{"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":"2025-10-21T16:09:48","modified_gmt":"2025-10-21T07:09:48","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>\uc774 \uc7a5\uc5d0\uc11c\ub294 \ubca1\ud130\ud574\uc11d\uc758 \uae30\ubcf8 \uac1c\ub150\uacfc \uadf8\ub9b0\uc758 \uc815\ub9ac, \ubc1c\uc0b0 \uc815\ub9ac, \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac \ub4f1 \uc911\uc694\ud55c \uc801\ubd84\uc815\ub9ac\ub4e4\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc774\ub7ec\ud55c \uc815\ub9ac\ub4e4\uc740 \ubb3c\ub9ac\ud559\uacfc \uacf5\ud559 \ub4f1 \uc751\uc6a9 \ubd84\uc57c\uc5d0\uc11c \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud55c\ub2e4.<\/p>\n<h3>\ubca1\ud130\uc7a5\uc758 \uae30\ucd08<\/h3>\n<p>\\(\\mathbb{R}^n\\)\uc758 \uc601\uc5ed \\(D\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 \\(f: D \\to \\mathbb{R}\\)\uc744 <span class=\"defined\">\uc2a4\uce7c\ub77c\uc7a5<\/span>(scalar field)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \\(F: 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>\uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uba87 \uac00\uc9c0 \uc5f0\uc0b0\uc744 \uc815\uc758\ud558\uc790.<\/p>\n<ul>\n<li>\uc2a4\uce7c\ub77c\uc7a5 \\(f\\)\uc758 <span class=\"defined\">\uae30\uc6b8\uae30<\/span>(gradient)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\nabla f = \\left(\\frac{\\partial f}{\\partial x_1},\\, \\frac{\\partial f}{\\partial x_2},\\, \\ldots,\\, \\frac{\\partial f}{\\partial x_n}\\right).\\]<br \/>\n\uc774\uac83\uc740 \uc2a4\uce7c\ub77c\uc7a5\uc73c\ub85c\ubd80\ud130 \ubca1\ud130\uc7a5\uc744 \ub9cc\ub4e0\ub2e4.<\/li>\n<li>\ubca1\ud130\uc7a5 \\(F\\)\uc758 <span class=\"defined\">\ubc1c\uc0b0<\/span>(divergence)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\nabla \\cdot F = \\operatorname{div} F = \\sum_{i=1}^{n} \\frac{\\partial F_i}{\\partial x_i}.\\]<br \/>\n\uc774\uac83\uc740 \ubca1\ud130\uc7a5\uc73c\ub85c\ubd80\ud130 \uc2a4\uce7c\ub77c\uc7a5\uc744 \ub9cc\ub4e0\ub2e4.<\/li>\n<li>\\(\\mathbb{R}^3\\)\uc5d0\uc11c \ubca1\ud130\uc7a5 \\(F = (F_1,\\, F_2,\\, F_3)\\)\uc758 <span class=\"defined\">\ud68c\uc804<\/span>(curl)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\nabla \\times F = \\operatorname{curl} F = \\left(\\frac{\\partial F_3}{\\partial y} &#8211; \\frac{\\partial F_2}{\\partial z},\\,\\, \\frac{\\partial F_1}{\\partial z} &#8211; \\frac{\\partial F_3}{\\partial x},\\,\\, \\frac{\\partial F_2}{\\partial x} &#8211; \\frac{\\partial F_1}{\\partial y}\\right).\\]<br \/>\n\uc5ec\uae30\uc11c \\((x,\\, y,\\, z)\\)\ub294 \\(\\mathbb{R}^3\\)\uc758 \uc88c\ud45c\uc774\ub2e4. \\(F\\)\uc758 \ud68c\uc804\uc744 \ub2e4\uc74c\uacfc \uac19\uc740 \ud589\ub82c\uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<br \/>\n\\[\\nabla \\times F = \\begin{vmatrix} \\mathbf{i} &#038; \\mathbf{j} &#038; \\mathbf{k} \\\\ \\frac{\\partial}{\\partial x} &#038; \\frac{\\partial}{\\partial y} &#038; \\frac{\\partial}{\\partial z} \\\\ F_1 &#038; F_2 &#038; F_3 \\end{vmatrix}\\]<\/li>\n<li>\uc2a4\uce7c\ub77c\uc7a5 \\(f\\)\uc5d0 \ub300\ud55c <span class=\"defined\">\ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790<\/span>(Laplacian)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\Delta f = \\nabla^2 f = \\nabla \\cdot (\\nabla f) = \\sum_{i=1}^{n} \\frac{\\partial^2 f}{\\partial x_i^2}\\]<\/li>\n<\/ul>\n<p>\uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \uc5f0\uc0b0\uacfc \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(\\nabla \\times (\\nabla f) = 0\\)&nbsp; (\uae30\uc6b8\uae30\uc758 \ud68c\uc804\uc740 0\uc774\ub2e4.)<\/li>\n<li>\\(\\nabla \\cdot (\\nabla \\times F) = 0\\)&nbsp; (\ud68c\uc804\uc758 \ubc1c\uc0b0\uc740 0\uc774\ub2e4.)<\/li>\n<li>\\(\\nabla \\times (\\nabla \\times F) = \\nabla(\\nabla \\cdot F) &#8211; \\Delta F\\)&nbsp; (\ub2e8, \\(\\Delta F = (\\Delta F_1 ,\\, \\Delta F_2 ,\\, \\Delta F_3 )\\)\uc774\ub2e4.)<\/li>\n<\/ul>\n<p>\\(F\\)\uac00 \ubca1\ud130\uc7a5\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc2a4\uce7c\ub77c\uc7a5 \\(\\phi\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(F = \\nabla \\phi\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\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.<\/p>\n<p>\ubca1\ud130\uc7a5 \\(F\\)\uac00 \\(\\nabla \\cdot F = 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c \\(F\\)\ub97c <span class=\"defined\">\uc194\ub808\ub178\uc774\ub4dc\uc7a5<\/span>(solenoidal field)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ubca1\ud130 \ud56d\ub4f1\uc2dd\uc5d0 \uc758\ud574, \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\uc6d0\uc18c\ub97c \\(\\mathbf{i}=(1,\\,0)\\), \\(\\mathbf{j}=(0,\\,1)\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c, \uacf5\uc5ed\uc774 \\(\\mathbb{R}^2\\)\uc778 \ud568\uc218 \\(F\\)\ub294 \ub450 \uc2a4\uce7c\ub77c\ud568\uc218 \\(P\\), \\(Q\\)\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[F = P \\,\\mathbf{i} + Q \\,\\mathbf{j}.