{"id":2605,"date":"2022-03-26T13:58:30","date_gmt":"2022-03-26T04:58:30","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2605"},"modified":"2024-10-28T10:48:06","modified_gmt":"2024-10-28T01:48:06","slug":"%e3%82%b9%e3%82%ab%e3%83%a9%e3%83%bc%e5%a0%b4%e3%83%bb%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e5%a0%b4%e3%81%ae%e5%be%ae%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%82%b9%e3%82%ab%e3%83%a9%e3%83%bc%e5%a0%b4%e3%83%bb%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e5%a0%b4%e3%81%ae%e5%be%ae%e5%88%86\/","title":{"rendered":"\u30b9\u30ab\u30e9\u30fc\u5834\u30fb\u30d9\u30af\u30c8\u30eb\u5834\u306e\u5fae\u5206"},"content":{"rendered":"

<\/p>\n

\u504f\u5fae\u5206\uff1a\u591a\u5909\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n

\u30d9\u30af\u30c8\u30eb\u89e3\u6790\uff08\u30d9\u30af\u30c8\u30eb\u306e\u5fae\u5206\uff09\u306b\u5411\u3051\u3066\uff0c\u591a\u5909\u6570\u95a2\u6570\u306e\u5fae\u5206\u3067\u3042\u308b\u504f\u5fae\u5206\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n

\uff081\u7de8\u6570\u95a2\u6570\u306e\u300c\u666e\u901a\u306e\u300d\uff09\u5fae\u5206\u306e\u307e\u3068\u3081<\/h4>\n

 <\/p>\n

\u9ad8\u6821\u3067\u7fd2\u3063\u305f\u300c\u5fae\u5206\u300d\u306f1\u5909\u6570\u95a2\u6570 \\(f(x)\\) \u306b\u5bfe\u3059\u308b\u5fae\u5206\uff1a
\n$$\\frac{df}{dx} \\equiv \\lim_{h\\rightarrow 0}\\frac{f(x+h) – f(x)}{h}$$<\/p>\n

\u6211\u3005\u306e\u4f4f\u3080\u4e16\u754c\u306f3\u6b21\u5143\u7a7a\u9593\u3067\u3042\u308a\uff0c\uff08\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u7cfb\u3092\u3068\u308b\u3068\uff09 \\(x, y, z\\) \u306e3\u3064\u306e\u7a7a\u9593\u5ea7\u6a19\uff08\u5909\u6570\uff09\u3067\u8a18\u8ff0\u3055\u308c\u308b\u3002\u307e\u305f\uff0c\u4e00\u822c\u306b\u7269\u7406\u91cf\u306f\u6642\u9593\u3068\u3068\u3082\u306b\u5909\u5316\u3059\u308b\u304b\u3089\uff0c\u6642\u9593\u5ea7\u6a19 \\(t\\) \u306b\u3082\u4f9d\u5b58\u3059\u308b\u3002\u3064\u307e\u308a\uff0c\u6211\u3005\u304c\u96fb\u78c1\u6c17\u5b66\u3067\u6271\u3046\u91cf\u306f \\(f(x, y, z, t)\\) \u306e\u3088\u3046\u306b\u4e00\u822c\u7684\u306b\u306f4\u3064\u306e\u5909\u6570\u306b\u4f9d\u5b58\u3059\u308b\u591a\u5909\u6570\u95a2\u6570\u3067\u3042\u308b<\/strong><\/span>\u3002\u591a\u5909\u6570\u95a2\u6570\u306e\u5fae\u5206\u3068\u3057\u3066\u300c\u504f\u5fae\u5206\u300d\u304c\u3042\u308b<\/strong><\/span>\u3002<\/p>\n

\u504f\u5fae\u5206\uff1a\u591a\u5909\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h4>\n

\u96fb\u78c1\u6c17\u5b66\u3067\u6271\u3046\u7269\u7406\u91cf\u306f\uff0c\u6700\u3082\u4e00\u822c\u306b\u306f3\u6b21\u5143\u7a7a\u9593\u306e\u5ea7\u6a19 \\(x, y, z\\) \u304a\u3088\u3073\u6642\u9593 \\(t\\) \u306e4\u3064\u306e\u5909\u6570\u306b\u4f9d\u5b58\u3057\u305f\u591a\u5909\u6570\u95a2\u6570\u3067\u3042\u308b\u3002\u591a\u5909\u6570\u95a2\u6570 \\(f(x, y, z, t)\\) \u3092\u5fae\u5206\u3059\u308b\u3068\u304d\uff0c\u4f8b\u3048\u3070 \\(x\\) \u4ee5\u5916\u306e\u5909\u6570\u3092\u3042\u305f\u304b\u3082\u5b9a\u6570\u3068\u307f\u306a\u3057\u3066\u5909\u5316\u3055\u305b\u305a\uff0c\\(x\\) \u306b\u3064\u3044\u3066\u306e\u307f\u5fae\u5c0f\u5909\u5316\u3055\u305b\u308b\u3068\u304d\uff0c\\(x\\) \u3067\u300c\u504f\u5fae\u5206\u3059\u308b\u300d\u3068\u8a00\u3044\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3002<\/b><\/u><\/span><\/p>\n

$$ \\frac{\\partial f}{\\partial x} \\equiv \\lim_{h\\rightarrow 0} \\frac{f(x+h, y, z, t) – f(x, y, z, t)}{h} $$<\/p>\n

\u4ed6\u306e\u5909\u6570\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u3067\uff0c$$\\frac{\\partial f}{\\partial y}, \\\u00a0 \\frac{\\partial f}{\\partial z}, \\ \\frac{\\partial f}{\\partial t}$$<\/p>\n

\\( \\partial \\) \u306e\u8aad\u307f\u65b9\u306b\u3064\u3044\u3066\u306f\uff0c\u8cc7\u6599\u3092\u78ba\u8a8d\u3059\u308b\u3053\u3068\u3002\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u306e\u65e5\u672c\u8a9e IME\uff08\u304b\u306a\u3092\u6f22\u5b57\u7b49\u306b\u5909\u63db\u3059\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\uff09\u3067\u306f\u300c\u3067\u308b\u300d\u3068\u6253\u3063\u3066\u5909\u63db\u3059\u308b\u3068\u300c\u2202\u300d\u304c\u9078\u629e\u3067\u304d\u308b\u3002<\/p>\n

