{"id":2105,"date":"2019-10-13T13:52:28","date_gmt":"2019-10-13T04:52:28","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=2105"},"modified":"2020-11-24T15:00:12","modified_gmt":"2020-11-24T06:00:12","slug":"calculus-stokes-theorem-and-divergence-theorem","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-stokes-theorem-and-divergence-theorem\/","title":{"rendered":"\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\uc640 \ubc1c\uc0b0 \uc815\ub9ac"},"content":{"rendered":"<style type=\"text\/css\">\ndiv.imgs-center { text-align: center; margin-bottom: 1.5em; line-height: 0.1em; }\ndiv.imgs-center .caption { text-align: center; font-size: 0.85em; }\n<\/style>\n<p>\ud3c9\uba74\uc5d0\uc11c \uc120\uc801\ubd84\uacfc \uc774\uc911\uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\ub294 \uadf8\ub9b0 \uc815\ub9ac\uac00 \uc788\ub294 \uac83\ucc98\ub7fc \uacf5\uac04\uc5d0\uc11c\ub3c4 \uc120\uc801\ubd84\uacfc \uba74\uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\ub294 \uc815\ub9ac, \uba74\uc801\ubd84\uacfc \uc0bc\uc911\uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\ub294 \uc815\ub9ac\uac00 \uc788\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uadf8\ub9b0 \uc815\ub9ac\ub97c 3\ucc28\uc6d0\uc73c\ub85c \ud655\uc7a5\ud55c \uc801\ubd84 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<div class=\"box contentindex\">\n<h3 class=\"indextitle\">\ub0b4\uc6a9 \uc21c\uc11c<\/h3>\n<ul>\n<li><a href=\"#curl\">\ud68c\uc804\ubca1\ud130\uc7a5<\/a><\/li>\n<li><a href=\"#stokestheorem\">\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac<\/a><\/li>\n<li><a href=\"#stokestheoremapp\">\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\uc758 \uc751\uc6a9<\/a><\/li>\n<li><a href=\"#divergencetheorem\">\ubc1c\uc0b0 \uc815\ub9ac<\/a><\/li>\n<li><a href=\"#divergencetheoremapp\">\ubc1c\uc0b0 \uc815\ub9ac\uc758 \uc751\uc6a9<\/a><\/li>\n<\/ul>\n<h3 class=\"pretitle\">\ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9<\/h3>\n<ul>\n<li>Line integral (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Line_integral\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>Green&#8217;s theorem (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Green%27s_theorem\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>Surface integral (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Surface_integral\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<div style=\"display: none; visibility: hidden;\">\n\\[<br \/>\n\\newcommand{\\curl}{{\\operatorname{curl}}}<br \/>\n\\newcommand{\\grad}{{\\operatorname{grad}}}<br \/>\n\\newcommand{\\opdiv}{{\\operatorname{div}}}<br \/>\n\\]\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<a name=\"curl\"><\/a><\/p>\n<h2 class=\"margintop2\">\ud68c\uc804\ubca1\ud130\uc7a5<\/h2>\n<p>\uc55e\uc5d0\uc11c \uadf8\ub9b0 \uc815\ub9ac\ub97c \uacf5\ubd80\ud558\uba74\uc11c \ud3c9\uba74\ubca1\ud130\uc7a5 \\(\\mathbf{F}(x,\\,y) = M(x,\\,y)\\mathbf{i} + N(x,\\,y)\\mathbf{j}\\)\uc758 \uc21c\ud658\ubc00\ub3c4\uac00 \ub2e4\uc74c\uacfc \uac19\uc74c\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4.<br \/>\n\\[\\frac{\\partial N}{\\partial x} &#8211; \\frac{\\partial M}{\\partial y}.\\tag{1.1}\\]<br \/>\n\uc774 \uac12\uc740 \ud3c9\uba74\uc5d0\uc11c \uc720\uccb4\uc758 \ud750\ub984\uc774 \\(\\mathbf{F}\\)\uc640 \uac19\uc744 \ub54c, \uc720\uccb4\uac00 \ud750\ub974\ub294 \uc9c0\uc810 \\((x,\\,y)\\)\uc5d0 \ub193\uc778 \uc678\ub95c(paddle wheel)\uc774 \uc5bc\ub9c8\ub098 \ube60\ub974\uac8c \ud68c\uc804\ud558\ub294\uc9c0\ub97c \ub098\ud0c0\ub0b8\ub2e4. \uc774 \uac12\uc774 \uc591\uc218\uc774\uba74 \uc678\ub95c\uc740 \uc704\uc5d0\uc11c \ubcf4\uc558\uc744 \ub54c \uc591\uc758 \ubc29\ud5a5(\ubc18\uc2dc\uacc4\ubc29\ud5a5)\uc73c\ub85c \ud68c\uc804\ud558\uace0, \uc774 \uac12\uc774 \uc74c\uc218\uc774\uba74 \uc678\ub95c\uc740 \uc74c\uc758 \ubc29\ud5a5\uc73c\ub85c \ud68c\uc804\ud558\uba70, \uc774 \uac12\uc774 \\(0\\)\uc774\uba74 \uc678\ub95c\uc740 \ud68c\uc804\ud558\uc9c0 \uc54a\ub294\ub2e4(\uadf8\ub9bc 1).<\/p>\n<div class=\"imgs-center\">\n<img fetchpriority=\"high\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-01.png\" alt=\"\" width=\"494\" height=\"219\" class=\"aligncenter size-full wp-image-5945\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-01.png 988w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-01-300x133.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-01-768x340.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-01-585x259.png 585w\" sizes=\"(max-width: 494px) 100vw, 494px\" \/><\/p>\n<div class=\"caption\">[\uadf8\ub9bc 1]<\/div>\n<\/div>\n<p>\uacf5\uac04\ubca1\ud130\uc7a5\uc5d0 \ub300\ud574\uc11c\ub3c4 \uc774\uc640 \ube44\uc2b7\ud55c \uc815\uc758\ub97c \ub3c4\uc785\ud560 \uc218 \uc788\ub2e4. \uacf5\uac04\ubca1\ud130\uc7a5 \\[\\mathbf{F} (x,\\,y,\\,z) = M(x,\\,y,\\,z)\\mathbf{i} + N(x,\\,y,\\,z)\\mathbf{j} + P(x,\\,y,\\,z)\\mathbf{k}\\]\uc758 \ubaa8\ub4e0 \uc131\ubd84\uc774 \ubaa8\ub4e0 \ubcc0\uc218\uc5d0 \ub300\ud558\uc5ec \ud3b8\ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c, \\(\\mathbf{F}\\)\uc758 <span class=\"defined\">\ud68c\uc804\ubca1\ud130\uc7a5<\/span>(curl)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\curl \\,\\mathbf{F} = \\left( \\frac{\\partial P}{\\partial y} &#8211; \\frac{\\partial N}{\\partial z} \\right) \\mathbf{i} + \\left(\\frac{\\partial M}{\\partial z} &#8211; \\frac{\\partial P}{\\partial x}\\right) \\mathbf{j} + \\left(\\frac{\\partial N}{\\partial x} &#8211; \\frac{\\partial M}{\\partial y}\\right) \\mathbf{k}.\\tag{1.2}\\]<br \/>\n\uc704 \uc2dd\uc744 \uac04\ub2e8\ud558\uac8c \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[\\curl \\,\\mathbf{F} = \\nabla \\times \\mathbf{F} .