064\u7a4d\u5206\u3057\u3066\u3044\u304f\u3002\u307e\u305f\uff0c\u7a4d\u5206\u5909\u6570\u3068\u305d\u306e\u7a4d\u5206\u7bc4\u56f2\uff08\u4e0b\u7aef\u30fb\u4e0a\u7aef\uff09\u3092\u660e\u78ba\u306b\u3059\u308b\u305f\u3081\u306b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8a18\u6cd5\u3082\u4f7f\u308f\u308c\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\iint_D f(x, y) \\ dS &amp;=&amp;\\iint_D f(x, y) \\\u00a0 dx\\, dy \\\\<br \/>\n&amp;=&amp; \\int_{y_1}^{y_2} dy \\int_{x_1}^{x_2} dx\u00a0 \\, f(x, y)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u305f\u5834\u5408\uff0c\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u7a4d\u5206\u8a18\u53f7 $\\int$ \u3068 $d$ \u7a4d\u5206\u5909\u6570 \u3067\u631f\u307e\u308c\u3066\u3044\u306a\u3044\u306e\u3067\u6163\u308c\u306a\u3044\u3046\u3061\u306f\u306a\u3093\u3068\u306a\u304f\u843d\u3061\u7740\u304b\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u7a4d\u5206\u306f\u5fc5\u305a\u88ab\u7a4d\u5206\u95a2\u6570 $f(x, y)$ \u306b\u8fd1\u3044\u65b9\u306e\u7a4d\u5206\u5909\u6570\uff08\u4e0a\u8a18\u306e\u5834\u5408\u306f $dx$\uff09\u304b\u3089\u7a4d\u5206\u3059\u308b\u3053\u3068\u3002<\/p>\n<h4>2\u6b21\u5143\u6975\u5ea7\u6a19\u7cfb\u306e\u9762\u7a4d\u8981\u7d20<\/h4>\n<p>\u9818\u57df $D$ \u306b\u3088\u3063\u3066\u306f\uff0c\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y$ \u3088\u308a\u3082\u4ee5\u4e0b\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u6975\u5ea7\u6a19 $r, \\, \\varphi$ \u3092\u4f7f\u3063\u3066\u7a4d\u5206\u3057\u305f\u307b\u3046\u304c\u4fbf\u5229\u306a\u5834\u5408\u304c\u3042\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\nx &amp;=&amp; r \\cos\\varphi \\\\<br \/>\ny &amp;=&amp; r \\sin\\varphi<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u306e\u5834\u5408\u306e\u9762\u7a4d\u8981\u7d20 $dS$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002\uff08\u300c<a href=\"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\/\">\u53c2\u8003\uff1a\u6975\u5ea7\u6a19\u306b\u3088\u308b2\u91cd\u7a4d\u5206\u3068\u30e4\u30b3\u30d3\u30a2\u30f3<\/a>\u300d\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\u3002\uff09<\/p>\n<p>$$dS = dx\\, dy \\Rightarrow r\\, dr \\,d\\varphi$$<\/p>\n<h4>3\u91cd\u7a4d\u5206\u30fb\u4f53\u7a4d\u7a4d\u5206<\/h4>\n<p>3\u5909\u6570\u95a2\u6570 $f(x, y, z)$ \u306e\u9818\u57df $V$ \u3067\u306e\u7a4d\u5206<\/p>\n<p>$$\\iiint_V f(x, y, z) \\ dx\\, dy\\, dz$$<\/p>\n<p>\u306f\uff0c\u591a\u91cd\u7a4d\u5206\u306e\u3046\u3061\u3067\u7279\u306b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>3\u91cd\u7a4d\u5206<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u3002<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4f53\u7a4d\u8981\u7d20<\/strong><\/span> $dx\\, dy\\, dz$ \u306f\u4e00\u822c\u306b$ dx\\, dy\\, dz \\ \\rightarrow \\ dV$ \u3068\u66f8\u3044\u3066<\/p>\n<p>$$\\iiint_V