\u504f\u5fae\u5206\u306e\u521d\u6b69<\/h5>\n

\u305f\u3068\u3048\u3070\uff0c\\(C\\) \u3092\u5b9a\u6570\u3068\u3057\u305f1\u5909\u6570\u95a2\u6570 \\(f(x) = C x^2\\) \u3092 \\(x\\) \u3067\u5fae\u5206\u3059\u308b\u3068
\n$$\\frac{d}{dx} f(x) = \\frac{d}{dx} (C x^2) = C \\frac{d}{dx} x^2 = 2 C x$$ \u3067\u3042\u3063\u305f\u3002<\/p>\n

2\u5909\u6570\u95a2\u6570 \\(f(x, y) = (y^3 + 2y) x^2 \\) \u3092 \\(x\\) \u3067\u504f\u5fae\u5206\u3059\u308b\u3068\u306f\u3069\u3046\u3044\u3046\u3053\u3068\u304b\u3068\u3044\u3046\u3068\uff0c\\(x\\) \u4ee5\u5916\u306e\u5909\u6570\uff08\u3053\u3053\u3067\u306f \\(y\\)\uff09\u3092\u3042\u305f\u304b\u3082\u5b9a\u6570\u306e\u3088\u3046\u306b\u307f\u306a\u3057\u3066\u5fae\u5206\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u3060\u304b\u3089
\n$$\\frac{\\partial}{\\partial x} f(x, y) = \\frac{\\partial}{\\partial x}\\left\\{ (y^3 + 2y) x^2\\right\\} = (y^3 + 2y) \\frac{\\partial}{\\partial x} x^2 = 2 (y^3 + 2y) x$$
\n\u6700\u5f8c\u306e\u90e8\u5206\u3067\u306f\uff0c\\(x^2\\) \u3092\\(x\\) \u3067\u504f\u5fae\u5206\u3057\u3066\u3044\u308b\u304c\uff0c\\(x^2\\) \u306f1\u5909\u6570\u95a2\u6570\u306a\u306e\u3067
\n$$\\frac{\\partial}{\\partial x} x^2 = \\frac{d}{dx} x^2 = 2x$$ \u3067\u3042\u308b\u3002<\/p>\n

\u9ad8\u968e\u306e\u504f\u5fae\u5206<\/h4>\n

1\u5909\u6570\u95a2\u6570 \\(y = f(x)\\) \u306e\u9ad8\u968e\u5fae\u5206\uff082\u968e\u5fae\u5206\uff0c3\u968e\u5fae\u5206, …\uff09\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u306e\u3067\u3042\u3063\u305f\u3002
\n$$\\frac{d^2 y}{dx^2}, \\\u00a0 \\frac{d^2}{dx^2}f(x), \\\u00a0 \\frac{d^2 y}{dx^3}, \\\u00a0 \\frac{d^3}{dx^3} f(x), \\dots$$<\/p>\n

\u9ad8\u968e\uff08\u4f8b\u3048\u30702\u968e\uff09\u306e\u504f\u5fae\u5206\u306e\u66f8\u304d\u65b9\u306f…<\/p>\n

$$\\frac{\\partial^2 f}{\\partial x^2}, \\ \\frac{\\partial^2}{\\partial x^2} f, \\\u00a0 \\frac{\\partial^2 f}{\\partial x \\partial y}, \\ \\frac{\\partial^2}{\\partial x \\partial y} f, \\dots$$<\/p>\n

\u9ad8\u968e\u306e\u504f\u5fae\u5206\u306e\u91cd\u8981\u306a\u6027\u8cea<\/h5>\n

\u504f\u5fae\u5206\u306f\u4ea4\u63db\u53ef\u80fd<\/strong><\/span>\u3067\u3059\u3002<\/p>\n

$$ \\frac{\\partial^2 f}{\\partial x \\partial y} = \\frac{\\partial^2 f}{\\partial y \\partial x}, \\dots$$ \\(x, y\\) \u306b\u3064\u3044\u3066\u3060\u3051\u3067\u306a\u304f\uff0c\u4ed6\u306e\u5909\u6570\u306b\u3088\u308b\u504f\u5fae\u5206\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u3067\u3059\u3002<\/p>\n

\u30b9\u30ab\u30e9\u30fc\u5834\u3084\u30d9\u30af\u30c8\u30eb\u5834\u306e\u5fae\u5206<\/h3>\n

\u300c\u5834\uff08\u3070\uff09\u300d\u3068\u306f<\/h4>\n

\u3042\u308b\u91cf \\(X\\) \u306e\u5024\u304c\u7a7a\u9593\u5185\u306e\u5404\u70b9 \\(\\boldsymbol{r}\\) \u306e\u95a2\u6570\u3068\u3057\u3066\u4e00\u7fa9\u7684\u306b\u6c7a\u307e\u308b\u3068\u304d\uff0c\\(X\\) \u306f\u5834\u3067\u3042\u308b\u3068\u3044\u3044\uff0c\\( X(\\boldsymbol{r})\\) \u3068\u66f8\u304f\u3002\u4e00\u822c\u7684\u306b\u306f\u6642\u9593\u306e\u95a2\u6570\u3067\u3082\u3042\u308b\u3060\u308d\u3046\u304b\u3089 \\( X(\\boldsymbol{r}, t) \\) \u3068\u66f8\u3044\u3066\uff08\u7279\u306b\u6642\u9593\u5909\u52d5\u3059\u308b\uff09\u5834\u3068\u3044\u3046\u3002<\/p>\n

\u96fb\u78c1\u6c17\u5b66\u3067\u3044\u3046\u300c\u96fb\u5834\u300d\u3084\u300c\u78c1\u5834\u300d\u3068\u306f\uff0c\u8377\u96fb\u7c92\u5b50\u304c\u53d7\u3051\u308b\u96fb\u6c17\u7684\u529b\u3084\u78c1\u6c17\u7684\u529b\u304c\u5834\u6240\uff08\u7a7a\u9593\u5ea7\u6a19\uff09\u306e\u95a2\u6570\u3067\u3042\u308b\uff08\u3088\u308a\u4e00\u822c\u7684\u306b\u306f\u6642\u9593\u7684\u306b\u3082\u5909\u52d5\u3059\u308b\u306e\u3067\u6642\u9593\u5ea7\u6a19\u306e\u95a2\u6570\u3067\u3082\u3042\u308b\uff09\u3068\u3044\u3046\u3053\u3068\u3002\u306a\u304a\u5de5\u5b66\u7cfb\u306e\u6559\u79d1\u66f8\u3084\u9ad8\u6821\u306e\u6559\u79d1\u66f8\u3067\u306f\u300c\u96fb\u5834\u300d\u3084\u300c\u78c1\u5834\u300d\u306e\u304b\u308f\u308a\u306b\u300c\u96fb\u754c\u300d\u300c\u78c1\u754c\u300d\u3068\u66f8\u3044\u3066\u3044\u308b\u5834\u5408\u3082\u3042\u308b\u3002\u300c\u5834\u300d\u3082\u300c\u754c\u300d\u3082\u300cfield\u300d\u306e\u8a33\u3067\u3042\u308b\uff0c\u78ba\u304b\u3002<\/p>\n