\\tag{1.3}\\]<br \/>\n\ud3c9\uba74\ubca1\ud130\uc7a5\uc758 \uc21c\ud658\ubc00\ub3c4\uac00 \uc2a4\uce7c\ub77c\uc778 \uac83\uacfc\ub294 \ub2ec\ub9ac \uacf5\uac04\ubca1\ud130\uc7a5\uc758 \ud68c\uc804\ubca1\ud130\uc7a5\uc740 \ubca1\ud130\uc774\ub2e4. \ud2b9\ud788 (2)\uc758 \\(\\mathbf{k}\\) \uc131\ubd84\uc740 (1)\uacfc \uac19\ub2e4. \uc989<br \/>\n\\[(\\curl \\,\\mathbf{F})\\cdot \\mathbf{k} = \\frac{\\partial N}{\\partial x} &#8211; \\frac{\\partial M}{\\partial y}\\tag{1.4}\\]<br \/>\n\uc774\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \uc21c\ud658\ubc00\ub3c4\ub97c \u2018\ud68c\uc804\ubca1\ud130\uc7a5\uc758 \\(\\mathbf{k}\\) \uc131\ubd84\u2019\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1<\/span><br \/>\n\ubca1\ud130\uc7a5 \\(\\mathbf{F} = (x^2 &#8211; z)\\mathbf{i} + xe^z\\,\\mathbf{j} + xy\\,\\mathbf{k}\\)\uc758 \ud68c\uc804\ubca1\ud130\uc7a5\uc744 \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[\\begin{align}<br \/>\n\\curl \\,\\mathbf{F}<br \/>\n&#038;= \\nabla \\times \\mathbf{F} \\\\[5pt]<br \/>\n&#038;= \\left\\lvert \\begin{array}{ccc} \\mathbf{i} &#038; \\mathbf{j} &#038; \\mathbf{k} \\\\ \\frac{\\partial}{\\partial x} &#038; \\frac{\\partial}{\\partial y} &#038; \\frac{\\partial}{\\partial z} \\\\ x^2 -z &#038; xe^z &#038; xy  \\end{array} \\right\\vert \\\\[5pt]<br \/>\n&#038;= (x-xe^z )\\mathbf{i} &#8211; (y+1)\\mathbf{j} + (e^z-0)\\mathbf{k} \\\\[5pt]<br \/>\n&#038;= x(1-e^z)\\mathbf{i} &#8211; (y+1)\\mathbf{j} + e^z\\,\\mathbf{k}.<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p>\ud68c\uc804\ubca1\ud130\uc7a5\uc740 \ud06c\uae30\uc640 \ubc29\ud5a5\uc744 \uac00\uc9c4\ub2e4. \uc774\ub97c \ubc14\ud0d5\uc73c\ub85c \ud68c\uc804\ubca1\ud130\uc7a5\uc758 \uc758\ubbf8\ub97c \ubb3c\ub9ac\uc801\uc73c\ub85c \uc124\uba85\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"imgs-center\">\n<img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-02.png\" alt=\"\" width=\"250\" height=\"176\" class=\"aligncenter size-full wp-image-5946\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-02.png 500w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-02-300x211.png 300w\" sizes=\"(max-width: 250px) 100vw, 250px\" \/><\/p>\n<div class=\"caption\">[\uadf8\ub9bc 2]<\/div>\n<\/div>\n<p>\\(\\mathbf{F}\\)\uac00 \uacf5\uac04\uc5d0\uc11c \uc720\uccb4\uc758 \ud750\ub984\uc744 \ub098\ud0c0\ub0b4\ub294 \uacf5\uac04\ubca1\ud130\uc7a5\uc774\ub77c\uace0 \ud558\uc790. \uace0\uc815\ub41c \uc9c0\uc810 \\((x,\\,y,\\,z)\\)\uc5d0 \uc678\ub95c\uc744 \ub450\uc5c8\uc744 \ub54c, \uc678\ub95c\uc774 \\(\\mathbf{F}\\)\uc758 \uc601\ud5a5\uc744 \ubc1b\uc544 \ud68c\uc804\ud558\ub294 \uc18d\ub3c4\ub294 \uc678\ub95c\uc758 \ucd95\uc758 \ubc29\ud5a5\uc5d0 \uc758\ud558\uc5ec \uc815\ub41c\ub2e4. \ub9cc\uc57d \\(\\nabla \\times \\mathbf{F} \\ne \\mathbf{0}\\)\uc774\ub77c\uba74, \uc678\ub95c\uc758 \ucd95\uc774 \\(\\nabla \\times \\mathbf{F}\\)\uc640 \ud3c9\ud589\ud560 \ub54c \uc678\ub95c\uc774 \uac00\uc7a5 \ube60\ub974\uac8c \ud68c\uc804\ud558\uba70, \uc774\ub54c \uc678\ub95c\uc758 \ud68c\uc804 \ubc29\ud5a5\uc740 \\(\\nabla \\times \\mathbf{F}\\)\uc758 \ubc29\ud5a5\uc744 \uae30\uc900\uc73c\ub85c \ud558\ub294 \uc624\ub978\uc190 \ubc95\uce59\uc744 \ub530\ub974\ub294 \ubc29\ud5a5\uc774\ub2e4(\uadf8\ub9bc 2).\n<\/p>\n<p>\ud2b9\ud788, \\(\\mathbf{n}\\)\uc774 \ub2e8\uc704\ubca1\ud130\uc77c \ub54c \\((\\nabla \\times \\mathbf{F})\\cdot\\mathbf{n}\\)\uc740 \uc810 \\((x,\\,y,\\,z)\\)\uc5d0 \ub193\uc5ec \uc788\uace0 \ucd95\uc758 \ubc29\ud5a5\uc774 \\(\\mathbf{n}\\)\uacfc \uc77c\uce58\ud558\ub294 \uc678\ub95c\uc774 \uc5bc\ub9c8\ub098 \ube60\ub974\uac8c \ud68c\uc804\ud558\ub294\uc9c0\ub97c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p><!-- ##################################################################### --><br \/>\n<a name=\"stokestheorem\"><\/a><\/p>\n<h2 class=\"margintop2\">\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac<\/h2>\n<p>\uadf8\ub9b0 \uc815\ub9ac\ub294 \ud3c9\uba74\uc601\uc5ed\uc758 \uacbd\uacc4\uc120\uc5d0\uc11c\uc758 \uc120\uc801\ubd84\uacfc \ud3c9\uba74\uc601\uc5ed\uc5d0\uc11c\uc758 \uc911\uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud574 \uc900\ub2e4. \uc989 \ud3c9\uba74\uace1\uc120 \\(C\\)\uac00 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b0\uba70 \ub2eb\ud600 \uc788\ub294 \ub2e8\uc21c\uace1\uc120\uc774\uace0, \\(C\\)\uc5d0 \uc758\ud558\uc5ec \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed\uc744 \\(R\\)\ub77c\uace0 \ud560 \ub54c, \\(C\\)\uc640 \\(R\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc5f4\ub9b0 \uc601\uc5ed\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud3b8\ub3c4\ud568\uc218\ub97c \uac16\ub294 \ud3c9\uba74\ubca1\ud130\uc7a5 \\(\\mathbf{F} = M\\,\\mathbf{i} + N\\,\\mathbf{j}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\oint_C M\\,dx + N\\,dy = \\iint_R \\left( \\frac{\\partial N}{\\partial x} &#8211; \\frac{\\partial M}{\\partial y} \\right) dx\\,dy .\\tag{2.1}\\]<br \/>\n\uc774 \uc2dd\uc758 \uc88c\ubcc0\uc740 \\(C\\)\ub97c \ub530\ub77c \ud750\ub974\ub294 \\(\\mathbf{F}\\)\uc758 \ubc18\uc2dc\uacc4\ubc29\ud5a5 \uc21c\ud658\uc774\uba70, \uc6b0\ubcc0\uc740 \uc601\uc5ed \\(R\\)\uc5d0\uc11c \uc21c\ud658\ubc00\ub3c4 \\((\\curl\\,\\mathbf{F})\\cdot\\mathbf{k}\\)\uc758 \uc774\uc911\uc801\ubd84\uc774\ub2e4.<\/p>\n<p>\uc774\ub7ec\ud55c \uc815\ub9ac\ub97c 3\ucc28\uc6d0 \uacf5\uac04\uc73c\ub85c \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4. \uc774\uac83\uc744 \uc704\ud574\uc11c \uba3c\uc800 \u2018\ub2e4\uc591\uccb4\uacbd\uacc4\u2019\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud574\uc57c \ud55c\ub2e4. \ub2e4\uc74c\uacfc \uac19\uc740 \uc9d1\ud569\uc744 \uc0dd\uac01\ud574 \ubcf4\uc790.<br \/>\n\\[S_1 = \\left\\{ (x,\\,y,\\,z) \\,\\vert\\, x^2 + y^2 \\le 1 ,\\,\\, z=0 \\right\\}\\tag{2.2}\\]<br \/>\n\uc9d1\ud569 \\(S_1\\)\uc740 \\(\\mathbb{R}^3\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uba70, \uc774 \uc9d1\ud569\uc758 \ubaa8\ub4e0 \uc810\uc740 \uc790\uae30 \uc790\uc2e0\uc758 \uacbd\uacc4\uc810\uc774\ub2e4. \ud558\uc9c0\ub9cc \\(S_1\\)\uc744 \\(xy\\) \ud3c9\uba74\uc5d0 \ub193\uc778 \uc870\uac01\uc774\ub77c \uc0dd\uac01\ud558\uba74 \uc774 \uc9d1\ud569\uc758 \uacbd\uacc4\ub97c \uc6d0<br \/>\n\\[C = \\left\\{ (x,\\,y) \\,\\vert\\, x^2 + y^2 = 1\\right\\}\\tag{2.3}\\]<br \/>\n\uc774\ub77c\uace0 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774\ub54c \\(C\\)\ub97c \\(S_1\\)\uc758 <span class=\"defined\">\ub2e4\uc591\uccb4\uacbd\uacc4<\/span>(manifold boundary)\ub77c\uace0 \ubd80\ub978\ub2e4. \ub2e4\uc591\uccb4\uacbd\uacc4\uc640 \uad6c\ubd84\ud558\uae30 \uc704\ud558\uc5ec \uae30\uc874\uc758 \uacbd\uacc4\ub97c <span class=\"defined\">\uc704\uc0c1\uacbd\uacc4<\/span>(topological boundary)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc774\ud574\ub97c \ub3d5\uae30 \uc704\ud574 \ub2e4\uc591\uccb4\uacbd\uacc4\uc758 \uc608\ub97c \ud558\ub098 \ub354 \uc0b4\ud3b4\ubcf4\uc790. \ub2e4\uc74c\uacfc \uac19\uc740 \ubc18\uad6c\uba74 \\(S_2\\)\ub97c \uc0dd\uac01\ud558\uc790.