{"id":9499,"date":"2025-10-20T19:00:50","date_gmt":"2025-10-20T10:00:50","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9499"},"modified":"2026-09-27T16:32:47","modified_gmt":"2026-09-27T07:32:47","slug":"ch11-vector-field-and-fundamental-theorems","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch11-vector-field-and-fundamental-theorems\/","title":{"rendered":"\ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac"},"content":{"rendered":"<div class=\"analysis2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>\ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac<\/h2>\n\n --><\/p>\n<p><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\">9\uc7a5<\/a>\uc5d0\uc11c\ub294 \ub2e4\ubcc0\uc218\ud568\uc218\uc758 \ubbf8\ubd84\uc744, <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\">10\uc7a5<\/a>\uc5d0\uc11c\ub294 \ub2e4\uc911\uc801\ubd84\uc744 \ub2e4\ub8e8\uc5c8\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \ub450 \uc774\ub860\uc744 \uacb0\ud569\ud558\uc5ec \ubca1\ud130\uc7a5\uc758 \uc120\uc801\ubd84\uacfc \uba74\uc801\ubd84\uc744 \uc815\uc758\ud558\uace0, \uadf8\ub9b0\uc758 \uc815\ub9ac, \ubc1c\uc0b0 \uc815\ub9ac, \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. \ub9c8\uc9c0\ub9c9\uc5d0\ub294 \uc774 \uc801\ubd84\uc815\ub9ac\ub4e4\uc744 \ubbf8\ubd84\ud615\uc2dd\uc758 \uc5b8\uc5b4\ub85c \ud558\ub098\uc758 \uc815\ub9ac\ub85c \ud1b5\ud569\ud55c\ub2e4.<\/p>\n<h3>\ubca1\ud130\uc7a5\uc758 \uae30\ucd08<\/h3>\n<p>\\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569 \\(D\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 \\(f\\colon D\\to\\mathbb{R}\\)\uc744 <span class=\"defined\">\uc2a4\uce7c\ub77c\uc7a5<\/span>(scalar field)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \\(F\\colon D\\to\\mathbb{R}^n\\)\uc744 <span class=\"defined\">\ubca1\ud130\uc7a5<\/span>(vector field)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud604\uc2e4 \uc138\uacc4\uc5d0\uc11c \uc628\ub3c4\ub098 \uc555\ub825 \ubd84\ud3ec\ub294 \uc2a4\uce7c\ub77c\uc7a5\uc774\uace0, \uc18d\ub3c4\uc7a5\uc774\ub098 \uc804\uae30\uc7a5\uc740 \ubca1\ud130\uc7a5\uc774\ub2e4.<\/p>\n<p><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\">9\uc7a5<\/a>\uc5d0\uc11c \uc815\uc758\ud55c \uae30\uc6b8\uae30\uc640 \ud568\uaed8 \ub2e4\uc74c \uc5f0\uc0b0\uc744 \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(C^1\\) \uc2a4\uce7c\ub77c\uc7a5 \\(f\\)\uc758 \uae30\uc6b8\uae30\ub294<br \/>\n\\[<br \/>\n\\nabla f<br \/>\n=<br \/>\n\\left(<br \/>\n\\frac{\\partial f}{\\partial x_1},\\,<br \/>\n\\frac{\\partial f}{\\partial x_2},\\,<br \/>\n\\ldots,\\,<br \/>\n\\frac{\\partial f}{\\partial x_n}<br \/>\n\\right)<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<li>\\(C^1\\) \ubca1\ud130\uc7a5 \\(F=(F_1,\\ldots,F_n)\\)\uc758 <span class=\"defined\">\ubc1c\uc0b0<\/span>(divergence)\uc740<br \/>\n\\[<br \/>\n\\nabla\\cdot F<br \/>\n=<br \/>\n\\operatorname{div}F<br \/>\n=<br \/>\n\\sum_{i=1}^n\\frac{\\partial F_i}{\\partial x_i}<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<li>\\(\\mathbb{R}^3\\)\uc5d0\uc11c \\(C^1\\) \ubca1\ud130\uc7a5 \\(F=(F_1,F_2,F_3)\\)\uc758 <span class=\"defined\">\ud68c\uc804<\/span>(curl)\uc740<br \/>\n\\[<br \/>\n\\nabla\\times F<br \/>\n=<br \/>\n\\operatorname{curl}F<br \/>\n=<br \/>\n\\left(<br \/>\n\\frac{\\partial F_3}{\\partial y}-\\frac{\\partial F_2}{\\partial z},\\,<br \/>\n\\frac{\\partial F_1}{\\partial z}-\\frac{\\partial F_3}{\\partial x},\\,<br \/>\n\\frac{\\partial F_2}{\\partial x}-\\frac{\\partial F_1}{\\partial y}<br \/>\n\\right)<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ub2e4\uc74c \ud589\ub82c\uc2dd \ud45c\uae30\ub294 \ud68c\uc804\uc744 \uae30\uc5b5\ud558\uae30 \uc704\ud55c \ud615\uc2dd\uc801\uc778 \ud45c\uae30\uc774\ub2e4.<br \/>\n\\[<br \/>\n\\nabla\\times F<br \/>\n=<br \/>\n\\begin{vmatrix}<br \/>\n\\mathbf{i}&#038;\\mathbf{j}&#038;\\mathbf{k}\\\\<br \/>\n\\frac{\\partial}{\\partial x}&#038;\\frac{\\partial}{\\partial y}&#038;\\frac{\\partial}{\\partial z}\\\\<br \/>\nF_1&#038;F_2&#038;F_3<br \/>\n\\end{vmatrix}.<br \/>\n\\]<\/li>\n<li>\\(C^2\\) \uc2a4\uce7c\ub77c\uc7a5 \\(f\\)\uc758 <span class=\"defined\">\ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790<\/span>(Laplacian)\ub294<br \/>\n\\[<br \/>\n\\Delta f<br \/>\n=<br \/>\n\\nabla\\cdot(\\nabla f)<br \/>\n=<br \/>\n\\sum_{i=1}^n\\frac{\\partial^2f}{\\partial x_i^2}<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ubca1\ud130\uc7a5 \\(F=(F_1,F_2,F_3)\\)\uc5d0\ub294 \uc131\ubd84\ubcc4\ub85c<br \/>\n\\[<br \/>\n\\Delta F=(\\Delta F_1,\\Delta F_2,\\Delta F_3)<br \/>\n\\]<br \/>\n\ub77c\uace0 \uc4f4\ub2e4.<\/li>\n<\/ul>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.1. (\ubca1\ud130 \ubbf8\ubd84\uc758 \uae30\ubcf8 \ud56d\ub4f1\uc2dd)<\/span><\/p>\n<p>\\(f\\)\uac00 \\(\\mathbb{R}^3\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc5d0\uc11c \\(C^2\\)\uc774\uace0 \\(F\\)\uac00 \uac19\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc5d0\uc11c \\(C^2\\)\uc778 \ubca1\ud130\uc7a5\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[<br \/>\n\\nabla\\times(\\nabla f)=0,\\quad<br \/>\n\\nabla\\cdot(\\nabla\\times F)=0,<br \/>\n\\]<br \/>\n\\[<br \/>\n\\nabla\\times(\\nabla\\times F)<br \/>\n=<br \/>\n\\nabla(\\nabla\\cdot F)-\\Delta F.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uccab \ubc88\uc9f8\uc640 \ub450 \ubc88\uc9f8 \ub4f1\uc2dd\uc740 <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\">\uc815\ub9ac 9.5\uc758 \ud074\ub808\ub85c \uc815\ub9ac<\/a>\uc5d0 \uc758\ud574 \ud63c\ud569\ud3b8\ubbf8\ubd84\uc758 \uc21c\uc11c\ub97c \ubc14\uafb8\uba74 \ubc14\ub85c \uc5bb\ub294\ub2e4. \uc138 \ubc88\uc9f8 \ub4f1\uc2dd\uc758 \uccab\uc9f8 \uc131\ubd84\uc740<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\n[\\nabla\\times(\\nabla\\times F)]_1<br \/>\n&#038;=<br \/>\n\\frac{\\partial}{\\partial y}<br \/>\n\\left(<br \/>\n\\frac{\\partial F_2}{\\partial x}<br \/>\n&#8211;<br \/>\n\\frac{\\partial F_1}{\\partial y}<br \/>\n\\right)<br \/>\n&#8211;<br \/>\n\\frac{\\partial}{\\partial z}<br \/>\n\\left(<br \/>\n\\frac{\\partial F_1}{\\partial z}<br \/>\n&#8211;<br \/>\n\\frac{\\partial F_3}{\\partial x}<br \/>\n\\right)\\\\<br \/>\n&#038;=<br \/>\n\\frac{\\partial}{\\partial x}<br \/>\n\\left(<br \/>\n\\frac{\\partial F_2}{\\partial y}<br \/>\n+<br \/>\n\\frac{\\partial F_3}{\\partial z}<br \/>\n\\right)<br \/>\n&#8211;<br \/>\n\\frac{\\partial^2F_1}{\\partial y^2}<br \/>\n&#8211;<br \/>\n\\frac{\\partial^2F_1}{\\partial z^2}\\\\<br \/>\n&#038;=<br \/>\n[\\nabla(\\nabla\\cdot F)-\\Delta F]_1.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\ub098\uba38\uc9c0 \ub450 \uc131\ubd84\ub3c4 \uac19\uc740 \uacc4\uc0b0\uc73c\ub85c \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ubca1\ud130\uc7a5 \\(F\\)\uc5d0 \ub300\ud558\uc5ec \uc2a4\uce7c\ub77c\uc7a5 \\(\\phi\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(F=\\nabla\\phi\\)\uc774\uba74 \\(F\\)\ub97c <span class=\"defined\">\ubcf4\uc874\uc7a5<\/span>(conservative field) \ub610\ub294 <span class=\"defined\">\uacbd\uc0ac\uc7a5<\/span>(gradient field)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \\(\\phi\\)\ub97c <span class=\"defined\">\ud37c\ud150\uc15c \ud568\uc218<\/span>(potential function)\ub77c\uace0 \ubd80\ub978\ub2e4. \ubca1\ud130\uc7a5 \\(F\\)\uac00 \\(\\nabla\\cdot F=0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(F\\)\ub97c <span class=\"defined\">\uc194\ub808\ub178\uc774\ub4dc\uc7a5<\/span>(solenoidal field)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc815\ub9ac 11.1\uc5d0 \uc758\ud574 \\(C^2\\) \ubca1\ud130\uc7a5 \\(G\\)\uc758 \ubaa8\ub4e0 \ud68c\uc804\uc7a5 \\(\\nabla\\times G\\)\ub294 \uc194\ub808\ub178\uc774\ub4dc\uc7a5\uc774\ub2e4.<\/p>\n<p>\uc720\ud074\ub9ac\ub4dc \uacf5\uac04 \\(\\mathbb{R}^2\\)\uc5d0\uc11c \ud45c\uc900\uae30\uc800\ub97c \\(\\mathbf{i}=(1,0)\\), \\(\\mathbf{j}=(0,1)\\)\ub85c \ub098\ud0c0\ub0b4\uba74 \ubca1\ud130\uc7a5\uc740<br \/>\n\\[<br \/>\nF=P\\mathbf{i}+Q\\mathbf{j}<br \/>\n\\]<br \/>\n\ub85c \uc4f8 \uc218 \uc788\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(\\mathbb{R}^3\\)\uc5d0\uc11c\ub294<br \/>\n\\[<br \/>\nF=P\\mathbf{i}+Q\\mathbf{j}+R\\mathbf{k}<br \/>\n\\]<br \/>\n\ub85c \uc4f4\ub2e4.