\u30b9\u30ab\u30e9\u30fc\u5834<\/h5>\n

\u30b9\u30ab\u30e9\u30fc\u91cf \\(\\varphi\\) \u304c\u7a7a\u9593\u5ea7\u6a19 \\(\\boldsymbol{r} = (x, y, z)\\) \u306e\u95a2\u6570\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3068\u304d\uff0c\\(\\varphi(\\boldsymbol{r})\\) \u3092\u30b9\u30ab\u30e9\u30fc\u5834\u3068\u3044\u3046\u3002<\/p>\n

 <\/p>\n

\u30d9\u30af\u30c8\u30eb\u5834<\/h5>\n

\u30d9\u30af\u30c8\u30eb\u91cf \\(\\boldsymbol{a}\\) \u304c\u7a7a\u9593\u5ea7\u6a19 \\(\\boldsymbol{r}\\) \u306e\u95a2\u6570\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3068\u304d\uff0c\\(\\boldsymbol{a}(\\boldsymbol{r})\\) \u3092\u30d9\u30af\u30c8\u30eb\u5834\u3068\u3044\u3046\u3002<\/p>\n

 <\/p>\n

\u30ca\u30d6\u30e9\uff1a\u30d9\u30af\u30c8\u30eb\u5fae\u5206\u6f14\u7b97\u5b50<\/h4>\n

\u300c\u5834\u300d\u306e\u91cf\u3092\u5fae\u5206\u3059\u308b\u6f14\u7b97\u5b50\u3068\u3057\u3066\u300c\u30ca\u30d6\u30e9<\/strong><\/span>\u300d\\(\\nabla\\) \u3068\u547c\u3070\u308c\u308b\u30d9\u30af\u30c8\u30eb\u5fae\u5206\u6f14\u7b97\u5b50\u3092\u5b9a\u7fa9\u3059\u308b\u3002
\n$$\\nabla \\equiv \\boldsymbol{i}\\frac{\\partial}{\\partial x} + \\boldsymbol{j}\\frac{\\partial}{\\partial y} +\\boldsymbol{k}\\frac{\\partial}{\\partial z} $$<\/p>\n

\u30ca\u30d6\u30e9 \\(\\nabla \\) \u305d\u306e\u3082\u306e\u306f\u30d9\u30af\u30c8\u30eb\u3067\u306f\u306a\u3044\u3002\u30b9\u30ab\u30e9\u30fc\u5834\u3084\u30d9\u30af\u30c8\u30eb\u5834\u306b\u4f5c\u7528\u3057\u3066\u521d\u3081\u3066\u5024\u3092\u6301\u3064\u3068\u3044\u3046\u610f\u5473\u3067\u300c\u6f14\u7b97\u5b50\u300d\u3002<\/p>\n

\u4ee5\u4e0b\u3067\u5b9a\u7fa9\u3059\u308b\u30d9\u30af\u30c8\u30eb\u306e\u767a\u6563\u3084\u56de\u8ee2\u306f\uff0c\u3042\u305f\u304b\u3082\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u3084\u5916\u7a4d\u306e\u3088\u3046\u306b\u899a\u3048\u3066\u304a\u3051\u3070\u3044\u3044\u306e\u3067\uff0c\u305d\u306e\u305f\u3081\u306b\u30ca\u30d6\u30e9\u306e\u5c11\u3057\u7c21\u6613\u7684\u306a\u8868\u8a18\u3082\u66f8\u3044\u3066\u304a\u3053\u3046\u3002
\n$$\\nabla = \\boldsymbol{i}\\,\\partial_x+ \\boldsymbol{j}\\,\\partial _y+\\boldsymbol{k}\\,\\partial_z $$ \u3064\u307e\u308a\uff0c
\n$$\\partial_x = \\frac{\\partial}{\\partial x}, \\ \\\u00a0 \\partial_y = \\frac{\\partial}{\\partial y}, \\ \\ \\partial_z = \\frac{\\partial}{\\partial z}$$<\/p>\n

\\( \\partial_x, \\partial_y, \\partial_z \\) \u306e\u8868\u8a18\u306f\u5927\u5b66\u521d\u7d1a\u306e\u30c6\u30ad\u30b9\u30c8\u3067\u306f\u3042\u307e\u308a\u898b\u3089\u308c\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u504f\u5fae\u5206\u306e\u7701\u30b9\u30da\u30fc\u30b9\u306a\u8868\u8a18\u6cd5\u3068\u3057\u3066\uff08\u79c1\u306e\u7814\u7a76\u5206\u91ce\u3067\u306f\uff0c\u3088\u304f\uff09\u4f7f\u308f\u308c\u308b\u3002<\/p>\n

\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d grad<\/h4>\n

\u30b9\u30ab\u30e9\u30fc\u5834 \\(\\varphi(\\boldsymbol{r})\\) \u306b\u30ca\u30d6\u30e9 \\(\\nabla\\) \u3092\u4f5c\u7528\u3055\u305b\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30d9\u30af\u30c8\u30eb\u5834\u306b\u306a\u308b\u3002\u3053\u308c\u3092\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d<\/strong><\/span> (gradient<\/strong><\/span>) \u3068\u547c\u3076\u3002<\/p>\n

$$\\nabla \\varphi = \\boldsymbol{i}\\partial_x\\varphi + \\boldsymbol{j}\\partial _y\\varphi+\\boldsymbol{k}\\partial_z\\varphi = \\frac{\\partial \\varphi}{\\partial x} \\,\\boldsymbol{i} + \\frac{\\partial \\varphi}{\\partial y} \\,\\boldsymbol{j} +\\frac{\\partial \\varphi}{\\partial z} \\,\\boldsymbol{k}$$<\/p>\n