<br \/>\n\\[S_2 = \\left\\{ (x,\\,y,\\,z) \\,\\vert\\, x^2 + y^2 + z^2 = 1 ,\\,\\, z \\ge 0 \\right\\}\\tag{2.4}\\]<br \/>\n\\(S_2\\)\uc758 \uc704\uc0c1\uacbd\uacc4\ub294 \uc790\uae30 \uc790\uc2e0\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(S_2\\)\uc758 \ub2e4\uc591\uccb4\uacbd\uacc4\ub294 (2.2)\uc758 \ub2e4\uc591\uccb4\uacbd\uacc4\uc640 \uac19\uc740 (2.3)\uc758 \uc6d0 \\(C\\)\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \uadf8\ub9b0 \uc815\ub9ac\ub97c 3\ucc28\uc6d0 \uacf5\uac04\uc73c\ub85c \ud655\uc7a5\ud55c \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uc5ec\uae30\uc11c\ub294 \uc99d\uba85\uc740 \ud558\uc9c0 \uc54a\uace0 \uc2dd\uc758 \uc758\ubbf8\ub97c \uc0b4\ud3b4\ubcf4\uaca0\ub2e4. \uc99d\uba85\uc774 \uad81\uae08\ud558\uba74 \ubbf8\uc801\ubd84\ud559 \uad50\uc7ac\ub97c \ucc38\uace0\ud558\uae30 \ubc14\ub780\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 1. (Stokes \uc815\ub9ac)<\/span><br \/>\n\\(S\\)\uac00 \uacf5\uac04\uc5d0 \ub193\uc778 \uace1\uba74\uc774\uba70 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7fd\uace0 \ubc29\ud5a5\uc744 \uac00\uc9c4 \uace1\uba74\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(S\\)\uc758 \ub2e4\uc591\uccb4\uacbd\uacc4 \\(C\\)\uac00 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc774\ub77c\uace0 \ud558\uc790(\uadf8\ub9bc 3).<\/p>\n<div class=\"imgs-center\">\n<img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-11-stokes-theorem-m1.png\" alt=\"\" width=\"244\" height=\"158\" class=\"aligncenter size-full wp-image-5961\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-11-stokes-theorem-m1.png 488w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-11-stokes-theorem-m1-300x194.png 300w\" sizes=\"(max-width: 244px) 100vw, 244px\" \/><\/p>\n<div class=\"caption\">[\uadf8\ub9bc 3]<\/div>\n<\/div>\n<p>\ubca1\ud130\uc7a5 \\(\\mathbf{F} = M\\,\\mathbf{i} + N\\,\\mathbf{j} + P\\,\\mathbf{k}\\)\uac00 \\(S\\)\ub97c \ud3ec\ud568\ud558\ub294 \ud55c \uc5f4\ub9b0 \uc601\uc5ed\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0, \uadf8 \uc601\uc5ed\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud3b8\ub3c4\ud568\uc218\ub97c \uac00\uc9c0\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\oint_C \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times \\mathbf{F})\\cdot \\mathbf{n} \\,d\\sigma .\\tag{2.5}\\]<br \/>\n\uc774 \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc740 \uace1\uba74\uc758 \ub2e8\uc704\ubc95\uc120\ubca1\ud130 \\(\\mathbf{n}\\)\uc5d0 \uad00\ud558\uc5ec \uc624\ub978\uc190 \ubc95\uce59\uc5d0 \ub530\ub77c \ubc18\uc2dc\uacc4\ubc29\ud5a5\uc73c\ub85c \\(C\\)\ub97c \ub530\ub77c \ud68c\uc804\ud558\ub294 \\(\\mathbf{F}\\)\uc758 \uc21c\ud658\uc774\uba70, \uc6b0\ubcc0\uc740 \uace1\uba74 \\(S\\) \uc704\uc5d0\uc11c \uc21c\ud658\ubc00\ub3c4 \\((\\nabla \\times \\mathbf{F})\\cdot \\mathbf{n}\\)\uc758 \uc801\ubd84\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\uc640 \uadf8\ub9b0 \uc815\ub9ac\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4 \ubcf4\uc790.<\/p>\n<div class=\"imgs-center\">\n<img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-12-green_and_stokes-theorem.png\" alt=\"\" width=\"483\" height=\"147\" class=\"aligncenter size-full wp-image-5948\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-12-green_and_stokes-theorem.png 966w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-12-green_and_stokes-theorem-300x91.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-12-green_and_stokes-theorem-768x234.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-12-green_and_stokes-theorem-585x178.png 585w\" sizes=\"(max-width: 483px) 100vw, 483px\" \/><\/p>\n<div class=\"caption\">[\uadf8\ub9bc 4]<\/div>\n<\/div>\n<p>\\(S\\)\uac00 \\(xy\\) \ud3c9\uba74\uc5d0 \ub193\uc778 \uc601\uc5ed\uc774\uace0 \uadf8\ub9b0 \uc815\ub9ac\ub97c \uc801\uc6a9\ud560 \uc218 \uc788\ub294 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(d\\sigma = dx\\,dy\\)\uc774\uace0 (1.4)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[(\\nabla \\times \\mathbf{F})\\cdot\\mathbf{n} = (\\nabla \\times \\mathbf{F})\\cdot\\mathbf{k} = \\frac{\\partial N}{\\partial x} &#8211; \\frac{\\partial M}{\\partial y}\\tag{2.6}\\]<br \/>\n\uc774\ubbc0\ub85c \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\uc758 \uc2dd (2.5)\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\oint_C \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times \\mathbf{F})\\cdot \\mathbf{n} \\,d\\sigma =\\iint_S \\left( \\frac{\\partial N}{\\partial x} &#8211; \\frac{\\partial M}{\\partial y}\\right) dx\\,dy.\\tag{2.7}\\]<br \/>\n\uc774\uac83\uc740 \uace7 \uadf8\ub9b0 \uc815\ub9ac\uc640 \uac19\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span><br \/>\n\uc0c1\ubc18\uad6c\uba74<br \/>\n\\[S = \\left\\{ (x,\\,y,\\,z) \\,\\vert\\, x^2 + y^2 + z^2 = 9 ,\\,\\, z\\ge 0\\right\\}\\tag{2.8}\\]<br \/>\n\uc5d0 \uc6d0\uc810\uc73c\ub85c\ubd80\ud130 \uba40\uc5b4\uc9c0\ub294 \ubc29\ud5a5(\\(z\\)\ucd95\uc758 \uc591\uc758 \ubc29\ud5a5\uc5d0 \uac00\uae4c\uc6b4 \ubc29\ud5a5)\uc73c\ub85c \ub2e8\uc704\ubc95\uc120\ubca1\ud130 \\(\\mathbf{n}\\)\uc774 \uc8fc\uc5b4\uc838 \uc788\uace0, \\(\\mathbf{n}\\)\uc744 \uae30\uc900\uc73c\ub85c \uc624\ub978\uc190 \ubc95\uce59\uc744 \ub530\ub77c \uc591\uc758 \ubc29\ud5a5\uc73c\ub85c \ud68c\uc804\uc774 \uc8fc\uc5b4\uc838 \uc788\uc73c\uba70 \\(S\\)\uc758 \ub2e4\uc591\uccb4\uacbd\uacc4\uc5d0 \ud574\ub2f9\ud558\ub294 \uc6d0<br \/>\n\\[C = \\left\\{(x,\\,y) \\,\\vert\\, x^2 + y^2 = 9\\right\\}\\tag{2.9}\\]<br \/>\n\uc5d0 \\(S\\)\uc758 \ud68c\uc804\ubc29\ud5a5\uacfc \uc77c\uce58\ud558\ub294 \ubc29\ud5a5\uc774 \uc8fc\uc5b4\uc838 \uc788\ub2e4\uace0 \ud558\uc790(\uadf8\ub9bc 5).<\/p>\n<div class=\"imgs-center\">\n<img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-13-ex2.png\" alt=\"\" width=\"217\" height=\"176\" class=\"aligncenter size-full wp-image-5949\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-13-ex2.png 434w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-13-ex2-300x243.png 300w\" sizes=\"(max-width: 217px) 100vw, 217px\" \/><\/p>\n<div class=\"caption\">[\uadf8\ub9bc 5]<\/div>\n<\/div>\n<p>\ubca1\ud130\uc7a5 \\(\\mathbf{F} = y\\,\\mathbf{i} &#8211; x\\,\\mathbf{j}\\)\uc5d0 \ub300\ud558\uc5ec \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub97c \ud655\uc778\ud574 \ubcf4\uc790.