<\/p>\n<h3>\uc120\uc801\ubd84<\/h3>\n<p>\\(\\mathbb{R}^n\\)\uc758 \uace1\uc120 \\(C\\)\uac00 \\(C^1\\) \ud568\uc218 \\(r\\colon[a,b]\\to\\mathbb{R}^n\\)\ub85c \ub9e4\uac1c\ubcc0\uc218\ud654\ub418\uace0 \\(r'(t)\\ne0\\)\uc774\uba74 \\(r\\)\uc744 <span class=\"defined\">\uc815\uce59 \ub9e4\uac1c\ud654<\/span>(regular parametrization)\ub77c\uace0 \ud558\uace0 \\(C\\)\ub97c <span class=\"defined\">\ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120<\/span>(smooth curve)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(t\\)\uac00 \uc99d\uac00\ud558\ub294 \ubc29\ud5a5\uc744 \\(C\\)\uc758 \ubc29\ud5a5\uc73c\ub85c \uc815\ud55c\ub2e4. \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc744 \uc720\ud55c \uac1c \uc774\uc5b4 \ubd99\uc778 \uace1\uc120\uc744 <span class=\"defined\">\uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120<\/span>(piecewise smooth curve)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120 \\(C\\)\uc5d0 \ub300\ud55c \uc5f0\uc18d \uc2a4\uce7c\ub77c\uc7a5 \\(f\\)\uc758 <span class=\"defined\">\uc120\uc801\ubd84<\/span>(line integral)\uc744<br \/>\n\\[<br \/>\n\\int_C f\\,ds<br \/>\n=<br \/>\n\\int_a^b f(r(t))\\|r'(t)\\|\\,dt.<br \/>\n\\tag{11.1}<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(ds\\)\ub294 \ud638\uc758 \uae38\uc774\uc5d0 \ub300\ud55c \ubbf8\ubd84\uc18c\uc774\ub2e4.<\/p>\n<p>\ub9e4\ub044\ub7ec\uc6b4 \uc720\ud5a5\uace1\uc120 \\(C\\)\uc5d0 \ub300\ud55c \uc5f0\uc18d \ubca1\ud130\uc7a5 \\(F\\)\uc758 \uc120\uc801\ubd84\uc740<br \/>\n\\[<br \/>\n\\int_C F\\cdot dr<br \/>\n=<br \/>\n\\int_C F\\cdot T\\,ds<br \/>\n=<br \/>\n\\int_a^bF(r(t))\\cdot r'(t)\\,dt.<br \/>\n\\tag{11.2}<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc5ec\uae30\uc11c<br \/>\n\\[<br \/>\nT=\\frac{r'(t)}{\\|r'(t)\\|}<br \/>\n\\]<br \/>\n\ub294 \ub2e8\uc704\uc811\uc120\ubca1\ud130\uc774\ub2e4. \uc774 \uc801\ubd84\uc744 <span class=\"defined\">\ubc29\ud5a5\uc120\uc801\ubd84<\/span>\uc774\ub77c\uace0\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\\(C\\)\uac00 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc774\uace0<br \/>\n\\[<br \/>\nC=C_1\\cup\\cdots\\cup C_k<br \/>\n\\]<br \/>\n\ub85c \uc774\uc5b4 \ubd99\uc5ec\uc84c\uc73c\uba74 \uac01 \uc870\uac01\uc758 \uc801\ubd84\uc744 \ub354\ud558\uc5ec \uc120\uc801\ubd84\uc744 \uc815\uc758\ud55c\ub2e4. \ud3d0\uace1\uc120 \uc704\uc758 \ubc29\ud5a5\uc120\uc801\ubd84\uc740<br \/>\n\\[<br \/>\n\\oint_C F\\cdot dr<br \/>\n\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ubb3c\ub9ac\ud559\uc5d0\uc11c \\(\\int_CF\\cdot dr\\)\uc740 \ud798 \\(F\\)\uac00 \uacbd\ub85c \\(C\\)\ub97c \ub530\ub77c \ud55c \uc77c\uc5d0 \ud574\ub2f9\ud558\uba70, \ud3d0\uace1\uc120\uc5d0\uc11c\ub294 \uc21c\ud658(circulation)\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 11.1.<\/span><br \/>\n\ub450 \uc815\uce59 \ub9e4\uac1c\ud654 \\(r_1\\colon[a,b]\\to\\mathbb{R}^n\\), \\(r_2\\colon[c,d]\\to\\mathbb{R}^n\\)\uac00 \uac19\uc740 \uc810\uc9d1\ud569\uc744 \ub098\ud0c0\ub0b8\ub2e4\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\ubc29\ud5a5\uc744 \ubcf4\uc874\ud558\ub294 \\(C^1\\) \uc77c\ub300\uc77c\ub300\uc751 \\(\\varphi\\colon[c,d]\\to[a,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(r_2=r_1\\circ\\varphi\\)\uc774\uba74 \ub450 \uc120\uc801\ubd84 (11.1), (11.2)\uc758 \uac12\uc774 \ubaa8\ub450 \uac19\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(\\varphi\\)\uac00 \ubc29\ud5a5\uc744 \ub4a4\uc9d1\uc73c\uba74 \uc2a4\uce7c\ub77c \uc120\uc801\ubd84\uc740 \ubcc0\ud558\uc9c0 \uc54a\uace0 \ubc29\ud5a5\uc120\uc801\ubd84\uc758 \ubd80\ud638\uac00 \ubc14\ub01c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\ub530\ub77c\uc11c \uc120\uc801\ubd84\uc758 \uad00\uc810\uc5d0\uc11c \uace1\uc120\uc744 \ub2e8\uc21c\ud55c \uc810\uc9d1\ud569\uc774 \uc544\ub2c8\ub77c \ub9e4\uac1c\ud654\uc640 \ubc29\ud5a5\uc744 \uace0\ub824\ud55c \ub300\uc0c1\uc73c\ub85c \ubcf4\uc544\uc57c \ud558\ub294 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.2. (\uc120\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac)<\/span><\/p>\n<p>\\(f\\)\uac00 \uc5f4\ub9b0\uc9d1\ud569 \\(D\\subseteq\\mathbb{R}^n\\)\uc5d0\uc11c \\(C^1\\)\uc774\uace0 \\(F=\\nabla f\\)\ub77c\uace0 \ud558\uc790. \\(C\\subseteq D\\)\uac00 \uc810 \\(P\\)\uc5d0\uc11c \\(Q\\)\ub85c \uac00\ub294 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc774\uba74<br \/>\n\\[<br \/>\n\\int_CF\\cdot dr=f(Q)-f(P)<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\ud55c \uac1c\uc758 \ub9e4\ub044\ub7ec\uc6b4 \uc870\uac01 \\(r\\colon[a,b]\\to D\\)\uc5d0 \ub300\ud574\uc11c\ub294 <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\">\uc815\ub9ac 9.3\uc758 \uc5f0\uc1c4\ubc95\uce59<\/a>\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\n\\frac{d}{dt}f(r(t))<br \/>\n=<br \/>\n\\nabla f(r(t))\\cdot r'(t)<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch06-the-riemann-integral\">\uc815\ub9ac 6.6\uc758 \ubbf8\uc801\ubd84\uc758 \uc81c2\uae30\ubcf8\uc815\ub9ac<\/a>\ub97c \uc801\uc6a9\ud558\uba74<br \/>\n\\[<br \/>\n\\int_C\\nabla f\\cdot dr<br \/>\n=<br \/>\n\\int_a^b\\frac{d}{dt}f(r(t))\\,dt<br \/>\n=<br \/>\nf(r(b))-f(r(a)).<br \/>\n\\]<br \/>\n\uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc5d0\uc11c\ub294 \uac01 \uc870\uac01\uc5d0 \uc801\uc6a9\ud55c \ub4a4 \ub354\ud558\uba74 \uc911\uac04\uc810\uc758 \uac12\uc774 \uc18c\uac70\ub41c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub450 \uc810\uc744 \uc787\ub294 \uc784\uc758\uc758 \ub450 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc5d0 \ub300\ud55c \uc120\uc801\ubd84\uc758 \uac12\uc774 \uac19\uc73c\uba74 \uc120\uc801\ubd84\uc774 <span class=\"defined\">\uacbd\ub85c\ub3c5\ub9bd<\/span>(path independent)\uc774\ub77c\uace0 \ud55c\ub2e4. \uc5f4\ub9b0\uc9d1\ud569 \\(D\\)\uac00 <span class=\"defined\">\ub2e8\uc21c\uc5f0\uacb0<\/span>(simply connected)\uc774\ub77c\ub294 \uac83\uc740 \\(D\\) \uc548\uc758 \ubaa8\ub4e0 \ud3d0\uace1\uc120\uc774 \\(D\\) \uc548\uc5d0\uc11c \ud55c \uc810\uc73c\ub85c \uc5f0\uc18d\uc801\uc73c\ub85c \uc218\ucd95\ub420 \uc218 \uc788\ub2e4\ub294 \ub73b\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 11.3. (\ubcf4\uc874\uc801 \ubca1\ud130\uc7a5\uc758 \uc131\uc9c8)<\/span><\/p>\n<p>\\(D\\subseteq\\mathbb{R}^3\\)\uac00 \uc5f0\uacb0\ub41c \uc5f4\ub9b0\uc9d1\ud569\uc774\uace0 \\(F\\colon D\\to\\mathbb{R}^3\\)\uac00 \\(C^1\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \uc138 \uc870\uac74\uc740 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(F\\)\ub294 \ubcf4\uc874\uc7a5\uc774\ub2e4.<\/li>\n<li>\\(D\\) \uc548\uc758 \ubaa8\ub4e0 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \ud3d0\uace1\uc120 \\(C\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\oint_CF\\cdot dr=0<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<li>\\(F\\)\uc758 \uc120\uc801\ubd84\uc740 \uacbd\ub85c\ub3c5\ub9bd\uc774\ub2e4.<\/li>\n<\/ol>\n<p>\ub610\ud55c \\(D\\)\uac00 \ub2e8\uc21c\uc5f0\uacb0\uc774\uba74 \uc704 \uc870\uac74\ub4e4\uc740 \ub2e4\uc74c \uc870\uac74\uacfc\ub3c4 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\" start=\"4\">\n<li>\\(\\nabla\\times F=0\\).<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n(1)\\(\\Rightarrow\\)(2)\ub294 \uc815\ub9ac 11.2\uc5d0\uc11c \ubc14\ub85c \ub530\ub978\ub2e4. (2)\\(\\Rightarrow\\)(3)\uc744 \ubcf4\uc774\uae30 \uc704\ud574 \uac19\uc740 \ub450 \ub05d\uc810\uc744 \uac00\uc9c0\ub294 \ub450 \uace1\uc120 \\(C_1,C_2\\)\ub97c \ud0dd\ud558\uace0 \\(C_2^{-1}\\)\uc744 \\(C_2\\)\uc758 \ubc29\ud5a5\uc744 \ub4a4\uc9d1\uc740 \uace1\uc120\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[<br \/>\n0<br \/>\n=<br \/>\n\\oint_{C_1\\cup C_2^{-1}}F\\cdot dr<br \/>\n=<br \/>\n\\int_{C_1}F\\cdot dr-\\int_{C_2}F\\cdot dr.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \uacbd\ub85c\ub3c5\ub9bd\uc774\ub2e4.<\/p>\n<p>(3)\\(\\Rightarrow\\)(1)\uc744 \ubcf4\uc774\uc790. \uc5f4\ub9b0 \uc5f0\uacb0\uc9d1\ud569 \\(D\\subseteq\\mathbb R^3\\)\uc758 \ub450 \uc810\uc740 \uc870\uac01\ub9c8\ub2e4 \uc120\ud615\uc778 \uacbd\ub85c\ub85c \uc5f0\uacb0\ud560 \uc218 \uc788\ub2e4. \ud55c \uc810 \\(p\\in D\\)\ub97c \uace0\uc815\ud558\uace0<br \/>\n\\[<br \/>\n\\phi(x)=\\int_{C_{p,x}}F\\cdot dr<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(C_{p,x}\\)\ub294 \\(p\\)\uc5d0\uc11c \\(x\\)\ub85c \uac00\ub294 \uc784\uc758\uc758 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc774\ub2e4. \uacbd\ub85c\ub3c5\ub9bd\uc131 \ub54c\ubb38\uc5d0 \\(\\phi\\)\ub294 \uc798 \uc815\uc758\ub41c\ub2e4. \\(x\\)\uc5d0\uc11c \\(x+he_i\\)\uae4c\uc9c0\uc758 \uc9e7\uc740 \uc120\ubd84\uc774 \\(D\\) \uc548\uc5d0 \ub4e4\uc5b4\uac00\ub3c4\ub85d \\(h\\)\ub97c \ud0dd\ud558\uba74<br \/>\n\\[<br \/>\n\\frac{\\phi(x+he_i)-\\phi(x)}{h}<br \/>\n=<br \/>\n\\frac1h\\int_0^hF_i(x+te_i)\\,dt<br \/>\n\\longrightarrow F_i(x).