\\]<br \/>\n\ub9c8\ucc2c\uac00\uc9c0\ub85c \uc720\ud074\ub9ac\ub4dc \uacf5\uac04 \\(\\mathbb{R}^3\\)\uc5d0\uc11c \ud45c\uc900\uae30\uc800\uc6d0\uc18c\ub97c \\(\\mathbf{i}\\), \\(\\mathbf{j}\\), \\(\\mathbf{k}\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \uc774\ub54c \uacf5\uc5ed\uc774 \\(\\mathbb{R}^3\\)\uc778 \ud568\uc218 \\(F\\)\ub294 \uc138 \uc2a4\uce7c\ub77c\ud568\uc218 \\(P\\), \\(Q\\), \\(R\\)\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[F = P \\,\\mathbf{i} + Q \\,\\mathbf{j} + R\\,\\mathbf{k}.\\]<\/p>\n<h3>\uc120\uc801\ubd84<\/h3>\n<p>\\(\\mathbb{R}^n\\)\uc5d0\uc11c \uace1\uc120 \\(C\\)\uac00 \ub9e4\uac1c\ubcc0\uc218\ud568\uc218 \\(r: [a,\\, b] \\to \\mathbb{R}^n\\)\uc5d0 \uc758\ud558\uc5ec \ud45c\ud604\ub418\uace0, \\(r\\)\uc774 \\(C^1\\)\uc774\uba70 \\(r &#8216; (t) \\ne 0\\)\uc77c \ub54c, \\(C\\)\ub97c <span class=\"defined\">\ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120<\/span>(smooth curve)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uace1\uc120 \\(C\\)\uc5d0\ub294 \\(t\\)\uc758 \uac12\uc774 \\(a\\)\uc5d0\uc11c \\(b\\)\uae4c\uc9c0 \uc99d\uac00\ud560 \ub54c \uace1\uc120 \uc704\uc758 \uc810\uc774 \uc6c0\uc9c1\uc774\ub294 \ubc29\ud5a5\uacfc \uac19\uc740 <span class=\"defined\">\ubc29\ud5a5<\/span>\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4. \uc774\uc640 \uac19\uc774 \ubc29\ud5a5\uc774 \uc8fc\uc5b4\uc9c4 \uace1\uc120\uc744 <span class=\"defined\">\uc720\ud5a5\uace1\uc120<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc744 \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 \uc2a4\uce7c\ub77c\uc7a5 \\(f\\)\uc758 <span class=\"defined\">\uc120\uc801\ubd84<\/span>(line integral)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\int_C f\\, ds = \\int_a^b f(r(t)) \\|r'(t)\\| dt.\\tag{11.1}\\]<br \/>\n\uc5ec\uae30\uc11c \\(ds\\)\ub294 \ud638\uc758 \uae38\uc774\uc5d0 \ub300\ud55c \ubbf8\ubd84\uc18c\uc774\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \uc801\ubd84 (11.1)\uc744 <span class=\"defined\">\uace1\uc120 \uae38\uc774\uc5d0 \ub300\ud55c \uc120\uc801\ubd84<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120 \\(C\\)\uc5d0 \ub300\ud55c \ubca1\ud130\uc7a5 \\(F\\)\uc758 \uc120\uc801\ubd84\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\int_C F \\cdot dr = \\int_C F\\cdot T \\,ds = \\int_a^b F(r(t)) \\cdot r'(t) dt.\\tag{11.2}\\]<br \/>\n\uc5ec\uae30\uc11c \\(T\\)\ub294 \uace1\uc120 \\(C\\) \uc704\uc758 \uc810\uc5d0\uc11c \uc774 \uace1\uc120\uc5d0 \uc811\ud558\ub294 \ub2e8\uc704\uc811\uc120\ubca1\ud130 \\(T=r'(t) \/ \\lVert r&#8217; (t) \\rVert\\)\uc774\ub2e4.<\/p>\n<p>\uace1\uc120 \\(C\\)\uc758 \ubaa8\uc591\uc774 \ubcc0\ud558\uc9c0 \uc54a\ub354\ub77c\ub3c4 \\(C\\)\uc758 \ubc29\ud5a5\uc774 \ub2ec\ub77c\uc9c0\uba74 (11.2)\uc758 \uac12\uc774 \ub2ec\ub77c\uc9c4\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \uc801\ubd84 (11.2)\ub97c <span class=\"defined\">\ubc29\ud5a5\uc120\uc801\ubd84<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\uc801\ubd84 (11.2)\ub294 \ubb3c\ub9ac\ud559\uc5d0\uc11c \ud798 \\(F\\)\uac00 \uacbd\ub85c \\(C\\)\ub97c \ub530\ub77c \ud55c \uc77c\uc744 \ub098\ud0c0\ub0b8\ub2e4. \ub610\ud55c \\(F\\)\uac00 \uace1\uc120\uc744 \ud3ec\ud568\ud558\ub294 \uc601\uc5ed\uc5d0\uc11c \uc815\uc758\ub41c \uc18d\ub3c4\ubca1\ud130\uc7a5\uc77c \ub54c \uc801\ubd84 (11.2)\ub294 \uc810 \\(r(a)\\)\uc5d0\uc11c \uc810 \\(r(b)\\)\uae4c\uc9c0 \uace1\uc120\uc744 \ub530\ub77c \ud750\ub974\ub294 <span class=\"defined\">\uc720\ub3d9<\/span>(flow)\uc774\ub2e4. \ub9cc\uc57d \uace1\uc120 \\(C\\)\uc758 \uc2dc\uc791\uc810\uacfc \ub05d\uc810\uc774 \uc77c\uce58\ud558\uba74, \uc801\ubd84 (11.2)\ub97c \uace1\uc120\uc744 \ub530\ub978 <span class=\"defined\">\uc21c\ud658<\/span>(circulation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.1.<\/span><br \/>\n\\(\\mathbb{R}^n\\)\uc5d0\uc11c \ub450 \uace1\uc120 \\(C_1\\)\uacfc \\(C_2\\)\uac00 \uc11c\ub85c \ub2e4\ub978 \ub9e4\uac1c\ubcc0\uc218\ud568\uc218 \\(r_1\\)\uacfc \\(r_2\\)\uc5d0 \uc758\ud558\uc5ec \ud45c\ud604\ub418\uace0, \uc810\uc9d1\ud569\uc73c\ub85c\uc11c \\(C_1\\)\uacfc \\(C_2\\)\uac00 \uc77c\uce58\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(C_1\\)\uacfc \\(C_2\\)\ub294 \uac19\uc740 \uace1\uc120\uc778\uac00 \uc544\ub2c8\uba74 \ub2e4\ub978 \uace1\uc120\uc778\uac00? \uc120\uc801\ubd84\uc758 \uc815\uc758 (11.1)\uacfc (11.2)\uc758 \uad00\uc810\uc5d0\uc11c \ubcf4\uc558\uc744 \ub54c, \uc5b4\ub5a0\ud55c \uace1\uc120\uc744 \uc11c\ub85c \uac19\uc740 \uace1\uc120\uc73c\ub85c \uac04\uc8fc\ud560 \uc218 \uc788\ub294\uac00?<\/p>\n<\/div>\n<p>\ub9cc\uc57d \\(C\\)\uac00 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc774\uace0,<br \/>\n\\[C = C_1 \\cup C_2 \\cup \\cdots \\cup  C_k\\]<br \/>\n\uc640 \uac19\uc774 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120 \\(C_j\\)\ub4e4\uc744 \uc774\uc5b4 \ubd99\uc5ec \ub9cc\ub4e0 \uace1\uc120\uc77c \ub54c, \uace1\uc120 \\(C\\)\uc5d0 \ub300\ud55c \uc2a4\uce7c\ub77c\uc7a5 \\(f\\)\uc758 \uc120\uc801\ubd84\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\int_C f \\,ds = \\sum_{j=1}^{k} \\int_{C_j} f \\,ds .\\]<br \/>\n\uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120 \\(C\\)\uc5d0 \ub300\ud55c \ubca1\ud130\uc7a5 \\(F\\)\uc758 \uc120\uc801\ubd84\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\uace1\uc120 \\(C\\)\uac00 \ud3d0\uace1\uc120\uc77c \ub54c \\(C\\) \uc704\uc5d0\uc11c\uc758 \uc120\uc801\ubd84\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \\(\\oint\\) \uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[\\oint_C F \\cdot dr.