\u52fe\u914d\u306e\u8a08\u7b97\u4f8b<\/h5>\n

\\( r = \\sqrt{x^2 + y^2 + z^2} \\) \u306e\u3068\u304d\uff0c\u4f8b\u3048\u3070\u4ee5\u4e0b\u306e\u8a08\u7b97\u3092\u3057\u3066\u307f\u3088\u3046\u3002<\/p>\n

$$\\nabla r = \\frac{\\partial r}{\\partial x} \\,\\boldsymbol{i} + \\frac{\\partial r}{\\partial y} \\,\\boldsymbol{j} + \\frac{\\partial r}{\\partial z} \\,\\boldsymbol{k}$$ \u307e\u305a\u306f \\(x\\) \u6210\u5206\u306e\u8a08\u7b97\uff1a
\n\\begin{eqnarray}
\n\\frac{\\partial r}{\\partial x}&=& \\frac{\\partial }{\\partial x} \\left(x^2 + y^2 + z^2\\right)^{\\frac{1}{2}} \\\\
\n&=& \\frac{1}{2} \\left(x^2 + y^2 + z^2\\right)^{-\\frac{1}{2}} \\frac{\\partial }{\\partial x} \\left(x^2 + y^2 + z^2\\right) \\\\
\n&=& \\frac{1}{2} \\left(x^2 + y^2 + z^2\\right)^{-\\frac{1}{2}}\\ 2 x \\\\
\n&=& \\frac{x}{r}
\n\\end{eqnarray} \u540c\u69d8\u306b\u3057\u3066
\n$$\\frac{\\partial r}{\\partial y} = \\frac{y}{r}, \\quad \\frac{\\partial r}{\\partial z} = \\frac{z}{r} $$ \u3057\u305f\u304c\u3063\u3066
\n$$\\nabla r =\\frac{x}{r} \\,\\boldsymbol{i} + \\frac{y}{r} \\,\\boldsymbol{j} + \\frac{z}{r} \\,\\boldsymbol{k} = \\frac{\\boldsymbol{r}}{r}$$<\/p>\n

\u307e\u305f\uff0c\u96fb\u78c1\u6c17\u5b66\u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8a08\u7b97\u3092\u3059\u308b\u5fc5\u8981\u3082\u51fa\u3066\u304f\u308b\u306e\u3067\uff0c\u3053\u3053\u3067\u3084\u3063\u3066\u304a\u3053\u3046\u3002<\/p>\n

\\begin{eqnarray}
\n\\nabla \\left(\\frac{1}{r}\\right) &=& \\boldsymbol{i} \\frac{\\partial}{\\partial x}\\frac{1}{r} + \\boldsymbol{j} \\frac{\\partial}{\\partial y}\\frac{1}{r} + \\boldsymbol{k} \\frac{\\partial}{\\partial z}\\frac{1}{r} \\\\
\n&=& – \\frac{1}{r^2} \\frac{\\partial r}{\\partial x} \\,\\boldsymbol{i}\u00a0 -\\frac{1}{r^2} \\frac{\\partial r}{\\partial y} \\,\\boldsymbol{j}\u00a0 -\\frac{1}{r^2} \\frac{\\partial r}{\\partial z} \\,\\boldsymbol{k} \\\\
\n&=& – \\frac{1}{r^2} \\left( \\frac{\\partial r}{\\partial x} \\,\\boldsymbol{i} + \\frac{\\partial r}{\\partial y} \\,\\boldsymbol{j} + \\frac{\\partial r}{\\partial z} \\,\\boldsymbol{k}\\right) \\\\
\n&=& – \\frac{1}{r^2} \\nabla r = -\\frac{1}{r^2} \\frac{\\boldsymbol{r}}{r} \\\\
\n&=&\u00a0 – \\frac{\\boldsymbol{r}}{r^3}
\n\\end{eqnarray} \u3064\u307e\u308a\uff0c$$\\nabla r^{-1} = \u00a0 – \\frac{\\boldsymbol{r}}{r^3} $$ \u3058\u3083\u3042\uff0c\u30bc\u30ed\u3067\u306a\u3044\u4efb\u610f\u306e\u6574\u6570 \\(n\\) \u306b\u3064\u3044\u3066 \\(\\nabla r^{n} \\) \u306e\u8a08\u7b97\u3082\u3067\u304d\u307e\u3059\u3088\u306d\u3002<\/p>\n

\u30d9\u30af\u30c8\u30eb\u5834\u306e\u767a\u6563 div<\/h4>\n

\u30d9\u30af\u30c8\u30eb\u5834 \\(\\boldsymbol{a}(\\boldsymbol{r})\\) \u306b\u30ca\u30d6\u30e9 \\(\\nabla\\) \u3092\u5de6\u304b\u3089\u300c\u5185\u7a4d\u306e\u3088\u3046\u306b\u300d\u4f5c\u7528\u3055\u305b\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30b9\u30ab\u30e9\u30fc\u5834\u306b\u306a\u308b\u3002\u3053\u308c\u3092\u30d9\u30af\u30c8\u30eb\u5834\u306e\u767a\u6563<\/strong><\/span> (divergence<\/strong><\/span>) \u3068\u547c\u3076\u3002
\n$$ \\nabla\\cdot\\boldsymbol{a} =\\partial_x a_x + \\partial_y a_y + \\partial_z a_z =\u00a0 \\frac{\\partial a_x}{\\partial x} + \\frac{\\partial a_y}{\\partial y} + \\frac{\\partial a_z}{\\partial z}$$<\/p>\n

\u767a\u6563\u306e\u8a08\u7b97\u4f8b<\/h5>\n

\\( \\boldsymbol{r} = (x, y, z) \\) \u306e\u3068\u304d
\n$$\\nabla\\cdot \\boldsymbol{r} = \\frac{\\partial x}{\\partial x} + \\frac{\\partial y}{\\partial y} +\\frac{\\partial z}{\\partial z} = 1+1+1 = 3$$ \uff08\u7b54\u3048\u306f\u7a7a\u9593\u306e\u6b21\u5143\u306e\u6570\uff09<\/p>\n

\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2 rot \u307e\u305f\u306f curl<\/h4>\n