<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(\\nabla\\phi=F\\)\uc774\ub2e4.<\/p>\n<p>(1)\\(\\Rightarrow\\)(4)\ub294 \uc815\ub9ac 11.1\uc5d0\uc11c \ub530\ub978\ub2e4. \ub2e8\uc21c\uc5f0\uacb0 \uc601\uc5ed\uc5d0\uc11c (4)\\(\\Rightarrow\\)(1)\uc740 \ub2eb\ud78c \\(1\\)-\ud615\uc2dd\uc758 \uc120\uc801\ubd84\uc774 \ub05d\uc810\uc744 \uace0\uc815\ud55c \ud638\ubaa8\ud1a0\ud53c \uc544\ub798 \ubd88\ubcc0\uc774\ub77c\ub294 \ud45c\uc900 \uacb0\uacfc\uc5d0\uc11c \ub530\ub978\ub2e4. \uc774 \ud638\ubaa8\ud1a0\ud53c \ubd88\ubcc0\uc131\uc740 \ub4a4\uc758 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac \ub610\ub294 \ubbf8\ubd84\ud615\uc2dd\uc758 \uc5b8\uc5b4\ub85c \uc99d\uba85\ud560 \uc218 \uc788\uc73c\uba70, \uc5ec\uae30\uc11c\ub294 \uc77c\ubc18\uc801\uc778 \uc99d\uba85\uc744 \uc0dd\ub7b5\ud55c\ub2e4. \ub2e4\ub9cc \\(D\\)\uac00 \ud55c \uc810 \\(a\\)\uc5d0 \ub300\ud558\uc5ec <span class=\"defined\">\ubcc4\ubaa8\uc591<\/span>(star-shaped), \uc989 \ubaa8\ub4e0 \\(x\\in D\\)\uc640 \\(0\\le t\\le1\\)\uc5d0 \ub300\ud558\uc5ec \\(a+t(x-a)\\in D\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \uc9c1\uc811 \ud655\uc778\ud560 \uc218 \uc788\ub2e4. \uc2e4\uc81c\ub85c<br \/>\n\\[<br \/>\n\\phi(x)<br \/>\n=<br \/>\n\\int_0^1F(a+t(x-a))\\cdot(x-a)\\,dt<br \/>\n\\]<br \/>\n\ub85c \ub450\uba74 \\(\\nabla\\times F=0\\)\uc5d0 \uc758\ud574 \\(DF\\)\uac00 \ub300\uce6d\uc774\uace0, \uc801\ubd84 \uc548\uc744 \ud3b8\ubbf8\ubd84\ud558\uc5ec<br \/>\n\\[<br \/>\n\\frac{\\partial\\phi}{\\partial x_j}(x)<br \/>\n=<br \/>\n\\int_0^1<br \/>\n\\frac{d}{dt}<br \/>\n\\left[<br \/>\ntF_j(a+t(x-a))<br \/>\n\\right]dt<br \/>\n=<br \/>\nF_j(x)<br \/>\n\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c \ubcc4\ubaa8\uc591 \uc601\uc5ed\uc5d0\uc11c\ub294 \\(F=\\nabla\\phi\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.2.<\/span><br \/>\n\ubca1\ud130\uc7a5<br \/>\n\\[<br \/>\nF(x,y,z)=(yz,\\,xz+z,\\,xy+y)<br \/>\n\\]<br \/>\n\uac00 \ubcf4\uc874\uc7a5\uc784\uc744 \ubcf4\uc774\uace0 \ud37c\ud150\uc15c \ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 11.4.<\/span><\/p>\n<p>\n\\[<br \/>\nF(x,y)<br \/>\n=<br \/>\n\\left(<br \/>\n-\\frac{y}{x^2+y^2},\\,<br \/>\n\\frac{x}{x^2+y^2}<br \/>\n\\right)<br \/>\n\\]<br \/>\n\ub97c \\(\\mathbb{R}^2\\setminus\\{(0,0)\\}\\)\uc5d0\uc11c \uc0dd\uac01\ud558\uc790. \uc774 \ubca1\ud130\uc7a5\uc740 \ubaa8\ub4e0 \uc810\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\frac{\\partial F_2}{\\partial x}<br \/>\n&#8211;<br \/>\n\\frac{\\partial F_1}{\\partial y}<br \/>\n=<br \/>\n0<br \/>\n\\]<br \/>\n\uc774\uc9c0\ub9cc, \uc6d0\uc810\uc744 \uc911\uc2ec\uc73c\ub85c \ud558\ub294 \uc591\uc758 \ubc29\ud5a5\uc758 \ub2e8\uc704\uc6d0 \\(C\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\oint_CF\\cdot dr=2\\pi<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uc815\uc758\uc5ed\uc5d0 \uc6d0\uc810\uc744 \ub458\ub7ec\uc2fc \uad6c\uba4d\uc774 \uc788\uc73c\ubbc0\ub85c \ud3d0\uace1\uc120\uc744 \ud55c \uc810\uc73c\ub85c \uc218\ucd95\uc2dc\ud0ac \uc218 \uc5c6\ub2e4. \ud55c\ud3b8 \uc6d0\uc810\uc744 \ub458\ub7ec\uc2f8\uc9c0 \uc54a\ub294 \uc870\uac01\ub9c8\ub2e4 \\(C^1\\)\uc778 \ub2e8\uc21c\ud3d0\uace1\uc120\uc5d0 \ub300\ud574\uc11c\ub294 \uadf8\ub9b0\uc758 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uc5ec \uc801\ubd84\uac12\uc774 \\(0\\)\uc784\uc744 \ud655\uc778\ud560 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.3.<\/span><br \/>\n\uc815\ub9ac 11.3\uc758 (1)\u2013(3)\uc758 \ub3d9\uce58\ub97c \uc9c1\uc811 \uc99d\uba85\ud558\uc2dc\uc624. \ub610\ud55c \\(D\\)\uac00 \uc810 \\(a\\)\uc5d0 \ub300\ud558\uc5ec \ubcc4\ubaa8\uc591\uc774\uace0 \\(\\nabla\\times F=0\\)\uc77c \ub54c<br \/>\n\\[<br \/>\n\\phi(x)<br \/>\n=<br \/>\n\\int_0^1F(a+t(x-a))\\cdot(x-a)\\,dt<br \/>\n\\]<br \/>\n\uac00 \\(F=\\nabla\\phi\\)\ub97c \ub9cc\uc871\uc2dc\ud0b4\uc744 \uc9c1\uc811 \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\ubca1\ud130\uc7a5\uc758 \uc120\uc801\ubd84\uc744 \ubbf8\ubd84\ud615\uc2dd\uc73c\ub85c \ud45c\ud604\ud560 \uc218\ub3c4 \uc788\ub2e4. \\(\\mathbb{R}^2\\)\uc5d0\uc11c \ubca1\ud130\uc7a5<br \/>\n\\[<br \/>\nF=P\\mathbf{i}+Q\\mathbf{j}<br \/>\n\\]<br \/>\n\uc5d0 \ub300\uc751\ud558\ub294 \\(1\\)-\ud615\uc2dd\uc744<br \/>\n\\[<br \/>\n\\omega=P\\,dx+Q\\,dy<br \/>\n\\]<br \/>\n\ub85c \uc4f4\ub2e4. \uc5ec\uae30\uc11c \\(dx,dy\\)\ub294 \uc811\ubca1\ud130\uc5d0 \uc791\uc6a9\ud558\ub294 \uc88c\ud45c \uc120\ud615\ud568\uc218\uc774\ub2e4. \\(r(t)=(x(t),y(t))\\)\uc774\uba74<br \/>\n\\[<br \/>\n\\int_C\\omega<br \/>\n=<br \/>\n\\int_a^b<br \/>\n\\left[<br \/>\nP(r(t))x'(t)+Q(r(t))y'(t)<br \/>\n\\right]dt<br \/>\n=<br \/>\n\\int_CF\\cdot dr.<br \/>\n\\]<br \/>\n\\(\\mathbb{R}^3\\)\uc5d0\uc11c\ub3c4<br \/>\n\\[<br \/>\nF=P\\mathbf{i}+Q\\mathbf{j}+R\\mathbf{k}<br \/>\n\\]<br \/>\n\uc5d0 \ub300\uc751\ud558\ub294 \\(1\\)-\ud615\uc2dd\uc740<br \/>\n\\[<br \/>\n\\omega=P\\,dx+Q\\,dy+R\\,dz<br \/>\n\\]<br \/>\n\uc774\uace0 \\(\\int_C\\omega=\\int_CF\\cdot dr\\)\uc774\ub2e4.<\/p>\n<h3>\uadf8\ub9b0\uc758 \uc815\ub9ac<\/h3>\n<p>\uc774 \uc808\uc5d0\uc11c\ub294 \ud544\uc694\uc5d0 \ub530\ub77c \uc720\ud55c \uac1c\uc758 \\(x\\)-\ub2e8\uc21c\uc601\uc5ed \ub610\ub294 \\(y\\)-\ub2e8\uc21c\uc601\uc5ed\uc73c\ub85c \ubd84\ud560\ud560 \uc218 \uc788\uace0 \uacbd\uacc4\uac00 \uc870\uac01\ub9c8\ub2e4 \\(C^1\\)\uc778 \uc720\uacc4 \ud3c9\uba74\uc601\uc5ed\uc744 <span class=\"defined\">\uc815\uce59 \ud3c9\uba74\uc601\uc5ed<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uacbd\uacc4\uc758 \uc591\uc758 \ubc29\ud5a5\uc740 \uc601\uc5ed\uc744 \uc9c4\ud589\ubc29\ud5a5\uc758 \uc67c\ucabd\uc5d0 \ub450\ub294 \ubc29\ud5a5\uc774\ub2e4. \ub530\ub77c\uc11c \uc678\ubd80\uacbd\uacc4\ub294 \ubc18\uc2dc\uacc4\ubc29\ud5a5\uc774\uace0 \uad6c\uba4d\uc758 \uacbd\uacc4\ub294 \uc2dc\uacc4\ubc29\ud5a5\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.5. (\uadf8\ub9b0\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\\(D\\subseteq\\mathbb{R}^2\\)\uac00 \uc815\uce59 \ud3c9\uba74\uc601\uc5ed\uc774\uace0 \\(P,Q\\)\uac00 \\(\\overline D\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc5d0\uc11c \\(C^1\\)\uc774\uba74<br \/>\n\\[<br \/>\n\\oint_{\\partial D}P\\,dx+Q\\,dy<br \/>\n=<br \/>\n\\iint_D<br \/>\n\\left(<br \/>\n\\frac{\\partial Q}{\\partial x}<br \/>\n&#8211;<br \/>\n\\frac{\\partial P}{\\partial y}<br \/>\n\\right)dA.<br \/>\n\\tag{11.3}<br \/>\n\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\partial D\\)\uc5d0\ub294 \uc591\uc758 \ubc29\ud5a5\uc744 \uc900\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800<br \/>\n\\[<br \/>\nD<br \/>\n=<br \/>\n\\{(x,y)\\mid a\\le x\\le b,\\ \\phi(x)\\le y\\le\\psi(x)\\}<br \/>\n\\]<br \/>\n\uc778 \\(y\\)-\ub2e8\uc21c\uc601\uc5ed\uc744 \uc0dd\uac01\ud558\uc790. \uc218\uc9c1\uc778 \ub450 \uacbd\uacc4\uc870\uac01\uc5d0\uc11c\ub294 \\(dx=0\\)\uc774\uace0, \uc544\ub798\ucabd \uadf8\ub798\ud504\ub294 \\(x=a\\)\uc5d0\uc11c \\(b\\)\ub85c, \uc704\ucabd \uadf8\ub798\ud504\ub294 \\(x=b\\)\uc5d0\uc11c \\(a\\)\ub85c \uc9c4\ud589\ud558\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\n\\oint_{\\partial D}P\\,dx<br \/>\n&#038;=<br \/>\n\\int_a^bP(x,\\phi(x))\\,dx<br \/>\n&#8211;<br \/>\n\\int_a^bP(x,\\psi(x))\\,dx\\\\<br \/>\n&#038;=<br \/>\n-\\int_a^b<br \/>\n\\int_{\\phi(x)}^{\\psi(x)}<br \/>\n\\frac{\\partial P}{\\partial y}(x,y)\\,dy\\,dx\\\\<br \/>\n&#038;=<br \/>\n-\\iint_D\\frac{\\partial P}{\\partial y}\\,dA.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\ub9c8\ucc2c\uac00\uc9c0\ub85c<br \/>\n\\[<br \/>\nD<br \/>\n=<br \/>\n\\{(x,y)\\mid c\\le y\\le d,\\ \\alpha(y)\\le x\\le\\beta(y)\\}<br \/>\n\\]<br \/>\n\uc778 \\(x\\)-\ub2e8\uc21c\uc601\uc5ed\uc5d0\uc11c\ub294<br \/>\n\\[<br \/>\n\\oint_{\\partial D}Q\\,dy<br \/>\n=<br \/>\n\\iint_D\\frac{\\partial Q}{\\partial x}\\,dA.