\\]<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.1. (\uc120\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac)<\/span><\/p>\n<p>\\(F = \\nabla f\\)\uac00 \uc5f0\uc18d\uc774\uace0 \\(C\\)\uac00 \uc810 \\(P\\)\uc5d0\uc11c \\(Q\\)\ub85c \uac00\ub294 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\int_C F \\cdot dr = f(Q) &#8211; f(P).\\]<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc5f0\uc1c4\ubc95\uce59\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \\(\\frac{d}{dt}f(r(t)) = \\nabla f(r(t)) \\cdot r'(t)\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\int_C \\nabla f \\cdot dr = \\int_a^b \\frac{d}{dt}f(r(t)) dt = f(r(b)) &#8211; f(r(a)) = f(Q) &#8211; f(P)\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc704 \uc815\ub9ac\ub294 \ubcf4\uc874\uc7a5\uc758 \uc120\uc801\ubd84\uc774 \uacbd\ub85c\uc5d0 \ubb34\uad00\ud558\uace0 \uc2dc\uc791\uc810\uacfc \ub05d\uc810\uc5d0\ub9cc \uc758\uc874\ud55c\ub2e4\ub294 \uc131\uc9c8\uc744 \uc124\uba85\ud55c\ub2e4. \uc774\ub7ec\ud55c \uc131\uc9c8\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.2. (\ubcf4\uc874\uc801 \ubca1\ud130\uc7a5\uc758 \uc131\uc9c8)<\/span><\/p>\n<p>\ubca1\ud130\uc7a5 \\(F\\)\uac00 \ub2e8\uc21c\uc5f0\uacb0\uc601\uc5ed \\(D\\)\uc5d0\uc11c \\(C^1\\)\uc77c \ub54c, \ub2e4\uc74c \uc870\uac74\ub4e4\uc740 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(F\\)\ub294 \ubcf4\uc874\uc7a5\uc774\ub2e4.<\/li>\n<li>\ubaa8\ub4e0 \ud3d0\uace1\uc120 \\(C\\)\uc5d0 \ub300\ud574 \\(\\oint_C F \\cdot dr = 0\\)\uc774\ub2e4.<\/li>\n<li>\uc120\uc801\ubd84\uc774 \uacbd\ub85c\uc5d0 \ub300\ud558\uc5ec \ub3c5\ub9bd\uc774\ub2e4.<\/li>\n<li>\\(\\nabla \\times F = 0\\)\uc774\ub2e4. (\\(D\\subseteq \\mathbb{R}^3\\)\uc77c \ub54c)<\/li>\n<\/ol>\n<\/div>\n<p>\uc704 \uc815\ub9ac\uc5d0\uc11c \uc601\uc5ed \\(D\\)\uac00 <span class=\"defined\">\ub2e8\uc21c\uc5f0\uacb0\uc601\uc5ed<\/span>(simply connected region)\uc774\ub77c\ub294 \uac83\uc740 \\(D\\) \uc548\uc5d0 \uc788\ub294 \ubaa8\ub4e0 \ud3d0\uace1\uc120\uc744 \uc5f0\uc18d\uc801\uc73c\ub85c \ud55c \uc810\uc73c\ub85c \uc218\ucd95\uc2dc\ud0ac \uc218 \uc788\ub2e4\ub294 \uc758\ubbf8\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.2.<\/span><br \/>\n\ubca1\ud130\uc7a5 \\(F(x,\\, y,\\, z) = (yz,\\, xz + z,\\, xy + y)\\)\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 1.<\/span><\/p>\n<p>\ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ubca1\ud130\uc7a5 \\(F\\)\ub97c \uc0dd\uac01\ud558\uc790.<br \/>\n\\[F(x,\\, y) = \\left(-\\frac{y}{x^2 + y^2},\\, \\frac{x}{x^2 + y^2}\\right).\\]<br \/>\n\uc774 \ubca1\ud130\uc7a5\uc740 \uc6d0\uc810\uc744 \uc81c\uc678\ud55c \\(\\mathbb{R}^2\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(\\nabla \\times F = 0\\)\uc774\uc9c0\ub9cc, \uc6d0\uc810\uc744 \uc911\uc2ec\uc73c\ub85c \ud558\ub294 \ub2e8\uc704\uc6d0 \\(C\\)\uc5d0 \ub300\ud574<br \/>\n\\[\\oint_C F \\cdot dr = 2\\pi\\]<br \/>\n\uc774\ub2e4. \uc774 \uc801\ubd84\uac12\uc774 \\(0\\)\uc774 \uc544\ub2cc \uc774\uc720\ub294 \\(F\\)\uc758 \uc815\uc758\uc5ed\uc774 \ub2e8\uc21c\uc5f0\uacb0\uc601\uc5ed\uc774 \uc544\ub2c8\uae30 \ub54c\ubb38\uc774\ub2e4. \ub9cc\uc57d \uace1\uc120 \\(C\\)\uc758 \uc548\ucabd\uc5d0 \uc6d0\uc810\uc774 \uc5c6\ub2e4\uba74, \\(C\\)\uc5d0 \ub300\ud55c \\(F\\)\uc758 \uc120\uc801\ubd84\uc758 \uac12\uc740 \\(0\\)\uc774 \ub41c\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.3.<\/span><br \/>\n\uc815\ub9ac 11.2\ub97c \uc99d\uba85\ud558\uc2dc\uc624. (\ub4a4\uc5d0 \ub4f1\uc7a5\ud558\ub294 \uadf8\ub9b0 \uc815\ub9ac, \ubc1c\uc0b0 \uc815\ub9ac, \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.)<\/p>\n<\/div>\n<p>\ubca1\ud130\uc7a5\uc758 \uc120\uc801\ubd84\uc744 <span class=\"defined\">\ubbf8\ubd84\ud615\uc2dd<\/span>(differential form)\uc73c\ub85c \ud45c\ud604\ud560 \uc218\ub3c4 \uc788\ub2e4. \\(\\mathbb{R}^2\\)\uc5d0\uc11c \ubca1\ud130\uc7a5 \\(F = P\\,\\mathbf{i} + Q\\,\\mathbf{j}\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uc774\uc5d0 \ub300\uc751\ud558\ub294 <span class=\"defined\">1-\ud615\uc2dd<\/span>(1-form)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\omega = P\\, dx + Q\\, dy.\\]<\/p>\n<p>\uc5ec\uae30\uc11c \\(dx\\), \\(dy\\)\ub294 \uc88c\ud45c\ud568\uc218\uc758 \ubbf8\ubd84\uc744 \ub098\ud0c0\ub0b4\ub294 \uae30\ud638\ub85c\uc11c, \uc811\ubca1\ud130\uc5d0 \uc791\uc6a9\ud558\ub294 \uc120\ud615\ud568\uc218\ub85c \uc774\ud574\ud560 \uc218 \uc788\ub2e4. \uace1\uc120 \\(C\\)\uac00 \\(r(t) = x(t)\\mathbf{i} + y(t)\\mathbf{j}\\)\ub85c \ub9e4\uac1c\ubcc0\uc218\ud654\ub420 \ub54c, 1-\ud615\uc2dd \\(\\omega\\)\uc758 \uc120\uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ub41c\ub2e4.<br \/>\n\\[\\int_C \\omega = \\int_C P\\, dx + Q\\, dy = \\int_a^b \\left[P(r(t))\\frac{dx}{dt} + Q(r(t))\\frac{dy}{dt}\\right] dt.