f(x, y, z) \\, dV$$<\/p>\n<p>\u3092\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4f53\u7a4d\u7a4d\u5206<\/strong><\/span>\u300d\u3068\u3044\u3046\u3002<\/p>\n<p>\u5177\u4f53\u7684\u306a\u8a08\u7b97\u306f\uff0c\u305f\u3068\u3048\u3070\u9818\u57df $V$ \u304c<br \/>\n$$V: x_1 \\leq x \\leq x_2, \\ y_1 \\leq y \\leq y_2, \\ z_1 \\leq z \\leq z_2$$<br \/>\n\u3067\u3042\u308c\u3070<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\iiint_V f(x, y, z) \\, dV &amp;=&amp;\\iiint_V f(x, y, z) \\\u00a0 dx\\, dy\\,\u00a0 dz\\\\<br \/>\n&amp;=&amp; \\int_{z_1}^{z_2} \\left\\{ \\int_{y_1}^{y_2} \\left\\{\\int_{x_1}^{x_2} f(x, y, z) \\,dx \\right\\} dy\\right\\} dz<br \/>\n\\end{eqnarray}<\/p>\n<p>\u306e\u3088\u3046\u306b1\u5909\u6570\u305a\u3064\u7a4d\u5206\u3057\u3066\u3044\u304f\u3002\u307e\u305f\uff0c\u7a4d\u5206\u5909\u6570\u3068\u305d\u306e\u7a4d\u5206\u7bc4\u56f2\uff08\u4e0b\u7aef\u30fb\u4e0a\u7aef\uff09\u3092\u660e\u78ba\u306b\u3059\u308b\u305f\u3081\u306b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8a18\u6cd5\u3082\u4f7f\u308f\u308c\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\iiint_V f(x, y, z) \\\u00a0 dV &amp;=&amp;\\iiint_V f(x, y, z) \\ dx\\, dy\\,\u00a0 dz\\\\<br \/>\n&amp;=&amp; \\int_{z_1}^{z_2} dz \\int_{y_1}^{y_2} dy \\int_{x_1}^{x_2} dx\u00a0 \\, f(x, y, z)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u305f\u5834\u5408\uff0c\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u7a4d\u5206\u8a18\u53f7 $\\int$ \u3068 $d$ \u7a4d\u5206\u5909\u6570 \u3067\u631f\u307e\u308c\u3066\u3044\u306a\u3044\u306e\u3067\u6163\u308c\u306a\u3044\u3046\u3061\u306f\u306a\u3093\u3068\u306a\u304f\u843d\u3061\u7740\u304b\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u7a4d\u5206\u306f\u5fc5\u305a\u88ab\u7a4d\u5206\u95a2\u6570 $f(x, y, z)$ \u306b\u8fd1\u3044\u65b9\u306e\u7a4d\u5206\u5909\u6570\uff08\u4e0a\u8a18\u306e\u5834\u5408\u306f $dx$\uff09\u304b\u3089\u7a4d\u5206\u3059\u308b\u3053\u3068\u3002<\/p>\n<h4>3\u6b21\u5143\u5186\u7b52\u5ea7\u6a19\u7cfb\u306e\u4f53\u7a4d\u8981\u7d20<\/h4>\n<p>\u9818\u57df $V$ \u306b\u3088\u3063\u3066\u306f\uff0c\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y, z$ \u3088\u308a\u3082\u4ee5\u4e0b\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u5186\u7b52\u5ea7\u6a19 $\\rho, \\, \\varphi, \\,z$ \u3092\u4f7f\u3063\u3066\u7a4d\u5206\u3057\u305f\u307b\u3046\u304c\u4fbf\u5229\u306a\u5834\u5408\u304c\u3042\u308b\u3002\uff08$r$ \u306f3\u6b21\u5143\u6975\u5ea7\u6a19\u306e\u5834\u5408\u306b\u3068\u3063\u3066\u304a\u304f\u3002\uff09<\/p>\n<p>\\begin{eqnarray}<br \/>\nx &amp;=&amp; \\rho \\cos\\varphi \\\\<br \/>\ny &amp;=&amp; \\rho \\sin\\varphi \\\\<br \/>\nz &amp;=&amp; z<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u306e\u5834\u5408\u306e\u4f53\u7a4d\u8981\u7d20 $dV$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002\uff08\u300c<a