<br \/>\n\\]<br \/>\n\ub450 \uc2dd\uc744 \ub354\ud558\uba74 \ub450 \ubc29\ud5a5\uc73c\ub85c \ubaa8\ub450 \ub2e8\uc21c\ud55c \uc601\uc5ed\uc5d0\uc11c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4. \uc77c\ubc18\uc801\uc778 \uc815\uce59 \ud3c9\uba74\uc601\uc5ed\uc740 \ud544\uc694\ud55c \ubc29\ud5a5\uc758 \ub2e8\uc21c\uc601\uc5ed\ub4e4\ub85c \uc720\ud55c\ud558\uac8c \ubd84\ud560\ud560 \uc218 \uc788\uace0, \uc870\uac01\uc758 \ub0b4\ubd80\uacbd\uacc4\uc5d0\uc11c\ub294 \uc11c\ub85c \ubc18\ub300 \ubc29\ud5a5\uc758 \uc120\uc801\ubd84\uc774 \uc18c\uac70\ub41c\ub2e4. \ub530\ub77c\uc11c \uc601\uc5ed \uac00\ubc95\uc131\uacfc <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\">10\uc7a5\uc758 \ud478\ube44\ub2c8 \uc815\ub9ac<\/a>\uc5d0 \uc758\ud574 \uc77c\ubc18\uc801\uc778 \uacbd\uc6b0\ub3c4 \uc131\ub9bd\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ubca1\ud130\uc7a5 \\(F=P\\mathbf{i}+Q\\mathbf{j}\\)\uc5d0 \ub300\ud558\uc5ec \uadf8\ub9b0\uc758 \uc815\ub9ac\ub294<br \/>\n\\[<br \/>\n\\oint_{\\partial D}F\\cdot dr<br \/>\n=<br \/>\n\\iint_D(\\nabla\\times(P,Q,0))\\cdot\\mathbf{k}\\,dA<br \/>\n\\tag{11.4}<br \/>\n\\]<br \/>\n\ub85c \uc4f8 \uc218 \uc788\ub2e4. \ub610\ud55c \ubc14\uae65\ucabd \ub2e8\uc704\ubc95\uc120\ubca1\ud130\ub97c \\(\\mathbf n\\)\uc774\ub77c \ud558\uba74<br \/>\n\\[<br \/>\n\\oint_{\\partial D}F\\cdot\\mathbf n\\,ds<br \/>\n=<br \/>\n\\iint_D<br \/>\n\\left(<br \/>\n\\frac{\\partial P}{\\partial x}<br \/>\n+<br \/>\n\\frac{\\partial Q}{\\partial y}<br \/>\n\\right)dA<br \/>\n=<br \/>\n\\iint_D\\nabla\\cdot F\\,dA.<br \/>\n\\tag{11.5}<br \/>\n\\]<br \/>\n\uccab \ubc88\uc9f8 \ud615\ud0dc\ub97c <span class=\"defined\">\uc21c\ud658-\ud68c\uc804 \ud615\uc2dd<\/span>, \ub450 \ubc88\uc9f8 \ud615\ud0dc\ub97c <span class=\"defined\">\uc720\ucd9c-\ubc1c\uc0b0 \ud615\uc2dd<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.4.<\/span><br \/>\n\uadf8\ub9b0\uc758 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec<br \/>\n\\[<br \/>\n\\oint_C(x^2-y)\\,dx+(x+y^2)\\,dy<br \/>\n\\]<br \/>\n\ub97c \uacc4\uc0b0\ud558\uc2dc\uc624. \uc5ec\uae30\uc11c \\(C\\)\ub294 \uc6d0 \\(x^2+y^2=4\\)\uc5d0 \ubc18\uc2dc\uacc4\ubc29\ud5a5\uc744 \uc900 \uace1\uc120\uc774\ub2e4.<\/p>\n<\/div>\n<p>\uadf8\ub9b0\uc758 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uba74 \ud3c9\uba74\ub3c4\ud615\uc758 \ub113\uc774\ub97c<br \/>\n\\[<br \/>\n\\operatorname{Area}(D)<br \/>\n=<br \/>\n\\iint_DdA<br \/>\n=<br \/>\n\\frac12<br \/>\n\\oint_{\\partial D}<br \/>\nx\\,dy-y\\,dx<br \/>\n\\tag{11.6}<br \/>\n\\]<br \/>\n\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 11.6.<\/span><\/p>\n<p>\\(a,b&gt;0\\)\uc77c \ub54c \ud0c0\uc6d0<br \/>\n\\[<br \/>\n\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1<br \/>\n\\]<br \/>\n\uc744 \\(x=a\\cos t\\), \\(y=b\\sin t\\), \\(0\\le t\\le2\\pi\\)\ub85c \ub9e4\uac1c\ud654\ud558\uba74<br \/>\n\\[<br \/>\n\\operatorname{Area}<br \/>\n=<br \/>\n\\frac12<br \/>\n\\int_0^{2\\pi}<br \/>\n\\left[<br \/>\n(a\\cos t)(b\\cos t)<br \/>\n&#8211;<br \/>\n(b\\sin t)(-a\\sin t)<br \/>\n\\right]dt<br \/>\n=<br \/>\n\\pi ab.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.5.<\/span><br \/>\n\ud3c9\uba74\ub3c4\ud615\uc758 \ub113\uc774 \uacf5\uc2dd (11.6)\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>\uba74\uc801\ubd84\uacfc \ubc1c\uc0b0 \uc815\ub9ac<\/h3>\n<p>\\(\\mathbb{R}^3\\)\uc758 \uace1\uba74 \\(S\\)\uac00 \\(D\\subseteq\\mathbb{R}^2\\)\uc5d0\uc11c \uc815\uc758\ub41c \\(C^1\\) \ub9e4\uac1c\ud654<br \/>\n\\[<br \/>\nr\\colon D\\to\\mathbb{R}^3<br \/>\n\\]<br \/>\n\ub97c \uac00\uc9c0\uace0, \uc5ec\uae30\uc11c\ub294 \\(r\\)\uc774 \\(D\\)\uc758 \ub0b4\ubd80\uc5d0\uc11c \uc77c\ub300\uc77c\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \ub610\ud55c<br \/>\n\\[<br \/>\nN(u,v)<br \/>\n=<br \/>\n\\frac{\\partial r}{\\partial u}<br \/>\n\\times<br \/>\n\\frac{\\partial r}{\\partial v}<br \/>\n\\ne0<br \/>\n\\tag{11.7}<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(r\\)\uc744 <span class=\"defined\">\uc815\uce59 \ub9e4\uac1c\ud654<\/span>(regular parametrization)\ub77c\uace0 \ud55c\ub2e4. \ub2e8\uc704\ubc95\ubca1\ud130\ub294<br \/>\n\\[<br \/>\n\\mathbf n=\\frac{N}{\\|N\\|}<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uace1\uba74 \uc804\uccb4\uc5d0\uc11c \uc5f0\uc18d\uc801\uc73c\ub85c \ub2e8\uc704\ubc95\ubca1\ud130\ub97c \ud558\ub098 \uc120\ud0dd\ud560 \uc218 \uc788\uc73c\uba74 \uace1\uba74\uc774 <span class=\"defined\">\ubc29\ud5a5\uc744 \uac00\uc9c4\ub2e4<\/span>(orientable)\uace0 \ud558\uace0, \uadf8 \uc120\ud0dd\uc744 \uace1\uba74\uc758 \ubc29\ud5a5\uc774\ub77c\uace0 \ud55c\ub2e4.<\/p>\n<p>\uc5f0\uc18d \uc2a4\uce7c\ub77c\uc7a5 \\(f\\)\uc758 <span class=\"defined\">\uba74\uc801\ubd84<\/span>(surface integral)\uc740<br \/>\n\\[<br \/>\n\\iint_Sf\\,dS<br \/>\n=<br \/>\n\\iint_D<br \/>\nf(r(u,v))<br \/>\n\\left\\|<br \/>\n\\frac{\\partial r}{\\partial u}<br \/>\n\\times<br \/>\n\\frac{\\partial r}{\\partial v}<br \/>\n\\right\\|du\\,dv<br \/>\n\\tag{11.8}<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \uace1\uba74\uc774 \\(z=g(x,y)\\)\uc758 \uadf8\ub798\ud504\uc774\uba74<br \/>\n\\[<br \/>\n\\iint_Sf\\,dS<br \/>\n=<br \/>\n\\iint_D<br \/>\nf(x,y,g(x,y))<br \/>\n\\sqrt{<br \/>\n1+\\left(\\frac{\\partial g}{\\partial x}\\right)^2<br \/>\n+\\left(\\frac{\\partial g}{\\partial y}\\right)^2<br \/>\n}\\,dx\\,dy.<br \/>\n\\tag{11.9}<br \/>\n\\]<\/p>\n<p>\ubc29\ud5a5\uc744 \uac00\uc9c4 \uace1\uba74 \\(S\\)\uc5d0 \ub300\ud55c \ubca1\ud130\uc7a5 \\(F\\)\uc758 \uc720\ud5a5\uba74\uc801\ubd84 \ub610\ub294 <span class=\"defined\">\uc720\ucd9c<\/span>(flux)\uc740<br \/>\n\\[<br \/>\n\\iint_SF\\cdot\\mathbf n\\,dS<br \/>\n=<br \/>\n\\iint_D<br \/>\nF(r(u,v))<br \/>\n\\cdot<br \/>\n\\left(<br \/>\n\\frac{\\partial r}{\\partial u}<br \/>\n\\times<br \/>\n\\frac{\\partial r}{\\partial v}<br \/>\n\\right)du\\,dv.<br \/>\n\\tag{11.10}<br \/>\n\\]<br \/>\n\uc624\ub978\ucabd \uc2dd\uc5d0\uc11c\ub294 \ub9e4\uac1c\ud654\uc758 \ubc29\ud5a5\uc774 \uc120\ud0dd\ud55c \\(\\mathbf n\\)\uacfc \uc77c\uce58\ud55c\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 11.6.<\/span><br \/>\n\uace1\uba74\uc758 \ub450 \uc815\uce59 \ub9e4\uac1c\ud654\uac00 \\(C^1\\) \ubbf8\ubd84\ub3d9\ud615\uc0ac\uc0c1\uc73c\ub85c \uc5f0\uacb0\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\">\uc815\ub9ac 10.9\uc758 \ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac<\/a>\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc2a4\uce7c\ub77c \uba74\uc801\ubd84\uc774 \ub9e4\uac1c\ud654\uc5d0 \ubb34\uad00\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\ub9e4\uac1c\ubcc0\uc218\ubcc0\ud658\uc774 \ubc29\ud5a5\uc744 \ubcf4\uc874\ud558\uba74 \uc720\ud5a5\uba74\uc801\ubd84\uc774 \ubcc0\ud558\uc9c0 \uc54a\uace0, \ubc29\ud5a5\uc744 \ub4a4\uc9d1\uc73c\uba74 \ubd80\ud638\uac00 \ubc14\ub01c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\uc774 \uc808\uc5d0\uc11c \\(3\\)\ucc28\uc6d0 <span class=\"defined\">\uc815\uce59 \uc601\uc5ed<\/span>\uc740 \uac01 \uc88c\ud45c\ubc29\ud5a5\uc5d0 \ub300\ud558\uc5ec \ud544\uc694\ud55c \ub9cc\ud07c \uc720\ud55c \uac1c\uc758 \ub2e8\uc21c\uc601\uc5ed\uc73c\ub85c \ubd84\ud560\ud560 \uc218 \uc788\uace0 \uacbd\uacc4\uac00 \uc720\ud55c \uac1c\uc758 \\(C^1\\) \uace1\uba74\uc870\uac01\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc720\uacc4 \uc601\uc5ed\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.7. (\ubc1c\uc0b0 \uc815\ub9ac\/\uac00\uc6b0\uc2a4 \uc815\ub9ac)<\/span><\/p>\n<p>\\(V\\subseteq\\mathbb{R}^3\\)\uac00 \uc815\uce59 \uc601\uc5ed\uc774\uace0 \\(F\\)\uac00 \\(\\overline V\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc5d0\uc11c \\(C^1\\)\uc778 \ubca1\ud130\uc7a5\uc774\uba74<br \/>\n\\[<br \/>\n\\iint_{\\partial V}F\\cdot\\mathbf n\\,dS<br \/>\n=<br \/>\n\\iiint_V\\nabla\\cdot F\\,dV.<br \/>\n\\tag{11.11}<br \/>\n\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\mathbf n\\)\uc740 \\(\\partial V\\)\uc758 \ubc14\uae65\ucabd \ub2e8\uc704\ubc95\ubca1\ud130\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800 \\(F=(F_1,0,0)\\)\uc774\uace0<br \/>\n\\[<br \/>\nV<br \/>\n=<br \/>\n\\{<br \/>\n(x,y,z)\\mid<br \/>\n(y,z)\\in D,\\<br \/>\n\\alpha(y,z)\\le x\\le\\beta(y,z)<br \/>\n\\}<br \/>\n\\]<br \/>\n\uc778 \\(x\\)-\ub2e8\uc21c\uc601\uc5ed\uc744 \uc0dd\uac01\ud558\uc790. \ud478\ube44\ub2c8 \uc815\ub9ac\uc640 \ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\n\\iiint_V\\frac{\\partial F_1}{\\partial x}\\,dV<br \/>\n=<br \/>\n\\iint_D<br \/>\n\\left[<br \/>\nF_1(\\beta(y,z),y,z)<br \/>\n&#8211;<br \/>\nF_1(\\alpha(y,z),y,z)<br \/>\n\\right]dy\\,dz.