\u30d9\u30af\u30c8\u30eb\u5834 \\(\\boldsymbol{a}(\\boldsymbol{r})\\) \u306b\u30ca\u30d6\u30e9 \\(\\nabla\\) \u3092\u5de6\u304b\u3089\u300c\u5916\u7a4d\u306e\u3088\u3046\u306b\u300d\u4f5c\u7528\u3055\u305b\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30d9\u30af\u30c8\u30eb\u5834\u306b\u306a\u308b\u3002\u3053\u308c\u3092\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2<\/strong><\/span> (rotation<\/strong><\/span>) \u3068\u547c\u3076\u3002
\n\\begin{eqnarray}
\n\\nabla\\times\\boldsymbol{a} &=& \\left(\u00a0 \\partial_y a_z -\\partial_z a_y\\right) \\boldsymbol{i} + \\left(\u00a0 \\partial_z a_x -\\partial_x a_z\\right) \\boldsymbol{j} +\\left(\u00a0 \\partial_x a_y -\\partial_y a_x\\right) \\boldsymbol{k} \\\\
\n&=&\\left(\u00a0 \\frac{\\partial}{\\partial y} a_z -\\frac{\\partial}{\\partial z} a_y\\right) \\boldsymbol{i} + \\left(\u00a0 \\frac{\\partial}{\\partial z} a_x -\\frac{\\partial}{\\partial x} a_z\\right) \\boldsymbol{j} +\\left(\u00a0 \\frac{\\partial}{\\partial x} a_y -\\frac{\\partial}{\\partial yz} a_x\\right) \\boldsymbol{k}
\n\\end{eqnarray}<\/p>\n

\u56de\u8ee2\u306e\u8a08\u7b97\u4f8b<\/h5>\n

$$\\nabla\\times \\left(\\frac{\\boldsymbol{r}}{r}\\right)$$ \u3053\u3053\u3067\uff0c\\(\\boldsymbol{r} = (x, y, z), \\ r = |\\boldsymbol{r}| \\)<\/p>\n

\u307e\u305a \\(x\\) \u6210\u5206\u3092\u8a08\u7b97\u3059\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\left(\\nabla\\times \\left(\\frac{\\boldsymbol{r}}{r}\\right)\\right)_x &=& \\frac{\\partial}{\\partial y} \\left(\\frac{z}{r}\\right) -\\frac{\\partial}{\\partial z} \\left(\\frac{y}{r}\\right) \\\\
\n&=& z \\frac{\\partial}{\\partial y} \\left(x^2+y^2+z^2\\right)^{-\\frac{1}{2}} -y \\frac{\\partial}{\\partial z} \\left(x^2+y^2+z^2\\right)^{-\\frac{1}{2}} \\\\
\n&=& z \\left( – \\frac{y}{r^3} \\right) -y \\left( – \\frac{z}{r^3} \\right) \\\\
\n&=& 0
\n\\end{eqnarray} \u4ed6\u306e\u6210\u5206\u3082\u540c\u69d8\u306b…<\/p>\n

\u30b9\u30ab\u30e9\u30fc\u5834\u3084\u30d9\u30af\u30c8\u30eb\u5834\u306e2\u968e\u5fae\u5206\u3068\u6052\u7b49\u5f0f<\/h3>\n

\u30b9\u30ab\u30e9\u30fc\u5834\u3084\u30d9\u30af\u30c8\u30eb\u5834\u306b\u30d9\u30af\u30c8\u30eb\u5fae\u5206\u6f14\u7b97\u5b50\u300c\u30ca\u30d6\u30e9\u300d\\(\\nabla\\) \u304c2\u968e\u4f5c\u7528\u3059\u308b\u3044\u304f\u3064\u304b\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u308b\u3002\u3044\u304f\u3064\u304b\u306e\u53ef\u80fd\u306a\u7d44\u307f\u5408\u308f\u305b\u3068\u3057\u3066\u306f<\/p>\n

    \n
  • $\\nabla\\cdot(\\nabla \\varphi)$<\/li>\n
  • $\\nabla\\times (\\nabla \\varphi)$<\/li>\n
  • $\\nabla (\\nabla\\cdot \\boldsymbol{a})$<\/li>\n
  • $\\nabla\\cdot (\\nabla\\times \\boldsymbol{a})$<\/li>\n
  • $\\nabla\\times (\\nabla\\times \\boldsymbol{a})$<\/li>\n
  • $(\\nabla\\cdot\\nabla) \\boldsymbol{a}$, etc…<\/li>\n<\/ul>\n

    \u52fe\u914d\u306e\u767a\u6563<\/h4>\n

    \u30b9\u30ab\u30e9\u30fc\u5834 \\(\\varphi\\) \u306e\u52fe\u914d \\(\\nabla \\varphi\\) \u306e\u767a\u6563 \\(\\nabla\\cdot(\\nabla \\varphi) \\)<\/p>\n

    \\begin{eqnarray}
    \n\\nabla\\cdot(\\nabla \\varphi) &=&
    \n\\left(\\boldsymbol{i}\\frac{\\partial}{\\partial x} + \\boldsymbol{j}\\frac{\\partial}{\\partial y} + \\boldsymbol{k}\\frac{\\partial}{\\partial z}\\right) \\cdot
    \n\\left(\\frac{\\partial \\varphi }{\\partial x} \\boldsymbol{i} + \\frac{\\partial \\varphi}{\\partial y} \\boldsymbol{j} + \\frac{\\partial \\varphi}{\\partial z} \\boldsymbol{k}\\right)\\\\
    \n&=& \\frac{\\partial^2 \\varphi}{\\partial x^2} + \\frac{\\partial^2\\varphi}{\\partial y^2} + \\frac{\\partial^2 \\varphi}{\\partial z^2} \\\\
    \n&=& \\left(\\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2}+\\frac{\\partial^2}{\\partial z^2}\\right) \\varphi \\\\
    \n&\\equiv& \\nabla^2\\varphi
    \n\\end{eqnarray} \u3053\u3053\u3067
    \n\\[ \\nabla^2 \\equiv \\nabla\\cdot\\nabla =\\left(\\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2}+\\frac{\\partial^2}{\\partial z^2}\\right) \\] \u306f\u7279\u306b\u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\uff08\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\uff09\u3068\u547c\u3070\u308c\u308b\u30b9\u30ab\u30e9\u30fc\uff082\u968e\uff09\u5fae\u5206\u6f14\u7b97\u5b50\u3002<\/p>\n

    \u3064\u307e\u308a\uff0c\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d\u306e\u767a\u6563\u306f\uff0c\u30b9\u30ab\u30e9\u30fc\u5834\u306b\u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\u3068\u547c\u3070\u308c\u308b\u30b9\u30ab\u30e9\u30fc\u5fae\u5206\u6f14\u7b97\u5b50\u3092\u4f5c\u7528\u3055\u305b\u308b\u3053\u3068\u3067\u3042\u308a\uff0c\u7d50\u679c\u306f\u30b9\u30ab\u30e9\u30fc\u306b\u306a\u308b\u3002<\/p>\n

