{"id":26613,"date":"2025-11-25T12:24:15","date_gmt":"2025-11-25T04:24:15","guid":{"rendered":"http:\/\/www.jdcui.com\/?p=26613"},"modified":"2026-02-15T10:39:33","modified_gmt":"2026-02-15T02:39:33","slug":"%e6%95%a3%e5%ba%a6%e7%9a%84%e7%89%a9%e7%90%86%e6%84%8f%e4%b9%89%e5%92%8c%e6%95%a3%e5%ba%a6%e5%ae%9a%e7%90%86-the-physical-meaning-of-divergence-and-the-divergence-theorem","status":"publish","type":"post","link":"http:\/\/www.jdcui.com\/?p=26613","title":{"rendered":"\u6563\u5ea6\u7684\u7269\u7406\u610f\u4e49\u548c\u6563\u5ea6\u5b9a\u7406 [The Physical Meaning of Divergence and the Divergence Theorem]"},"content":{"rendered":"<p><script><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span>\nMathJax = {\n  tex: {\n    inlineMath: [['$', '$'], ['\\\\(', '\\\\)']]\n  }\n};<span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span>\n<\/script><br \/>\n<script id=\"MathJax-script\" src=\"http:\/\/www.jdcui.com\/wp-content\/MathJax_3_1_2\/MathJax-master\/es5\/tex-mml-chtml.js\" async=\"\"><\/script><\/p>\n<p><em><span style=\"color: #ff00ff;\"><strong>\u5b9e\u5e72\u3001\u5b9e\u8df5\u3001\u79ef\u7d2f\u3001\u601d\u8003\u3001\u521b\u65b0\uff01<\/strong><\/span><\/em><\/p>\n<hr \/>\n<p>\u6700\u8fd1\u770b\u6570\u5b66\u4e66\uff0c\u6574\u7406\u4e00\u4e0b\u6563\u5ea6\u7684\u77e5\u8bc6\u70b9\uff0c\u65b9\u4fbf\u8bb0\u5fc6\u3002\u4e0b\u9762\u4ee5\u76f4\u89d2\u5750\u6807\u7cfb (x, y, z) \u4e2d\u7684\u5411\u91cf\u573a\u4e3a\u4f8b\u8fdb\u884c\u9610\u8ff0\u3002<\/p>\n<p><span style=\"font-size: 12pt;\"><strong>\u7b2c\u4e00\uff1a\u6563\u5ea6\u7684\u7269\u7406\u610f\u4e49<\/strong><\/span><\/p>\n<p><strong><span style=\"color: #008080;\">\u6563\u5ea6 (Divergence)<\/span><\/strong>\u662f\u63cf\u8ff0<span style=\"color: #008080;\"><strong>\u5411\u91cf\u573a (Vector Field) \u6e90\u5f3a\u5ea6 (Source Strength)<\/strong> <\/span>\u7684\u6807\u91cf\u51fd\u6570\u3002\u5728\u7269\u7406\u4e0a\uff0c\u5b83\u53ef\u4ee5\u88ab\u7cbe\u786e\u5b9a\u4e49\u4e3a<strong><span style=\"color: #008080;\">\u5355\u4f4d\u4f53\u79ef\u7684\u901a\u91cf(Flux per Unit Volume)<\/span>\u3002<\/strong><\/p>\n<p>\u8003\u8651\u4e00\u4e2a\u5411\u91cf\u573a \\(\\vec{F}(x, y, z) = P\\mathbf{i} + Q\\mathbf{j} + R\\mathbf{k}\uff0c\u5176\u4e2d P, Q, R\\)\u662f\u5404\u65b9\u5411\u7684\u5206\u91cf\u3002\u5728\u573a\u4e2d\u4efb\u53d6\u4e00\u70b9 \\(M(x, y, z)\\)\uff0c\u5176\u6563\u5ea6\u5b9a\u4e49\u4e3a\uff1a<\/p>\n<p>$$<br \/>\n\\text{div} \\, \\vec{F} = \\nabla \\cdot \\vec{F} = \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y} + \\frac{\\partial