{"id":74787,"date":"2016-12-24T08:05:41","date_gmt":"2016-12-24T00:05:41","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=74787"},"modified":"2021-10-06T16:03:41","modified_gmt":"2021-10-06T08:03:41","slug":"%e6%95%a3%e5%ba%a6%e5%92%8c%e6%97%8b%e5%ba%a6","status":"publish","type":"post","link":"http:\/\/localhost\/%e6%95%a3%e5%ba%a6%e5%92%8c%e6%97%8b%e5%ba%a6\/","title":{"rendered":"\u6563\u5ea6\u548c\u65cb\u5ea6"},"content":{"rendered":"

\u6563\u5ea6\u548c\u65cb\u5ea6 (Divergence and Curl)<\/strong><\/span>
\n\u570b\u7acb\u81fa\u7063\u5927\u5b78\u7269\u7406\u5b78\u7cfb \u9673\u54c1\u5168<\/strong><\/span><\/p>\n

\u6563\u5ea6 (Divergence)\uff1a<\/span><\/strong><\/p>\n

\"74787_p1\"<\/a>

\u5716\u4e00\u3001\u4e00\u500b\u767c\u6563\u5834 (A divergent field)\u3002\uff08\u672c\u6587\u4f5c\u8005\u9673\u54c1\u5168\u7e6a\uff09<\/p><\/div>\n

Divergence \u4e2d\u6587\u8b6f\u4f5c\u6563\u5ea6\uff0c\u662f\u5728\u5f62\u5bb9\u67d0\u4e00\u500b\u5411\u91cf\u7684\u767c\u6563\u7a0b\u5ea6\u70ba\u4f55\u3002\u5728\u4e09\u7dad\u7684\u7b1b\u5361\u5152\u5ea7\u6a19\u4e2d\uff0c\u4e00\u500b\u5411\u91cf\u5834 $$\\vec{v}$$ \u7684\u6563\u5ea6\u662f\uff1a<\/p>\n

$$\\displaystyle \\vec{\\nabla}\\cdot\\vec{v}\\equiv\\frac{\\partial v_x}{\\partial x}+\\frac{\\partial v_y}{\\partial y}+\\frac{\\partial v_z}{\\partial z}$$<\/p>\n

\u5229\u7528\u6563\u5ea6\u5b9a\u7406 (Divergence theorem)\uff0c\u6211\u5011\u53ef\u4ee5\u5f9e\u53e6\u4e00\u500b\u89c0\u9ede\u4f86\u770b\u6563\u5ea6\u7684\u6578\u5b78\u610f\u7fa9\uff1a<\/p>\n

$$\\displaystyle \\int_V(\\vec{\\nabla}\\cdot\\vec{v})dV=\\oint_{\\partial V}\\vec{v}\\cdot d\\vec{a}$$<\/p>\n

\u5982\u679c\u8981\u8a08\u7b97\u67d0\u500b\u7a7a\u9593 $$V$$ \u5167\u5411\u91cf $$\\vec{v}$$ \u7684\u6563\u5ea6\uff0c\u4e00\u500b\u53d6\u800c\u4ee3\u4e4b\u7684\u65b9\u6cd5\u662f\u5148\u53bb\u8a08\u7b97\u5411\u91cf $$\\vec{v}$$ \u901a\u904e\u9019\u500b\u7a7a\u9593\u7684\u8868\u9762 $$\\partial V$$ \u7684\u7e3d\u901a\u91cf\u70ba\u4f55\uff08\u9762\u7a4d\u5143\u5411\u91cf $$d\\vec{a}$$ \u7684\u65b9\u5411\u70ba\u8868\u9762\u4e0a\u67d0\u500b\u9ede\u7684\u6cd5\u5411\u91cf\uff0c\u4e26\u4e14\u901a\u5e38\u5b9a\u7fa9\u5411\u5916\u70ba\u6b63\uff09\u3002\u5982\u679c\u5c07\u9019\u500b\u7a7a\u9593\u7e2e\u5230\u7121\u9650\u5c0f\uff0c\u90a3\u9ebc\u6211\u5011\u5247\u53ef\u4ee5\u7531\u6563\u5ea6\u5b9a\u7406\u91cd\u65b0\u5beb\u51fa\u6563\u5ea6\uff0c\u70ba\uff1a<\/p>\n

$$\\displaystyle \\vec{\\nabla}\\cdot\\vec{v}=\\lim_{V\\to 0}\\frac{\\oint_{\\partial V}\\vec{v}\\cdot d\\vec{a}}{V}$$<\/p>\n

