{"id":9264,"date":"2022-12-07T09:45:26","date_gmt":"2022-12-07T00:45:26","guid":{"rendered":"http:\/\/web.tohoku.ac.jp\/yusa\/?p=9264"},"modified":"2022-12-23T18:25:16","modified_gmt":"2022-12-23T09:25:16","slug":"helmholtztheorem","status":"publish","type":"post","link":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/2022\/helmholtztheorem\/","title":{"rendered":"\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u306e\u5b9a\u7406"},"content":{"rendered":"\n<p>\u96fb\u78c1\u6c17\u5b66I\u3067\u3061\u3087\u3053\u3061\u3087\u3053\u540d\u524d\u304c\u51fa\u3066\u304d\u307e\u3057\u305f\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u306e\u5b9a\u7406\u306b\u3064\u3044\u3066\u306e\u3001\u3068\u308a\u3042\u3048\u305a\u306e\u8aac\u660e\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u203b\\vec\u304c\u898b\u3065\u3089\u3044\u306e\u3067\\boldsymbol\u306b\u7f6e\u63db\u4e2d\u3002\uff12\u3064\u306e\u30d9\u30af\u30c8\u30eb\u8868\u8a18\u304c\u6df7\u5728\u3057\u3066\u3044\u307e\u3059\u304c\u3001\u6c17\u306b\u3057\u306a\u3044\u3067\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n\n\n\n<h2 class=\"wp-block-heading\">\u6982\u8981<\/h2>\n\n\n\n<p>\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u306e\u5b9a\u7406\u3068\u306f\u3001\u3042\u308b\u9818\u57df<span class=\"wp-katex-eq\" data-display=\"false\">V<\/span>\u306b\u304a\u3044\u3066\u4efb\u610f\u306e\u30d9\u30af\u30c8\u30eb<span class=\"wp-katex-eq\" data-display=\"false\">\\boldsymbol{X}<\/span>\u306f\u3001<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\boldsymbol{X}(\\boldsymbol{r})=\\frac{1}{4\\pi}\\int_V \\left( \\nabla&#039; \\cdot  \\boldsymbol{X}(\\boldsymbol{r^{\\prime}}) +\\nabla&#039; \\times \\boldsymbol{X}(\\boldsymbol{r&#039;})  \\times \\right) \\nabla&#039;  \\frac{1}{\\left| \\boldsymbol{r}-\\boldsymbol{r&#039;} \\right|}dV&#039; -\\frac{1}{4\\pi}\\int_S \\left( \\boldsymbol{n} \\cdot \\boldsymbol{X}(\\boldsymbol{r&#039;}) +\\boldsymbol{n} \\times \\boldsymbol{X}(\\boldsymbol{r&#039;})  \\times \\right) \\nabla&#039;\\frac{1}{\\left|\\boldsymbol{r}-\\boldsymbol{r&#039;}\\right|}dS&#039; <\/span>\n\n\n\n<p>\u3068\u8868\u3055\u308c\u308b\u3001\u3068\u3044\u3046\u3082\u306e\u3067\u3059<span id='easy-footnote-1-9264' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/2022\/helmholtztheorem\/#easy-footnote-bottom-1-9264' title='\u3053\u3053\u3067&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;\\boldsymbol{r}&lt;\/span&gt;\u3068&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;\\boldsymbol{r^{\\prime}}\u306f[latex]V&lt;\/span&gt;\u5185\u306e\u70b9\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u3001\u307e\u305f&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;S&lt;\/span&gt;\u306f&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;V&lt;\/span&gt;\u306e\u8868\u9762\u3067\u3059\u3002'><sup>1<\/sup><\/a><\/span>\uff08\u4ed6\u306e\u8868\u73fe\u3082\u3042\u308b\u3068\u601d\u3044\u307e\u3059\u304c\uff09\u3002<\/p>\n\n\n\n<p>\u306a\u306e\u3067\u3001<span class=\"wp-katex-eq\" data-display=\"false\">\\boldsymbol{X}<\/span>\u304c\u305d\u306e\u6e90\u304b\u3089\u306e\u8ddd\u96e2\u3068\u5171\u306b\u5341\u5206\u306b\u306f\u3084\u304f\u6e1b\u8870\u3059\u308b\u306e\u3067\u3042\u308c\u3070\u3001\u9818\u57df<span