<\/p>\n<p>\uba3c\uc800 \\(C\\)\ub97c \ub530\ub974\ub294 \uc120\uc801\ubd84, \uc989 (2.5)\uc758 \uc88c\ubcc0\uc744 \uacc4\uc0b0\ud574 \ubcf4\uc790. \uc6d0 \\(C\\)\ub97c \ub9e4\uac1c\ud654\ud558\uba74<br \/>\n\\[\\mathbf{r}(\\theta) = (3\\cos\\theta) \\mathbf{i} + (3\\sin\\theta )\\mathbf{j} ,\\,\\, 0\\le \\theta\\le 2\\pi\\]<br \/>\n\uc774\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nd\\mathbf{r} &#038;= (-3\\sin\\theta \\,d\\theta )\\mathbf{i} + (3\\cos\\theta \\,d\\theta )\\mathbf{j}, \\\\[5pt]<br \/>\n\\mathbf{F} &#038;= y\\,\\mathbf{i} &#8211; x\\,\\mathbf{j} = (3\\sin\\theta )\\mathbf{i} &#8211; (3\\cos\\theta )\\mathbf{j}, \\\\[5pt]<br \/>\n\\mathbf{F} \\cdot d\\mathbf{r} &#038;= -9 \\sin^2 \\theta \\,d\\theta &#8211; 9\\cos^2 \\theta \\,d\\theta =-9 \\,d\\theta, \\\\[5pt]<br \/>\n\\oint_C \\mathbf{F} \\cdot d\\mathbf{r} &#038;= \\int_0^{2\\pi} (-9)d\\theta = &#8211; 18 \\pi . \\tag{2.10}<br \/>\n\\end{align}\\]<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(S\\) \uc704\uc5d0\uc11c \uc21c\ud658\ubc00\ub3c4\uc758 \uc801\ubd84, \uc989 (2.5)\uc758 \uc6b0\ubcc0\uc744 \uacc4\uc0b0\ud574 \ubcf4\uc790. \uace1\uba74 \\(S\\)\ub97c<br \/>\n\\[f(x,\\,y,\\,z) = x^2 + y^2 + z^2 = 9\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b4\uba74<br \/>\n\\[\\begin{align}<br \/>\n\\lvert \\nabla f \\rvert &#038;= \\lvert 2x\\,\\mathbf{i} + 2y\\,\\mathbf{j} + 2z\\,\\mathbf{k} \\rvert = 2\\sqrt{x^2 + y^2 + z^2} = 6 ,\\\\[5pt]<br \/>\n\\lvert \\nabla f \\cdot \\mathbf{k} \\rvert &#038;= \\lvert 2z \\rvert = 2z ,\\\\[5pt]<br \/>\nd\\sigma &#038;= \\frac{\\lvert \\nabla f \\rvert}{\\lvert \\nabla f \\cdot \\mathbf{k} \\rvert} dA = \\frac{3}{z} dA<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(dA\\)\ub294 \\(xy\\) \ud3c9\uba74 \uc704\uc758 \uc6d0\ud310 \\(x^2 + y^2 \\le 9\\)\uc758 \ubbf8\ubd84\uc18c\ub97c \ub098\ud0c0\ub0b8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\nabla \\times \\mathbf{F} &#038;= (0-0) \\mathbf{i} + (0-0)\\mathbf{j} + (-1-1)\\mathbf{k} = -2\\mathbf{k} ,\\\\[5pt]<br \/>\n\\mathbf{n} &#038;= \\frac{x\\,\\mathbf{i} + y\\,\\mathbf{j} + z\\,\\mathbf{k}}{\\sqrt{x^2 + y^2 + z^2}} = \\frac{x\\,\\mathbf{i} + y\\,\\mathbf{j} + z\\,\\mathbf{k}}{3},\\\\[5pt]<br \/>\nd\\sigma &#038;= \\frac{3}{z} dA ,\\\\[5pt]<br \/>\n(\\nabla \\times \\mathbf{F}) \\cdot \\mathbf{n} &#038;= &#8211; \\frac{2z}{3} \\frac{3}{z} dA = -2\\,dA,\\\\[5pt]<br \/>\n\\iint_S (\\nabla \\times \\mathbf{F} )\\cdot \\mathbf{n} \\,d\\sigma &#038;= \\iint_{x^2 +y^2 \\le 9} -2 \\,dA = -18 \\pi . \\tag{2.11}<br \/>\n\\end{align}\\]<br \/>\n\uacc4\uc0b0 \uacb0\uacfc (2.10)\uacfc (2.11)\uc744 \ube44\uad50\ud558\uba74 \uc77c\uce58\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span><br \/>\n\ubcf4\uae30 2\uc758 \ubca1\ud130\uc7a5 \\(\\mathbf{F} = y\\,\\mathbf{i} &#8211; x\\,\\mathbf{j}\\)\uac00 \\(xy\\) \ud3c9\uba74\uc758 \uc6d0\ud310 \\(x^2 + y^2 \\le 9\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc601\uc5ed\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud3b8\ub3c4\ud568\uc218\ub97c \uac00\uc9c0\ubbc0\ub85c, \ubcf4\uae30 2\uc758 \uc801\ubd84\uc740 \uadf8\ub9b0 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uc5ec \ub354 \uc27d\uac8c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \\(xy\\) \ud3c9\uba74\uc5d0\uc11c (2.9)\uc758 \uc6d0 \\(C\\)\ub97c \ub458\ub808\ub85c \uac16\ub294 \uc6d0\ud310 \uc601\uc5ed\uc744 \\(A\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<div class=\"imgs-center\">\n<img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-13-ex2.png\" alt=\"\" width=\"217\" height=\"176\" class=\"aligncenter size-full wp-image-5949\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-13-ex2.png 434w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-13-ex2-300x243.png 300w\" sizes=\"(max-width: 217px) 100vw, 217px\" \/><\/p>\n<div class=\"caption\">[\uadf8\ub9bc 5]<\/div>\n<\/div>\n<p>\uadf8\ub7ec\uba74 \\(A\\) \uc704\uc5d0\uc11c \\(d\\sigma = dA\\)\uc774\uace0 \\(\\mathbf{n} = \\mathbf{k}\\)\uc774\ubbc0\ub85c<br \/>\n\\[(\\nabla \\times \\mathbf{F})\\cdot \\mathbf{n} \\,d\\sigma = -2 \\,\\mathbf{k} \\cdot \\mathbf{k} \\,dA = -2 \\,dA\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uadf8\ub9b0 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\oint_C \\mathbf{F} \\cdot d\\mathbf{r} =<br \/>\n\\iint_A ( \\nabla \\times \\mathbf{F})\\cdot \\mathbf{k} \\,dA = \\iint_A (-2)dA = -18 \\pi\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<a name=\"stokestheoremapp\"><\/a><\/p>\n<h2 class=\"margintop2\">\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\uc758 \uc751\uc6a9<\/h2>\n<p>\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74 \uc55e\uc5d0\uc11c \uc0dd\ub7b5\ud558\uc600\ub358 \ubcf4\uc874\uc801 \ubca1\ud130\uc7a5\uc758 \uc131\uc9c8\uc758 \uc99d\uba85\uc744 \ud560 \uc218 \uc788\ub2e4. \uba3c\uc800 \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud568\uc744 \ud655\uc778\ud558\uc790.<\/p>\n<div class=\"box oneline\">\n\\[\\curl \\,\\grad f = \\mathbf{0}\\tag{3.1}\\]\n<\/div>\n<p>\ub4f1\uc2dd (3.1)\uc740 \uc815\uc758\uc5d0 \ub530\ub77c \uacc4\uc0b0\ud558\uba74 \ubc14\ub85c \uc5bb\uc744 \uc218 \uc788\ub2e4. \uc989<br \/>\n\\[\\begin{align}<br \/>\n\\curl \\,\\grad f<br \/>\n&#038;= \\nabla \\times \\nabla f \\\\[5pt]<br \/>\n&#038;= (f_{zy} &#8211; f_{yz})\\mathbf{i} &#8211; (f_{zx} &#8211; f_{xz})\\mathbf{j} + (f_{yx} &#8211; f_{xy})\\mathbf{k} \\tag{3.2} \\\\[5pt]<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \ud074\ub798\ub85c\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \ud63c\ud569\ud3b8\ub3c4\ud568\uc218\uc758 \ud3b8\ubbf8\ubd84 \uc21c\uc11c\ub97c \ubc14\uafb8\uc5b4\ub3c4 \uacb0\uacfc\uac00 \ub3d9\uc77c\ud558\ubbc0\ub85c, (3.2)\uc758 \ub9c8\uc9c0\ub9c9 \uc2dd\uc758 \uac01 \uc131\ubd84\uc740 \ubaa8\ub450 \\(0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (3.1)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uc5ec\uae30\uc11c \uba48\ucd94\uc9c0 \ub9d0\uace0 (3.2)\uc758 \uc2dd\uc744 \uc798 \uc0b4\ud3b4\ubcf4\uc790. \ub9cc\uc57d<br \/>\n\\[\\mathbf{F} = M\\,\\mathbf{i} + N\\,\\mathbf{j} + P\\,\\mathbf{k} = \\nabla f\\tag{3.3}\\]<br \/>\n\ub77c\uba74, (3.2)\uc758  \uc6b0\ubcc0\uc758 \uac01 \uc131\ubd84\uc774 \\(0\\)\uc774\ub77c\ub294 \uac83\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[\\frac{\\partial P}{\\partial y} = \\frac{\\partial N}{\\partial z}  ,\\,\\,<br \/>\n\\frac{\\partial M}{\\partial z} = \\frac{\\partial P}{\\partial x}  ,\\,\\,<br \/>\n\\frac{\\partial N}{\\partial x} = \\frac{\\partial M}{\\partial y}  .