\\]<\/p>\n<p>\uc774\uac83\uc740 \ubca1\ud130\uc7a5 \\(F\\)\uc758 \uc120\uc801\ubd84 \\(\\int_C F \\cdot dr\\)\uacfc \uc77c\uce58\ud55c\ub2e4. \ub530\ub77c\uc11c \ubca1\ud130\uc7a5\uc758 \uc120\uc801\ubd84\uacfc 1-\ud615\uc2dd\uc758 \uc801\ubd84\uc740 \ubcf8\uc9c8\uc801\uc73c\ub85c \uac19\uc740 \uac1c\ub150\uc774\uba70, \uc0c1\ud669\uc5d0 \ub530\ub77c \ud3b8\ub9ac\ud55c \ud45c\uae30\ub97c \uc120\ud0dd\ud558\uc5ec \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<p>\ub9c8\ucc2c\uac00\uc9c0\ub85c \\(\\mathbb{R}^3\\)\uc5d0\uc11c \ubca1\ud130\uc7a5 \\(F = P\\,\\mathbf{i} + Q\\,\\mathbf{j} +R\\,\\mathbf{k}\\)\uc5d0 \ub300\uc751\ud558\ub294 1-\ud615\uc2dd\uc740<br \/>\n\\[\\omega = P\\, dx + Q\\, dy + R\\, dz\\]<br \/>\n\uc774\uace0, \uace1\uc120 \\(C\\)\uc5d0 \ub300\ud55c \\(F\\)\uc758 \uc801\ubd84\uc740<br \/>\n\\[\\int_C \\omega = \\int_C P\\, dx + Q\\, dy + R\\, dz = \\int_C F \\cdot dr\\]<br \/>\n\uc774\ub2e4.<\/p>\n<h3>\uadf8\ub9b0\uc758 \uc815\ub9ac<\/h3>\n<p>\ud3c9\uba74\uc5d0\uc11c\uc758 \uc911\uc694\ud55c \uc801\ubd84\uc815\ub9ac\uc778 \uadf8\ub9b0\uc758 \uc815\ub9ac\ub294 \uc774\uc911\uc801\ubd84\uacfc \uc120\uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.3. (\uadf8\ub9b0\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\\(D\\)\uac00 \\(\\mathbb{R}^2\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ub2e8\uc21c\uc5f0\uacb0\uc601\uc5ed\uc774\uace0 \uadf8 \uacbd\uacc4 \\(\\partial D\\)\uac00 \uc870\uac01\ub9c8\ub2e4 \\(C^1\\)\uc778 \ub2e8\uc21c\ud3d0\uace1\uc120\uc774\uba70, \\(P,\\, Q\\)\uac00 \\(\\overline{D}\\)\uc5d0\uc11c \\(C^1\\)\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\oint_{\\partial D} P\\, dx + Q\\, dy = \\iint_D \\left(\\frac{\\partial Q}{\\partial x} &#8211; \\frac{\\partial P}{\\partial y}\\right) dA.\\tag{11.3}\\]<br \/>\n\uc5ec\uae30\uc11c \uacbd\uacc4\uc120 \uc704\uc758 \ubc29\ud5a5\uc740 \ubc18\uc2dc\uacc4\ubc29\ud5a5\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\uba3c\uc800 \\(D\\)\uac00 \\(x\\)\ucd95\uacfc \\(y\\)\ucd95\uc5d0 \ud3c9\ud589\ud55c \uacbd\uacc4\ub97c \uac00\uc9c4 \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud55c\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(D\\)\uac00 \\(D = \\{(x,\\, y) \\mid a \\leq x \\leq b,\\, \\phi(x) \\leq y \\leq \\psi(x)\\}\\) \ud615\ud0dc\uc77c \ub54c<br \/>\n\\[\\iint_D \\frac{\\partial P}{\\partial y} dA = \\int_a^b \\int_{\\phi(x)}^{\\psi(x)} \\frac{\\partial P}{\\partial y} dy\\, dx = \\int_a^b [P(x,\\, \\psi(x)) &#8211; P(x,\\, \\phi(x))] dx\\]<br \/>\n\uc774\ub2e4. \ud55c\ud3b8, \\(D\\)\uc758 \uacbd\uacc4\uc120 \uc704\uc5d0\uc11c \uc120\uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uba74 \uac19\uc740 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<p>\ub9c8\ucc2c\uac00\uc9c0\ub85c \\(\\iint_D \\frac{\\partial Q}{\\partial x} dA = \\oint_{\\partial D} Q\\, dy\\)\ub97c \ubcf4\uc778\ub2e4.<\/p>\n<p>\\(D\\)\uac00 \uc77c\ubc18\uc801\uc778 \uc601\uc5ed\uc778 \uacbd\uc6b0\ub294 \\(D\\)\ub97c \ub2e8\uc21c\uc601\uc5ed\uc73c\ub85c \ubd84\ud560\ud558\uc5ec \uc99d\uba85\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uadf8\ub9b0\uc758 \uc815\ub9ac\ub97c \ubca1\ud130 \ud615\uc2dd\uc73c\ub85c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4. \uc989 \ubca1\ud130\uc7a5 \\(F = P\\,\\mathbf{i} + Q\\,\\mathbf{j} + 0\\mathbf{k}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\oint_{\\partial D} F \\cdot dr = \\iint_D (\\nabla \\times F) \\cdot \\mathbf{k}\\, dA.\\tag{11.4}\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\nabla \\times F = (0,\\,0,\\,Q_x &#8211; P_y )\\)\uc774\uace0, \\(\\mathbf{k}\\)\ub294 \\(z\\)\ucd95 \ubc29\ud5a5 \ub2e8\uc704\ubca1\ud130\uc774\ub2e4. \ub610\ud55c \\(F = P\\,\\mathbf{i} + Q\\,\\mathbf{j}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\oint_{\\partial D} F \\cdot \\mathbf{n} \\,ds = \\iint_D \\left( \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y}\\right)dA = \\iint_D (\\nabla \\cdot F) dA\\tag{11.5}\\]<br \/>\n\ub85c \ub098\ud0c0\ub0bc \uc218\ub3c4 \uc788\ub2e4. \uc5ec\uae30\uc11c \\(\\mathbf{n}\\)\uc740 \uacbd\uacc4\uc5d0\uc11c \ubc14\uae65\ucabd\uc744 \ud5a5\ud558\ub294 \ub2e8\uc704\ubc95\uc120\ubca1\ud130\uc774\ub2e4. \ub4f1\uc2dd (11.5)\uc758 \uccab \ubc88\uc9f8 \uc801\ubd84\uc744 \uace1\uc120 \\(\\partial D\\)\ub97c \uac00\ub85c\uc9c0\ub974\ub294 \\(F\\)\uc758 <span class=\"defined\">\uc720\ucd9c<\/span>(flux)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ub4f1\uc2dd (11.3)\uacfc (11.4)\ub97c \uadf8\ub9b0 \uc815\ub9ac\uc758 <span class=\"defined\">\uc21c\ud658-\ud68c\uc804 \ud615\uc2dd<\/span> \ub610\ub294 <span class=\"defined\">\uc811\uc120 \ud615\uc2dd<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0, \ub4f1\uc2dd (11.5)\ub97c \uadf8\ub9b0 \uc815\ub9ac\uc758 <span class=\"defined\">\uc720\ucd9c-\ubc1c\uc0b0 \ud615\uc2dd<\/span> \ub610\ub294 <span class=\"defined\">\ubc95\uc120 \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 \\(\\oint_C (x^2 &#8211; y)\\, dx + (x + y^2)\\, dy\\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. \uc5ec\uae30\uc11c \\(C\\)\ub294 \uc6d0 \\(x^2 + y^2 = 4\\)\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 \ub2e4\uc74c\uacfc \uac19\uc774 \uc120\uc801\ubd84\uc73c\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[\\text{Area}(D) = \\iint_D dA = \\frac{1}{2} \\oint_{\\partial D} x\\, dy &#8211; y\\, dx.