class=\"wp-katex-eq\" data-display=\"false\">V<\/span>\u3092\u5341\u5206\u306b\u5927\u304d\u304f\u53d6\u308b\u3053\u3068\u306b\u3088\u308a\u9762\u7a4d\u5206\u306e\u9805\u306f\u6d88\u3048\u3066\u304f\u308c\u3066 <span id='easy-footnote-2-9264' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/2022\/helmholtztheorem\/#easy-footnote-bottom-2-9264' title='\u3064\u307e\u308a\u3001&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;S&lt;\/span&gt;\u306e\u8868\u9762\u7a4d\u304c\u5897\u5927\u3059\u308b\u5ea6\u5408\u3044\u306b\u6bd4\u3079\u3066\u9762\u7a4d\u5206\u306e\u4e2d\u306e\u9805\u306e\u6e1b\u5c11\u306e\u5ea6\u5408\u3044\u304c\u65e9\u3044\u306e\u3067\u3042\u308c\u3070\u3001\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002'><sup>2<\/sup><\/a><\/span>\u3001<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\boldsymbol{X}(\\boldsymbol{r})=\\frac{1}{4\\pi}\\int_V \\left( \\nabla&#039; \\cdot \\boldsymbol{X}(\\boldsymbol{r^{\\prime}}) +\\nabla&#039; \\times \\boldsymbol{X}(\\boldsymbol {r&#039;})  \\times \\right) \\nabla&#039;  \\frac{1}{\\left| \\boldsymbol{r}- \\boldsymbol{r&#039;} \\right|}dV&#039;  <\/span>\n\n\n\n<p>\u3068\u66f8\u3051\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u3001\u53f3\u8fba\u7b2c\uff11\u9805\u306b\u3064\u3044\u3066\u306f<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla \\left( \\frac{\\nabla&#039; \\cdot \\boldsymbol{X}(\\boldsymbol{r&#039;})}{\\left| \\boldsymbol{r}- \\boldsymbol{r&#039;} \\right| } \\right) =\\left( \\nabla&#039;\\cdot \\boldsymbol{X}(\\boldsymbol{r&#039;})\\right)\\nabla\\frac{1}{\\left| \\boldsymbol{r}-\\boldsymbol{r&#039;} \\right|} + \\frac{1}{\\left| \\boldsymbol{r}- \\boldsymbol{r&#039;} \\right|}\\nabla \\left( \\nabla&#039;\\cdot \\boldsymbol{X}(\\boldsymbol{r&#039;})\\right)= - \\left( \\nabla&#039;\\cdot \\boldsymbol{X}(\\boldsymbol{r&#039;})\\right)\\nabla&#039;\\frac{1}{\\left| \\boldsymbol{r}-\\boldsymbol{r&#039;} \\right|} <\/span>\n\n\n\n<p>\u3068\u306a\u308a\u3001\u7b2c\uff12\u9805\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306e\u5909\u5f62\u304c\u53ef\u80fd\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\boldsymbol{X}(\\boldsymbol{r})=-\\nabla \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\nabla&#039; \\cdot  \\boldsymbol{X}(\\boldsymbol{r^{\\prime}})}{ \\left| \\boldsymbol{r}-\\boldsymbol{r&#039;} \\right|  }\\right) dV&#039; \\right\\} + \\nabla \\times \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\nabla&#039; \\times  \\boldsymbol{X}(\\boldsymbol{r^{\\prime}})}{ \\left| \\boldsymbol{r}-\\boldsymbol{r&#039;} \\right|  }\\right) dV&#039; \\right\\}  <\/span>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059<span id='easy-footnote-3-9264' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/2022\/helmholtztheorem\/#easy-footnote-bottom-3-9264' title='&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;\\nabla&lt;\/span&gt;\u3068&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;\\nabla&amp;#039;&lt;\/span&gt;\u306e\u9055\u3044\u306b\u6ce8\u610f\u3002'><sup>3<\/sup><\/a><\/span>\u3002<\/p>\n\n\n\n<p>\u3053\u308c\u306f\u5373\u3061\u3001<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u3042\u308b\u30d9\u30af\u30c8\u30eb\u306f\u30b9\u30ab\u30e9\u30fc\u95a2\u6570\u306e\u52fe\u914d\u3068\u30d9\u30af\u30c8\u30eb\u95a2\u6570\u306e\u56de