\\tag{3.4}\\]<br \/>\n\uc774\uac83\uc740 \\(\\mathbf{F}\\)\uac00 \ubcf4\uc874\uc801 \ubca1\ud130\uc7a5\uc778\uc9c0\ub97c \ud310\ubcc4\ud558\ub294 \uc131\ubd84 \ud310\uc815\ubc95\uc758 \uc2dd\uacfc \uc77c\uce58\ud55c\ub2e4. 16.6\uc808\uc5d0\uc11c \\(\\mathbf{F}\\)\uac00 \ubcf4\uc874\uc801 \ubca1\ud130\uc7a5\uc77c \ub54c (3.4)\uac00 \uc131\ub9bd\ud55c\ub2e4\ub294 \uc0ac\uc2e4\ub9cc \uc99d\uba85\ud558\uace0 \uadf8 \uc5ed\uc740 \uc99d\uba85\ud558\uc9c0 \uc54a\uc558\ub2e4. \uc774\uc81c \uadf8 \uc5ed\uc744 \uc99d\uba85\ud558\uc790.<\/p>\n<p>\ub2e8\uc21c\uc5f0\uacb0\uc601\uc5ed \\(D\\)\uc5d0\uc11c (3.4)\uac00 \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\nabla \\times \\mathbf{F} = \\mathbf{0}\\tag{3.5}\\]<br \/>\n\uc774\ub2e4. \\(C\\)\uac00 \\(D\\)\uc5d0 \ud3ec\ud568\ub418\uace0 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \ub2eb\ud78c \ub2e8\uc21c\uace1\uc120\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(C\\)\ub97c \ub2e4\uc591\uccb4\uacbd\uacc4\ub85c \uac16\uace0 \\(D\\)\uc5d0 \ud3ec\ud568\ub418\uba70 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uc591\uba74\uace1\uba74 \ud558\ub098\ub97c \ud0dd\ud558\uc5ec \\(S\\)\ub77c\uace0 \ud558\uc790. \\(S\\)\uc758 \ub450 \ubc29\ud5a5 \uc911 \ud558\ub098\ub97c \uc784\uc758\ub85c \ud0dd\ud558\uace0 \uadf8 \ubc29\ud5a5\uacfc \uc77c\uce58\ud558\ub294 \\(S\\) \uc704\uc758 \ub2e8\uc704\ubc95\uc120\ubca1\ud130\ub97c \\(\\mathbf{n}\\)\uc774\ub77c \ud558\uba70, \\(\\mathbf{n}\\)\uc744 \uae30\uc900\uc73c\ub85c \uc624\ub978\uc190 \ubc95\uce59\uc744 \ub530\ub974\ub294 \ud68c\uc804\ubc29\ud5a5\uacfc \uc77c\uce58\ud558\ub3c4\ub85d \\(C\\)\uc5d0 \ubc29\ud5a5\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\oint_C \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times \\mathbf{F}) \\cdot \\mathbf{n} \\,d\\sigma = \\iint_S \\mathbf{0} \\cdot \\mathbf{n} \\,d\\sigma = 0.\\]<br \/>\n\uc5ec\uae30\uc11c \\(C\\)\uac00 \\(D\\)\uc5d0 \ud3ec\ud568\ub418\uace0 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \u2018\uc784\uc758\uc758\u2019 \ub2eb\ud78c \ub2e8\uc21c\uace1\uc120\uc774\ubbc0\ub85c \\(\\mathbf{F}\\)\ub294 \\(R\\)\uc5d0\uc11c \ubcf4\uc874\uc801\uc774\ub2e4.<\/p>\n<p><!-- ##################################################################### --><br \/>\n<a name=\"divergencetheorem\"><\/a><\/p>\n<h2 class=\"margintop2\">\ubc1c\uc0b0 \uc815\ub9ac<\/h2>\n<p>\uadf8\ub9b0 \uc815\ub9ac\ub97c 3\ucc28\uc6d0 \uacf5\uac04\uc73c\ub85c \ud655\uc7a5\ud55c \ub610 \ub2e4\ub978 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ud3c9\uba74\uace1\uc120 \\(C\\)\uac00 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b0\uba70 \ub2eb\ud600 \uc788\ub294 \ub2e8\uc21c\uace1\uc120\uc774\uace0 \\(C\\)\uc5d0 \uc758\ud558\uc5ec \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed\uc744 \\(R\\)\ub77c\uace0 \ud560 \ub54c, \\(C\\)\uc640 \\(R\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc5f4\ub9b0 \uc601\uc5ed\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud3b8\ub3c4\ud568\uc218\ub97c \uac16\ub294 \ud3c9\uba74\ubca1\ud130\uc7a5 \\(\\mathbf{F} = M\\,\\mathbf{i} + N \\,\\mathbf{j}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\oint_C M\\,dy &#8211; N\\,dx = \\iint_R \\left( \\frac{\\partial M}{\\partial x} + \\frac{\\partial N}{\\partial y}\\right) dx\\,dy.\\tag{4.1}\\]<br \/>\n\uc774 \uc2dd\uc758 \uc88c\ubcc0\uc740 \uace1\uc120 \\(C\\)\uc758 \ubc14\uae65\ucabd\uc73c\ub85c \ud758\ub7ec \ub098\uac00\ub294 \\(\\mathbf{F}\\)\uc758 \uc720\ucd9c(flux)\uc774\uba70, \uc6b0\ubcc0\uc740 \uc601\uc5ed \\(R\\)\uc5d0\uc11c \uc720\ucd9c\ubc00\ub3c4 \\(\\opdiv \\,\\mathbf{F}\\)\uc758 \uc774\uc911\uc801\ubd84\uc774\ub2e4.<\/p>\n<p>\uc720\uccb4\uc758 \ud750\ub984\uc744 \uc774\uc6a9\ud558\uc5ec \uc2dd (4.1)\uc744 \uc124\uba85\ud560 \uc218 \uc788\ub2e4. \ud3c9\ud3c9\ud55c \uc9c0\ud45c\uba74 \ubc14\ub2e5\uc5d0\uc11c \ubb3c\uc774 \ud758\ub7ec\ub098\uc624\uace0 \uc788\ub294\ub370, \uadf8 \uc601\uc5ed\uc758 \uace1\uc120 \\(C\\)\uc5d0 \ud574\ub2f9\ud558\ub294 \ubd80\ubd84\uc5d0 \uc6b8\ud0c0\ub9ac\ub97c \uccd0 \ub450\uc5c8\ub2e4. \\(C\\) \uc548\ucabd \ubc14\ub2e5\uba74\uc5d0\uc11c \ub2e8\uc704\uc2dc\uac04\ub3d9\uc548 \ubb3c\uc774 \uc5bc\ub9c8\ub098 \ud758\ub7ec\ub098\uc624\ub294\uc9c0 \uad6c\ud558\uace0 \uc2f6\ub2e4\uba74 (4.1)\uc758 \uc6b0\ubcc0\uc744 \uacc4\uc0b0\ud558\uba74 \ub41c\ub2e4. \uadf8\ub7f0\ub370 \uc774\uac83\uc740 \ub2e8\uc704\uc2dc\uac04\ub3d9\uc548 \uc6b8\ud0c0\ub9ac \\(C\\)\ub97c \ub118\uc5b4\uc11c \ud758\ub7ec\ub098\uc624\ub294 \ubb3c\uc758 \uc591\uc744 \uad6c\ud558\uba74 \ub418\ubbc0\ub85c (4.1)\uc758 \uc88c\ubcc0\uc744 \uacc4\uc0b0\ud558\uba74 \ub41c\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c (4.1)\uc758 \uc88c\ubcc0\uacfc \uc6b0\ubcc0\uc740 \uac19\uc740 \uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<p>\uc720\ucd9c\ubc00\ub3c4 \\(\\opdiv\\,\\mathbf{F}\\)\ub294 \ud1b5\uc0c1\uc801\uc73c\ub85c \u2018\ubc1c\uc0b0\u2019\uc774\ub77c\ub294 \uc774\ub984\uc73c\ub85c \ub354 \ub9ce\uc774 \ubd88\ub9ac\ubbc0\ub85c, \uc5ec\uae30\uc11c\ub3c4 \u2018\ubc1c\uc0b0\u2019\uc774\ub77c\ub294 \uc774\ub984\uc744 \uc0ac\uc6a9\ud558\uaca0\ub2e4. \ud3c9\uba74\ubca1\ud130\uc7a5\uc758 \ubc1c\uc0b0\uc740 \uc790\uc5f0\uc2a4\ub7fd\uac8c \uacf5\uac04\ubca1\ud130\uc7a5\uc73c\ub85c \ud655\uc7a5\ub420 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"imgs-center\">\n<img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-41-divergence.png\" alt=\"\" width=\"438\" height=\"164\" class=\"aligncenter size-full wp-image-5950\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-41-divergence.png 875w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-41-divergence-300x112.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-41-divergence-768x288.