R}{\\partial z}<br \/>\n$$<\/p>\n<p><span style=\"color: #008080;\"><strong>\u7269\u7406\u8be0\u91ca\uff1a<\/strong><\/span><\/p>\n<p><strong>&#8211;<span style=\"color: #008080;\"> \u6b63\u6563\u5ea6 (Positive Divergence)<\/span><\/strong> \u8868\u793a\u8be5\u70b9\u662f\u4e00\u4e2a \u201c<span style=\"color: #008080;\"><strong>\u6e90 (Source)\u201d<\/strong> <\/span>\uff0c\u6709\u51c0\u6d41\u4f53\uff08\u6216\u573a\u7ebf\uff09\u4ece\u6b64\u70b9\u4ea7\u751f\u6216\u53d1\u6563\u51fa\u53bb\u3002<\/p>\n<p><strong><span style=\"color: #008080;\">&#8211; \u8d1f\u6563\u5ea6 (Negative Divergence)<\/span> <\/strong>\u8868\u793a\u8be5\u70b9\u662f\u4e00\u4e2a <span style=\"color: #008080;\"><strong>\u201c\u6c47 (Sink)\u201d<\/strong><\/span> \uff0c\u6709\u51c0\u6d41\u4f53\uff08\u6216\u573a\u7ebf\uff09\u5728\u6b64\u70b9\u88ab\u5438\u6536\u6216\u6c47\u805a\u3002<\/p>\n<p><strong><span style=\"color: #008080;\">&#8211; \u96f6\u6563\u5ea6 (Zero Divergence)<\/span> <\/strong>\u8868\u793a\u8be5\u70b9\u65e0\u6e90\u65e0\u6c47\uff0c\u6d41\u5165\u4e0e\u6d41\u51fa\u8be5\u70b9\u7684\u901a\u91cf\u76f8\u4e92\u62b5\u6d88\u3002<\/p>\n<p>\u56e0\u6b64\uff0c\u6563\u5ea6 \\(\\nabla \\cdot \\vec{F} \u91cf\u5316\u4e86\u5728\u70b9 M\\) \u5904\uff0c\u4ece\u8be5\u70b9\u65e0\u9650\u5c0f\u4f53\u79ef\u5185<span style=\"color: #008080;\"><strong>\u51c0\u6d41\u51fa<\/strong><\/span>\u7684<span style=\"color: #008080;\"><strong>\u901a\u91cf\u5bc6\u5ea6(Flux Density)<\/strong><\/span>\u3002<\/p>\n<p><span style=\"font-size: 12pt;\"><strong>\u00a0\u7b2c\u4e8c\uff1a\u6563\u5ea6\u5b9a\u7406<\/strong><\/span><\/p>\n<p><span style=\"color: #008080;\"><strong>\u6563\u5ea6\u5b9a\u7406 (Divergence Theorem)<\/strong><\/span>\uff0c\u4e5f\u5e38\u79f0\u4e3a<span style=\"color: #008080;\"><strong>\u9ad8\u65af\u5b9a\u7406 (Gauss&#8217;s Theorem)<\/strong><\/span>\uff0c\u662f\u6c9f\u901a\u5411\u91cf\u573a\u5185\u90e8\u7279\u6027\u4e0e\u8fb9\u754c\u884c\u4e3a\u7684\u6865\u6881\u3002\u5b83\u5efa\u7acb\u4e86<span style=\"color: #008080;\"><strong>\u4f53\u79ef\u5206 (Volume Integral)<\/strong><\/span> \u4e0e<span style=\"color: #008080;\"><strong>\u66f2\u9762\u79ef\u5206 (Surface Integral)<\/strong> <\/span>\u4e4b\u95f4\u7684\u8054\u7cfb\u3002<\/p>\n<p>\u5bf9\u4e8e\u4e00\u4e2a\u7531\u95ed\u5408\u66f2\u9762 (Closed Surface) \\(S\\) \u6240\u56f4\u6210\u7684\u7a7a\u95f4\u533a\u57df \\(\\Omega\\)\uff0c\u6563\u5ea6\u5b9a\u7406\u8868\u8ff0\u4e3a\uff1a<\/p>\n<p>$$<br \/>\n\\iiint\\limits_{\\Omega} (\\nabla \\cdot \\vec{F}) \\, dV = \\oint\\limits_{S} \\vec{F} \\cdot d\\vec{A}<br \/>\n$$<\/p>\n<p><span style=\"color: #008080;\"><strong>\u4e13\u4e1a\u8868\u8ff0\uff1a<\/strong><\/span><\/p>\n<p><strong><span style=\"color: #008080;\">-\u5de6\u4fa7\uff1a<\/span><\/strong>\\(\\iiint_{\\Omega} (\\nabla \\cdot \\vec{F}) \\, dV\uff0c\u8868\u793a\u5bf9\u6574\u4e2a\u533a\u57df \\Omega\\ \\)\u5185\u6240\u6709\u70b9\u7684<strong><span style=\"color: #008080;\">\u6563\u5ea6\uff08\u6e90\u5f3a\u5ea6\uff09\u8fdb\u884c\u79ef\u5206<\/span><\/strong>\u3002<\/p>\n<p>\u5176\u7269\u7406\u610f\u4e49\u662f\u533a\u57df\\( \\Omega\\)\u5185\u6240\u6709<strong>\u201c\u6e90<\/strong>\u201d\u548c<strong>\u201c\u6c47\u201d<\/strong>\u4ea7\u751f\u7684<span style=\"color: #008080;\"><strong>\u51c0\u901a\u91cf (Net Flux)<\/strong><\/span> \u603b\u548c\u3002<\/p>\n<p><strong><span style=\"color: #008080;\">-\u53f3\u4fa7\uff1a<\/span><\/strong>\\(\\oint_{S} \\vec{F} \\cdot d\\vec{A}\uff0c\u8868\u793a\u5411\u91cf\u573a \\vec{F}\u7a7f\u8fc7\u6574\u4e2a\u8fb9\u754c\u66f2\u9762 S\\) \u7684<span style=\"color: #008080;\"><strong>\u51c0\u901a\u91cf<\/strong><\/span>\u3002<\/p>\n<p><strong><span style=\"font-size: 12pt;\">\u4e3b\u8981\u7ed3\u8bba<\/span><\/strong>\uff1a<br \/>\n<span style=\"color: #008080;\"><strong>\u533a\u57df\u5185\u6240\u6709\u6e90\u548c\u6c47\u4ea7\u751f\u7684\u603b\u51c0\u901a\u91cf\uff0c\u5fc5\u7136\u7b49\u4e8e\u7a7f\u8fc7\u8be5\u533a\u57df\u8fb9\u754c\u66f2\u9762\u7684\u603b\u51c0\u901a\u91cf\u3002<\/strong><\/span>\u8fd9\u70b9\u5728\u7406\u89e3\u4e86\u6563\u5ea6\u7684\u7269\u7406\u610f\u4e49\u540e\u5f88\u5bb9\u6613\u660e\u767d\uff0c\u516c\u5f0f\u5bb9\u6613\u8bb0\u5f97\u4e86\u3002\u6563\u5ea6\u5b9a\u7406\u63ed\u793a\u4e86\u573a\u5728\u533a\u57df\u5185\u90e8\u7684\u201c\u4ea7\u751f\u201d\u4e0e\u5728\u8fb9\u754c\u4e0a\u7684\u201c\u6d41\u52a8\u201d\u662f\u5b88\u6052\u7684\u3002<\/p>\n<section>\n<section>\n<section>\n<section>\n<section><\/section>\n<\/section>\n<\/section>\n<\/section>\n<section>\n<section>\n<hr \/>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; background-color: #999999; color: #000000;\"><strong>\u5173\u4e8e\u6211\u4eec<\/strong><\/span><\/p>\n<p style=\"text-align: center;\">\u7ed3\u6784\u52a8\u529b\u5b66\u4e0e\u6297\u9707 | \u590d\u6742\u7ed3\u6784\u5206\u6790\u4e0e\u8bbe\u8ba1 | \u5730\u94c1\u51cf\u632f\u63a7\u5236 | \u4eba\u884c\u53ca\u98ce\u81f4\u632f\u52a8\u63a7\u5236 | \u7a0b\u5e8f\u5f00\u53d1\u4e0e\u8f6f\u4ef6\u5b9a\u5236 | \u6709\u9650\u5143\u4eff\u771f | \u6709\u9650\u5143\u6c42\u89e3\u5668\u5f00\u53d1 | \u56fa\u4f53\u529b\u5b66<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2017\/01\/QRCODE.