\u8ee2\u306e\u548c\u306b\u5206\u89e3\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/li><li>\u3042\u308b\u30d9\u30af\u30c8\u30eb\u3092\u4e00\u610f\u306b\u6c7a\u3081\u308b\u305f\u3081\u306b\u306f\u3001\u305d\u306e\u30d9\u30af\u30c8\u30eb\u306e\u767a\u6563\u3068\u56de\u8ee2\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/li><\/ul>\n\n\n\n<p>\u3068\u3044\u3046\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3068\u3044\u3048\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u8a3c\u660e<\/h2>\n\n\n\n<p>\u304a\u305d\u3089\u304f\u8272\u3005\u306a\u8a3c\u660e\u65b9\u6cd5\u304c\u3042\u308b\u3068\u601d\u3044\u307e\u3059\u304c\u3001\u305d\u308c\u306a\u308a\u306b\u6d41\u308c\u304c\u660e\u77ad\u306a\u65b9\u6cd5\u3067\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u30d9\u30af\u30c8\u30eb\u30b0\u30ea\u30fc\u30f3\u306e\u516c\u5f0f\u304b\u3089\u30b9\u30bf\u30fc\u30c8\u3059\u308b\u65b9\u6cd5<\/h3>\n\n\n\n<p>2\u3064\u306e\u4efb\u610f\u306e\u30d9\u30af\u30c8\u30eb<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{X}<\/span>\u3068<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{Y}<\/span>\u306b\u3064\u3044\u3066\u3001 \u30d9\u30af\u30c8\u30eb\u6052\u7b49\u5f0f<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla\\cdot\\left(\\vec{X}\\times\\nabla\\times\\vec{Y}\\right)=\\left( \\nabla\\times\\vec{X}\\right) \\cdot\\left( \\nabla\\times\\vec{Y}\\right) -\\vec{X}\\cdot \\left( \\nabla\\times\\nabla\\times\\vec{Y} \\right) <\/span>\n\n\n\n<p>\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304b\u3089\u3001\u3053\u308c\u3092\u9818\u57df<span class=\"wp-katex-eq\" data-display=\"false\">V&#039;<\/span>\u3067\u4f53\u7a4d\u5206\u3057\u3001\u3055\u3089\u306b\u30ac\u30a6\u30b9\u306e\u767a\u6563\u5b9a\u7406\u3092\u4f7f\u3046\u3053\u3068\u3067<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\int_V \\left[  \\left( \\nabla\\times\\vec{X}\\right) \\cdot\\left( \\nabla\\times\\vec{Y}\\right) - \\vec{X}\\cdot \\left( \\nabla\\times\\nabla\\times\\vec{Y} \\right) \\right]dV&#039;=\\int_S \\left(\\vec{X}\\times\\nabla\\times\\vec{Y}\\right)\\cdot\\vec{n}dS&#039; <\/span>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u304c\u51fa\u767a\u70b9\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u6700\u7d42\u7684\u306b<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{X}<\/span>\u304c<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{X}<\/span>\u306e\u767a\u6563\u3068\u56de\u8ee2\u306e\u4f53\u7a4d\u5206\u3067\u8868\u3055\u308c\u308b\u3088\u3046\u306b\u3057\u305f\u3044\u306e\u3067\u3001\u9069\u5f53\u306a\u5b9a\u30d9\u30af\u30c8\u30eb<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{a}<\/span><span id='easy-footnote-4-9264' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/2022\/helmholtztheorem\/#easy-footnote-bottom-4-9264' title='\u306a\u306e\u3067&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;\\nabla\\cdot\\vec{a}=\\nabla\\times\\vec{a}=0&lt;\/span&gt;\u3068\u306a\u308a\u307e\u3059\u3002'><sup>4<\/sup><\/a><\/span>\u3001<span class=\"wp-katex-eq\" data-display=\"false\">R=\\left|\\vec{r}-\\vec{r&#039;}\\right|<\/span>\u3092\u7528\u3044\u3066<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{Y}=\\frac{\\vec{a}}{R}<\/span>\u3068\u304a\u304d\u3001\u6700\u7d42\u7684\u306b\u4e21\u8fba\u3092<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{a}<\/span>\u3067\u5272\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n\n\n\n<p> <span