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-41-divergence-585x219.png 585w\" sizes=\"(max-width: 438px) 100vw, 438px\" \/><\/p>\n<div class=\"caption\">[\uadf8\ub9bc 6]<\/div>\n<\/div>\n<p>3\ucc28\uc6d0 \uacf5\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \ubaa8\ub4e0 \uc131\ubd84\uc774 \ud3b8\ub3c4\ud568\uc218\ub97c \uac16\ub294 \ubca1\ud130\uc7a5<br \/>\n\\[\\mathbf{F} = M\\,\\mathbf{i} + N\\,\\mathbf{j} + P\\,\\mathbf{k}\\]<br \/>\n\uc758 <span class=\"defined\">\ubc1c\uc0b0<\/span>(divergence)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\opdiv\\,\\mathbf{F} = \\nabla \\cdot \\mathbf{F} = \\frac{\\partial M}{\\partial x} + \\frac{\\partial N}{\\partial y} + \\frac{\\partial P}{\\partial z}.\\tag{4.2}\\]<br \/>\n\uc720\uccb4\uc758 \ud750\ub984\uc774 \\(\\mathbf{F}\\)\uc640 \uac19\uc744 \ub54c, \\(\\opdiv \\,\\mathbf{F} > 0\\)\uc774\uba74 \uc720\uccb4\uac00 \ud33d\ucc3d\ud558\ub294 \uac83\uc774\uba70, \\(\\opdiv \\,\\mathbf{F} < 0\\)\uc774\uba74 \uc720\uccb4\uac00 \uc555\ucd95\ub418\uace0 \uc788\ub294 \uac83\uc774\ub2e4(\uadf8\ub9bc 6).<\/p>\n<p>\uadf8\ub9b0 \uc815\ub9ac\uc758 \uacf5\uc2dd (4.1)\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc790\uc5f0\uc2a4\ub7fd\uac8c 3\ucc28\uc6d0 \uacf5\uac04\uc73c\ub85c \ud655\uc7a5\ub41c\ub2e4. \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c, \uc5ec\uae30\uc11c\ub294 \uc99d\uba85\uc740 \ud558\uc9c0 \uc54a\uace0 \uc2dd\uc758 \uc758\ubbf8\ub97c \uc0b4\ud3b4\ubcf4\uaca0\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\ubc1c\uc0b0 \uc815\ub9ac; Gauss \uc815\ub9ac)<\/span><br \/>\n\\(S\\)\uac00 \uacf5\uac04\uc5d0 \ub193\uc778 \uace1\uba74\uc774\uba70 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7fd\uace0 \ubc29\ud5a5\uc744 \uac00\uc9c4 \ub2eb\ud78c\uace1\uba74\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(S\\) \uc704\uc5d0 \ubc14\uae65\ucabd\uc744 \ud5a5\ud558\ub294 \ub2e8\uc704\ubc95\uc120\ubca1\ud130\ub97c \\(\\mathbf{n}\\)\uc774\ub77c\uace0 \ud558\uc790. \\(S\\)\ub97c \uacbd\uacc4\ub85c \uac16\ub294 \ub2eb\ud78c\uc601\uc5ed\uc744 \\(D\\)\ub77c\uace0 \ud558\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\iint_S \\mathbf{F} \\cdot \\mathbf{n} \\,d\\sigma = \\iiint_D \\nabla \\cdot \\mathbf{F} \\,dV . \\tag{4.3}\\]<br \/>\n\uc774 \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc740 \uace1\uba74 \\(S\\)\ub97c \ud1b5\uacfc\ud558\uc5ec \ubc14\uae65\ucabd\uc73c\ub85c \ub098\uac00\ub294 \ubca1\ud130\uc7a5 \\(\\mathbf{F}\\)\uc758 \uc720\ucd9c(flux)\uc774\uba70, \uc6b0\ubcc0\uc740 \uc601\uc5ed \\(D\\)\uc5d0\uc11c \uc720\ucd9c\ubc00\ub3c4 \\(\\opdiv \\,\\mathbf{F}\\)\uc758 \uc801\ubd84\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc608\ub97c \ud1b5\ud574 \ubc1c\uc0b0 \uc815\ub9ac\ub97c \ud655\uc778\ud574 \ubcf4\uc790.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span><br \/>\n\ubc18\uc9c0\ub984\uc774 \\(a > 0\\)\uc778 \uad6c\uba74 \\(x^2 + y^2 + z^2 = a^2\\)\uc5d0 \ubc14\uae65\ucabd\uc744 \ud5a5\ud558\ub294 \ubc29\ud5a5\uc774 \uc8fc\uc5b4\uc9c4 \uace1\uba74\uc744 \\(S\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(S\\)\uc5d0 \uc758\ud558\uc5ec \ub458\ub7ec\uc2f8\uc778 \ub2eb\ud78c \uc601\uc5ed\uc744 \\(D\\)\ub77c\uace0 \ud558\uc790. \uc989 \\(D\\)\ub294 \\(x^2 + y^2 + z^2 \\le a^2\\)\uc778 \uc601\uc5ed\uc774\ub2e4. \ubca1\ud130\uc7a5 \\(\\mathbf{F} = x\\,\\mathbf{i} + y\\,\\mathbf{j} + z\\,\\mathbf{k}\\)\uc5d0 \ub300\ud558\uc5ec \ubc1c\uc0b0 \uc815\ub9ac\ub97c \ud655\uc778\ud574 \ubcf4\uc790.<\/p>\n<p>\uba3c\uc800 \uace1\uba74 \\(S\\)\ub97c \ud1b5\uacfc\ud558\uc5ec \ubc14\uae65\ucabd\uc73c\ub85c \ub098\uac00\ub294 \\(\\mathbf{F}\\)\uc758 \uc720\ucd9c, \uc989 (4.3)\uc758 \uc88c\ubcc0\uc744 \uacc4\uc0b0\ud574 \ubcf4\uc790. \uad6c\uba74 \uc704\uc758 \uc810 \\(P(x,\\,y,\\,z)\\)\uc5d0\uc11c \uc774 \uad6c\uba74\uc5d0 \uc218\uc9c1\uc774\uace0 \ubc14\uae65\ucabd\uc744 \ud5a5\ud558\ub294 \ubc95\uc120\ubca1\ud130\ub294 \\(\\overrightarrow{OP}\\)\uc640 \ud3c9\ud589\ud558\ubbc0\ub85c<br \/>\n\\[\\mathbf{n} = \\frac{\\overrightarrow{OP}}{\\lvert\\,\\overrightarrow{OP}\\,\\rvert} = \\frac{x\\,\\mathbf{i} + y\\,\\mathbf{j} + z\\,\\mathbf{k}}{a}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\mathbf{F} \\cdot \\mathbf{n} \\,d\\sigma = \\frac{x^2 + y^2 + z^2}{a} d\\sigma = \\frac{a^2}{a} d\\sigma= a\\,d\\sigma \\]<br \/>\n\uc774\uba70<br \/>\n\\[\\iint_S \\mathbf{F} \\cdot \\mathbf{n} \\,d\\sigma = \\iint_S a\\,d\\sigma = a \\cdot (\\text{area of } S) = 4\\pi a^3\\tag{4.4}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(D\\)\uc5d0\uc11c\uc758 \uc720\ucd9c\ubc00\ub3c4\uc758 \uc801\ubd84, \uc989 (4.3)\uc758 \uc6b0\ubcc0\uc744 \uacc4\uc0b0\ud574 \ubcf4\uc790.<br \/>\n\\[\\nabla \\cdot \\mathbf{F} = \\frac{\\partial}{\\partial x}x + \\frac{\\partial}{\\partial y}y + \\frac{\\partial}{\\partial z}z = 3\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\iiint_D \\nabla \\cdot \\mathbf{F} \\,dV = \\iiint_D 3 \\,dV = 3\\cdot (\\text{volume of }D) = 4\\pi a^3\\tag{4.5}<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uacc4\uc0b0 \uacb0\uacfc (4.4)\uc640 (4.5)\ub97c \ube44\uad50\ud558\uba74 \uc77c\uce58\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<a name=\"divergencetheoremapp\"><\/a><\/p>\n<h2 class=\"margintop2\">\ubc1c\uc0b0 \uc815\ub9ac\uc758 \uc751\uc6a9<\/h2>\n<p>(3.1)\uacfc \ube44\uc2b7\ud55c \ub4f1\uc2dd\uc774 \ubc1c\uc0b0\uacfc \ud68c\uc804\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud55c\ub2e4. \uc989 \ubca1\ud130\uc7a5 \\(\\mathbf{F} = M\\,\\mathbf{i} + N\\,\\mathbf{j} + P\\,\\mathbf{k}\\)\uac00 \uc5f0\uc18d\uc778 2\uacc4 \ud3b8\ub3c4\ud568\uc218\ub97c \uac00\uc9c0\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<div class=\"box oneline\">\n\\[\\opdiv (\\curl \\,\\mathbf{F} ) = 0. \\tag{5.1}\\]\n<\/div>\n<p>\ub4f1\uc2dd (5.1)\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. \ubc1c\uc0b0\uacfc \ud68c\uc804\uc758 \uc815\uc758\ub97c \ub530\ub77c \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\opdiv (\\curl \\,\\mathbf{F} )<br \/>\n&#038;= \\nabla \\cdot (\\nabla \\times \\mathbf{F} )\\\\[5pt]<br \/>\n&#038;=<br \/>\n\\frac{\\partial}{\\partial x}\\left( \\frac{\\partial P}{\\partial y} &#8211; \\frac{\\partial N}{\\partial z}\\right) +<br \/>\n\\frac{\\partial}{\\partial y}\\left( \\frac{\\partial