<br \/>\n\\]<br \/>\n\uacbd\uacc4\uc758 \ub450 \\(x\\)-\uadf8\ub798\ud504\uc5d0\uc11c \ubc14\uae65\ucabd \ubca1\ud130\uba74\uc801\uc18c\uc758 \\(x\\)-\uc131\ubd84\uc740 \uac01\uac01 \\(+1\\)\uacfc \\(-1\\)\uc774\uace0, \\(D\\)\uc758 \uacbd\uacc4\uc5d0\uc11c \uc0dd\uae30\ub294 \uc606\uba74\uc758 \ubc95\ubca1\ud130\ub294 \\(x\\)-\uc131\ubd84\uc774 \\(0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \uc6b0\ubcc0\uc740 \uc815\ud655\ud788<br \/>\n\\[<br \/>\n\\iint_{\\partial V}(F_1,0,0)\\cdot\\mathbf n\\,dS<br \/>\n\\]<br \/>\n\uc774\ub2e4. \\(F_2,F_3\\)\uc5d0 \ub300\ud574\uc11c\ub3c4 \uac01\uac01 \\(y\\)-\ub2e8\uc21c, \\(z\\)-\ub2e8\uc21c \uc601\uc5ed\uc5d0\uc11c \uac19\uc740 \ub17c\ub9ac\ub97c \uc801\uc6a9\ud55c\ub2e4. \uc77c\ubc18\uc801\uc778 \uc815\uce59 \uc601\uc5ed\uc744 \uc774\ub7ec\ud55c \uc601\uc5ed\ub4e4\ub85c \ubd84\ud560\ud558\uba74 \ub0b4\ubd80\uba74\uc5d0\uc11c\uc758 \uc720\ucd9c\uc740 \ubc18\ub300 \ubc29\ud5a5\uc73c\ub85c \ub450 \ubc88 \ub098\ud0c0\ub098 \uc18c\uac70\ub41c\ub2e4. \uc138 \uc131\ubd84\uc758 \uacb0\uacfc\ub97c \ub354\ud558\uba74 \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ubc1c\uc0b0 \uc815\ub9ac\ub97c \\(F=r\/3=(x,y,z)\/3\\)\uc5d0 \uc801\uc6a9\ud558\uba74<br \/>\n\\[<br \/>\n\\operatorname{Vol}(V)<br \/>\n=<br \/>\n\\frac13<br \/>\n\\iint_{\\partial V}<br \/>\nr\\cdot\\mathbf n\\,dS.<br \/>\n\\tag{11.12}<br \/>\n\\]<\/p>\n<p>\ub610\ud55c \\(f\\)\uac00 \\(\\overline V\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc5d0\uc11c \\(C^1\\), \\(g\\)\uac00 \uac19\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc5d0\uc11c \\(C^2\\)\uc774\uba74<br \/>\n\\[<br \/>\n\\nabla\\cdot(f\\nabla g)<br \/>\n=<br \/>\n\\nabla f\\cdot\\nabla g+f\\Delta g<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \ubc1c\uc0b0 \uc815\ub9ac\uc5d0 \uc758\ud574 <span class=\"defined\">\uadf8\ub9b0\uc758 \uc81c1\ud56d\ub4f1\uc2dd<\/span><br \/>\n\\[<br \/>\n\\iiint_V<br \/>\n\\left(<br \/>\nf\\Delta g+\\nabla f\\cdot\\nabla g<br \/>\n\\right)dV<br \/>\n=<br \/>\n\\iint_{\\partial V}<br \/>\nf\\frac{\\partial g}{\\partial n}\\,dS<br \/>\n\\tag{11.13}<br \/>\n\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \uc5ec\uae30\uc11c<br \/>\n\\[<br \/>\n\\frac{\\partial g}{\\partial n}<br \/>\n=<br \/>\n\\nabla g\\cdot\\mathbf n<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 11.8.<\/span><\/p>\n<p>\ubc18\uc9c0\ub984\uc774 \\(R\\)\uc778 \uacf5 \\(B\\)\uc5d0 \ub300\ud558\uc5ec \\(F=(x,y,z)\/3\\)\uc774\uba74 \\(\\nabla\\cdot F=1\\)\uc774\ub2e4. \uad6c\uba74\uc5d0\uc11c \\(r\\cdot\\mathbf n=R\\)\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\operatorname{Vol}(B)<br \/>\n=<br \/>\n\\frac13<br \/>\n\\iint_{\\partial B}R\\,dS<br \/>\n=<br \/>\n\\frac13R\\cdot4\\pi R^2<br \/>\n=<br \/>\n\\frac{4\\pi R^3}{3}.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.7.<\/span><br \/>\n\ubc1c\uc0b0 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \ubca1\ud130\uc7a5<br \/>\n\\[<br \/>\nF=(x^3,y^3,z^3)<br \/>\n\\]<br \/>\n\uc758 \ub2e8\uc704\uad6c\uba74\uc744 \ubc14\uae65\ucabd\uc73c\ub85c \ud1b5\uacfc\ud558\ub294 \uc720\ub7c9\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.8.<\/span><br \/>\n3\ucc28\uc6d0 \uc601\uc5ed\uc758 \ubd80\ud53c \uacf5\uc2dd (11.12)\ub97c \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.9.<\/span><br \/>\n\uadf8\ub9b0\uc758 \ud56d\ub4f1\uc2dd (11.13)\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac<\/h3>\n<p>\uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uc720\ud5a5\uace1\uba74\uc740 \uc720\ud55c \uac1c\uc758 \uc815\uce59 \uace1\uba74\uc870\uac01\uc744 \uc774\uc5b4 \ubd99\uc5ec \uc5bb\ub294 \uace1\uba74\uc73c\ub85c \uc0dd\uac01\ud55c\ub2e4. \uc11c\ub85c \ub9cc\ub098\ub294 \ub0b4\ubd80\uacbd\uacc4\uc5d0\ub294 \ub450 \uc870\uac01\uc5d0\uc11c \uc11c\ub85c \ubc18\ub300 \ubc29\ud5a5\uc774 \uc720\ub3c4\ub41c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.9. (\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac)<\/span><\/p>\n<p>\\(S\\)\uac00 \ucef4\ud329\ud2b8\ud55c \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uc720\ud5a5\uace1\uba74\uc774\uace0 \\(\\partial S\\)\uac00 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uc720\ud5a5\uace1\uc120\uc774\ub77c\uace0 \ud558\uc790. \\(F\\)\uac00 \\(S\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc5d0\uc11c \\(C^1\\)\uc774\uba74<br \/>\n\\[<br \/>\n\\oint_{\\partial S}F\\cdot dr<br \/>\n=<br \/>\n\\iint_S(\\nabla\\times F)\\cdot\\mathbf n\\,dS.<br \/>\n\\tag{11.14}<br \/>\n\\]<br \/>\n\uacbd\uacc4\uc758 \ubc29\ud5a5\uc740 \uc120\ud0dd\ud55c \ubc95\ubca1\ud130\uc640 \uc624\ub978\uc190 \ubc95\uce59\uc774 \ub9de\ub3c4\ub85d \uc815\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800 \\(S\\)\uac00 \uadf8\ub798\ud504<br \/>\n\\[<br \/>\nr(x,y)=(x,y,g(x,y)),<br \/>\n\\quad (x,y)\\in D<br \/>\n\\]<br \/>\n\uc774\uace0 \uc704\ucabd \ubc29\ud5a5\uc744 \uac00\uc9c0\ub294 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud558\uc790. \uc774\ub54c<br \/>\n\\[<br \/>\nN=(-g_x,-g_y,1)<br \/>\n\\]<br \/>\n\uc774\uace0, \\(F=(P,Q,R)\\)\ub97c \uadf8\ub798\ud504 \uc704\uc5d0 \uc81c\ud55c\ud558\uba74<br \/>\n\\[<br \/>\nF\\cdot dr<br \/>\n=<br \/>\n(P+Rg_x)\\,dx+(Q+Rg_y)\\,dy.<br \/>\n\\]<br \/>\n\uadf8\ub9b0\uc758 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\n\\oint_{\\partial S}F\\cdot dr<br \/>\n&#038;=<br \/>\n\\iint_D<br \/>\n\\left[<br \/>\n\\frac{\\partial}{\\partial x}(Q+Rg_y)<br \/>\n&#8211;<br \/>\n\\frac{\\partial}{\\partial y}(P+Rg_x)<br \/>\n\\right]dx\\,dy\\\\<br \/>\n&#038;=<br \/>\n\\iint_D<br \/>\n(\\nabla\\times F)(r(x,y))<br \/>\n\\cdot(-g_x,-g_y,1)\\,dx\\,dy\\\\<br \/>\n&#038;=<br \/>\n\\iint_S(\\nabla\\times F)\\cdot\\mathbf n\\,dS.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\uac00\uc6b4\ub370 \ub4f1\uc2dd\uc5d0\uc11c\ub294 \ud569\uc131\ud568\uc218\uc758 \ud3b8\ubbf8\ubd84\uc5d0 \uc5f0\uc1c4\ubc95\uce59\uc744 \uc0ac\uc6a9\ud588\ub2e4. \uc544\ub798\ucabd \ubc29\ud5a5\uc5d0\uc11c\ub294 \uc591\ubcc0\uc758 \ubd80\ud638\uac00 \ub3d9\uc2dc\uc5d0 \ubc14\ub010\ub2e4.<\/p>\n<p>\uc815\uce59 \uace1\uba74\uc758 \uac01 \uc810\uc5d0\uc11c\ub294 \uc801\uc5b4\ub3c4 \ud558\ub098\uc758 \uc88c\ud45c\ud3c9\uba74\uc73c\ub85c\uc758 \uc0ac\uc601\uc774 \uac00\uc5ed \ubbf8\ubd84\uc744 \uac00\uc9c0\ubbc0\ub85c <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\">9\uc7a5\uc758 \uc5ed\ud568\uc218 \uc815\ub9ac<\/a>\uc5d0 \uc758\ud574 \uace1\uba74\uc744 \uad6d\uc18c\uc801\uc73c\ub85c \uadf8\ub798\ud504\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \ucef4\ud329\ud2b8\uc131\uc73c\ub85c \uc720\ud55c \uac1c\uc758 \uadf8\ub798\ud504 \uc870\uac01\uc744 \ud0dd\ud560 \uc218 \uc788\uc73c\ubbc0\ub85c, \uc77c\ubc18\uc801\uc778 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uba74\uc740 \uc720\ud55c \uac1c\uc758 \uc774\ub7ec\ud55c \uadf8\ub798\ud504 \uc870\uac01\uc73c\ub85c \ubd84\ud560\ud560 \uc218 \uc788\ub2e4. \uac01 \uc870\uac01\uc5d0 \uc704 \uacb0\uacfc\ub97c \uc801\uc6a9\ud558\uc5ec \ub354\ud558\uba74 \ub0b4\ubd80\uacbd\uacc4\uc758 \uc120\uc801\ubd84\uc740 \ubc18\ub300 \ubc29\ud5a5\uc73c\ub85c \ub450 \ubc88 \ub098\ud0c0\ub098 \uc18c\uac70\ub418\uace0 \uc678\ubd80\uacbd\uacc4\ub9cc \ub0a8\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ud2b9\ud788 \\(S\\)\uac00 \uacbd\uacc4\uac00 \uc5c6\ub294 \ud3d0\uace1\uba74\uc774\uba74<br \/>\n\\[<br \/>\n\\iint_S<br \/>\n(\\nabla\\times F)\\cdot\\mathbf n\\,dS<br \/>\n=<br \/>\n0.<br \/>\n\\tag{11.15}<br \/>\n\\]<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.10.<\/span><br \/>\n\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec<br \/>\n\\[<br \/>\n\\oint_CF\\cdot dr<br \/>\n\\]<br \/>\n\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624. \uc5ec\uae30\uc11c \\(F=(z,x,y)\\)\uc774\uace0 \\(C\\)\ub294 \ud3c9\uba74 \\(x+y+z=1\\)\uacfc \uc138 \uc88c\ud45c\ud3c9\uba74\uc774 \ub9cc\ub4dc\ub294 \uc0bc\uac01\ud615\uc758 \uacbd\uacc4\uc774\ub2e4. \\(C\\)\uc758 \ubc29\ud5a5\uc740 \ubc95\ubca1\ud130 \\((1,1,1)\\)\uacfc \uc624\ub978\uc190 \ubc95\uce59\uc774 \ub9de\ub3c4\ub85d \uc815\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 11.11.<\/span><br \/>\n\\(D\\subseteq\\mathbb{R}^3\\)\uac00 \uacf5 \ub610\ub294 \uc9c1\uc721\uba74\uccb4\uc758 \ub0b4\ubd80\uc774\uace0 \\(F\\colon D\\to\\mathbb{R}^3\\)\uac00 \\(C^2\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \uc870\uac74\uc774 \ub3d9\uce58\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(D\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\nabla\\times G=F<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(C^2\\) \ubca1\ud130\uc7a5 \\(G\\colon D\\to\\mathbb{R}^3\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\ud3d0\ud3ec\uac00 \\(D\\)\uc5d0 \ud3ec\ud568\ub418\uace0 \ubc1c\uc0b0 \uc815\ub9ac\ub97c \uc801\uc6a9\ud560 \uc218 \uc788\ub294 \uc784\uc758\uc758 \uc815\uce59 \uc601\uc5ed \\(V\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\iint_{\\partial V}F\\cdot\\mathbf n\\,dS=0.