\\tag{11.6}\\]<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span><\/p>\n<p>\\(a>0\\), \\(b>0\\)\uc77c \ub54c, \ud0c0\uc6d0 \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\)\uc758 \ub113\uc774\ub97c \uad6c\ud574 \ubcf4\uc790. \ub9e4\uac1c\ubcc0\uc218 \ud45c\ud604 \\(x = a\\cos t\\), \\(y = b\\sin t\\), \\(0\\le t\\le 2\\pi\\)\ub97c \uc0ac\uc6a9\ud558\uc5ec \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\text{Area} = \\frac{1}{2} \\int_0^{2\\pi} \\left[ (a\\cos t)(b\\cos t) &#8211; (b\\sin t)(-a\\sin t) \\right] dt = \\pi ab.\\]<\/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<p>\uadf8\ub9b0\uc758 \uc815\ub9ac\ub97c \ub2e4\uc911\uc5f0\uacb0\uc601\uc5ed\uc5d0\ub3c4 \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4. \uc989 \\(D\\)\uac00 \uad6c\uba4d\uc774 \uc788\ub294 \uc601\uc5ed\uc774\uba74, \uc120\uc801\ubd84\uc744 \uacc4\uc0b0\ud560 \ub54c \uc678\ubd80\uacbd\uacc4\uc120 \uc704\uc5d0\uc11c\ub294 \ubc18\uc2dc\uacc4\ubc29\ud5a5\uc73c\ub85c, \ub0b4\ubd80\uacbd\uacc4\uc120 \uc704\uc5d0\uc11c\ub294 \uc2dc\uacc4\ubc29\ud5a5\uc73c\ub85c \uc801\ubd84\ud55c\ub2e4.<\/p>\n<h3>\uba74\uc801\ubd84\uacfc \ubc1c\uc0b0 \uc815\ub9ac<\/h3>\n<p>\\(\\mathbb{R}^3\\)\uc5d0\uc11c \uace1\uba74 \\(S\\)\uac00 \\(D \\subseteq \\mathbb{R}^2\\)\uc778 \uc601\uc5ed \\(D\\)\uc5d0 \ub300\ud558\uc5ec \ub9e4\uac1c\ubcc0\uc218 \ud45c\ud604 \\(r: D \\to \\mathbb{R}^3\\)\ub97c \uac00\uc9c4\ub2e4\uace0 \ud558\uc790. \uc810 \\((u,\\, v) \\in D\\)\uc5d0 \ub300\uc751\ub418\ub294 \uace1\uba74 \uc704\uc758 \uc810\uc5d0\uc11c <span class=\"defined\">\ubc95\ubca1\ud130<\/span>(normal vector)\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4.<br \/>\n\\[\\mathbf{n} = \\frac{\\partial r}{\\partial u} \\times \\frac{\\partial r}{\\partial v}.\\tag{11.7}\\]<br \/>\n\uc774 \ubc95\ubca1\ud130\uac00 \\(0\\)\uc774 \uc544\ub2c8\uba74\uc11c, \\((u,\\,v)\\)\uc5d0 \ub300\ud55c \uc5f0\uc18d\ud568\uc218\uc77c \ub54c, \\(S\\)\ub97c <span class=\"defined\">\ub9e4\ub044\ub7ec\uc6b4 \uace1\uba74<\/span>(smooth surface)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uace1\uba74 \\(S\\)\uc5d0 \ub300\ud55c \uc2a4\uce7c\ub77c\uc7a5 \\(f\\)\uc758 <span class=\"defined\">\uba74\uc801\ubd84<\/span>(surface integral)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\iint_S f\\, dS = \\iint_D f(r(u,\\, v)) \\left\\|\\frac{\\partial r}{\\partial u} \\times \\frac{\\partial r}{\\partial v}\\right\\| du\\, dv.\\tag{11.8}\\]<br \/>\n\uc704 \uba74\uc801\ubd84\uc744 <span class=\"defined\">\uace1\uba74 \ub113\uc774\uc5d0 \ub300\ud55c \uba74\uc801\ubd84<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\uace1\uba74\uc774 \\(z = g(x,\\, y)\\) \ud615\ud0dc\uc758 \ud568\uc218\uc758 \uadf8\ub798\ud504\ub85c \uc8fc\uc5b4\uc9c0\uba74 \uba74\uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ub41c\ub2e4.<br \/>\n\\[\\iint_S f\\, dS = \\iint_D f(x,\\, y,\\, g(x,\\, y)) \\sqrt{1 + \\left(\\frac{\\partial g}{\\partial x}\\right)^2 + \\left(\\frac{\\partial g}{\\partial y}\\right)^2} dx\\, dy.\\tag{11.9}\\]<\/p>\n<p>\uace1\uc120 \uae38\uc774\uc5d0 \ub300\ud55c \uc120\uc801\ubd84\uacfc \ubc29\ud5a5\uc120\uc801\ubd84\uc774 \uc874\uc7ac\ud55c \uac83\ucc98\ub7fc, \uba74\uc801\ubd84\uc5d0\uc11c\ub3c4 \uace1\uba74\uc758 \ubc29\ud5a5\uc744 \uace0\ub824\ud558\ub294 \ubca1\ud130\uc7a5\uc758 \uba74\uc801\ubd84\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uace1\uba74 \\(S\\)\uac00 \ub450 \uac1c\uc758 \ubc29\ud5a5\uc744 \uac00\uc9c0\uace0, \uadf8 \uc911 \uace1\uba74 \uc704\uc758 \ubc95\ubca1\ud130 \\(\\mathbf{n}\\)\uacfc \uac19\uc740 \ubc29\ud5a5\uc744 \uae30\uc900\uc73c\ub85c \ud558\uc600\uc744 \ub54c, \\(S\\)\uc5d0 \ub300\ud55c \ubca1\ud130\uc7a5 \\(F\\)\uc758 \uba74\uc801\ubd84\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\iint_S F \\cdot \\mathbf{n}\\, dS = \\iint_D F(r(u,\\, v)) \\cdot \\left(\\frac{\\partial r}{\\partial u} \\times \\frac{\\partial r}{\\partial v}\\right) du\\, dv.\\tag{11.10}\\]<br \/>\n\uc704 \uba74\uc801\ubd84\uc744 <span class=\"defined\">\uc720\ud5a5\uba74\uc801\ubd84<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\ubb3c\ub9ac\ud559\uc5d0\uc11c \uc774 \uc801\ubd84\uc740 \ubca1\ud130\uc7a5 \\(F\\)\uac00 \uace1\uba74 \\(S\\)\ub97c \ud1b5\uacfc\ud558\ub294 \uc720\ub7c9\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.6.<\/span><br \/>\n\uace1\uba74\uc5d0 \ub300\ud558\uc5ec \ubb38\uc81c 11.1\uacfc \uac19\uc740 \uc9c8\ubb38\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.4. (\ubc1c\uc0b0 \uc815\ub9ac\/\uac00\uc6b0\uc2a4 \uc815\ub9ac)<\/span><\/p>\n<p>\\(V\\)\uac00 \\(\\mathbb{R}^3\\)\uc758 \uc720\uacc4\uc778 \uc601\uc5ed\uc774\uace0 \uadf8 \uacbd\uacc4 \\(\\partial V\\)\uac00 \uc870\uac01\ub9c8\ub2e4 \\(C^1\\)\uc778 \ud3d0\uace1\uba74\uc774\uba70, \\(F\\)\uac00 \\(\\overline{V}\\)\uc5d0\uc11c \\(C^1\\)\uc778 \ubca1\ud130\uc7a5\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\iint_{\\partial V} F \\cdot \\mathbf{n}\\, dS = \\iiint_V \\nabla \\cdot F\\, dV.