href=\"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\/\">\u53c2\u8003\uff1a\u6975\u5ea7\u6a19\u306b\u3088\u308b2\u91cd\u7a4d\u5206\u3068\u30e4\u30b3\u30d3\u30a2\u30f3<\/a>\u300d\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\u3002\uff09<\/p>\n<p>$$dV = dx\\, dy\\,dz \\Rightarrow \\rho\\, d\\rho \\,d\\varphi\\, dz$$<\/p>\n<h4>3\u6b21\u5143\u6975\u5ea7\u6a19\u7cfb\u306e\u4f53\u7a4d\u8981\u7d20<\/h4>\n<p>\u9818\u57df $V$ \u306b\u3088\u3063\u3066\u306f\uff0c\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y, z$ \u3088\u308a\u3082\u4ee5\u4e0b\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u6975\u5ea7\u6a19 $r, \\,\\theta,\\, \\varphi$ \u3092\u4f7f\u3063\u3066\u7a4d\u5206\u3057\u305f\u307b\u3046\u304c\u4fbf\u5229\u306a\u5834\u5408\u304c\u3042\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\nx &amp;=&amp; r \\sin\\theta \\cos\\varphi \\\\<br \/>\ny &amp;=&amp; r \\sin\\theta\u00a0 \\sin\\varphi \\\\<br \/>\nz &amp;=&amp; r \\cos\\theta<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u306e\u5834\u5408\u306e\u4f53\u7a4d\u8981\u7d20 $dV$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n<p>$$dV = dx\\, dy\\, dz \\Rightarrow r^2 dr\\, \\sin\\theta\\, d\\theta\\, d\\varphi$$<\/p>\n<p>&nbsp;<\/p>\n<h3 id=\"yui_3_17_2_1_1648272986075_1410\">\u52fe\u914d grad \u306e\u7dda\u7a4d\u5206<\/h3>\n<p>1\u5909\u6570\u95a2\u6570 \\( f(x) \\) \u306b\u3064\u3044\u3066\uff0c\\( x \\) \u304b\u3089 \\(x + dx \\) \u307e\u3067\u306e\u7121\u9650\u5c0f\u5909\u5316\u5206\u306f\uff08 \\(dx \\) \u306e1\u6b21\u307e\u3067\u3068\u308b\u3068\u3057\u3066\uff09<br \/>\n$$df \\equiv f(x + dx) -f(x) \\simeq \\frac{df}{dx} dx$$<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\uff0c\\( x = x_1 \\) \u304b\u3089 \\( x = x_2 \\) \u307e\u3067\u306e \\( f(x) \\) \u306e\u5909\u5316\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7a4d\u5206\u3067\u66f8\u3051\u308b\u3002<\/p>\n<p>$$ \\int_{x_1}^{x_2} \\frac{df}{dx} dx = f(x_2)\u00a0 -f(x_1)$$<\/p>\n<p>\u591a\u5909\u6570\u95a2\u6570\u3067\u3042\u308b\u30b9\u30ab\u30e9\u30fc\u5834 \\( \\psi(x, y, z)\u00a0 \\) \u306e\u5834\u5408\u3082\u540c\u69d8\u306b<br \/>\n\\begin{eqnarray}<br \/>\nd\\psi &amp;\\equiv&amp; \\psi(x+dx, y+dy, z + dz)\u00a0 -\\psi(x, y, z) \\\\<br \/>\n&amp;\\simeq&amp; \\frac{\\partial \\psi}{\\partial x} dx + \\frac{\\partial \\psi}{\\partial y} dy + \\frac{\\partial \\psi}{\\partial z} dz \\\\<br \/>\n&amp;=&amp; (\\nabla \\psi)\\cdot \\frac{d\\boldsymbol{r}}{dv} \\,dv \\end{eqnarray}<br \/>\n\u3053\u3053\u3067 $$\\frac{d\\boldsymbol{r}}{dv}<br \/>\n= \\frac{dx}{dv} \\,\\boldsymbol{i} + \\frac{dy}{dv} \\,\\boldsymbol{j} + \\frac{dz}{dv} \\,\\boldsymbol{k}$$<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\uff0c\u7a7a\u9593\u306e\u70b9 \\( \\boldsymbol{r}_1 \\) \u304b\u3089\u70b9 \\( \\boldsymbol{r}_2 \\) \u307e\u3067\u306e \\( \\psi(\\boldsymbol{r}) \\) \u306e\u5909\u5316\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7dda\u7a4d\u5206\u3067\u66f8\u3051\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} d\\psi &amp;=&amp; \\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} \\nabla \\psi\\cdot \\frac{d\\boldsymbol{r}}{dv}\\, dv \\\\<br \/>\n&amp;=&amp; \\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} \\nabla \\psi\\cdot d\\boldsymbol{r} \\\\<br \/>\n&amp;=&amp; \\psi(\\boldsymbol{r}_2) -\\psi(\\boldsymbol{r}_1)<br \/>\n\\end{eqnarray}<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-8925\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7711-640x589.png\" alt=\"\" width=\"480\" height=\"442\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7711-640x589.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7711-300x276.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7711-750x691.png 750w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7711.png 1151w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/>$d\\boldsymbol{r}$ \u306f $\\boldsymbol{r}_1$ \u304b\u3089 $\\boldsymbol{r}_2$ \u307e\u3067\u3092\u7d50\u3076\u66f2\u7dda $C$ \u306b\u305d\u3063\u305f\u5fae\u5c0f\u63a5\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u304b\u3089\uff0c\u3053\u306e\u7dda\u7a4d\u5206\u306e\u5024\u306f\u66f2\u7dda $C$ \u306e\u53d6\u308a\u65b9\u306b\u4f9d\u5b58\u3057\u305d\u3046\u306a\u3082\u306e\u3060\u304c\uff0c\u7d50\u679c\u306f\uff0c$\\psi(\\boldsymbol{r}_2) -\\psi(\\boldsymbol{r}_1)$ \u3068\u306a\u308a\uff0c2\u70b9\u306e\u4f4d\u7f6e\u306e\u307f\u3067\u6c7a\u307e\u308a\uff0c2\u70b9\u3092\u7d50\u3076\u66f2\u7dda\u306e\u53d6\u308a\u65b9\u306b\u306f\u4f9d\u5b58\u3057\u306a\u3044\u3002\u3053\u306e\u7406\u7531\u306f\uff0c\u3082\u3046\u5c11\u3057\u5f8c\u3067\u30b9\u30c8\u30fc\u30af\u30b9\u306e\u5b9a\u7406\u3092\u77e5\u308b\u3068\u7406\u89e3\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n<h3>\u767a\u6563 div \u306e\u4f53\u7a4d\u7a4d\u5206\u3068\u9589\u66f2\u9762\u3092\u8cab\u304f\u5916\u5411\u304d\u306e\u5168\u6d41\u901f\uff1a\u30ac\u30a6\u30b9\u306e\u5b9a\u7406<\/h3>\n<p>\u30d9\u30af\u30c8\u30eb\u5834 \\( \\boldsymbol{a} \\) \u304c\uff0c\u4f55\u304b\uff08\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u304b\u71b1\u91cf\u3068\u304b\uff09\u306e\u5358\u4f4d\u9762\u7a4d\u3042\u305f\u308a\u306e\u6d41\u308c\u3092\u8868\u3059\u3082\u306e\uff08\u6d41\u675f\u5bc6\u5ea6\u30d9\u30af\u30c8\u30eb\uff0c\u3053\u306e\u5834\u5408\u306e\u300c\u5bc6\u5ea6\u300d\u3068\u306f\u300c\u9762\u5bc6\u5ea6\u300d\u306e\u3053\u3068\uff09\u3068\u60f3\u50cf\u3057\u3088\u3046\u3002<\/p>\n<p>\u5fae\u5c0f\u9762\u7a4d \\( dS \\) \u3092\u5782\u76f4\u306b\u8cab\u3044\u3066\u6d41\u308c\u308b\u91cf\uff08\u6d41\u675f\uff09\u306f\uff0c\\( dS \\) \u306b\u5782\u76f4\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\( \\boldsymbol{n} \\) \u3092\u4f7f\u3063\u3066 \\( \\boldsymbol{a}\\cdot\\boldsymbol{n}\\\u00a0 dS \\) \u3068\u306a\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c\u9762\u7a4d \\( S \\) \u3092\u5782\u76f4\u306b\u8cab\u3044\u3066\u6d41\u308c\u308b\u91cf\uff08\u5916\u5411\u304d\u306e\u5168\u6d41\u901f\uff09\u306f<br \/>\n$$ \\iint_S \\boldsymbol{a}\\cdot\\boldsymbol{n} \\ dS $$<\/p>\n<p>\u30ac\u30a6\u30b9\u306e\u5b9a\u7406\u3068\u306f\uff0c\u30d9\u30af\u30c8\u30eb\u5834\u306e\u767a\u6563\u306e\u4f53\u7a4d\u306b\u3064\u3044\u3066\u306e\u7a4d\u5206\u306b\u95a2\u3059\u308b\u5b9a\u7406\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<br \/>\n$$ \\iiint_V \\nabla\\cdot \\boldsymbol{a}\\, dV = \\iint_S \\boldsymbol{a}\\cdot\\boldsymbol{n} \\ dS $$ \u3053\u3053\u3067\uff0c\\( S \\) \u306f\u4f53\u7a4d \\( V \\) \u3092\u56f2\u3080\u9589\u66f2\u9762\u3067\u3042\u308b\u3002<\/p>\n<p>\u7279\u306b\uff0c\uff08\u4ee5\u4e0b\u306b\u8ff0\u3079\u308b\u30b9\u30c8\u30fc\u30af\u30b9\u306e\u5b9a\u7406\u3067\u306f\u9589\u66f2\u7dda\u306b\u6cbf\u3063\u305f1\u5468\u7a4d\u5206\u306f\u666e\u901a\u306e\u7dda\u7a4d\u5206\u306e\u4ee3\u308f\u308a\u306b $\\displaystyle\\oint$ \u3092\u4f7f\u3046\u306e\u3067\uff0c\u3053\u308c\u306b\u5023\u3063\u3066\uff09<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u9589\u66f2\u9762\u4e0a\u3067\u306e\u9762\u7a4d\u5206<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u3092\u5f37\u8abf\u3059\u308b\u305f\u3081\u306b\u30ac\u30a6\u30b9\u306e\u5b9a\u7406\u306e\u53f3\u8fba\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3079\u304d\u3060\u3068\u601d\u3046\u3002<\/p>\n<p>$$ \\iiint_V \\nabla\\cdot \\boldsymbol{a}\\, dV = {\\color{red}\\int\\!\\!\\!\\!\\!\\!\\bigcirc\\!\\!\\!\\!\\!\\!\\int}_{\\!\\!S}\\\u00a0 \\boldsymbol{a}\\cdot\\boldsymbol{n} \\ dS $$<\/p>\n<p>\uff08\u3067\u3082 $\\displaystyle {\\color{red}\\int\\!\\!\\!\\!\\!\\!\\bigcirc\\!\\!\\!\\!\\!\\!\\int}$ \u3092\u66f8\u304f\u306e\u306f\u5927\u5909\u306a\u3093\u3060\u3088\u306a\u3041\uff0cMathjax \u3067\u306f <code>\\userpackage{esint}<\/code> \u304c\u4f7f\u3048\u306a\u3044\u3088\u3046\u3067\uff0c<code>$\\oiint$<\/code> \u3084 <code>$\\varoiint$<\/code> \u304c\u52b9\u304b\u306a\u3044\u3002\uff09<\/p>\n<p>\u30ac\u30a6\u30b9\u306e\u5b9a\u7406\u3092\u8a00\u8449\u3067\u8868\u3059\u3068\uff0c\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><u><b>\u30d9\u30af\u30c8\u30eb\u5834\u306e\u767a\u6563\u306e\u4f53\u7a4d\u7a4d\u5206\uff08\u3044\u308f\u3070\u4f53\u7a4d $V$ \u5185\u306e\u5168\u767a\u6563\uff09\u306f\uff0c\u305d\u306e\u4f53\u7a4d\u3092\u56f2\u3080\u9589\u66f2\u9762\u3092\u8cab\u304f\u5916\u5411\u304d\u306e\u5168\u6d41\u901f\u3067\u3042\u308b<\/b><\/u><\/span>\u300d\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n<p>\u8a3c\u660e\u306f\u5225\u30da\u30fc\u30b8\u300c<a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e5%a0%b4%e3%81%ae%e7%a9%8d%e5%88%86\/%e5%8f%82%e8%80%83%ef%bc%9a%e3%82%ac%e3%82%a6%e3%82%b9%e3%81%ae%e5%ae%9a%e7%90%86%e3%81%ae%e8%a8%bc%e6%98%8e\/\">\u53c2\u8003\uff1a\u30ac\u30a6\u30b9\u306e\u5b9a\u7406\u306e\u8a3c\u660e<\/a>\u300d\u3067\u3002<\/p>\n<p>\u3061\u306a\u307f\u306b\uff0c\u300c\u7dda\u7a4d\u5206\u300d\u306f\u300c\u7dda\u7a4d\u5206\u300d\u3067\u554f\u984c\u306a\u3044\u304c\uff0c\u3042\u308b\u91cf \\( f(x,y,z) \\) \u306e\u4f53\u7a4d \\( V \\) \u5185\u3067\u306e\u7a4d\u5206<br \/>\n$$ \\iiint_{V}\u00a0 f\\; dx\\,dy\\,dz, \\quad\u00a0 \\iiint_{V} f \\;dV$$ \u306a\u3069\u3092\u300c\u4f53\u7a4d\u5206\u300d\u3068\u547c\u3076\u3079\u304d\u304b\u300c\u4f53\u7a4d\u7a4d\u5206\u300d\u3068\u547c\u3076\u3079\u304d\u304b\u60a9\u3080\u3002\u540c\u69d8\u306b\uff0c\u3042\u308b\u91cf \\( f(x,y,z) \\) \u306e\u9762\u7a4d \\( S \\) \u5185\u3067\u306e\u7a4d\u5206<br \/>\n$$ \\iint_{S} f\\ dx\\, dy, \\quad \\iint_{S} f \\ dS$$ \u306a\u3069\u3092\u300c\u9762\u7a4d\u5206\u300d\u3068\u547c\u3076\u3079\u304d\u304b\u300c\u9762\u7a4d\u7a4d\u5206\u300d\u3068\u547c\u3076\u3079\u304d\u304b\u3082\u5c11\u3057\u60a9\u3080\u3002<\/p>\n<p>&nbsp;<\/p>\n<h3>\u56de\u8ee2 rot \u306e\u9762\u7a4d\u5206\u3068\u9589\u66f2\u7dda\u306b\u6cbf\u3063\u305f\u5faa\u74b0\uff1a\u30b9\u30c8\u30fc\u30af\u30b9\u306e\u5b9a\u7406<\/h3>\n<p>\u30d9\u30af\u30c8\u30eb\u5834 \\(\\boldsymbol{a} \\) \u306e\u9589\u66f2\u7dda \\(C \\) \u306b\u6cbf\u3063\u305f\u6210\u5206\u3092\u3061\u3087\u3046\u30691\u5468\uff0c\u7dda\u7a4d\u5206\u3059\u308b\u3053\u3068\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3002<br \/>\n$$\\oint_{C} \\boldsymbol{a}\\cdot \\frac{d\\boldsymbol{r}}{d\\ell}\\, d\\ell\u00a0 = \\oint_{C} \\boldsymbol{a}\\cdot d\\boldsymbol{r} $$ \u3053\u306e\u91cf\u3092\u9589\u66f2\u7dda\uff08\u3042\u308b\u3044\u306f\u30eb\u30fc\u30d7\uff09\\( C \\) \u306e\u307e\u308f\u308a\u306e\u5faa\u74b0\u3068\u547c\u3076\u3002<\/p>\n<p>&nbsp;<\/p>\n<p>\u30b9\u30c8\u30fc\u30af\u30b9\u306e\u5b9a\u7406\u3068\u306f\uff0c\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2\u306e\u5782\u76f4\u6210\u5206\u306e\u9762\u7a4d\u5206\u306b\u95a2\u3059\u308b\u5b9a\u7406\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<br \/>\n$$ \\iint_{S} (\\nabla\\times \\boldsymbol{a})\\cdot \\boldsymbol{n}\\ dS = \\oint_{C} \\boldsymbol{a}\\cdot d\\boldsymbol{r} $$ \u3053\u3053\u3067\uff0c\\( S \\) \u306f\u9589\u66f2\u7dda \\( C \\) \u3092\u7e01\u3068\u3059\u308b\u4efb\u610f\u306e\u66f2\u9762\u3067\u3042\u308b\u3002\u5358\u306b\u9589\u66f2\u7dda \\( C \\) \u3092\u7e01\u3068\u3059\u308b\u66f2\u9762\u3068\u3044\u3063\u3066\u3082\u7121\u6570\u306b\u3042\u308b\u3068\u601d\u3046\u304c\uff0c\u5b9f\u306f\u5de6\u8fba\u306e\u9762\u7a4d\u5206\u306f\u305d\u306e\u66f2\u9762\u306e\u53d6\u308a\u65b9\u306b\u3088\u3089\u306a\u3044\u3053\u3068\u3082\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n<p>\u30b9\u30c8\u30fc\u30af\u30b9\u306e\u5b9a\u7406\u3092\u8a00\u8449\u3067\u8868\u3059\u3068\uff0c\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><u><b>\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2\u306e\u5782\u76f4\u6210\u5206\u306e\u9762\u7a4d\u5206\u306f\uff0c\u305d\u306e\u9762\u7a4d\u306e\u5883\u754c\u3067\u3042\u308b\u9589\u66f2\u7dda\u306b\u6cbf\u3063\u305f\u5faa\u74b0\u3067\u3042\u308b<\/b><\/u><\/span>\u300d\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n<p