class=\"wp-katex-eq\" data-display=\"false\">\\vec{Y}=\\frac{\\vec{a}}{R}<\/span>\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001\u4e0a\u8a18\u306e\u30d9\u30af\u30c8\u30eb\u30b0\u30ea\u30fc\u30f3\u306e\u516c\u5f0f\u306f<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\int_V \\left[  \\left( \\nabla\\times\\vec{X}\\right) \\cdot\\left( \\nabla\\times\\frac{\\vec{a}}{R}\\right) - \\vec{X}\\cdot \\left( \\nabla\\times\\nabla\\times \\frac{\\vec{a}}{R}\\right) \\right]dV&#039;=\\int_S \\left(\\vec{X}\\times\\nabla\\times \\frac{\\vec{a}}{R} \\right)\\cdot\\vec{n}dS&#039; <\/span>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30d9\u30af\u30c8\u30eb\u89e3\u6790\u306e\u516c\u5f0f<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{X}\\cdot\\left(\\vec{Y}\\times\\vec{Z}\\right)= \\vec{Z}\\cdot\\left(\\vec{X}\\times\\vec{Y}\\right) <\/span>\u3092\u4f7f\u3063\u3066<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{a}<\/span>\u304c\u5916\u306b\u51fa\u308b\u3088\u3046\u306b\u4e0a\u5f0f\u3092\u5f0f\u5909\u5f62\u3057\u3066\u3086\u304f\u3068\u3001\u4f53\u7a4d\u5206\u306e\u4e2d\u306f<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\"> \\left\\{ \\vec{a}\\cdot\\left(\\nabla&#039;\\times\\vec{X}\\times\\nabla&#039;\\frac{1}{R}\\right)\\right\\} +\\left\\{ \\nabla&#039;\\cdot\\left\\{ \\vec{X}\\left(\\vec{a}\\cdot\\nabla&#039;\\frac{1}{R}\\right)\\right\\} -\\left(\\vec{a}\\cdot\\nabla&#039;\\frac{1}{R}\\right)\\nabla&#039;\\cdot\\vec{X}  -\\vec{a}\\cdot\\vec{X}\\nabla&#039;^2\\frac{1}{R}\\right\\} + \\left\\{ \\vec{a}\\cdot\\left\\{ \\left( \\vec{n}\\times\\vec{X}\\right) \\times\\nabla&#039;\\frac{1}{R} \\right\\} \\right\\}<\/span>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059<span id='easy-footnote-5-9264' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/2022\/helmholtztheorem\/#easy-footnote-bottom-5-9264' title='\u975e\u5e38\u306b\u9762\u5012\u3067\u3059\u304c\u3002'><sup>5<\/sup><\/a><\/span>\u3002<\/p>\n\n\n\n<p>\u3053\u3053\u3067\u3001\uff12\u3064\u76ee\u306e{}\u306e\u4e2d\u306e\uff11\u3064\u76ee\u306e\u9805\u306b\u3064\u3044\u3066\u306f\u3001\u30ac\u30a6\u30b9\u306e\u767a\u6563\u5b9a\u7406\u3092\u4f7f\u3046\u3053\u3068\u3067\u4f53\u7a4d\u5206\u3092\u9762\u7a4d\u5206\u306b\u5909\u63db\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u307e\u305f\u3001\u540c\u3058\u304f\uff12\u3064\u76ee\u306e{}\u306e\u4e2d\u306e\uff13\u3064\u76ee\u306e\u9805\u306f\u3001<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\int_V \\vec{a}\\cdot\\vec{X}\\nabla&#039;^2\\frac{1}{R} dV&#039;=\\vec{a}\\cdot\\left(-4\\pi\\vec{X}(\\vec{r})\\right)<\/span>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059<span id='easy-footnote-6-9264' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/2022\/helmholtztheorem\/#easy-footnote-bottom-6-9264' title=' &lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;\\delta(\\vec{r}-\\vec{r&amp;#039;})=-\\frac{1}{4\\pi}\\nabla^2\\frac{1}{|\\vec{r}-\\vec{r&amp;#039;}|}&lt;\/span&gt;\u3092\u4f7f\u3044\u307e\u3057\u305f\u3002'><sup>6<\/sup><\/a><\/span>\u3002<\/p>\n\n\n\n<p>\u4ee5\u4e0a\u3092\u307e\u3068\u3081\u3066\u3001\u5171\u901a\u306e<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{a}<\/span>\u3067\u5272\u308b\u3068\u3001\u6700\u521d\u306b\u66f8\u3044\u305f\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u306e\u516c\u5f0f\u304c\u5f97\u3089\u308c\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u30c7\u30eb\u30bf\u95a2\u6570\u304b\u3089\u30b9\u30bf\u30fc\u30c8\u3059\u308b\u65b9\u6cd5<\/h3>\n\n\n\n<p>\u4e0a\u3067\u3082\u4f7f\u3063\u305f<span