     <\/p>\n

    \u52fe\u914d\u306e\u56de\u8ee2<\/h4>\n

    \u30b9\u30ab\u30e9\u30fc\u5834 \\(\\varphi\\) \u306e\u52fe\u914d \\(\\nabla \\varphi\\) \u306e\u56de\u8ee2 \\(\\nabla\\times(\\nabla \\varphi) \\)\u3002\u307e\u305a \\(x\\) \u6210\u5206\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n

     <\/p>\n

    \\begin{eqnarray}
    \n\\left\\{\\nabla\\times(\\nabla\\varphi) \\right\\}_x &=& \\partial_y (\\partial_z \\varphi) -\\partial_z (\\partial_y \\varphi) \\\\
    \n&=& \\frac{\\partial^2 \\varphi}{\\partial y \\partial z} -\\frac{\\partial^2 \\varphi}{\\partial z \\partial y} = 0
    \n\\end{eqnarray} \u504f\u5fae\u5206\u306f\u4ea4\u63db\u53ef\u80fd\u3060\u3063\u305f\u304b\u3089\u3002\\(y, z\\) \u6210\u5206\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u3002\u3086\u3048\u306b\uff0c<\/p>\n

    $$\\nabla\\times(\\nabla \\varphi) = \\boldsymbol{0}$$ \u4efb\u610f\u306e\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d\u306e\u56de\u8ee2\u306f\uff0c\u6052\u7b49\u7684\u306b\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b<\/strong><\/span>\u3002<\/p>\n

    \u3053\u306e\u6052\u7b49\u5f0f\u306f\uff0c\u96fb\u78c1\u6c17\u5b66\u306e\u4e2d\u3067\u3088\u304f\u4f7f\u308f\u308c\u308b\u3002\u7279\u306b\uff0c\u3053\u306e\u6052\u7b49\u5f0f\u3092\u9006\u624b\u306b\u3068\u3063\u3066\uff0c\u4f55\u304b\u30d9\u30af\u30c8\u30eb\u304c\u3042\u3063\u3066\uff0c\u305d\u306e\u30d9\u30af\u30c8\u30eb\u306e\u56de\u8ee2\u304c\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb\u306b\u306a\u308b\u306e\u3067\u3042\u308c\u3070\uff0c\u305d\u306e\u30d9\u30af\u30c8\u30eb\u306f\u5fc5\u305a\uff0c\u3042\u308b\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d\u3068\u3057\u3066\u66f8\u3051\u308b<\/strong><\/span>\u3002\u3053\u308c\u4f7f\u3044\u307e\u3059\u304b\u3089\uff0c\u899a\u3048\u3066\u304a\u3044\u3066\u3002<\/p>\n

     <\/p>\n

    \u56de\u8ee2\u306e\u767a\u6563<\/h4>\n

    \u30d9\u30af\u30c8\u30eb\u5834 \\(\\boldsymbol{a}\\) \u306e\u56de\u8ee2 \\(\\nabla \\times\\boldsymbol{a} \\) \u306e\u767a\u6563 \\(\\nabla\\cdot(\\nabla \\times\\boldsymbol{a}) \\)\u3002<\/p>\n

     <\/p>\n

    \\begin{eqnarray}
    \n\\nabla\\cdot(\\nabla \\times\\boldsymbol{a}) &=&
    \n\\partial_x (\\nabla \\times\\boldsymbol{a})_x + \\partial_y (\\nabla \\times\\boldsymbol{a})_y + \\partial_z (\\nabla \\times\\boldsymbol{a})_z \\\\
    \n&=&\\partial_x (\\partial_y {\\color{red}{a_z}} -\\partial_z {\\color{blue}{a_y}}) + \\partial_y (\\partial_z {\\color{green}{a_x}} -\\partial_x {\\color{red}{a_z}}) +
    \n\\partial_z (\\partial_x {\\color{blue}{a_y}} -\\partial_y{\\color{green}{a_x}}) \\\\
    \n&=&(\\partial_y\\partial_z -\\partial_z\\partial_y){\\color{green}{a_x}} + (\\partial_z\\partial_x -\\partial_x\\partial_z){\\color{blue}{a_y}} + (\\partial_x\\partial_y -\\partial_y\\partial_x){\\color{red}{a_z}} \\\\
    \n&=& \\frac{\\partial^2 {\\color{green}{a_x}}}{\\partial y \\partial z} -\\frac{\\partial^2 {\\color{green}{a_x}}}{\\partial z \\partial y} + \\frac{\\partial^2 {\\color{blue}{a_y}}}{\\partial z \\partial x} -\\frac{\\partial^2 {\\color{blue}{a_y}}}{\\partial x \\partial z} + \\frac{\\partial^2 {\\color{red}{a_z}}}{\\partial x \\partial y} -\\frac{\\partial^2 {\\color{red}{a_z}}}{\\partial y \\partial x}\\\\
    \n&=& 0 \\end{eqnarray}<\/p>\n

    \u3075\u305f\u305f\u3073\uff0c\u504f\u5fae\u5206\u306f\u4ea4\u63db\u53ef\u80fd\u3060\u3063\u305f\u304b\u3089\u3002\u3068\u3044\u3046\u308f\u3051\u3067\uff0c\u4efb\u610f\u306e\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2\u306e\u767a\u6563\u306f\u6052\u7b49\u7684\u306b\u30bc\u30ed\u3067\u3042\u308b<\/strong><\/span>\u3002<\/p>\n

    \u3053\u306e\u6052\u7b49\u5f0f\u306f\uff0c\u96fb\u78c1\u6c17\u5b66\u306e\u4e2d\u3067\u3088\u304f\u4f7f\u308f\u308c\u308b\u3002\u7279\u306b\uff0c\u3053\u306e\u6052\u7b49\u5f0f\u3092\u9006\u624b\u306b\u3068\u3063\u3066\uff0c\u4f55\u304b\u30d9\u30af\u30c8\u30eb\u304c\u3042\u3063\u3066\uff0c\u305d\u306e\u30d9\u30af\u30c8\u30eb\u306e\u767a\u6563\u304c\u30bc\u30ed\u306b\u306a\u308b\u306e\u3067\u3042\u308c\u3070\uff0c\u305d\u306e\u30d9\u30af\u30c8\u30eb\u306f\u5fc5\u305a\uff0c\u3042\u308b\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2\u3068\u3057\u3066\u66f8\u3051\u308b<\/strong><\/span>\u3002\u3053\u308c\u4f7f\u3044\u307e\u3059\u304b\u3089\uff0c\u899a\u3048\u3066\u304a\u3044\u3066\u3002<\/p>\n