id=\"yui_3_17_2_1_1648272986075_1415\">\u3084\u306f\u308a\u8a3c\u660e\u306f\u5225\u30da\u30fc\u30b8\u300c<a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e5%a0%b4%e3%81%ae%e7%a9%8d%e5%88%86\/%e5%8f%82%e8%80%83%ef%bc%9a%e3%82%b9%e3%83%88%e3%83%bc%e3%82%af%e3%82%b9%e3%81%ae%e5%ae%9a%e7%90%86%e3%81%ae%e8%a8%bc%e6%98%8e\/\">\u53c2\u8003\uff1a\u30b9\u30c8\u30fc\u30af\u30b9\u306e\u5b9a\u7406\u306e\u8a3c\u660e<\/a>\u300d\u3067\u3002<\/p>\n<h4>\u53c2\u8003\uff1a\u52fe\u914d\u306e\u7dda\u7a4d\u5206\u304c\u7d4c\u8def\u306e\u53d6\u308a\u65b9\u306b\u4f9d\u5b58\u3057\u306a\u3044\u3053\u3068<\/h4>\n<p>\u3055\u3066\uff0c\u3082\u3057\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}$ \u304c\u3042\u308b\u30b9\u30ab\u30e9\u30fc\u95a2\u6570 $\\psi$ \u306e\u52fe\u914d grad \u3067\u66f8\u3051\u308b\u3068\u3057\u3088\u3046\u3002\u305d\u3046\u3059\u308b\u3068\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\boldsymbol{a} &amp;\\equiv&amp; \\nabla \\psi \\\\<br \/>\n\\therefore \\ \\ \\iint_{S} (\\nabla\\times (\\nabla \\psi))\\cdot \\boldsymbol{n}\\ dS &amp;=&amp; \\oint_{C} (\\nabla \\psi)\\cdot d\\boldsymbol{r}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u30d9\u30af\u30c8\u30eb\u89e3\u6790\u306e\u6052\u7b49\u5f0f<\/p>\n<p>$$\\nabla\\times (\\nabla \\psi) = \\boldsymbol{0}$$<\/p>\n<p>\u3092\u4f7f\u3046\u3068\uff0c<\/p>\n<p>$$\\oint_{C} (\\nabla \\psi)\\cdot d\\boldsymbol{r}\u00a0 = 0$$<\/p>\n<p>\u3053\u306e\u3053\u3068\u304b\u3089\uff0c\u30b9\u30ab\u30e9\u30fc\u5834 $\\psi(\\boldsymbol{r})$ \u306e\u52fe\u914d $\\nabla \\psi$ \u306e\u7dda\u7a4d\u5206\u304c\u7d4c\u8def\u306b\u3088\u3089\u305a\u306b\uff0c\u7aef\u70b9\u3067\u306e\u5024\u3060\u3051\u3067\u6c7a\u307e\u308b\u3053\u3068\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u308f\u304b\u308b\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-8936\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7712-640x640.png\" alt=\"\" width=\"480\" height=\"480\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7712-640x640.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7712-300x300.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7712-150x150.png 150w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7712-1536x1536.png 1536w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7712-750x750.png 750w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7712.png 1748w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/p>\n<p>\u56f3\u306e\u3088\u3046\u306b\uff0c$\\boldsymbol{r}_1$ \u304b\u3089\u70b9 $\\boldsymbol{r}_2$ \u307e\u3067\u306e\u66f2\u7dda ${\\color{red}{C_1}}$ \u3068\uff0c$\\boldsymbol{r}_2$ \u304b\u3089\u70b9 $\\boldsymbol{r}_1$ \u307e\u3067\u306e\u66f2\u7dda ${\\color{blue}{C_2}}$ \u304b\u3089\u306a\u308b\u9589\u66f2\u7dda $C$ \u3092\u8003\u3048\u308b\u3068\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n0 = \\oint_{C} (\\nabla \\psi)\\cdot d\\boldsymbol{r} &amp;=&amp;<br \/>\n{\\color{red}{\\int_{C_1}}} (\\nabla \\psi)\\cdot d\\boldsymbol{r} +<br \/>\n{\\color{blue}{\\int_{C_2}}} (\\nabla \\psi)\\cdot d\\boldsymbol{r} \\\\<br \/>\n\\therefore\\ \\ {\\color{red}{\\int_{C_1}}} (\\nabla \\psi)\\cdot d\\boldsymbol{r} &amp;=&amp;<br \/>\n-{\\color{blue}{\\int_{C_2}}} (\\nabla \\psi)\\cdot d\\boldsymbol{r} \\\\<br \/>\n\\therefore\\ \\ {\\color{red}{\\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2}}} (\\nabla \\psi)\\cdot d\\boldsymbol{r} &amp;=&amp;<br \/>\n-{\\color{blue}{\\int_{\\boldsymbol{r}_2}^{\\boldsymbol{r}_1}}} (\\nabla \\psi)\\cdot d\\boldsymbol{r} \\\\<br \/>\n\\therefore \\ \\<br \/>\n\\therefore\\ \\ {\\color{red}{\\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2}}} (\\nabla \\psi)\\cdot d\\boldsymbol{r} &amp;=&amp;<br \/>\n{\\color{blue}{\\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2}}} (\\nabla \\psi)\\cdot d\\boldsymbol{r}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u306e\u3053\u3068\u306f\uff0c$\\boldsymbol{r}_1$ \u304b\u3089 $\\boldsymbol{r}_2$ \u307e\u3067\u306e\u7dda\u7a4d\u5206\u306e\u5024\u304c\u66f2\u7dda ${\\color{red}{C_1}}$ \u306b\u6cbf\u3063\u3066\u8a08\u7b97\u3057\u3066\u3082\uff0c\u66f2\u7dda ${\\color{blue}{C_2}}$ \u306b\u6cbf\u3063\u3066\u8a08\u7b97\u3057\u3066\u3082\u540c\u3058\u3068\u3044\u3046\u3053\u3068\uff0c\u3064\u307e\u308a\u7dda\u7a4d\u5206 $\\displaystyle \\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} (\\nabla \\psi)\\cdot d\\boldsymbol{r}$ \u306f $\\boldsymbol{r}_1$ \u304b\u3089 $\\boldsymbol{r}_2$ \u307e\u3067\u306e\u7d4c\u8def\u306e\u53d6\u308a\u65b9\u306b\u4f9d\u5b58\u305b\u305a\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} d\\psi &amp;=&amp; \\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} \\nabla \\psi\\cdot \\frac{d\\boldsymbol{r}}{dv}\\, dv \\\\<br \/>\n&amp;=&amp; \\int_{\\boldsymbol{r}_1}^{\\boldsymbol{r}_2} \\nabla \\psi\\cdot d\\boldsymbol{r} \\\\<br \/>\n&amp;=&amp; \\psi(\\boldsymbol{r}_2) -\\psi(\\boldsymbol{r}_1)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u306e\u3088\u3046\u306b\uff0c\u7aef\u70b9\u3067\u306e\u5024\u3060\u3051\u3067\u66f8\u3051\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2561,"menu_order":8,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2615","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\/2615","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=2615"}],"version-history":[{"count":41,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2615\/revisions"}],"predecessor-version":[{"id":10547,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2615\/revisions\/10547"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2561"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2615"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}