class=\"wp-katex-eq\" data-display=\"false\">\\delta(\\vec{r}-\\vec{r&#039;})=-\\frac{1}{4\\pi}\\nabla^2\\frac{1}{|\\vec{r}-\\vec{r&#039;}|}<\/span>\u3092\u7528\u3044\u308b\u3068<span id='easy-footnote-7-9264' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/2022\/helmholtztheorem\/#easy-footnote-bottom-7-9264' title='\u306a\u305c\u3053\u3046\u66f8\u3051\u308b\u304b\u306f\u30b0\u30ea\u30d5\u30a3\u30b9\u96fb\u78c1\u6c17\u5b66\u306e1.5\u53c2\u7167'><sup>7<\/sup><\/a><\/span>\u3001<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{X}(\\vec{r})=\\int_v\\vec{X}(\\vec{r&#039;})\\left( -\\frac{1}{4\\pi}\\nabla^2\\frac{1}{|\\vec{r}-\\vec{r&#039;}|} \\right) dV&#039; <\/span>\n\n\n\n<p>\u3053\u308c\u3092<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla\\frac{1}{|\\vec{r}-\\vec{r&#039;}|}=-\\nabla&#039; \\frac{1}{|\\vec{r}-\\vec{r&#039;}|} <\/span>\n\n\n\n<p>\u3068\u30d9\u30af\u30c8\u30eb\u89e3\u6790\u306e\u516c\u5f0f\u3092\u5229\u7528\u3057\u3066\u304c\u308a\u304c\u308a\u5f0f\u5909\u5f62\u3057\u3066\u3044\u3051\u3070\u3088\u3044\u3002<\/p>\n\n\n\n<p>\u306e\u3060\u304c\u3061\u3087\u3063\u3068\u8abf\u3079\u305f\u3089\u3053\u306e\u624b\u9806\u306b\u3064\u3044\u3066\u306f<a rel=\"noreferrer noopener\" aria-label=\"\u82f1\u8a9e\u7248Wikipedia\u306b\u8a73\u7d30\u306a\u8aac\u660e (\u65b0\u3057\u3044\u30bf\u30d6\u3067\u958b\u304f)\" href=\"https:\/\/en.wikipedia.org\/wiki\/Helmholtz_decomposition\" target=\"_blank\">\u82f1\u8a9e\u7248Wikipedia\u306b\u8a73\u7d30\u306a\u8aac\u660e<\/a>\u304c\u3042\u3063\u305f\u306e\u3067\u3001\u7565\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u96fb\u78c1\u6c17\u5b66I\u306e\u5185\u5bb9\u306b\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u306e\u5b9a\u7406\u3092\u9069\u7528<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u9759\u96fb\u5834\u306e\u5834\u5408<\/h3>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{E}(\\vec{r})=-\\nabla \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\nabla&#039; \\cdot  \\vec{E}(\\vec{r^{\\prime}})}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039;\\right\\}  + \\nabla \\times \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\nabla&#039; \\times  \\vec{E}(\\vec{r^{\\prime}})}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039; \\right\\}  <\/span>\n\n\n\n<p>\u306b\u3001\u9759\u96fb\u5834\u306b\u304a\u3051\u308bMaxwell\u65b9\u7a0b\u5f0f<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla\\cdot\\vec{E}=\\rho\/\\varepsilon<\/span>\n\n\n\n<p><\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla\\times\\vec{E}=0<\/span>\n\n\n\n<p>\u3092\u4ee3\u5165\u3059\u308b\u3068<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{E}(\\vec{r})=-\\nabla \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\rho(\\vec{r^{\\prime}})\/\\varepsilon}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039;  \\right\\}  + \\nabla \\times \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{0}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039; \\right\\}     <\/span>\n\n\n\n<p>\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{E}(\\vec{r})=-\\nabla\\phi<\/span>\n\n\n\n<p><\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\"> \\phi= \\frac{1}{4\\pi\\varepsilon} \\int_V \\frac{\\rho(\\vec{r^{\\prime}})}{ \\left| \\vec{r}-\\vec{r&#039;} \\right| }dV&#039;  <\/span>\n\n\n\n<p>\u3068\u306a\u308a\u3001\u96fb\u5834\u3068\u96fb\u6c17\u30b9\u30ab\u30e9\u30fc\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u95a2\u4fc2\u3092\u793a\u3059\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u9759\u78c1\u5834\u306e\u5834\u5408<\/h3>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{B}(\\vec{r})=-\\nabla \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\nabla&#039; \\cdot  \\vec{B}(\\vec{r^{\\prime}})}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right)dV&#039; \\right\\}  + \\nabla \\times \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\nabla&#039; \\times  \\vec{B}(\\vec{r^{\\prime}})}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039; \\right\\}  <\/span>\n\n\n\n<p>\u306b\u9759\u78c1\u5834\u306b\u304a\u3051\u308bMaxwell\u65b9\u7a0b\u5f0f<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla\\cdot\\vec{B}=0<\/span>\n\n\n\n<p><\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla\\times\\vec{B}=\\mu\\vec{j}<\/span>\n\n\n\n<p>\u3092\u4ee3\u5165\u3059\u308b\u3068<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{B}(\\vec{r})=-\\nabla \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{0}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039;  \\right\\}  + \\nabla \\times \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\mu\\vec{j}}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039; \\right\\}     <\/span>\n\n\n\n<p>\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{B}=\\nabla\\times\\vec{A}<\/span>\n\n\n\n<p><\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{A}= \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\mu\\vec{j}}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039; <\/span>\n\n\n\n<p>\u3068\u306a\u308a\u3001\u78c1\u675f\u5bc6\u5ea6\u3068\u78c1\u6c17\u30d9\u30af\u30c8\u30eb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u95a2\u4fc2\u3092\u793a\u3059\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6642\u9593\u5909\u52d5\u3059\u308b\u5834\u5408<\/h3>\n\n\n\n<p>\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u306e\u5b9a\u7406\u306b\u306f\u6642\u9593\u304c\u542b\u307e\u308c\u3066\u3044\u306a\u3044\u305f\u3081\u3001\u3042\u308b\u70b9\u3067\u306e\u60c5\u5831\u304c\u77ac\u6642\u306b\u5168\u4f53\u306b\u4f1d\u308f\u308b\u3053\u3068\u3092\u524d\u63d0\u3068\u3057\u3066\u3044\u307e\u3059\u3002\u306a\u306e\u3067\u3001\u5b9f\u306f\u7269\u7406\u7684\u306b\u306f\u3061\u3087\u3063\u3068\u6b63\u3057\u304f\u306a\u3044\u306e\u3067\u3059\u304c\u3001\u3068\u308a\u3042\u3048\u305a\u96fb\u5834\u306b\u95a2\u3059\u308bMaxwell\u65b9\u7a0b\u5f0f<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla\\cdot\\boldsymbol{E}=\\rho\/\\varepsilon<\/span>\n\n\n\n<p><\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla\\times\\boldsymbol{E}=-\\frac{\\partial \\boldsymbol{B}}{\\partial