M}{\\partial z} &#8211; \\frac{\\partial P}{\\partial x}\\right) +<br \/>\n\\frac{\\partial}{\\partial z}\\left( \\frac{\\partial N}{\\partial x} &#8211; \\frac{\\partial M}{\\partial y}\\right) \\\\[5pt]<br \/>\n&#038;=<br \/>\n\\frac{\\partial^2 P}{\\partial x\\,\\partial y} &#8211; \\frac{\\partial^2 N}{\\partial x\\,\\partial z} +<br \/>\n\\frac{\\partial^2 M}{\\partial y\\,\\partial z} &#8211; \\frac{\\partial^2 P}{\\partial y \\,\\partial x} +<br \/>\n\\frac{\\partial^2 N}{\\partial z\\,\\partial x} &#8211; \\frac{\\partial^2 M}{\\partial z\\,\\partial y}\\\\[5pt]<br \/>\n&#038;=<br \/>\n\\left(\\frac{\\partial^2 P}{\\partial x\\,\\partial y} &#8211; \\frac{\\partial^2 P}{\\partial y \\,\\partial x}\\right) +<br \/>\n\\left(\\frac{\\partial^2 M}{\\partial y\\,\\partial z} &#8211; \\frac{\\partial ^2 M}{\\partial z\\,\\partial y}\\right) +<br \/>\n\\left(\\frac{\\partial^2 N}{\\partial z\\,\\partial x} &#8211; \\frac{\\partial^2 N}{\\partial x\\,\\partial z}\\right)<br \/>\n.\\tag{5.2}<br \/>\n\\end{align}\\]<br \/>\n\ud074\ub798\ub85c\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec, (5.2)\uc758 \ub9c8\uc9c0\ub9c9 \uc2dd\uc5d0\uc11c \uad04\ud638\ub85c \ubb36\uc778 \uac01 \uc2dd\uc758 \uac12\uc740 \\(0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (5.2)\uc758 \uacb0\uacfc\ub294 \\(0\\)\uc774\ub2e4. \uc989 \ub4f1\uc2dd (5.1)\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>\ub4f1\uc2dd (5.1)\uc758 \uacb0\uacfc\ub85c\uc11c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 3.<\/span><br \/>\n\\(\\opdiv \\,\\mathbf{G} \\ne 0\\)\uc774\uba74 \\(\\mathbf{G} = \\curl\\,\\mathbf{F}\\)\uc778 \ubca1\ud130\uc7a5 \\(\\mathbf{F}\\)\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.\n<\/p>\n<\/div>\n<p>\ubc1c\uc0b0 \uc815\ub9ac\uc758 \ub610 \ub2e4\ub978 \uc608\ub85c\uc11c \uc804\uc790\uae30 \uc774\ub860\uc758 <span class=\"defined\">\uac00\uc6b0\uc2a4 \ubc95\uce59<\/span>(Gauss&#8217; law)\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 4. (\uc804\uc790\uae30 \uc774\ub860\uc758 \uac00\uc6b0\uc2a4 \ubc95\uce59)<\/span><br \/>\n3\ucc28\uc6d0 \uacf5\uac04\uc5d0\uc11c \uc810\uc804\ud558 \\(q\\)\ub97c \uc548\ucabd\uc5d0 \ud488\ub294 \ub2eb\ud78c \ub2e8\uc21c\uace1\uba74 \\(S\\)\uac00 \uc788\uc744 \ub54c, \\(q\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uae30\ub294 \uc804\uae30\uc7a5\uc774 \\(S\\)\uc5d0 \uc791\uc6a9\ud558\ub294(\\(S\\)\ub97c \uac00\ub85c\uc9c8\ub7ec \ubc14\uae65\uc73c\ub85c \ub098\uac00\ub294) \ucd1d \uc804\uae30\ub825\uc740<br \/>\n\\[\\frac{q}{\\epsilon_0}\\]<br \/>\n\uc774\ub2e4. (\\(q\\)\uc640 \\(\\epsilon_0\\)\uc5d0 \ub300\ud55c \uc124\uba85\uc740 \uc774 \ubc95\uce59\uc744 \uc720\ub3c4\ud558\ub294 \uacfc\uc815\uc5d0\uc11c \uc124\uba85\ud55c\ub2e4.)\n<\/p>\n<\/div>\n<p>\uac00\uc6b0\uc2a4 \ubc95\uce59\uc744 \uc720\ub3c4\ud558\ub294 \uacfc\uc815\uc740 3\ub2e8\uacc4\ub85c \ub098\ub20c \uc218 \uc788\ub2e4.<\/p>\n<p><span class=\"definition\">1\ub2e8\uacc4.<\/span><br \/>\n\\(a\\)\uc640 \\(b\\)\uac00 \\(a > b\\)\uc778 \uc591\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \uc601\uc5ed<br \/>\n\\[D : \\,\\, b^2 \\le x^2 + y^2 + z^2 \\le a^2\\]<br \/>\n\uc758 \uacbd\uacc4\ub97c \ud1b5\uacfc\ud558\ub294 \ubca1\ud130\uc7a5<br \/>\n\\[\\mathbf{F} = \\frac{x\\,\\mathbf{i} + y\\,\\mathbf{j} + z\\,\\mathbf{k}}{\\rho^2} ,\\,\\, \\rho=\\sqrt{x^2 + y^2 + z^2}\\tag{5.3}\\]<br \/>\n\uc758 \uc678\ud5a5\uc720\ucd9c\uc744 \uad6c\ud558\uc790.<\/p>\n<p>\uba3c\uc800<br \/>\n\\[\\frac{\\partial \\rho}{\\partial x} = \\frac{1}{2} (x^2 +y^2 +z^2 )^{-1\/2} \\cdot 2x = \\frac{x}{\\rho}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\frac{\\partial M}{\\partial x} = \\frac{\\partial}{\\partial x}(x\\rho^{-3}) = \\rho^{-3} &#8211; 3x\\rho^{-4} \\frac{\\partial\\rho}{\\partial x} = \\frac{1}{\\rho^3} &#8211; \\frac{3x^2}{\\rho^5}\\]<br \/>\n\uc774\ub2e4. \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(N_y\\)\uc640 \\(P_z\\)\ub97c \uad6c\ud558\uba74<br \/>\n\\[\\frac{\\partial N}{\\partial y} = \\frac{1}{\\rho^3} &#8211; \\frac{3y^2}{\\rho^5} ,\\,\\, \\frac{\\partial P}{\\partial z} = \\frac{1}{\\rho^3} &#8211; \\frac{3z^2}{\\rho^5}\\]<br \/>\n\uc774\ub2e4. \uc774\ub85c\uc368 \\(\\mathbf{F}\\)\uc758 \ubc1c\uc0b0\uc774<br \/>\n\\[\\opdiv \\,\\mathbf{F} = \\frac{3}{\\rho^3} &#8211; \\frac{3}{\\rho^5} (x^2 + y^2 + z^2 ) = \\frac{3}{\\rho^3} &#8211; \\frac{3\\rho^2}{\\rho^5} = 0 \\tag{5.4}\\]<br \/>\n\uc774\ubbc0\ub85c, \\(D\\)\uc774 \uacbd\uacc4\ub97c \ud1b5\uacfc\ud558\ub294 \\(\\mathbf{F}\\)\uc758 \uc678\ud5a5 \uc720\ucd9c\uc740 \\(0\\)\uc774\ub2e4.<\/p>\n<p><span class=\"definition\">2\ub2e8\uacc4.<\/span><br \/>\n\uc911\uc2ec\uc774 \uc6d0\uc810\uc774\uace0 \ubc18\uc9c0\ub984\uc774 \\(a\\)\uc778 \uad6c\uba74 \\(S_a\\)\ub97c \ud1b5\uacfc\ud558\ub294 \\(\\mathbf{F}\\)\uc758 \uc678\ud5a5\uc720\ucd9c\uc744 \uad6c\ud558\uc790.<br \/>\n\ubcf4\uae30 4\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \ubc14\uc640 \uac19\uc774 \uad6c\uba74 \\(S_a\\) \ud45c\uba74\uc5d0\uc11c \ubc14\uae65\ucabd\uc744 \ud5a5\ud558\ub294 \ub2e8\uc704\ubc95\uc120\ubca1\ud130\ub294<br \/>\n\\[\\mathbf{n} = \\frac{x\\,\\mathbf{i} + y\\,\\mathbf{j} + z\\,\\mathbf{k}}{\\sqrt{x^2 + y^2 + z^2}} = \\frac{x\\,\\mathbf{i} + y\\,\\mathbf{j} + z\\,\\mathbf{k}}{a}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\n\\mathbf{F} \\cdot \\mathbf{n} &#038;= \\frac{x\\,\\mathbf{i} + y\\,\\mathbf{j} + z\\,\\mathbf{k}}{a^3} \\cdot \\frac{x\\,\\mathbf{i} + y\\,\\mathbf{j} + z\\,\\mathbf{k}}{a} \\\\[5pt]<br \/>\n&#038;= \\frac{x^2 + y^2 + z^2}{a^4} = \\frac{a^2}{a^4} = \\frac{1}{a^2},\\\\[5pt]<br \/>\n\\iint_{S_a} \\mathbf{F} \\cdot \\mathbf{n} \\,d\\sigma &#038;= \\frac{1}{a^2} \\iint_{S_a} d\\sigma = \\frac{1}{a^2} \\cdot 4\\pi a^2 = 4 \\pi\\tag{5.5}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p><span class=\"definition\">3\ub2e8\uacc4.<\/span><br \/>\n3\ucc28\uc6d0 \uc88c\ud45c\uacf5\uac04\uc5d0\uc11c \uc6d0\uc810\uc5d0 \ub193\uc778 \uc810\uc804\ud558 \\(q\\)\ub85c\ubd80\ud130 \uc0dd\uae30\ub294 \uc804\uae30\uc7a5\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\mathbf{E} (x,\\,y,\\,z)<br \/>\n&#038;= \\frac{1}{4\\pi \\epsilon_0} \\frac{q}{\\lvert\\mathbf{r}\\rvert^2} \\left( \\frac{\\mathbf{r}}{\\lvert\\mathbf{r}\\rvert}\\right) \\\\[7pt]<br \/>\n&#038;= \\frac{q}{4\\pi\\epsilon_0} \\frac{\\mathbf{r}}{\\lvert\\mathbf{r}\\rvert^3} \\\\[7pt]<br \/>\n&#038;= \\frac{q}{4\\pi\\epsilon_0}\\frac{x\\,\\mathbf{i} + y \\,\\mathbf{j} + z\\,\\mathbf{k}}{\\rho^3} .