<br \/>\n\\tag{11.16}<br \/>\n\\]<\/li>\n<li>\\(D\\)\uc5d0\uc11c \\(\\nabla\\cdot F=0\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<p>(3)\\(\\Rightarrow\\)(1)\uc5d0\uc11c\ub294 \\(a\\in D\\)\ub97c \uace0\uc815\ud558\uace0<br \/>\n\\[<br \/>\nG(x)<br \/>\n=<br \/>\n\\int_0^1<br \/>\nt\\,F(a+t(x-a))\\times(x-a)\\,dt<br \/>\n\\]<br \/>\n\ub97c \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/p>\n<\/div>\n<h3>\ubbf8\ubd84\ud615\uc2dd\uacfc \uc77c\ubc18\ud654\ub41c \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac<\/h3>\n<p>\uc801\ubd84 \uc815\ub9ac\ub4e4\uc744 \ud558\ub098\uc758 \uc5b8\uc5b4\ub85c \ud45c\ud604\ud558\uae30 \uc704\ud574 <span class=\"defined\">\ubbf8\ubd84\ud615\uc2dd<\/span>(differential form)\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \uc5f4\ub9b0\uc9d1\ud569 \\(U\\subseteq\\mathbb{R}^n\\)\uc5d0\uc11c \\(C^1\\) \\(k\\)-\ud615\uc2dd\uc740 \uac01 \uc810\uc5d0\uc11c \\(k\\)\uac1c\uc758 \ubca1\ud130\uc5d0 \uc791\uc6a9\ud558\ub294 \uad50\ub300 \\(k\\)-\uc911\uc120\ud615\ud568\uc218\uc774\uace0, \uc88c\ud45c\ub85c\ub294<br \/>\n\\[<br \/>\n\\omega<br \/>\n=<br \/>\n\\sum_{1\\le i_1&lt;\\cdots&lt;i_k\\le n}<br \/>\na_{i_1\\cdots i_k}<br \/>\n\\,dx_{i_1}\\wedge\\cdots\\wedge dx_{i_k}<br \/>\n\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p><span class=\"defined\">\uc410\uae30\uacf1<\/span>(wedge product)\uc740 \uc30d\uc120\ud615\uc774\uace0 \uacb0\ud569\uc801\uc774\uba70<br \/>\n\\[<br \/>\ndx_i\\wedge dx_j=-dx_j\\wedge dx_i,<br \/>\n\\quad<br \/>\ndx_i\\wedge dx_i=0<br \/>\n\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc77c\ubc18\uc801\uc73c\ub85c \\(p\\)-\ud615\uc2dd \\(\\alpha\\)\uc640 \\(q\\)-\ud615\uc2dd \\(\\beta\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\alpha\\wedge\\beta<br \/>\n=<br \/>\n(-1)^{pq}\\beta\\wedge\\alpha<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>\\(0\\)-\ud615\uc2dd\uc740 \uc2a4\uce7c\ub77c\ud568\uc218\uc774\uace0, \\(1\\)-\ud615\uc2dd\uc740<br \/>\n\\[<br \/>\n\\omega=\\sum_{i=1}^na_i\\,dx_i<br \/>\n\\]<br \/>\n\uaf34\uc774\ub2e4. \\(\\mathbb{R}^3\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\nP\\,dy\\wedge dz+Q\\,dz\\wedge dx+R\\,dx\\wedge dy<br \/>\n\\]<br \/>\n\ub294 \ubca1\ud130\uc7a5 \\((P,Q,R)\\)\uc758 \uc720\ud5a5\uba74\uc801\ubd84\uc5d0 \ub300\uc751\ud558\uace0,<br \/>\n\\[<br \/>\nf\\,dx\\wedge dy\\wedge dz<br \/>\n\\]<br \/>\n\ub294 \ubd80\ud53c\uc801\ubd84\uc5d0 \ub300\uc751\ud55c\ub2e4.<\/p>\n<p><span class=\"defined\">\uc678\ubbf8\ubd84<\/span>(exterior derivative) \\(d\\)\ub294 \\(k\\)-\ud615\uc2dd\uc744 \\((k+1)\\)-\ud615\uc2dd\uc73c\ub85c \ubcf4\ub0b8\ub2e4. \ud568\uc218 \\(f\\)\uc5d0 \ub300\ud574\uc11c\ub294<br \/>\n\\[<br \/>\ndf<br \/>\n=<br \/>\n\\sum_{i=1}^n<br \/>\n\\frac{\\partial f}{\\partial x_i}\\,dx_i.<br \/>\n\\tag{11.17}<br \/>\n\\]<br \/>\n\\(1\\)-\ud615\uc2dd \\(\\omega=\\sum_iP_i\\,dx_i\\)\uc5d0 \ub300\ud574\uc11c\ub294<br \/>\n\\[<br \/>\nd\\omega<br \/>\n=<br \/>\n\\sum_{i&lt;j}<br \/>\n\\left(<br \/>\n\\frac{\\partial P_j}{\\partial x_i}<br \/>\n&#8211;<br \/>\n\\frac{\\partial P_i}{\\partial x_j}<br \/>\n\\right)<br \/>\ndx_i\\wedge dx_j.<br \/>\n\\tag{11.18}<br \/>\n\\]<br \/>\n\ub610 \\(2\\)-\ud615\uc2dd<br \/>\n\\[<br \/>\n\\omega=\\sum_{i&lt;j}Q_{ij}\\,dx_i\\wedge dx_j<br \/>\n\\]<br \/>\n\uc5d0 \ub300\ud574\uc11c\ub294<br \/>\n\\[<br \/>\nd\\omega<br \/>\n=<br \/>\n\\sum_{i&lt;j}\\sum_{k=1}^n<br \/>\n\\frac{\\partial Q_{ij}}{\\partial x_k}<br \/>\n\\,dx_k\\wedge dx_i\\wedge dx_j,<br \/>\n\\tag{11.19}<br \/>\n\\]<br \/>\n\uc774\uba70 \uac19\uc740 \ubbf8\ubd84\uc18c\uac00 \ubc18\ubcf5\ub418\ub294 \ud56d\uc740 \\(0\\)\uc774 \ub41c\ub2e4. \ud2b9\ud788 \\(\\mathbb{R}^3\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\omega<br \/>\n=<br \/>\nP\\,dy\\wedge dz<br \/>\n+<br \/>\nQ\\,dz\\wedge dx<br \/>\n+<br \/>\nR\\,dx\\wedge dy<br \/>\n\\]<br \/>\n\uc774\uba74<br \/>\n\\[<br \/>\nd\\omega<br \/>\n=<br \/>\n\\left(<br \/>\nP_x+Q_y+R_z<br \/>\n\\right)<br \/>\ndx\\wedge dy\\wedge dz.<br \/>\n\\]<\/p>\n<p>\uc678\ubbf8\ubd84\uc740 \uc120\ud615\uc774\uace0, \\(\\alpha\\)\uac00 \\(p\\)-\ud615\uc2dd\uc774\uba74<br \/>\n\\[<br \/>\nd(\\alpha\\wedge\\beta)<br \/>\n=<br \/>\nd\\alpha\\wedge\\beta<br \/>\n+<br \/>\n(-1)^p\\alpha\\wedge d\\beta<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub610\ud55c \\(C^2\\) \ud615\uc2dd\uc5d0 \ub300\ud574\uc11c\ub294 \ud074\ub808\ub85c \uc815\ub9ac\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\nd(d\\omega)=0<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uc774 \uc131\uc9c8\uc744 \\(d^2=0\\)\uc774\ub77c\uace0 \uc4f4\ub2e4. \\(d^2=0\\) \uc790\uccb4\ub97c \ud3ec\uc559\uce74\ub808 \ubcf4\uc870\uc815\ub9ac\ub77c\uace0 \ubd80\ub974\ub294 \uac83\uc740 \uc815\ud655\ud558\uc9c0 \uc54a\ub2e4. <span class=\"defined\">\ud3ec\uc559\uce74\ub808 \ubcf4\uc870\uc815\ub9ac<\/span>(Poincar\u00e9 lemma)\ub294 \ubcc4\ubaa8\uc591 \uc5f4\ub9b0\uc9d1\ud569\uc5d0\uc11c \\(d\\omega=0\\)\uc778 \uc591\uc758 \ucc28\uc218\uc758 \ud615\uc2dd\uc774 \uc2e4\uc81c\ub85c \\(\\omega=d\\eta\\) \uaf34\uc774 \ub41c\ub2e4\ub294 \ub354 \uac15\ud55c \uba85\uc81c\uc774\ub2e4. \uadf8 \uc77c\ubc18\uc801\uc778 \uc99d\uba85\uc740 \uc774 \ucc45\uc758 \ubc94\uc704\ub97c \ub118\uc5b4\uac00\ubbc0\ub85c \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 11.10.<\/span><\/p>\n<p>\\(\\mathbb{R}^3\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\omega=x\\,dy-y\\,dx<br \/>\n\\]<br \/>\n\uc774\uba74<br \/>\n\\[<br \/>\nd\\omega<br \/>\n=<br \/>\ndx\\wedge dy-dy\\wedge dx<br \/>\n=<br \/>\n2\\,dx\\wedge dy.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.12.<\/span><br \/>\n\\(2\\)-\ud615\uc2dd<br \/>\n\\[<br \/>\n\\omega<br \/>\n=<br \/>\nx\\,dy\\wedge dz<br \/>\n+<br \/>\ny\\,dz\\wedge dx<br \/>\n+<br \/>\nz\\,dx\\wedge dy<br \/>\n\\]<br \/>\n\uc5d0 \ub300\ud558\uc5ec \\(d\\omega\\)\ub97c \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\ubbf8\ubd84\ud615\uc2dd\uc758 \uc801\ubd84\uc740 \ub9e4\uac1c\ud654\uc5d0 \ub300\ud55c <span class=\"defined\">\ub2f9\uae40<\/span>(pullback)\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ud55c\ub2e4. \ub9e4\ub044\ub7ec\uc6b4 \uc0ac\uc0c1 \\(\\Phi\\colon N\\to M\\)\uacfc \\(k\\)-\ud615\uc2dd \\(\\omega\\)\uc5d0 \ub300\ud558\uc5ec \\(\\Phi^*\\omega\\)\ub97c<br \/>\n\\[<br \/>\n(\\Phi^*\\omega)_p(v_1,\\ldots,v_k)<br \/>\n=<br \/>\n\\omega_{\\Phi(p)}<br \/>\n\\bigl(<br \/>\nD\\Phi_pv_1,\\ldots,D\\Phi_pv_k<br \/>\n\\bigr)<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \ubc29\ud5a5\uc744 \ubcf4\uc874\ud558\ub294 \ub9e4\uac1c\ud654 \\(\\Phi\\colon U\\subseteq\\mathbb{R}^k\\to M\\) \uc704\uc5d0\uc11c \\(k\\)-\ud615\uc2dd\uc758 \uc801\ubd84\uc740<br \/>\n\\[<br \/>\n\\int_{\\Phi(U)}\\omega<br \/>\n=<br \/>\n\\int_U\\Phi^*\\omega<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\">\uc815\ub9ac 10.9\uc758 \ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac<\/a>\uc5d0 \uc758\ud574 \uc774 \uac12\uc740 \ubc29\ud5a5\uc744 \ubcf4\uc874\ud558\ub294 \uc88c\ud45c\ubcc0\ud658\uc5d0 \ubb34\uad00\ud558\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 11.11.<\/span><\/p>\n<p>\\(\\mathbb{R}^2\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\omega=x\\,dy<br \/>\n\\]<br \/>\n\ub97c<br \/>\n\\[<br \/>\nr(t)=(\\cos t,\\sin t),<br \/>\n\\quad 0\\le t\\le\\frac{\\pi}{2}<br \/>\n\\]<br \/>\n\uc704\uc5d0\uc11c \uc801\ubd84\ud558\uba74<br \/>\n\\[<br \/>\nr^*\\omega<br \/>\n=<br \/>\n\\cos^2t\\,dt<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\int_C\\omega<br \/>\n=<br \/>\n\\int_0^{\\pi\/2}\\cos^2t\\,dt<br \/>\n=<br \/>\n\\frac{\\pi}{4}.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 11.12.<\/span><\/p>\n<p>\n\\[<br \/>\n\\omega=z\\,dx\\wedge dy<br \/>\n\\]<br \/>\n\ub97c<br \/>\n\\[<br \/>\nS=\\{(x,y,z)\\mid z=x+y,\\ 0\\le x,y\\le1\\}<br \/>\n\\]<br \/>\n\uc704\uc5d0\uc11c \uc704\ucabd \ubc29\ud5a5\uc73c\ub85c \uc801\ubd84\ud558\uc790. \\(r(x,y)=(x,y,x+y)\\)\uc774\uba74<br \/>\n\\[<br \/>\nr^*\\omega=(x+y)\\,dx\\wedge dy<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\iint_S\\omega<br \/>\n=<br \/>\n\\int_0^1\\int_0^1(x+y)\\,dx\\,dy<br \/>\n=<br \/>\n1.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 11.13.