    \u56de\u8ee2\u306e\u56de\u8ee2<\/h4>\n

    \u30d9\u30af\u30c8\u30eb\u5834 \\(\\boldsymbol{a}\\) \u306e\u56de\u8ee2 \\(\\nabla \\times\\boldsymbol{a} \\) \u306e\u56de\u8ee2 \\(\\nabla\\times(\\nabla \\times\\boldsymbol{a}) \\)\u3002\u307e\u305a \\(x\\) \u6210\u5206\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n

    \\begin{eqnarray}
    \n\\left\\{ \\nabla\\times(\\nabla \\times\\boldsymbol{a}) \\right\\}_x &=&
    \n\\partial_y (\\nabla \\times\\boldsymbol{a})_z -\\partial_z (\\nabla \\times\\boldsymbol{a})_y\\\\
    \n&=& \\partial_y ({\\color{red}{\\partial_x}}\u00a0 a_y -\\partial_y {\\color{blue}{a_x}} ) -\\partial_z ( \\partial_z {\\color{blue}{a_x}} -{\\color{red}{\\partial_x}} a_z ) \\\\
    \n&=& {\\color{red}{\\partial_x}} ({\\color{white}{\\partial_x a_x}\\ \\,} {\\color{white}{+}} \\partial_y a_y + \\partial_z a_z) -({\\color{white}{\\partial_x \\partial_x}\\ \\,} {\\color{white}{+}} \\partial_y \\partial_y + \\partial_z \\partial_z ) {\\color{blue}{a_x}} \\\\
    \n&=&\u00a0 \\partial_x\u00a0 (\\partial_x a_x\u00a0 + \\partial_y a_y + \\partial_z a_z) -( \\partial_x \\partial_x\u00a0 + \\partial_y \\partial_y + \\partial_z \\partial_z )\u00a0 a_x \\\\
    \n&=&\u00a0 \\partial_x\u00a0 (\\nabla\\cdot\\boldsymbol{a}) -\\nabla^2 a_x \\\\
    \n&=& \\left\\{ \\nabla\\left( \\nabla\\cdot\\boldsymbol{a} \\right) -\\nabla^2 \\boldsymbol{a} \\right\\}_x
    \n\\end{eqnarray} \\(y, z\\) \u6210\u5206\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u3002\u3086\u3048\u306b\uff0c
    \n$$ \\nabla\\times(\\nabla \\times\\boldsymbol{a}) = \\nabla\\left( \\nabla\\cdot\\boldsymbol{a} \\right) -\\nabla^2 \\boldsymbol{a} $$<\/p>\n

    \u88dc\u8db3\uff1a\u30d9\u30af\u30c8\u30eb\u5834\u306b\u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\u3092\u4f5c\u7528\u3055\u305b\u308b\u3068…<\/h4>\n

    \u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\u306f\u30b9\u30ab\u30e9\u30fc\u5fae\u5206\u6f14\u7b97\u5b50\u3067\u3042\u308b\u304b\u3089\uff0c\u30d9\u30af\u30c8\u30eb\u5834\u306b\u4f5c\u7528\u3055\u305b\u305f\u5834\u5408\u306f\uff0c\u6210\u5206\u305d\u308c\u305e\u308c\u306b\u4f5c\u7528\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002\u3064\u307e\u308a\uff0c
    \n$$\\nabla^2 \\boldsymbol{a} = (\\nabla^2 a_x)\\,\\boldsymbol{i}\u00a0 + (\\nabla^2 a_y)\\,\\boldsymbol{j}\u00a0 + (\\nabla^2 a_z)\\,\\boldsymbol{k} $$<\/p>\n

    \u307e\u3068\u3081<\/h3>\n

    \u52fe\u914d\uff0c\u767a\u6563\uff0c\u56de\u8ee2\u306a\u3069 \\(\\nabla \\) \u304c\u4f5c\u7528\u3059\u308b\u5fae\u5206\u306e\u307e\u3068\u3081\u3002<\/p>\n

    \u30ca\u30d6\u30e9\uff1a\u30d9\u30af\u30c8\u30eb\u5fae\u5206\u6f14\u7b97\u5b50 $\\nabla$<\/h4>\n

    $$\\nabla \\equiv \\boldsymbol{i}\\frac{\\partial}{\\partial x} + \\boldsymbol{j}\\frac{\\partial}{\\partial y} +\\boldsymbol{k}\\frac{\\partial}{\\partial z}\u00a0 = \\boldsymbol{i} \\partial_x + \\boldsymbol{j} \\partial_y + \\boldsymbol{k} \\partial_z$$<\/p>\n

    <\/h4>\n

    \u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d (grad)<\/h4>\n

    $$\\nabla \\varphi = \\boldsymbol{i}\\,\\partial_x\\varphi + \\boldsymbol{j}\\,\\partial _y\\varphi+\\boldsymbol{k}\\,\\partial_z\\varphi = \\frac{\\partial \\varphi}{\\partial x} \\,\\boldsymbol{i} + \\frac{\\partial \\varphi}{\\partial y} \\,\\boldsymbol{j} +\\frac{\\partial \\varphi}{\\partial z} \\,\\boldsymbol{k}$$<\/p>\n

     <\/p>\n

    \u30d9\u30af\u30c8\u30eb\u5834\u306e\u767a\u6563 (div)<\/h4>\n

    $$ \\nabla\\cdot\\boldsymbol{a} =\\partial_x a_x + \\partial_y a_y + \\partial_z a_z =\u00a0 \\frac{\\partial a_x}{\\partial x} + \\frac{\\partial a_y}{\\partial y} + \\frac{\\partial a_z}{\\partial z}$$<\/p>\n

    \u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2 (rot \u307e\u305f\u306f curl)<\/h4>\n

     <\/p>\n

    \\begin{eqnarray}
    \n\\nabla\\times\\boldsymbol{a} &=&
    \n\\left(\u00a0 \\partial_y a_z -\\partial_z a_y\\right) \\boldsymbol{i} +
    \n\\left(\u00a0 \\partial_z a_x -\\partial_x a_z\\right) \\boldsymbol{j} +
    \n\\left(\u00a0 \\partial_x a_y -\\partial_y a_x\\right) \\boldsymbol{k} \\\\
    \n&=&\\left(\u00a0 \\frac{\\partial a_z}{\\partial y}\u00a0 -\\frac{\\partial a_y}{\\partial z} \\right) \\boldsymbol{i} +
    \n\\left(\u00a0 \\frac{\\partial a_x}{\\partial z}\u00a0 -\\frac{\\partial a_z}{\\partial x} \\right) \\boldsymbol{j} +
    \n\\left(\u00a0 \\frac{\\partial a_y}{\\partial x}\u00a0 -\\frac{\\partial a_x}{\\partial yz} \\right) \\boldsymbol{k}
    \n\\end{eqnarray}<\/p>\n