t}<\/span>\n\n\n\n<p>\u3092\u4ee3\u5165\u3059\u308b\u3068<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{E}(\\vec{r})=-\\nabla \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\rho(\\vec{r^{\\prime}})\/\\varepsilon}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039;  \\right\\}  + \\nabla \\times \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{-\\partial \\vec{B}(\\vec{r&#039;})\/\\partial t}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039; \\right\\}  = -\\nabla \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{\\rho(\\vec{r^{\\prime}})\/\\varepsilon}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039;  \\right\\}  - \\frac{\\partial}{\\partial t}\\nabla \\times \\left\\{ \\frac{1}{4\\pi} \\int_V\\left( \\frac{ \\vec{B}(\\vec{r}&#039;)}{ \\left| \\vec{r}-\\vec{r&#039;} \\right|  }\\right) dV&#039; \\right\\}     <\/span>\n\n\n\n<p>\u53f3\u8fba\u7b2c\uff11\u9805\u306f\u4e0a\u306e\u96fb\u6c17\u30b9\u30ab\u30e9\u30fc\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092\u4f7f\u3063\u3066<span class=\"wp-katex-eq\" data-display=\"false\">-\\nabla\\phi<\/span>\u3068\u66f8\u3051\u307e\u3059\u3002\u7b2c\uff12\u9805\u306b\u3064\u3044\u3066\u306f<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla \\times \\left\\{ \\int_V\\left( \\frac{ \\boldsymbol{B}(\\vec{r}&#039;)}{ \\left| \\vec{r}-\\vec{r&#039;} \\right| }\\right) dV&#039; \\right\\}=\\int_V \\nabla\\frac{1}{\\left| \\vec{r}-\\vec{r&#039;} \\right|}\\times\\vec{B}(\\vec{r&#039;})dV&#039; =-\\int_V \\nabla&#039;\\frac{1}{\\left| \\vec{r}-\\vec{r&#039;} \\right|}\\times\\vec{B}(\\vec{r&#039;})dV&#039; =\\int_V\\frac{\\nabla&#039;\\times\\vec{B}(\\vec{r&#039;})}{\\left| \\vec{r}-\\vec{r&#039;} \\right|}dV&#039;+\\int_V \\nabla&#039;\\times\\frac{\\vec{B}(\\vec{r&#039;})}{\\left| \\vec{r}-\\vec{r&#039;} \\right|}dV&#039;  <\/span>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059\u304c\u3001\u6700\u5f8c\u306e\u9805\u306f\u9762\u7a4d\u5206\u306b\u5909\u3048\u308b\u3053\u3068\u304c\u3067\u304d<span id='easy-footnote-8-9264' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/2022\/helmholtztheorem\/#easy-footnote-bottom-8-9264' title='\u3042\u307e\u308a\u76ee\u306b\u3059\u308b\u3053\u3068\u304c\u306a\u3044\u5f0f\u3067\u3059\u304c\u3001&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;\\nabla\\times\\vec{C}=\\left(\\frac{\\partial C_z}{\\partial y}-\\frac{\\partial C_y}{\\partial z}, \\frac{\\partial C_x}{\\partial z}-\\frac{\\partial C_z}{\\partial x}, \\frac{\\partial C_y}{\\partial x}-\\frac{\\partial C_x}{\\partial y} \\right)=\\left(\\nabla\\cdot(0,C_z,-C_y), \\nabla\\cdot(-C_z,0,C_z), \\nabla\\cdot(C_y,-C_x,0) \\right)&lt;\/span&gt;\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001\u4f53\u7a4d\u5206\u3057\u305f\u5f8c\u5404\u6210\u5206\u306b\u5bfe\u3057\u3066\u767a\u6563\u5b9a\u7406\u3092\u4f7f\u3044\u3046\u3068\u3001&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;\\vec{n}\\times