\u65cb\u5ea6 (Curl)\uff1a<\/span><\/strong><\/p>\n

\"74787_p2\"<\/a>

\u5716\u4e8c\u3001\u4e00\u500b\u65cb\u8f49\u5834 (A rotational field)\u3002\uff08\u672c\u6587\u4f5c\u8005\u9673\u54c1\u5168\u7e6a\uff09<\/p><\/div>\n

Curl \u7684\u4e2d\u6587\u8b6f\u4f5c\u65cb\u5ea6\uff0c\u662f\u5728\u5f62\u5bb9\u67d0\u4e00\u500b\u5411\u91cf\u65cb\u8f49\u7684\u7a0b\u5ea6\u70ba\u4f55\u3002\u5728\u4e09\u7dad\u7684\u7b1b\u5361\u5152\u5ea7\u6a19\u4e2d\uff1a<\/p>\n

$$\\displaystyle\\vec{\\nabla}\\times\\vec{v}\\equiv\\left(\\frac{\\partial v_z}{\\partial y}-\\frac{\\partial v_y}{\\partial z}\\right)\\hat{i}+\\left(\\frac{\\partial v_x}{\\partial z}-\\frac{\\partial v_z}{\\partial x}\\right)\\hat{j}+\\left(\\frac{\\partial v_y}{\\partial x}-\\frac{\\partial v_x}{\\partial y}\\right)\\hat{k}$$<\/p>\n

\u5229\u7528\u65af\u6258\u514b\u65af\u5b9a\u7406 (Stokes’ theorem)\uff0c\u6211\u5011\u53ef\u4ee5\u5f9e\u53e6\u4e00\u500b\u89c0\u9ede\u4f86\u770b\u65cb\u5ea6\u7684\u6578\u5b78\u610f\u7fa9\uff1a<\/p>\n

$$\\displaystyle\\int_\\Sigma(\\vec{\\nabla}\\times\\vec{v})\\cdot d\\vec{a}=\\oint_{\\partial\\Sigma}\\vec{v}\\cdot d\\vec{s}$$<\/p>\n

\"74787_p3\"<\/a>

\u5716\u4e09\u3001\u65af\u6258\u514b\u65af\u5b9a\u7406 (Stokes\u2019 theorem)\u3002\uff08\u672c\u6587\u4f5c\u8005\u9673\u54c1\u5168\u7e6a\uff09<\/p><\/div>\n

\u5982\u679c\u8981\u8a08\u7b97\u67d0\u500b\u65cb\u5ea6 $$\\vec{\\nabla}\\times\\vec{v}$$ \u901a\u904e\u67d0\u500b\u66f2\u9762 $$\\Sigma$$\uff08\u4e0d\u4e00\u5b9a\u8981\u5c01\u9589\uff09\u7684\u7e3d\u901a\u91cf\uff0c\u90a3\u9ebc\u9019\u500b\u91cf\u6703\u7b49\u65bc\u8b93\u5411\u91cf $$\\vec{v}$$ \u6cbf\u8457\u66f2\u9762 $$\\Sigma$$ \u7684\u908a\u7de3\u5c01\u9589\u66f2\u7dda $$\\partial\\Sigma$$ \u7684\u8def\u5f91\u7a4d\u5206\u3002\u5982\u679c\u5c07\u9762\u7a4d $$\\Sigma$$ \u7e2e\u5230\u7121\u9650\u5c0f\uff0c\u90a3\u9ebc\u6211\u5011\u53ef\u4ee5\u5229\u7528\u65af\u6258\u514b\u65af\u5b9a\u7406\u91cd\u65b0\u5beb\u51fa\u65cb\u5ea6\u70ba\uff1a<\/p>\n

$$\\displaystyle\\vec{\\nabla}\\times\\vec{v}=\\lim_{A_{yz}\\to 0}\\frac{\\oint_{\\partial A_{yz}}\\vec{v}\\cdot d\\vec{s}}{A_{yz}}\\hat{i}+\\lim_{A_{xz}\\to 0}\\frac{\\oint_{\\partial A_{xz}}\\vec{v}\\cdot d\\vec{s}}{A_{xz}}\\hat{j}+\\lim_{A_{xy}\\to 0}\\frac{\\oint_{\\partial A_{xy}}\\vec{v}\\cdot d\\vec{s}}{A_{xy}}\\hat{k}$$<\/p>\n