<\/span><\/p>\n<p>\n\\[<br \/>\n\\omega=x\\,dy\\wedge dz<br \/>\n\\]<br \/>\n\ub97c \ub2e8\uc704 \uc717\ubc18\uad6c \\(S\\)\uc5d0 \ubc14\uae65\ucabd \ubc29\ud5a5\uc744 \uc8fc\uc5b4 \uc801\ubd84\ud558\uc790. \uc774 \ud615\uc2dd\uc740 \ubca1\ud130\uc7a5 \\(F=(x,0,0)\\)\uc758 \uc720\ucd9c\uc5d0 \ub300\uc751\ud55c\ub2e4. \ub2e8\uc704\uad6c\uc5d0\uc11c \\(\\mathbf n=(x,y,z)\\)\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\iint_S\\omega<br \/>\n=<br \/>\n\\iint_Sx^2\\,dS.<br \/>\n\\]<br \/>\n\uad6c\uba74\uc88c\ud45c<br \/>\n\\[<br \/>\nx=\\cos\\phi\\sin\\theta,\\quad<br \/>\n0\\le\\phi\\le2\\pi,\\quad<br \/>\n0\\le\\theta\\le\\frac{\\pi}{2}<br \/>\n\\]<br \/>\n\ub97c \uc0ac\uc6a9\ud558\uba74<br \/>\n\\[<br \/>\n\\iint_Sx^2\\,dS<br \/>\n=<br \/>\n\\int_0^{2\\pi}\\int_0^{\\pi\/2}<br \/>\n\\cos^2\\phi\\sin^3\\theta\\,d\\theta\\,d\\phi<br \/>\n=<br \/>\n\\frac{2\\pi}{3}.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 11.14.<\/span><\/p>\n<p>\\(3\\)-\ud615\uc2dd<br \/>\n\\[<br \/>\n\\omega=xyz\\,dx\\wedge dy\\wedge dz<br \/>\n\\]<br \/>\n\ub97c \\(V=[0,1]^3\\) \uc704\uc5d0\uc11c \uc801\ubd84\ud558\uba74<br \/>\n\\[<br \/>\n\\iiint_V\\omega<br \/>\n=<br \/>\n\\left(\\int_0^1x\\,dx\\right)<br \/>\n\\left(\\int_0^1y\\,dy\\right)<br \/>\n\\left(\\int_0^1z\\,dz\\right)<br \/>\n=<br \/>\n\\frac18.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 11.13.<\/span><br \/>\n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ubc14\uae65\ucabd \ubc29\ud5a5\uc744 \uac00\uc9c4 \ub2e8\uc704\uad6c\uba74 \\(S\\) \uc704\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\omega<br \/>\n=<br \/>\ny\\,dz\\wedge dx<br \/>\n+<br \/>\nz\\,dx\\wedge dy<br \/>\n\\]<br \/>\n\uc758 \uc801\ubd84\uc744 \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\ubc14\uae65\ucabd \ubc29\ud5a5\uc744 \uac00\uc9c4 \uc6d0\uae30\ub465\uc758 \uc606\uba74<br \/>\n\\[<br \/>\nS=\\{(x,y,z)\\mid x^2+y^2=1,\\ 0\\le z\\le2\\}<br \/>\n\\]<br \/>\n\uc704\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\omega=x\\,dy\\wedge dz<br \/>\n\\]<br \/>\n\uc758 \uc801\ubd84\uc744 \uad6c\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\uc774\uc81c \uc77c\ubc18\ud654\ub41c \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub97c \uc9c4\uc220\ud558\uae30 \uc704\ud574 \ud544\uc694\ud55c \ub2e4\uc591\uccb4\uc758 \ucd5c\uc18c\ud55c\uc758 \uac1c\ub150\uc744 \uc815\ub9ac\ud55c\ub2e4. \uc774 \ucc45\uc5d0\uc11c \\(k\\)\ucc28\uc6d0 <span class=\"defined\">\ub9e4\ub044\ub7ec\uc6b4 \ub2e4\uc591\uccb4<\/span>(smooth manifold) \\(M\\)\uc740 \ud558\uc6b0\uc2a4\ub3c4\ub974\ud504\uc774\uace0 \uc81c2\uac00\uc0b0\uc778 \uc704\uc0c1\uacf5\uac04\uc73c\ub85c\uc11c, \uac01 \uc810\uc774 \\(\\mathbb{R}^k\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uacfc \uc704\uc0c1\ub3d9\ud615\uc778 \uadfc\ubc29\uc744 \uac00\uc9c0\uba70 \uc88c\ud45c\ubcc0\ud658\uc774 \\(C^\\infty\\)\uc778 \uacf5\uac04\uc744 \ub73b\ud55c\ub2e4. \uc88c\ud45c\uc30d \\((U,\\phi)\\)\ub97c <span class=\"defined\">\ucc28\ud2b8<\/span>(chart), \ucc28\ud2b8\ub4e4\uc758 \ubaa8\uc784\uc744 <span class=\"defined\">\uc544\ud2c0\ub77c\uc2a4<\/span>(atlas)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc11c\ub85c \uacb9\uce58\ub294 \ub450 \ucc28\ud2b8\uc758 \uc804\ud658 \uc0ac\uc0c1\uc758 \uc57c\ucf54\ube44 \ud589\ub82c\uc2dd\uc774 \uc591\uc218\uac00 \ub418\ub3c4\ub85d \uc544\ud2c0\ub77c\uc2a4\ub97c \uc120\ud0dd\ud560 \uc218 \uc788\uc73c\uba74 \\(M\\)\uc774 <span class=\"defined\">\ubc29\ud5a5\uc744 \uac00\uc9c4\ub2e4<\/span>(oriented)\uace0 \ud55c\ub2e4. \uad6c\uba74\uc740 \ubc14\uae65\ucabd \ubc95\ubca1\ud130\ub85c \ubc29\ud5a5\uc744 \uc815\ud560 \uc218 \uc788\uc9c0\ub9cc \ubafc\ube44\uc6b0\uc2a4 \ub760\ub294 \ubc29\ud5a5\uc744 \uc815\ud560 \uc218 \uc5c6\ub2e4.<\/p>\n<p><span class=\"defined\">\uacbd\uacc4\ub97c \uac00\uc9c4 \ub2e4\uc591\uccb4<\/span>(manifold with boundary)\ub294 \uad6d\uc18c \uc88c\ud45c\uac00<br \/>\n\\[<br \/>\nH^k<br \/>\n=<br \/>\n\\{x\\in\\mathbb{R}^k\\mid x_k\\ge0\\}<br \/>\n\\]<br \/>\n\uc758 \uc0c1\ub300\uc801\uc73c\ub85c \uc5f4\ub9b0 \ubd80\ubd84\uc9d1\ud569\uc5d0 \uac12\uc744 \uac00\uc9c0\ub3c4\ub85d \ud5c8\uc6a9\ud55c \ub2e4\uc591\uccb4\uc774\ub2e4. \uc88c\ud45c\uc5d0\uc11c \\(x_k=0\\)\uc5d0 \ub193\uc774\ub294 \uc810\ub4e4\uc758 \uc9d1\ud569\uc744 \\(\\partial M\\)\uc774\ub77c\uace0 \ud55c\ub2e4. \ub9e4\ub044\ub7ec\uc6b4 \uc804\ud658 \uc0ac\uc0c1 \ub54c\ubb38\uc5d0 \uc774 \uc815\uc758\ub294 \ucc28\ud2b8\uc758 \uc120\ud0dd\uc5d0 \ubb34\uad00\ud558\ub2e4.<\/p>\n<p>\ud568\uc218 \\(f\\colon M\\to\\mathbb{R}\\)\uc758 <span class=\"defined\">\uc9c0\uc9c0\uc9d1\ud569<\/span>(support)\uc740<br \/>\n\\[<br \/>\n\\operatorname{supp}(f)<br \/>\n=<br \/>\n\\overline{\\{x\\in M\\mid f(x)\\ne0\\}}<br \/>\n\\]<br \/>\n\uc774\ub2e4. <span class=\"defined\">\ub2e8\uc704\ubd84\ud560<\/span>(partition of unity)\uc740 \uc5f4\ub9b0\ub36e\uac1c \\(\\{U_\\alpha\\}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n0\\le\\rho_\\alpha\\le1,\\quad<br \/>\n\\operatorname{supp}(\\rho_\\alpha)\\subseteq U_\\alpha,\\quad<br \/>\n\\sum_\\alpha\\rho_\\alpha=1<br \/>\n\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(C^\\infty\\) \ud568\uc218\ub4e4\uc758 \uad6d\uc18c\uc720\ud55c\ud55c \ubaa8\uc784\uc774\ub2e4. \ub9e4\ub044\ub7ec\uc6b4 \ub2e4\uc591\uccb4\uc758 \uc784\uc758\uc758 \uc5f4\ub9b0\ub36e\uac1c\uc5d0 \uc885\uc18d\ub41c \ub9e4\ub044\ub7ec\uc6b4 \ub2e8\uc704\ubd84\ud560\uc774 \uc874\uc7ac\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc740 \uc774 \uc7a5\uc5d0\uc11c \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<p>\ubc29\ud5a5\uc744 \uac00\uc9c4 \\(k\\)\ucc28\uc6d0 \ub2e4\uc591\uccb4\uc5d0\uc11c \ucef4\ud329\ud2b8 \uc9c0\uc9c0\ub97c \uac00\uc9c4 \\(k\\)-\ud615\uc2dd\uc758 \uc801\ubd84\uc740 \ubc29\ud5a5\uc744 \ubcf4\uc874\ud558\ub294 \ucc28\ud2b8\uc640 \ub2e8\uc704\ubd84\ud560\ub85c \uad6d\uc18c \uc801\ubd84\uc744 \ud569\ud558\uc5ec \uc815\uc758\ud55c\ub2e4. \ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc774 \uac12\uc740 \ucc28\ud2b8\uc640 \ub2e8\uc704\ubd84\ud560\uc758 \uc120\ud0dd\uc5d0 \ubb34\uad00\ud558\ub2e4. \ucef4\ud329\ud2b8 \ub2e4\uc591\uccb4\uc5d0\uc11c\ub294 \ubaa8\ub4e0 \uc5f0\uc18d \ud615\uc2dd\uc758 \uc9c0\uc9c0\uc9d1\ud569\uc774 \uc790\ub3d9\uc73c\ub85c \ucef4\ud329\ud2b8\ud558\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.15. (\uc77c\ubc18\ud654\ub41c \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac)<\/span><\/p>\n<p>\\(M\\)\uc774 \ubc29\ud5a5\uc744 \uac00\uc9c4 \\(k\\)\ucc28\uc6d0 \\(C^\\infty\\) \ucef4\ud329\ud2b8 \ub2e4\uc591\uccb4\uc774\uace0 \\(\\partial M\\)\uc774 \uadf8 \uacbd\uacc4\ub77c\uace0 \ud558\uc790. \\(\\omega\\)\uac00 \\(M\\)\uc5d0\uc11c \uc815\uc758\ub41c \\(C^1\\) \\((k-1)\\)-\ud615\uc2dd\uc774\uba74<br \/>\n\\[<br \/>\n\\int_{\\partial M}\\omega<br \/>\n=<br \/>\n\\int_Md\\omega.<br \/>\n\\tag{11.20}<br \/>\n\\]<br \/>\n\uacbd\uacc4\uc758 \ubc29\ud5a5\uc740 \ubc14\uae65\ucabd \ubc95\ubca1\ud130\uac00 \uba3c\uc800 \uc624\uace0 \uadf8 \ub4a4\uc5d0 \\(\\partial M\\)\uc758 \uc591\uc758 \ubc29\ud5a5\uae30\uc800\uac00 \uc62c \ub54c \\(M\\)\uc758 \uc591\uc758 \ubc29\ud5a5\uae30\uc800\uac00 \ub418\ub3c4\ub85d \uc815\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800 \\(M\\)\uc774 \uc0c1\ubc18\uacf5\uac04<br \/>\n\\[<br \/>\nH^k=\\{x_k\\ge0\\}<br \/>\n\\]<br \/>\n\uc758 \uc0c1\ub300\uc801\uc73c\ub85c \uc5f4\ub9b0 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(\\omega\\)\uac00 \ucef4\ud329\ud2b8 \uc9c0\uc9c0\ub97c \uac00\uc9c4\ub2e4\uace0 \ud558\uc790. \uacc4\uc218\ud568\uc218\ub4e4\uc740 \uc9c0\uc9c0\uc9d1\ud569 \ubc16\uc5d0\uc11c \\(0\\)\uc73c\ub85c \uc5f0\uc7a5\ud558\uc5ec \\(H^k\\) \uc804\uccb4\uc5d0\uc11c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc88c\ud45c\ub85c<br \/>\n\\[<br \/>\n\\omega<br \/>\n=<br \/>\n\\sum_{i=1}^k<br \/>\n(-1)^{i-1}<br \/>\na_i\\,<br \/>\ndx_1\\wedge\\cdots\\wedge\\widehat{dx_i}\\wedge\\cdots\\wedge dx_k<br \/>\n\\]<br \/>\n\ub77c\uace0 \uc4f0\uba74<br \/>\n\\[<br \/>\nd\\omega<br \/>\n=<br \/>\n\\left(<br \/>\n\\sum_{i=1}^k<br \/>\n\\frac{\\partial a_i}{\\partial x_i}<br \/>\n\\right)<br \/>\ndx_1\\wedge\\cdots\\wedge dx_k.<br \/>\n\\]<br \/>\n\\(i&lt;k\\)\uc778 \ud56d\uc740 \\(x_i\\) \ubc29\ud5a5\uc73c\ub85c \uc801\ubd84\ud558\uba74 \ucef4\ud329\ud2b8 \uc9c0\uc9c0 \ub54c\ubb38\uc5d0 \uc591 \ub05d\uac12\uc774 \\(0\\)\uc774\ubbc0\ub85c \uc801\ubd84\uc774 \\(0\\)\uc774\ub2e4. \\(i=k\\)\uc778 \ud56d\uc740<br \/>\n\\[<br \/>\n\\int_0^\\infty<br \/>\n\\frac{\\partial a_k}{\\partial x_k}\\,dx_k<br \/>\n=<br \/>\n-a_k(x_1,\\ldots,x_{k-1},0)<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\int_Md\\omega<br \/>\n=<br \/>\n&#8211;<br \/>\n\\int_{\\mathbb{R}^{k-1}}<br \/>\na_k(x_1,\\ldots,x_{k-1},0)\\,<br \/>\ndx_1\\cdots dx_{k-1}.