    \u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\uff08\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\uff09<\/h4>\n

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

    \u30d9\u30af\u30c8\u30eb\u5834\u306e\u5fae\u5206\u306e\u6052\u7b49\u5f0f<\/h4>\n

    \u52fe\u914d\u306e\u56de\u8ee2\u306f\u6052\u7b49\u7684\u306b\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb<\/h5>\n

    $$\\nabla\\times(\\nabla \\varphi) = \\boldsymbol{0}$$ \u4efb\u610f\u306e\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d\u306e\u56de\u8ee2\u306f\u6052\u7b49\u7684\u306b\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b<\/strong><\/span>\u3002<\/p>\n

    \u3053\u306e\u6052\u7b49\u5f0f\u306f\uff0c\u96fb\u78c1\u6c17\u5b66\u306e\u4e2d\u3067\u3088\u304f\u4f7f\u308f\u308c\u308b\u3002\u7279\u306b\uff0c\u3053\u306e\u6052\u7b49\u5f0f\u3092\u9006\u624b\u306b\u3068\u3063\u3066\uff0c\u4f55\u304b\u30d9\u30af\u30c8\u30eb\u304c\u3042\u3063\u3066\uff0c\u305d\u306e\u30d9\u30af\u30c8\u30eb\u306e\u56de\u8ee2\u304c\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb\u306b\u306a\u308b\u306e\u3067\u3042\u308c\u3070\uff0c\u305d\u306e\u30d9\u30af\u30c8\u30eb\u306f\u5fc5\u305a\uff0c\u3042\u308b\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d\u3068\u3057\u3066\u66f8\u3051\u308b<\/strong><\/span>\u3002<\/p>\n

    \u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2\u306e\u767a\u6563\u306f\u6052\u7b49\u7684\u306b\u30bc\u30ed<\/h5>\n

    $$ \\nabla\\cdot(\\nabla \\times\\boldsymbol{a})\u00a0 = 0$$ \u4efb\u610f\u306e\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2\u306e\u767a\u6563\u306f\u6052\u7b49\u7684\u306b\u30bc\u30ed\u3067\u3042\u308b<\/strong><\/span>\u3002<\/p>\n

    \u3053\u306e\u6052\u7b49\u5f0f\u306f\uff0c\u96fb\u78c1\u6c17\u5b66\u306e\u4e2d\u3067\u3088\u304f\u4f7f\u308f\u308c\u308b\u3002\u7279\u306b\uff0c\u3053\u306e\u6052\u7b49\u5f0f\u3092\u9006\u624b\u306b\u3068\u3063\u3066\uff0c\u4f55\u304b\u30d9\u30af\u30c8\u30eb\u304c\u3042\u3063\u3066\uff0c\u305d\u306e\u30d9\u30af\u30c8\u30eb\u306e\u767a\u6563\u304c\u30bc\u30ed\u306b\u306a\u308b\u306e\u3067\u3042\u308c\u3070\uff0c\u305d\u306e\u30d9\u30af\u30c8\u30eb\u306f\u5fc5\u305a\uff0c\u3042\u308b\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2\u3068\u3057\u3066\u66f8\u3051\u308b<\/strong><\/span>\u3002<\/p>\n

    \u56de\u8ee2\u306e\u56de\u8ee2<\/h4>\n

    $$ \\nabla\\times(\\nabla \\times\\boldsymbol{a}) = \\nabla\\left( \\nabla\\cdot\\boldsymbol{a} \\right) -\\nabla^2 \\boldsymbol{a} $$<\/p>\n

    \u8ffd\u8a18\uff1a\u30d9\u30af\u30c8\u30eb\u30fb\u30b9\u30ab\u30e9\u30fc\u306e\u7a4d\u306e\u767a\u6563<\/h4>\n

    $$\\nabla\\cdot\\left( \\boldsymbol{E}\\times\\boldsymbol{B}\\right)
    \n= \\left(\\nabla\\times\\boldsymbol{E} \\right)\\cdot\\boldsymbol{B}
    \n-\\boldsymbol{E}\\cdot\\left(\\nabla\\times\\boldsymbol{B} \\right)$$<\/p>\n

    $$\\nabla\\cdot\\left( \\phi\\,\\boldsymbol{D}\\right) = \\left(\\nabla\\phi\\right)\\cdot\\boldsymbol{D} +\\left(\\nabla\\cdot\\boldsymbol{D}\\right)\\,\\phi$$<\/p>\n

    \u3053\u3093\u306a\u95a2\u4fc2\u5f0f\uff0c\u3044\u3063\u305f\u3044\u3069\u3053\u3067\u4f7f\u308f\u308c\u308b\u3093\u3060\uff1f\u3068\u61a4\u308b\u4eba\u304c\u3044\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u96fb\u78c1\u5834\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6\u3068\u30dd\u30a4\u30f3\u30c6\u30a3\u30f3\u30b0\u30d9\u30af\u30c8\u30eb<\/strong><\/span><\/a>\u306e\u3068\u3053\u308d\u3067\u4f7f\u308f\u308c\u308b\uff0c\u305d\u308c\u306a\u308a\u306b\u5927\u4e8b\u306a\u5f0f\u306a\u3093\u3067\u3059\u3088\u3002\u5de6\u8fba\uff0c\u53f3\u8fba\u3092\u76f4\u63a5\u8a08\u7b97\u3057\u3066\u7b49\u3057\u3044\u3053\u3068\u3092\u8a3c\u660e\u3067\u304d\u307e\u3059\u304b\u3089\u304c\u3093\u3070\u3063\u3066\u3084\u3063\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002\u624b\u8a08\u7b97\u3059\u308b\u4f53\u529b\u304c\u306a\u3044\u4eba\u306f\uff0cSymPy \u3084 Maxima \u3067\u78ba\u304b\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\uff1a<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2561,"menu_order":6,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2605","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\/2605","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=2605"}],"version-history":[{"count":19,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2605\/revisions"}],"predecessor-version":[{"id":9485,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2605\/revisions\/9485"}],"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=2605"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}