\u5176\u4e2d $$A_{xy}$$ \u70ba x-y \u65b9\u5411\u4e0a\u7684\u4e00\u584a\u5c0f\u9762\u7a4d\uff0c$$A_{xz}$$ \u8207 $$A_{yz}$$ \u540c\u7406\u3002\u56e0\u70ba\u65cb\u5ea6\u662f\u5177\u6709\u65b9\u5411\u6027\u7684\uff0c\u4e00\u822c\u4f86\u8aaa\u5728\u7b1b\u5361\u5152\u5ea7\u6a19\u9700\u8981\u6709\u4e09\u500b\u4e0d\u540c\u65b9\u5411\u7684\u5206\u91cf\u624d\u80fd\u5b8c\u6574\u63cf\u8ff0 \u3002<\/p>\n

$$\\vec{\\nabla}\\times\\vec{v}$$ \u662f\u5177\u6709\u65b9\u5411\u6027\u7684\uff0c\u5b83\u7684\u65b9\u5411\u671d\u5411\u4f7f $$\\oint_{\\partial\\Sigma}\\vec{v}\\cdot d\\vec{s}$$ \u6700\u5927\u5316\u7684\u9762\u7a4d\u6cd5\u5411\u91cf $$\\hat{n}$$\u3002\u767d\u8a71\u4f86\u8aaa\uff0c\u5c31\u662f\u6211\u5011\u53ef\u4ee5\u9078\u64c7\u5f88\u591a\u5177\u6709\u4e0d\u540c\u6cd5\u5411\u91cf\u7684\u5c0f\u8ff4\u5708\u7576\u4f5c\u8def\u5f91\u74b0\u7a4d\u5206\u7684 $$\\partial \\Sigma$$\uff1b\u800c\u5728\u6211\u5011\u5f97\u5230\u6700\u5927\u503c\u7684\u6642\u5019\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u78ba\u5b9a\u65cb\u5ea6\u7684\u5411\u91cf\u7684\u65b9\u5411\u70ba\u5c0f\u8ff4\u5708\u7684\u6cd5\u5411\u91cf\uff0c\u800c $$\\vec{v}$$ \u4fbf\u662f\u6cbf\u8457\u9019\u500b\u6cd5\u5411\u91cf\u65cb\u8f49\uff08\u53f3\u624b\u5b9a\u5247\uff1a\u4ee5\u5927\u62c7\u6307\u70ba $$\\vec{\\nabla}\\times\\vec{v}$$ \uff0c\u5176\u9918\u56db\u6307\u70ba $$\\vec{v}$$ \u7684\u65b9\u5411\u3002)<\/p>\n

\"74787_p4\"<\/a>

\u5716\u56db\u3001\u53f3\u624b\u5b9a\u5247\u3002\uff08\u5716\u7247\u4f86\u6e90\uff1ahttp:\/\/en.wikipedia.org\/wiki\/File:Right-hand_grip_rule.svg\uff0c\u4f5c\u8005\uff1aSchorschi2\uff09<\/p><\/div>\n

\n

\u53c3\u8003\u6587\u737b<\/strong><\/p>\n

    \n
  1. Curl \u2014 http:\/\/en.wikipedia.org\/wiki\/Curl_(mathematics)<\/a><\/li>\n
  2. Divergence \u2014 http:\/\/en.wikipedia.org\/wiki\/Divergence<\/a><\/li>\n
  3. Stokes’ theorem \u2014 http:\/\/en.wikipedia.org\/wiki\/Stokes’_theorem<\/a><\/li>\n
  4. Divergence theorem \u2014 Wikipedia. http:\/\/en.wikipedia.org\/wiki\/Divergence_theorem<\/a><\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"

    \u6563\u5ea6\u548c\u65cb\u5ea6 (Divergence and Curl) \u570b\u7acb\u81fa\u7063\u5927\u5b78\u7269\u7406\u5b78\u7cfb \u9673\u54c1\u5168 \u6563\u5ea6 (Divergen…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[107,11002,27],"tags":[10998,10997,10999,11001,10996,11000,5736,461],"class_list":["post-74787","post","type-post","status-publish","format-standard","hentry","category-physics00","category-11002","category-physics06","tag-curl","tag-divergence","tag-gauss","tag-stokes","tag-10996","tag-11000","tag-5736","tag-461","loop-entry","cat-107","cat-11002","cat-27","no-thumbnail"],"views":13780,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/74787","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=74787"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/74787\/revisions"}],"predecessor-version":[{"id":85886,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/74787\/revisions\/85886"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=74787"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=74787"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=74787"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}