<br \/>\n\\]<br \/>\n\ud55c\ud3b8 \\(H^k\\)\uc758 \uacbd\uacc4\uc5d0\uc11c \ubc14\uae65\ucabd \ubc95\ubca1\ud130\ub294 \\(-\\partial\/\\partial x_k\\)\uc774\ub2e4. \ubc14\uae65\ucabd \ubc95\ubca1\ud130 \uc6b0\uc120 \uaddc\uc57d\uc73c\ub85c \uc720\ub3c4\ub41c \uacbd\uacc4 \ubc29\ud5a5\uc744 \uc0ac\uc6a9\ud558\uba74 \\(\\omega\\)\uc758 \uacbd\uacc4 \uc801\ubd84\ub3c4 \uc815\ud655\ud788<br \/>\n\\[<br \/>\n&#8211;<br \/>\n\\int_{\\mathbb{R}^{k-1}}<br \/>\na_k(x_1,\\ldots,x_{k-1},0)\\,<br \/>\ndx_1\\cdots dx_{k-1}<br \/>\n\\]<br \/>\n\uc774 \ub41c\ub2e4. \ub530\ub77c\uc11c \uad6d\uc18c \uc88c\ud45c\uc5d0\uc11c \uc815\ub9ac\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uc77c\ubc18\uc801\uc778 \\(M\\)\uc5d0\uc11c\ub294 \uc720\ud55c\ud55c \ubc29\ud5a5\ubcf4\uc874 \uc88c\ud45c\ub36e\uac1c\uc640 \uc774\uc5d0 \uc885\uc18d\ub41c \ub2e8\uc704\ubd84\ud560 \\(\\{\\rho_\\alpha\\}\\)\ub97c \ud0dd\ud55c\ub2e4. \uac01 \\(\\rho_\\alpha\\omega\\)\ub294 \ud558\ub098\uc758 \uc88c\ud45c\uadfc\ubc29 \uc548\uc5d0\uc11c \ucef4\ud329\ud2b8 \uc9c0\uc9c0\ub97c \uac00\uc9c0\ubbc0\ub85c \ubc29\uae08 \uc99d\uba85\ud55c \uad6d\uc18c \uacb0\uacfc\ub97c \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\n\\int_{\\partial M}\\omega<br \/>\n&#038;=<br \/>\n\\sum_\\alpha<br \/>\n\\int_{\\partial M}\\rho_\\alpha\\omega\\\\<br \/>\n&#038;=<br \/>\n\\sum_\\alpha<br \/>\n\\int_Md(\\rho_\\alpha\\omega)\\\\<br \/>\n&#038;=<br \/>\n\\int_M<br \/>\nd\\left(<br \/>\n\\sum_\\alpha\\rho_\\alpha\\omega<br \/>\n\\right)\\\\<br \/>\n&#038;=<br \/>\n\\int_Md\\omega.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\uc5ec\uae30\uc11c\ub294 \ub2f9\uae40\uacfc \uc678\ubbf8\ubd84\uc774 \uac00\ud658\ud558\uace0 \ubbf8\ubd84\ud615\uc2dd\uc758 \uc801\ubd84\uc774 \ubc29\ud5a5\ubcf4\uc874 \uc88c\ud45c\ubcc0\ud658\uc5d0 \ubd88\ubcc0\uc774\ub77c\ub294 \uc0ac\uc2e4\uc744 \uc0ac\uc6a9\ud588\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\\(k\\)-\ud615\uc2dd \\(\\omega\\)\uac00 \\(d\\omega=0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 <span class=\"defined\">\ub2eb\ud78c\ud615\uc2dd<\/span>(closed form)\uc774\ub77c\uace0 \ud55c\ub2e4. \uc5b4\ub5a4 \\((k-1)\\)-\ud615\uc2dd \\(\\eta\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\omega=d\\eta\\)\uc774\uba74 <span class=\"defined\">\uc644\uc804\ud615\uc2dd<\/span>(exact form)\uc774\ub77c\uace0 \ud55c\ub2e4. \\(d^2=0\\)\uc774\ubbc0\ub85c \uc644\uc804\ud615\uc2dd\uc740 \ud56d\uc0c1 \ub2eb\ud78c\ud615\uc2dd\uc774\ub2e4. \uc5ed\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uac70\uc9d3\uc774\uba70 \uc815\uc758\uc5ed\uc758 \uc704\uc0c1\uc5d0 \uc758\uc874\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 11.14.<\/span><br \/>\n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc644\uc804\ud615\uc2dd\uc740 \ud56d\uc0c1 \ub2eb\ud78c\ud615\uc2dd\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\\(\\mathbb{R}^2\\setminus\\{(0,0)\\}\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\omega<br \/>\n=<br \/>\n\\frac{-y\\,dx+x\\,dy}{x^2+y^2}<br \/>\n\\]<br \/>\n\uac00 \ub2eb\ud78c\ud615\uc2dd\uc784\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<li>\ub2e8\uc704\uc6d0 \\(C\\)\uc5d0 \ub300\ud558\uc5ec \\(\\int_C\\omega=2\\pi\\)\uc784\uc744 \uacc4\uc0b0\ud558\uace0, \uc120\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uc5ec \\(\\omega\\)\uac00 \uc644\uc804\ud615\uc2dd\uc774 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\uc77c\ubc18\ud654\ub41c \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub294 \uc55e\uc5d0\uc11c \ubcf8 \uc5ec\ub7ec \uc815\ub9ac\ub97c \ub3d9\uc2dc\uc5d0 \ud3ec\ud568\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(k=1\\), \\(M=[a,b]\\)\uc774\uba74 \ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac\uc774\ub2e4.<\/li>\n<li>\\(k=2\\), \\(M\\subseteq\\mathbb{R}^2\\)\uc774\uace0 \\(\\omega=P\\,dx+Q\\,dy\\)\uc774\uba74 \uadf8\ub9b0\uc758 \uc815\ub9ac\uc774\ub2e4.<\/li>\n<li>\\(k=2\\), \\(M\\subseteq\\mathbb{R}^3\\)\uc774\uace0 \\(\\omega=P\\,dx+Q\\,dy+R\\,dz\\)\uc774\uba74 \\(3\\)\ucc28\uc6d0 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\uc774\ub2e4.<\/li>\n<li>\\(k=3\\), \\(M\\subseteq\\mathbb{R}^3\\)\uc774\uace0<br \/>\n\\[<br \/>\n\\omega=P\\,dy\\wedge dz+Q\\,dz\\wedge dx+R\\,dx\\wedge dy<br \/>\n\\]<br \/>\n\uc774\uba74 \ubc1c\uc0b0 \uc815\ub9ac\uc774\ub2e4.<\/li>\n<\/ul>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.15.<\/span><br \/>\n\uc77c\ubc18\ud654\ub41c \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac(\uc815\ub9ac 11.15)\ub97c \uc0ac\uc6a9\ud558\uc5ec \uadf8\ub9b0\uc758 \uc815\ub9ac(\uc815\ub9ac 11.5), \ubc1c\uc0b0 \uc815\ub9ac(\uc815\ub9ac 11.7), \\(3\\)\ucc28\uc6d0 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac(\uc815\ub9ac 11.9)\ub97c \uac01\uac01 \uc720\ub3c4\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/\">\ud574\uc11d\ud559 \uac15\uc758\ub178\ud2b8<\/a><\/p>\n<ol class=\"contentboxorderedlist\">\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch01-real-number-system\">\uc2e4\uc218\uacc4\uc758 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch02-metric-spaces\">\uac70\ub9ac\uacf5\uac04<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch03-limit-of-sequences\">\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uc704\uc0c1\uc801 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch04-limit-of-functions-and-continuity\">\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch05-differentiation-of-functions-of-one-variable\">\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch06-the-riemann-integral\">\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch07-infinite-series\">\ubb34\ud55c\uae09\uc218<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch08-real-analytic-functions\">\uc2e4\ud574\uc11d\uc801 \ud568\uc218<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\">\ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\">\uc911\uc801\ubd84<\/a><\/li>\n<li class=\"contentboxthis\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch11-vector-field-and-fundamental-theorems\">\ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><!-- ################## --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>9\uc7a5\uc5d0\uc11c\ub294 \ub2e4\ubcc0\uc218\ud568\uc218\uc758 \ubbf8\ubd84\uc744, 10\uc7a5\uc5d0\uc11c\ub294 \ub2e4\uc911\uc801\ubd84\uc744 \ub2e4\ub8e8\uc5c8\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \ub450 \uc774\ub860\uc744 \uacb0\ud569\ud558\uc5ec \ubca1\ud130\uc7a5\uc758 \uc120\uc801\ubd84\uacfc \uba74\uc801\ubd84\uc744 \uc815\uc758\ud558\uace0, \uadf8\ub9b0\uc758 \uc815\ub9ac, \ubc1c\uc0b0 \uc815\ub9ac, \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. \ub9c8\uc9c0\ub9c9\uc5d0\ub294 \uc774 \uc801\ubd84\uc815\ub9ac\ub4e4\uc744 \ubbf8\ubd84\ud615\uc2dd\uc758 \uc5b8\uc5b4\ub85c \ud558\ub098\uc758 \uc815\ub9ac\ub85c \ud1b5\ud569\ud55c\ub2e4. \ubca1\ud130\uc7a5\uc758 \uae30\ucd08 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569 \\(D\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 \\(f\\colon D\\to\\mathbb{R}\\)\uc744 \uc2a4\uce7c\ub77c\uc7a5(scalar field)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \\(F\\colon D\\to\\mathbb{R}^n\\)\uc744 \ubca1\ud130\uc7a5(vector field)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud604\uc2e4 \uc138\uacc4\uc5d0\uc11c \uc628\ub3c4\ub098 \uc555\ub825 \ubd84\ud3ec\ub294 \uc2a4\uce7c\ub77c\uc7a5\uc774\uace0, \uc18d\ub3c4\uc7a5\uc774\ub098 \uc804\uae30\uc7a5\uc740 \ubca1\ud130\uc7a5\uc774\ub2e4. 9\uc7a5\uc5d0\uc11c \uc815\uc758\ud55c \uae30\uc6b8\uae30\uc640 \ud568\uaed8 \ub2e4\uc74c \uc5f0\uc0b0\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \\(C^1\\)&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9470,"menu_order":111,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9499","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9499","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"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=9499"}],"version-history":[{"count":18,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9499\/revisions"}],"predecessor-version":[{"id":10118,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9499\/revisions\/10118"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9470"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9499"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}