{"id":2619,"date":"2022-03-26T14:41:45","date_gmt":"2022-03-26T05:41:45","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2619"},"modified":"2025-07-15T17:25:15","modified_gmt":"2025-07-15T08:25:15","slug":"%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","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\/%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\/","title":{"rendered":"\u53c2\u8003\uff1a\u30ac\u30a6\u30b9\u306e\u5b9a\u7406\u306e\u8a3c\u660e"},"content":{"rendered":"

<\/p>\n

\u30ac\u30a6\u30b9\u306e\u5b9a\u7406<\/h3>\n

$$ \\iiint_V \\nabla\\cdot \\boldsymbol{a}\\ dV = \\iint_S \\boldsymbol{a}\\cdot\\boldsymbol{n}\\ dS$$<\/p>\n

\u3053\u3053\u3067\uff0c\\(S\\) \u306f\u4f53\u7a4d \\(V\\) \u306e\u7acb\u4f53\u3092\u56f2\u3080\u9589\u66f2\u9762<\/strong><\/span>\uff0c\\(dS\\) \u306f\u305d\u306e\u8868\u9762\u306e\u5fae\u5c0f\u9762\u7a4d\u90e8\u5206\uff0c\\(\\boldsymbol{n}\\) \u306f\u5fae\u5c0f\u9762\u7a4d \\(dS\\) \u306b\u5782\u76f4\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\uff08\u5411\u304d\u306f\u7acb\u4f53\u306e\u5916\u5074\u3092\u5411\u304f\uff09\u3002<\/p>\n

\u53f3\u8fba\u304c\u9589\u66f2\u9762 $S$ \u4e0a\u3067\u306e\u9762\u7a4d\u5206<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u3092\u5f37\u8abf\u3059\u308b\u305f\u3081\u306b\uff0c\u53f3\u8fba\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3079\u304d\u3060\u3068\u3044\u3046\u306e\u304c\u79c1\u306e\u610f\u898b\u3002<\/p>\n

$$\\iiint_V \\nabla\\cdot \\boldsymbol{a}\\, dV = {\\color{red}\\int\\!\\!\\!\\!\\!\\!\\bigcirc\\!\\!\\!\\!\\!\\!\\int}_{\\!\\!S} \\ \\boldsymbol{a}\\cdot\\boldsymbol{n} \\ dS$$\uff08\u3067\u3082 $\\color{red}\\displaystyle\\int\\!\\!\\!\\!\\!\\!\\bigcirc\\!\\!\\!\\!\\!\\!\\int$ \u3092\u66f8\u304f\u306e\u306f\u5927\u5909\u306a\u3093\u3060\u3088\u306a\u3041\uff0cMathjax \u3067\u306f \\userpackage{esint}<\/code> \u304c\u4f7f\u3048\u306a\u3044\u3088\u3046\u3067\uff0c$\\oiint$<\/code> \u3084 $\\varoiint$<\/code> \u304c\u52b9\u304b\u306a\u3044\u3002\uff09<\/p>\n

\"\"<\/p>\n

\u56f3\u306e\u3088\u3046\u306b\uff0c\u305d\u308c\u305e\u308c\u306e\u8fba\u306e\u9577\u3055\u304c \\(\\varDelta x, \\varDelta y, \\varDelta z\\) \u3067\u3042\u308b\u5fae\u5c0f\u76f4\u65b9\u4f53\u3092\u8003\u3048\uff0c\u3053\u306e\u4f53\u7a4d\u3092\u56f2\u3080\u9762\u304b\u3089\u306e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{a}\\) \u306e\u6d41\u675f<\/strong><\/span>\uff08 \\(\\boldsymbol{a}\\) \u306e\u9762\u306b\u5782\u76f4\u306a\u6210\u5206\u306e\u9762\u7a4d\u5206\uff0c\u5fae\u5c0f\u9762\u7a4d\u306e\u5834\u5408\u306f\u5358\u306b\u300c\u5782\u76f4\u6210\u5206\u300d\u304b\u3051\u308b\u300c\u9762\u7a4d\u300d\uff09\u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n

\u307e\u305a\uff0c\u9762\u2461\u304b\u3089\u51fa\u308b\u6d41\u675f\u306f $$(\\boldsymbol{a}\\cdot\\boldsymbol{n})_2\\, \\varDelta S_2 = a_z(\\varDelta z) \\varDelta x\\, \\varDelta y = \\left\\{a_z(0) + \\frac{\\partial a_z}{\\partial z} \\varDelta z \\right\\} \\varDelta x \\,\\varDelta y$$<\/p>\n

\u540c\u69d8\u306b\u3057\u3066\uff0c\u9762\u2460\u304b\u3089\u51fa\u308b\u6d41\u675f\u306f $$(\\boldsymbol{a}\\cdot\\boldsymbol{n})_1\\, \\varDelta S_1 = -a_z(0) \\,\\varDelta x\\, \\varDelta y $$ \u3053\u3053\u3067\u30de\u30a4\u30ca\u30b9\u304c\u3064\u304f\u306e\u306f\uff0c\\(\\boldsymbol{n}\\) \u304c\u7acb\u4f53\u306e\u8868\u9762\u5916\u5411\u304d\u306e\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u304b\u3089\u3002<\/p>\n