\\tag{11.11}\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\mathbf{n}\\)\uc740 \\(\\partial V\\)\uc758 \ubc14\uae65\ucabd\uc744 \ud5a5\ud558\ub294 \ub2e8\uc704\ubc95\ubca1\ud130\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\\(F = (F_1,\\, 0,\\, 0)\\) \ud615\ud0dc\uc758 \ubca1\ud130\uc7a5\uc5d0 \ub300\ud574 \uba3c\uc800 \uc99d\uba85\ud55c\ub2e4.<br \/>\n\uc601\uc5ed \\(V\\)\ub97c \\(z\\)\ucd95\uc5d0 \ud3c9\ud589\ud55c \uc9c1\uc120\uc774 \uacbd\uacc4\uc640 \ub450 \uc810\uc5d0\uc11c\ub9cc \ub9cc\ub098\ub294 \ub2e8\uc21c\ud55c \ud615\ud0dc\ub85c \uac00\uc815\ud55c\ub2e4.<br \/>\n\\(\\iiint_V \\frac{\\partial F_1}{\\partial x} dV\\)\ub97c \ubc18\ubcf5\uc801\ubd84\uc73c\ub85c \uacc4\uc0b0\ud558\uace0, \uc774\uac83\uc774 \uba74\uc801\ubd84\uacfc \uac19\uc74c\uc744 \ubcf4\uc778\ub2e4.<\/p>\n<p>\\(F_2\\), \\(F_3\\) \uc131\ubd84\uc5d0 \ub300\ud574\uc11c\ub3c4 \uac19\uc740 \ubc29\ubc95\uc744 \uc801\uc6a9\ud558\uc5ec \uacc4\uc0b0\ud55c\ub2e4.<\/p>\n<p>\\(V\\)\uac00 \uc77c\ubc18\uc801\uc778 \uc601\uc5ed\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(V\\)\ub97c \ub2e8\uc21c\uc601\uc5ed\uc73c\ub85c \ubd84\ud560\ud558\uc5ec \uc99d\uba85\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ubc1c\uc0b0 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc5bb\uc744 \uc218 \uc788\ub294 \uacb0\uacfc\ub294 \ub300\ud45c\uc801\uc73c\ub85c \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li><strong>\ubd80\ud53c \uacc4\uc0b0<\/strong>: \\(3\\)\ucc28\uc6d0 \uc601\uc5ed \\(V\\)\uc758 \ubd80\ud53c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\frac{1}{3} \\iint_{\\partial V} r \\cdot \\mathbf{n}\\, dS.\\tag{11.12}\\]<br \/>\n\uc5ec\uae30\uc11c \\(r = (x,\\, y,\\, z)\\)\uc774\ub2e4.<\/li>\n<li><strong>\uadf8\ub9b0\uc758 \uc81c1\ud56d\ub4f1\uc2dd<\/strong>: \\(f\\)\uc640 \\(g\\)\uac00 \uc2a4\uce7c\ub77c\uc7a5\uc774\uace0 \\(f\\)\uac00 \uc5f0\uc18d\uc778 \ub3c4\ud568\uc218\ub97c \uac00\uc9c0\uba70 \\(g\\)\uac00 \uc5f0\uc18d\uc778 \uc774\uacc4\ub3c4\ud568\uc218\ub97c \uac00\uc9c8 \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\iiint_V (f\\Delta g + \\nabla f \\cdot \\nabla g) dV = \\iint_{\\partial V} f\\frac{\\partial g}{\\partial n} dS.\\tag{11.13}\\]<\/li>\n<\/ul>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span><\/p>\n<p>\ubc18\uc9c0\ub984\uc758 \uae38\uc774\uac00 \\(R\\)\uc778 \uad6c\uc758 \ubd80\ud53c\ub97c \ubc1c\uc0b0 \uc815\ub9ac\ub85c \uacc4\uc0b0\ud574 \ubcf4\uc790. \\(F = \\frac{1}{3}(x,\\, y,\\, z)\\)\uc5d0 \ub300\ud574 \\(\\nabla \\cdot F = 1\\)\uc774\ubbc0\ub85c<br \/>\n\\[V = \\iiint_B dV = \\iint_{\\partial B} \\frac{1}{3}r \\cdot \\mathbf{n}\\, dS = \\frac{1}{3} \\iint_{\\partial B} R\\, dS = \\frac{1}{3} R \\cdot 4\\pi R^2 = \\frac{4\\pi R^3}{3}.\\]<\/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 \\(F = (x^3,\\, y^3,\\, z^3)\\)\uc758 \ub2e8\uc704\uad6c \ud45c\uba74\uc744 \ud1b5\uacfc\ud558\ub294 \uc720\ub7c9(flux)\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>\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub294 \uace1\uba74 \uc704\uc5d0\uc11c\uc758 \uba74\uc801\ubd84\uacfc \uadf8 \uace1\uba74\uc758 \uacbd\uacc4\uc5d0\uc11c\uc758 \uc120\uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.5. (\uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac)<\/span><\/p>\n<p>\\(S\\)\uac00 \\(\\mathbb{R}^3\\)\uc758 \ub2e8\uc21c\uc5f0\uacb0\ub41c \uace1\uba74\uc774\uace0 \ubc29\ud5a5\uc744 \uac00\uc9c0\uace0 \uc788\uc73c\uba70, \uadf8 \uacbd\uacc4 \\(\\partial S\\)\uac00 \uc870\uac01\ub9c8\ub2e4 \ub9e4\ub044\ub7ec\uc6b4 \ub2e8\uc21c\ud3d0\uace1\uc120\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(F\\)\uac00 \\(S\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc5d0\uc11c \\(C^1\\) \ubca1\ud130\uc7a5\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\oint_{\\partial S} F \\cdot dr = \\iint_S (\\nabla \\times F) \\cdot \\mathbf{n}\\, dS.\\tag{11.14}\\]<br \/>\n\uc5ec\uae30\uc11c \uacbd\uacc4 \\(\\partial S\\)\uc758 \ubc29\ud5a5\uacfc \ubc95\ubca1\ud130\uc758 \ubc29\ud5a5\uc740 \uc624\ub978\uc190 \ubc95\uce59\uc744 \ub530\ub978\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\uba3c\uc800 \uace1\uba74 \\(S\\)\uac00 \\(z = g(x,\\, y)\\) \ud615\ud0dc\uc774\uace0 \\(D\\)\uac00 \\(xy\\)-\ud3c9\uba74\uc758 \uc601\uc5ed\uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud558\uc790. \ub9e4\uac1c\ubcc0\uc218 \ud45c\ud604 \\[r(x,\\, y) = (x,\\, y,\\, g(x,\\, y))\\]\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc120\uc801\ubd84 \\(\\oint_{\\partial S} F \\cdot dr\\)\uc744 \\(\\partial D\\)\uc5d0\uc11c\uc758 \uc120\uc801\ubd84\uc73c\ub85c \ubcc0\ud658\ud55c\ub2e4. \uadf8\ub9b0\uc758 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(\\partial D\\)\uc5d0\uc11c\uc758 \uc120\uc801\ubd84\uc744 \uc774\uc911\uc801\ubd84\uc73c\ub85c \ubc14\uafb8\uace0, \uc774 \uc774\uc911\uc801\ubd84\uc774 \\(\\iint_S (\\nabla \\times F) \\cdot \\mathbf{n}\\, dS\\)\uc640 \uac19\uc74c\uc744 \ubcf4\uc778\ub2e4.<\/p>\n<p>\\(S\\)\uac00 \uc77c\ubc18\uc801\uc778 \uace1\uba74\uc778 \uacbd\uc6b0\uc5d0\ub294 \uace1\uba74\uc744 \uc5ec\ub7ec \uac1c\ub85c \ubd84\ud560\ud558\uc5ec \uc55e\uc758 \ubc29\ubc95\uc744 \ubc18\ubcf5\ud558\uc5ec \uc801\uc6a9\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc704 \uc815\ub9ac\uc5d0\uc11c \\(S\\)\uac00 \ud3c9\uba74\uc601\uc5ed\uc774\uba74 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub294 \uadf8\ub9b0\uc758 \uc815\ub9ac\uac00 \ub41c\ub2e4.