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3636 alignnone\" src=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2017\/01\/QRCODE.jpg\" alt=\"WeChat_QRCode\" width=\"250\" height=\"255\" \/><\/a><\/p>\n<p style=\"text-align: center;\">https:\/\/www.jdcui.com<\/p>\n<p style=\"text-align: center;\">\u5408\u4f5c\u53ca\u6280\u672f\u54a8\u8be2<\/p>\n<p style=\"text-align: center;\">COOPERATION &amp; CONTACT<\/p>\n<p style=\"text-align: center;\">E-mail\uff1ajidong_cui@163.com<\/p>\n<p 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R\\mathbf{k}\uff0c\u5176\u4e2d P, Q, &#8230;<\/p>\n","protected":false},"author":1,"featured_media":26637,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,447],"tags":[4449,4450,23,4451,4452,135,4453],"class_list":["post-26613","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-math-geometry","category-solid-mechanics-","tag-divergence","tag-divergence-theorem","tag-math","tag-4451","tag-4452","tag-135","tag-4453"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.10 - aioseo.com -->\n\t<meta name=\"description\" content=\"\u5b9e\u5e72\u3001\u5b9e\u8df5\u3001\u79ef\u7d2f\u3001\u601d\u8003\u3001\u521b\u65b0\uff01 \u6700\u8fd1\u770b\u6570\u5b66\u4e66\uff0c\u6574\u7406\u4e00\u4e0b\u6563\u5ea6\u7684\u77e5\u8bc6\u70b9\uff0c\u65b9\u4fbf\u8bb0\u5fc6\u3002\u4e0b\u9762\u4ee5\u76f4\u89d2\u5750\u6807\u7cfb (x, y, z) \u4e2d\u7684\u5411\u91cf\u573a\u4e3a\u4f8b\u8fdb\u884c\u9610\u8ff0\u3002 \u7b2c\u4e00\uff1a\u6563\u5ea6\u7684\u7269\u7406\u610f\u4e49 \u6563\u5ea6 (Divergence)\u662f\u63cf\u8ff0\u5411\u91cf\u573a (Vector Field) \u6e90\u5f3a\u5ea6 (Source Strength) \u7684\u6807\u91cf\u51fd\u6570\u3002\u5728\u7269\u7406\u4e0a\uff0c\u5b83\u53ef\u4ee5\u88ab\u7cbe\u786e\u5b9a\u4e49\u4e3a\u5355\u4f4d\u4f53\u79ef\u7684\u901a\u91cf(Flux per Unit Volume)\u3002 \u8003\u8651\u4e00\u4e2a\u5411\u91cf\u573a \\(\\vec{F}(x, y, z) = P\\mathbf{i} + Q\\mathbf{j} + R\\mathbf{k}\uff0c\u5176\u4e2d P, Q, R\\)\u662f\u5404\u65b9\u5411\u7684\u5206\u91cf\u3002\u5728\u573a\u4e2d\u4efb\u53d6\u4e00\u70b9 \\(M(x, y, z)\\)\uff0c\u5176\u6563\u5ea6\u5b9a\u4e49\u4e3a\uff1a $$ \\text{div} \\, \\vec{F} = \\nabla \\cdot \\vec{F} = \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y} + \\frac{\\partial R}{\\partial z} $$\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" 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