\\(z\\) \u8ef8\u306b\u76f4\u4ea4\u3059\u308b\u3053\u306e\u4e8c\u3064\u306e\u9762\u306e\u6d41\u675f\u3092\u8db3\u3059\u3068\uff0c$z$ \u65b9\u5411\u306e\u6b63\u5473\u306e\u6d41\u901f $(\\boldsymbol{a}\\cdot\\boldsymbol{n})_z\\, \\varDelta S_z$ \u306f<\/p>\n

\\begin{eqnarray}
\n(\\boldsymbol{a}\\cdot\\boldsymbol{n})_z\\, \\varDelta S_z &=& (\\boldsymbol{a}\\cdot\\boldsymbol{n})_1\\, \\varDelta S_1 + (\\boldsymbol{a}\\cdot\\boldsymbol{n})_2\\, \\varDelta S_2 \\\\
\n&=& \\frac{\\partial a_z}{\\partial z}\\\u00a0 \\varDelta x\\, \\varDelta y\\, \\varDelta z
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067 $z$ \u8ef8\u306b\u5782\u76f4\u306a\u5fae\u5c0f\u9762\u7a4d\u3068\u3057\u3066 $\\varDelta S_z \\equiv \\varDelta x\\, \\varDelta y$ \u3068\u5b9a\u7fa9\u3057\u305f\u3002<\/p>\n

$x$ \u65b9\u5411\uff0c$y$ \u65b9\u5411\uff0c$z$ \u65b9\u5411\u306e6\u9762\u5168\u90e8\u3092\u8003\u3048\u308b\u3068\uff0c\u3053\u306e\u5fae\u5c0f\u76f4\u65b9\u4f53\u304b\u3089\u51fa\u308b\u6d41\u675f\u306e\u548c\u306f\uff0c<\/p>\n

\\begin{eqnarray}
\n(\\boldsymbol{a}\\cdot\\boldsymbol{n}) \\,\\varDelta S
\n&=&
\n(\\boldsymbol{a}\\cdot\\boldsymbol{n})_x \\,\\varDelta S_x
\n+ (\\boldsymbol{a}\\cdot\\boldsymbol{n})_y \\,\\varDelta S_y
\n+ (\\boldsymbol{a}\\cdot\\boldsymbol{n})_z \\,\\varDelta S_z \\\\
\n&=& \\left( \\frac{\\partial a_x}{\\partial x} + \\frac{\\partial a_y}{\\partial y} + \\frac{\\partial a_z}{\\partial z} \\right) \\ \\varDelta x\\, \\varDelta y\\, \\varDelta z \\\\
\n&=& \\nabla\\cdot\\boldsymbol{a} \\ \\varDelta V \\\\ \\ \\\\
\n\\therefore\\ \\ (\\boldsymbol{a}\\cdot\\boldsymbol{n}) \\,\\varDelta S &=& \\nabla\\cdot\\boldsymbol{a} \\ \\varDelta V
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u304c\uff0c\u5fae\u5c0f\u76f4\u65b9\u4f53 $\\varDelta V$ \u3092\u56f2\u3080\u5fae\u5c0f\u9589\u66f2\u9762 $\\varDelta S$ \u304b\u3089\u51fa\u308b\u6d41\u901f $(\\boldsymbol{a}\\cdot\\boldsymbol{n}) \\,\\varDelta S$ \u3068\uff0c$\\varDelta V$ \u5185\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}$ \u306e\u767a\u6563\u3068\u306e\u9593\u306e\u95a2\u4fc2\u3067\u3042\u308b\u3002<\/p>\n

\u6709\u9650\u306e\u4f53\u7a4d \\(V\\) \u306e\u7acb\u4f53\u3068\u305d\u308c\u3092\u56f2\u3080\u8868\u9762 \\(S\\) \u306b\u3064\u3044\u3066\u306f\uff0c\u305d\u308c\u3089\u3092\u69cb\u6210\u3057\u3066\u3044\u308b\u5fae\u5c0f\u8981\u7d20\u306e\u7dcf\u548c\u3092\u53d6\u308b\u3053\u3068\u306b\u306a\u308b\u304c\uff0c\u7121\u9650\u5c0f\u9023\u7d9a\u6975\u9650\u3067\u306f $\\displaystyle \\sum_i\\, \\varDelta S_i \\Rightarrow \\iint_S \\,dS, \\ \\sum_i\\, \\varDelta V_i \\Rightarrow \\iiint_V \\, dV$ \u3068\u306a\u308b\u3053\u3068\u304b\u3089<\/p>\n

\\begin{eqnarray}
\n\\sum_i \\left\\{(\\boldsymbol{a}\\cdot\\boldsymbol{n}) \\,\\varDelta S\\right\\}_i &=& \\sum_i \\left\\{\\nabla\\cdot\\boldsymbol{a} \\,\\varDelta V\\right\\}_i \\\\
\n&\\Downarrow & \\\\
\n\\iint_S \\boldsymbol{a}\\cdot\\boldsymbol{n}\\\u00a0 dS &=& \\iiint_V \\nabla\\cdot\\boldsymbol{a} \\\u00a0 dV
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2615,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2619","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\/2619","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=2619"}],"version-history":[{"count":15,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2619\/revisions"}],"predecessor-version":[{"id":10548,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2619\/revisions\/10548"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2615"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2619"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}