<\/p>\n<p>\ub610\ud55c \\(S\\)\uac00 \ud3d0\uace1\uba74\uc774\uba74 \\(\\partial S = \\varnothing\\)\uc774\ubbc0\ub85c \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<br \/>\n\\[\\iint_S (\\nabla \\times F) \\cdot \\mathbf{n}\\, dS = 0.\\tag{11.15}\\]<\/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 \\(\\oint_C F \\cdot dr\\)\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624. \uc5ec\uae30\uc11c \\(F = (z,\\, x,\\, y)\\)\uc774\uace0 \\(C\\)\ub294 \ud3c9\uba74 \\(x + y + z = 1\\)\uacfc \uc88c\ud45c\ud3c9\uba74\ub4e4\uc758 \uad50\uc120\uc774\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\ubaa8\uc591\uc774\uac70\ub098 \uc9c1\uc0ac\uac01\ud615 \ubaa8\uc591\uc758 \uc9d1\ud569\uc774\uace0 \ub0b4\ubd80\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \\(F : D \\rightarrow\\mathbb{R}^3\\)\uac00 \\(D\\)\uc5d0\uc11c \\(C^1\\) \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \ubaa8\ub450 \ub3d9\uce58\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(D\\) \uc704\uc5d0\uc11c \\(\\operatorname{curl}G = F\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(C^2\\) \ud568\uc218 \\(G : D \\rightarrow\\mathbb{R}^3\\)\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\ubc1c\uc0b0 \uc815\ub9ac\uc758 \uac00\uc815\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc784\uc758\uc758 \uc9d1\ud569 \\(V\\subseteq D\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\iint_{\\partial V} F \\cdot \\mathrm{n}\\,dS = 0.\\tag{11.16}\\]<\/li>\n<li>\\(D\\) \uc704\uc5d0\uc11c \\(\\operatorname{div}F = 0\\)\uc774\ub2e4.<\/li>\n<\/ol>\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\ub97c \ud1b5\ud569\uc801\uc73c\ub85c \ub2e4\ub8e8\uae30 \uc704\ud55c \ubc29\ubc95\uc73c\ub85c <span class=\"defined\">\ubbf8\ubd84\ud615\uc2dd<\/span>(differential form)\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \\(\\mathbb{R}^n\\)\uc5d0\uc11c \\(k\\)-\ud615\uc2dd\uc740 \\(k\\)\uac1c\uc758 \ubca1\ud130\ub97c \uc2a4\uce7c\ub77c\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \uad50\ub300 \ub2e4\uc911\uc120\ud615\ud568\uc218\ub85c \uc815\uc758\ub41c\ub2e4.<\/p>\n<p>\uae30\ubcf8 \ubbf8\ubd84\ud615\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li><strong>0-\ud615\uc2dd<\/strong>: \uc2a4\uce7c\ub77c \ud568\uc218 \\(f(x_1,\\, \\ldots,\\, x_n)\\)<\/li>\n<li><strong>1-\ud615\uc2dd<\/strong>: \\(\\omega = \\sum_{i=1}^n a_i\\, dx_i\\) (\uc120\uc801\ubd84\uc758 \ud53c\uc801\ubd84\ud568\uc218)<\/li>\n<li><strong>2-\ud615\uc2dd<\/strong>: \\(\\omega = \\sum_{i < j} a_{ij}\\, dx_i \\wedge dx_j\\) (\uba74\uc801\ubd84\uc758 \ud53c\uc801\ubd84\ud568\uc218)<\/li>\n<li><strong>3-\ud615\uc2dd<\/strong>: \\(\\omega = \\sum_{i < j < k} a_{ijk}\\, dx_i \\wedge dx_j \\wedge dx_k\\) (\ubd80\ud53c\uc801\ubd84\uc758 \ud53c\uc801\ubd84\ud568\uc218)<\/li>\n<\/ul>\n<p>\uc5ec\uae30\uc11c \\(\\wedge\\)\ub294 <span class=\"defined\">\uc410\uae30\uacf1<\/span>(wedge product)\uc774\ub77c\uace0 \ubd80\ub974\ub294 \uc774\ud56d\uc5f0\uc0b0\uc790\ub85c\uc11c, \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\ubc18\uad50\ud658\uc131: \\(dx_i \\wedge dx_j = -dx_j \\wedge dx_i\\)<\/li>\n<li>\uba71\uc601\uc131: \\(dx_i \\wedge dx_i = 0\\)<\/li>\n<li>\uacb0\ud569\uc131: \\((dx_i \\wedge dx_j) \\wedge dx_k = dx_i \\wedge (dx_j \\wedge dx_k)\\)<\/li>\n<\/ul>\n<p>\\(\\mathbb{R}^3\\)\uc5d0\uc11c\uc758 \\(2\\)-\ud615\uc2dd\uacfc \\(3\\)-\ud615\uc2dd\uc740 \uad6c\uccb4\uc801\uc73c\ub85c \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>2-\ud615\uc2dd \\(\\omega = P\\, dy \\wedge dz + Q\\, dz \\wedge dx + R\\, dx \\wedge dy\\)\ub294 \ubca1\ud130\uc7a5 \\(F = (P,\\, Q,\\, R)\\)\uc758 \uc720\ucd9c\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>3-\ud615\uc2dd \\(\\omega = f\\, dx \\wedge dy \\wedge dz\\)\ub294 \ubc00\ub3c4\ud568\uc218 \\(f\\)\uc758 \ubd80\ud53c\uc801\ubd84\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ul>\n<p><span class=\"defined\">\uc678\ubbf8\ubd84<\/span>(exterior derivative) \\(d\\)\ub294 \\(k\\)-\ud615\uc2dd\uc744 \\((k+1)\\)-\ud615\uc2dd\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ubcc0\ud658\uc774\ub2e4.<\/p>\n<ul>\n<li>\\(f\\)\uac00 \\(0\\)-\ud615\uc2dd\uc77c \ub54c, \\(f\\)\uc758 \uc678\ubbf8\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4.<br \/>\n\\[df = \\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i} dx_i .\\tag{11.17}\\]<br \/>\n\uc774\uac83\uc740 \ud568\uc218\uc758 \uc804\ubbf8\ubd84\uc774\ub2e4.<\/li>\n<li>\\(\\omega = \\sum P_i\\, dx_i\\)\uac00 \\(1\\)-\ud615\uc2dd\uc77c \ub54c, \\(\\omega\\)\uc758 \uc678\ubbf8\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4.<br \/>\n\\[d\\omega = \\sum_{i < j} \\left(\\frac{\\partial P_j}{\\partial x_i} - \\frac{\\partial P_i}{\\partial x_j}\\right) dx_i \\wedge dx_j.\\tag{11.18}\\]\n\\(\\mathbb{R}^3\\)\uc5d0\uc11c \uc774\uac83\uc740 \ud68c\uc804(curl)\uc5d0 \ub300\uc751\ub41c\ub2e4.<\/li>\n<li>\\(\\omega = \\sum_{i < j} Q_{ij}\\, dx_i \\wedge dx_j\\)\uac00 \\(2\\)-\ud615\uc2dd\uc77c \ub54c, \\(\\omega\\)\uc758 \uc678\ubbf8\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4.