C&lt;\/span&gt;\u306e\u8868\u9762\u7a4d\u5206\u304c\u51fa\u3066\u304d\u307e\u3059\u3002'><sup>8<\/sup><\/a><\/span>\u3001\u7a4d\u5206\u9818\u57df\u3092\u5341\u5206\u306b\u5927\u304d\u304f\u3059\u308b\u3053\u3068\u30670\u306b\u306a\u308a\u307e\u3059\u3002\u306a\u306e\u3067\u3001\u7d50\u5c40<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\"> \\int_V\\frac{\\nabla&#039;\\times\\vec{B}(\\vec{r&#039;})}{\\left| \\vec{r}-\\vec{r&#039;} \\right|}dV&#039; = \\int_V\\frac{\\mu\\vec{j}(\\vec{r&#039;})}{\\left| \\vec{r}-\\vec{r&#039;} \\right|}dV&#039; <\/span>\n\n\n\n<p><\/p>\n\n\n\n<p>\u3068\u306a\u308a\u3001\u3053\u308c\u306f\u4e0a\u306e\u30d9\u30af\u30c8\u30eb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306b\u4e00\u81f4\u3002\u5373\u3061\u3001<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\vec{E}=-\\nabla\\phi -\\frac{\\partial \\vec{A}}{\\partial t}<\/span>\n\n\n\n<p>\u304c\u5f97\u3089\u308c\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u78c1\u5834\u306b\u3064\u3044\u3066\u306f\u3001\u4e0a\u306e\u5f0f\u5909\u5f62\u306e\u9014\u4e2d\u306e\u5f0f\u3092\u4f7f\u3063\u3066<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\nabla\\times\\vec{A}=\\vec{B}=\\nabla\\times \\frac{1}{4\\pi}\\int_V\\frac{\\nabla&#039;\\times\\vec{B}(\\vec{r&#039;})}{\\left| \\vec{r}-\\vec{r&#039;} \\right|}dV&#039; <\/span>\n\n\n\n<p>\u306a\u306e\u3067\u3001\u305d\u306e\u3082\u306e\u3067\u3059<span id='easy-footnote-9-9264' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/2022\/helmholtztheorem\/#easy-footnote-bottom-9-9264' title='&lt;span class=&quot;wp-katex-eq&quot; data-display=&quot;false&quot;&gt;\\nabla\\times\\vec{B}=\\mu\\vec{j}+\\mu\\varepsilon\\frac{\\partial\\vec{E}}{\\partial t}&lt;\/span&gt;\u3092\u4ee3\u5165\u3057\u3066\u5f0f\u5909\u5f62\u3057\u3066\u3082\u3067\u304d\u308b\u3068\u601d\u3046\u304c\u3001\u3084\u3063\u305f\u3053\u3068\u306a\u3044\u3002'><sup>9<\/sup><\/a><\/span>\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n","protected":false},"excerpt":{"rendered":"<p>\u96fb\u78c1\u6c17\u5b66I\u3067\u3061\u3087\u3053\u3061\u3087\u3053\u540d\u524d\u304c\u51fa\u3066\u304d\u307e\u3057\u305f\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u306e\u5b9a\u7406\u306b\u3064\u3044\u3066\u306e\u3001\u3068\u308a\u3042\u3048\u305a\u306e\u8aac\u660e\u3067\u3059\u3002 \u203b\\vec\u304c\u898b\u3065\u3089\u3044\u306e\u3067\\boldsymbol\u306b\u7f6e\u63db\u4e2d\u3002\uff12\u3064\u306e\u30d9 &#8230; <\/p>\n","protected":false},"author":1,"featured_media":3055,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[97,209],"tags":[],"class_list":["post-9264","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-lecture","category-under-construction"],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/web.tohoku.ac.jp\/yusa\/wp-content\/uploads\/2019\/07\/maxwell.png?fit=275%2C255","_links":{"self":[{"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/posts\/9264","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/comments?post=9264"}],"version-history":[{"count":188,"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/posts\/9264\/revisions"}],"predecessor-version":[{"id":9479,"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/posts\/9264\/revisions\/9479"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/media\/3055"}],"wp:attachment":[{"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/media?parent=9264"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/categories?post=9264"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/web.tohoku.ac.jp\/yusa\/index.php\/wp-json\/wp\/v2\/tags?post=9264"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}