<br \/>\n\\end{align}\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\epsilon_0\\)\ub294 \ubb3c\ub9ac\uc801 \uc0c1\uc218\uc774\uace0 \\(\\mathbf{r}\\)\ub294 \uc810 \\((x,\\,y,\\,z)\\)\uc758 \uc704\uce58\ubca1\ud130\uc774\uba70 \\(\\rho = \\lvert\\mathbf{r}\\rvert\\)\uc774\ub2e4. (5.3)\uc744 \uc774\uc6a9\ud558\uba74<br \/>\n\\[\\mathbf{E} = \\frac{q}{4\\pi\\epsilon_0}\\mathbf{F}\\]<br \/>\n\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<p>\\(S\\)\uac00 \uc6d0\uc810\uc744 \uc548\ucabd\uc5d0 \ud488\ub294 \ub2eb\ud78c \ub2e8\uc21c\uace1\uba74\uc774\uace0 \\(S_a\\)\uac00 \\(S\\)\ub97c \uc548\ucabd\uc5d0 \ud488\ub294 \uad6c\uba74\uc774\ub77c\uace0 \ud558\uc790(\uadf8\ub9bc 7).<\/p>\n<div class=\"imgs-center\">\n<img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-51-gauss-law.png\" alt=\"\" width=\"212\" height=\"256\" class=\"aligncenter size-full wp-image-5951\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-51-gauss-law.png 424w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/10\/stokes-divergence-51-gauss-law-248x300.png 248w\" sizes=\"(max-width: 212px) 100vw, 212px\" \/><\/p>\n<div class=\"caption\">[\uadf8\ub9bc 7]<\/div>\n<\/div>\n<p>(5.4)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\nabla\\cdot\\mathbf{E} = \\nabla \\cdot \\frac{q}{4\\pi\\epsilon_0}\\mathbf{F} = \\frac{q}{4\\pi\\epsilon_0} \\nabla \\cdot \\mathbf{F} = 0\\]<br \/>\n\uc774\ubbc0\ub85c \\(S\\)\uc640 \\(S_a\\) \uc0ac\uc774\uc758 \uc601\uc5ed \\(D\\)\uc5d0\uc11c \\(\\nabla \\cdot \\mathbf{E}\\)\uc758 \uc0bc\uc911\uc801\ubd84\uc740 \\(0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubc1c\uc0b0 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\iint_{\\partial D} \\mathbf{E} \\cdot \\mathbf{n} \\,d\\sigma = \\iiint_{D} \\nabla \\cdot \\mathbf{E} \\,dV = 0\\tag{5.6}\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\partial D\\)\ub294 \\(D\\)\uc758 \uacbd\uacc4, \uc989 \ub450 \uac1c\uc758 \uace1\uba74 \\(S\\)\uc640 \\(S_a\\)\uc758 \ud569\uc9d1\ud569\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uace1\uba74\uc744 \ub098\ud0c0\ub0b8\ub2e4. \uace1\uba74 \\(S\\)\uc758 \ubc14\uae65\ucabd \ubc29\ud5a5\uc740 \uc601\uc5ed \\(D\\)\uc758 \uc548\ucabd\uc744 \ud5a5\ud558\ub294 \ubc29\ud5a5\uc774\uba70, \uace1\uba74 \\(S_a\\)\uc758 \ubc14\uae65\ucabd \ubc29\ud5a5\uc740 \uc601\uc5ed \\(D\\)\uc758 \ubc14\uae65\uc744 \ud5a5\ud558\ub294 \ubc29\ud5a5\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\iint_{\\partial D} \\mathbf{E} \\cdot \\mathbf{n} \\,d\\sigma<br \/>\n=<br \/>\n&#8211; \\iint_{S} \\mathbf{E} \\cdot \\mathbf{n} \\,d\\sigma + \\iint_{S_a} \\mathbf{E} \\cdot \\mathbf{n} \\,d\\sigma\\tag{5.7}\\]<br \/>\n\uc774\ub2e4. \uc774 \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc740 (5.6)\uc5d0 \uc758\ud558\uc5ec \\(0\\)\uc774\uace0, \uc6b0\ubcc0\uc758 \ub450 \ubc88\uc9f8 \uc801\ubd84\uc758 \uac12\uc740 (5.5)\uc758 \uac12\uc5d0 \\(q \/ (4\\pi\\epsilon_0)\\)\ub97c \uacf1\ud55c \uac83\uacfc \uac19\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\iint_{S} \\mathbf{E} \\cdot \\mathbf{n} \\,d\\sigma = \\iint_{S_a} \\mathbf{E} \\cdot \\mathbf{n} \\,d\\sigma = \\frac{q}{\\epsilon_0}\\tag{5.8}\\]<br \/>\n\uc774\ub2e4. \uc774 \uac12\uc740 \uc810\uc804\ud558 \\(q\\)\uac00 \uc704\uce58\ud55c \uc9c0\uc810(\uc6d0\uc810)\uc744 \uc548\ucabd\uc5d0 \ud488\ub294 \uc784\uc758\uc758 \ub2eb\ud78c \ub2e8\uc21c\uace1\uba74 \\(S\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p><!--\n\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n\n\n\n<h2 class=\"margintop2\"><\/h2>\n\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\ud3c9\uba74\uc5d0\uc11c \uc120\uc801\ubd84\uacfc \uc774\uc911\uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\ub294 \uadf8\ub9b0 \uc815\ub9ac\uac00 \uc788\ub294 \uac83\ucc98\ub7fc \uacf5\uac04\uc5d0\uc11c\ub3c4 \uc120\uc801\ubd84\uacfc \uba74\uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\ub294 \uc815\ub9ac, \uba74\uc801\ubd84\uacfc \uc0bc\uc911\uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\ub294 \uc815\ub9ac\uac00 \uc788\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uadf8\ub9b0 \uc815\ub9ac\ub97c 3\ucc28\uc6d0\uc73c\ub85c \ud655\uc7a5\ud55c \uc801\ubd84 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. \ub0b4\uc6a9 \uc21c\uc11c \ud68c\uc804\ubca1\ud130\uc7a5 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\uc758 \uc751\uc6a9 \ubc1c\uc0b0 \uc815\ub9ac \ubc1c\uc0b0 \uc815\ub9ac\uc758 \uc751\uc6a9 \ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9 Line integral (\uad00\ub828 \uae00) Green&#8217;s theorem (\uad00\ub828 \uae00) Surface integral (\uad00\ub828 \uae00) \\( \\newcommand{\\curl}{{\\operatorname{curl}}} \\newcommand{\\grad}{{\\operatorname{grad}}} \\newcommand{\\opdiv}{{\\operatorname{div}}} \\)&hellip;<\/p>\n","protected":false},"author":1,"featured_media":5961,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[531,141,525,530,533,532,518,536,529,524,519,537,527,520,534,517,138,516,526,535,528],"class_list":["post-2105","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculus-ap","tag-curl","tag-divergence","tag-divergence-theorem","tag-gauss","tag-gauss-theorem","tag-green-theorem","tag-line-integral","tag-manifold-boundary","tag-stokes","tag-stokes-theorem","tag-surface-integral","tag-topological-boundary","tag-527","tag-520","tag-534","tag-517","tag-138","tag-516","tag-526","tag-535","tag-528"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2105","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=2105"}],"version-history":[{"count":75,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2105\/revisions"}],"predecessor-version":[{"id":6034,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2105\/revisions\/6034"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media\/5961"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2105"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2105"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2105"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}