\n\\[d\\omega = \\sum_{i < j < k} \\left(\\frac{\\partial Q_{jk}}{\\partial x_i} + \\frac{\\partial Q_{ki}}{\\partial x_j} + \\frac{\\partial Q_{ij}}{\\partial x_k}\\right) dx_i \\wedge dx_j \\wedge dx_k.\\tag{11.19}\\]\n\\(\\mathbb{R}^3\\)\uc5d0\uc11c \uc774\uac83\uc740 \ubc1c\uc0b0(divergence)\uc5d0 \ub300\uc751\ub41c\ub2e4.<\/li>\n<li>\\(k\\ge 3\\)\uc77c \ub54c, \\(\\mathbb{R}^3\\)\uc5d0\uc11c \\(k\\)-\ud615\uc2dd\uc758 \uc678\ubbf8\ubd84\uc740 \\(0\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc678\ubbf8\ubd84\uc758 \uc911\uc694\ud55c \uc131\uc9c8\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\uc120\ud615\uc131: \\(d(\\alpha\\omega + \\beta\\eta) = \\alpha\\, d\\omega + \\beta\\, d\\eta\\)<\/li>\n<li>\ub77c\uc774\ud504\ub2c8\uce20 \uacf5\uc2dd: \\(d(\\omega \\wedge \\eta) = d\\omega \\wedge \\eta + (-1)^k \\omega \\wedge d\\eta\\) (\uc5ec\uae30\uc11c \\(k\\)\ub294 \\(\\omega\\)\uc758 \ucc28\uc218\uc774\ub2e4.)<\/li>\n<li><span class=\"defined\">\ud3ec\uc559\uce74\ub808 \ubcf4\uc870\uc815\ub9ac<\/span>: \\(d(d\\omega) = 0\\). (\uc774\uac83\uc744 \\(d \\circ d = 0\\)\uc73c\ub85c \ud45c\ud604\ud558\uae30\ub3c4 \ud55c\ub2e4.)<\/li>\n<\/ul>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span><\/p>\n<p>\\(\\mathbb{R}^3\\)\uc5d0\uc11c 1-\ud615\uc2dd \\(\\omega = x\\, dy &#8211; y\\, dx\\)\uc758 \uc678\ubbf8\ubd84\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[d\\omega = \\frac{\\partial(-y)}{\\partial x} dx \\wedge dx + \\frac{\\partial x}{\\partial y} dy \\wedge dy + \\left(\\frac{\\partial x}{\\partial x} &#8211; \\frac{\\partial(-y)}{\\partial y}\\right) dx \\wedge dy = 2\\, dx \\wedge dy.\\]<br \/>\n\uc774\uac83\uc740 \uc6d0\uc810 \uc8fc\uc704\ub97c \ub3c4\ub294 \ubca1\ud130\uc7a5\uc758 \ud68c\uc804\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.12.<\/span><br \/>\n2-\ud615\uc2dd \\(\\omega = x\\, dy \\wedge dz + y\\, dz \\wedge dx + z\\, dx \\wedge dy\\)\uc5d0 \ub300\ud574 \\(d\\omega\\)\ub97c \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\ubbf8\ubd84\ud615\uc2dd\uc758 \uc801\ubd84\uc774 \uc2e4\uc81c\ub85c \uc5b4\ub5bb\uac8c \uacc4\uc0b0\ub418\ub294\uc9c0 \uad6c\uccb4\uc801\uc778 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.<\/span><\/p>\n<p>\\(\\mathbb{R}^2\\)\uc5d0\uc11c 1-\ud615\uc2dd \\(\\omega = x\\, dy\\)\ub97c \uace1\uc120 \\(C: r(t) = (\\cos t,\\, \\sin t)\\), \\(0 \\leq t \\leq \\pi\/2\\) \uc704\uc5d0\uc11c \uc801\ubd84\ud558\uc790. \ub9e4\uac1c\ubcc0\uc218 \ud45c\ud604\uc5d0\uc11c \\(x = \\cos t\\), \\(y = \\sin t\\)\uc774\ubbc0\ub85c \\(dy = \\cos t\\, dt\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[\\int_C \\omega = \\int_C x\\, dy = \\int_0^{\\pi\/2} \\cos t \\cdot \\cos t\\, dt = \\int_0^{\\pi\/2} \\cos^2 t\\, dt = \\frac{\\pi}{4}.\\]<br \/>\n\uc774\uac83\uc740 \ubca1\ud130\uc7a5 \\(F = (0,\\, x)\\)\uc758 \uc120\uc801\ubd84 \\(\\int_C F \\cdot dr\\)\uacfc \uac19\ub2e4.<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.<\/span><\/p>\n<p>\\(\\mathbb{R}^3\\)\uc5d0\uc11c 2-\ud615\uc2dd \\(\\omega = z\\, dx \\wedge dy\\)\ub97c \ud3c9\uba74 \\(S: z = x + y\\), \\(0 \\leq x \\leq 1\\), \\(0 \\leq y \\leq 1\\) \uc704\uc5d0\uc11c \uc801\ubd84\ud558\uc790. \uace1\uba74\uc758 \ub9e4\uac1c\ubcc0\uc218 \ud45c\ud604\uc740 \\(r(x,\\, y) = (x,\\, y,\\, x+y)\\)\uc774\ub2e4. 2-\ud615\uc2dd \\(z\\,dx \\wedge dy\\)\ub294 \ubca1\ud130\uc7a5 \\((0,\\,0,\\,z)\\)\uc758 \uc720\ucd9c\uc5d0 \ub300\uc751\ub418\ubbc0\ub85c<br \/>\n\\[\\iint_S \\omega = \\iint_D z\\, dx\\, dy = \\iint_D (x+y)\\, dx\\, dy\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(D = [0,\\,1] \\times [0,\\,1]\\)\uc774\ub2e4. \uc704 \uc911\uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\iint_D (x+y)\\, dx\\, dy = \\int_0^1 \\int_0^1 (x+y)\\, dx\\, dy = \\int_0^1 \\left[\\frac{x^2}{2} + xy\\right]_{x=0}^{x=1} dy = \\int_0^1 \\left(\\frac{1}{2} + y\\right) dy = 1.\\]<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 7.<\/span><\/p>\n<p>\\(\\mathbb{R}^3\\)\uc5d0\uc11c 2-\ud615\uc2dd \\(\\omega = x\\, dy \\wedge dz\\)\ub97c \ubc18\uad6c \\(S: x^2 + y^2 + z^2 = 1\\), \\(x \\geq 0\\) \uc704\uc5d0\uc11c \uc801\ubd84\ud558\uc790. \uad6c\uba74\uc88c\ud45c\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(x = \\cos\\phi\\sin\\theta\\), \\(y = \\sin\\phi\\sin\\theta\\), \\(z = \\cos\\theta\\)\ub85c \ub9e4\uac1c\ubcc0\uc218 \ud45c\ud604\ud558\uba74, \\(0 \\leq \\phi \\leq 2\\pi\\), \\(0 \\leq \\theta \\leq \\pi\/2\\)\uc774\ub2e4.<\/p>\n<p>2-\ud615\uc2dd \\(\\omega = x\\, dy \\wedge dz\\)\ub294 \ubca1\ud130\uc7a5 \\(F = (x,\\, 0,\\, 0)\\)\uc758 \uc720\ucd9c\uc5d0 \ub300\uc751\ub418\uace0, \uace1\uba74\uc5d0 \uc218\uc9c1\uc778 \ub2e8\uc704\ubc95\ubca1\ud130\ub294 \\(\\mathbf{n} = (x,\\, y,\\, z)\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\iint_S \\omega = \\iint_S F \\cdot \\mathbf{n}\\, dS = \\iint_S x^2\\, dS.\\]<br \/>\n\uad6c\uba74\uc5d0\uc11c \\(x^2 = \\cos^2\\phi\\sin^2\\theta\\)\uc774\uace0 \\(dS = \\sin\\theta\\, d\\theta\\, d\\phi\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\iint_S x^2\\, dS = \\int_0^{2\\pi} \\int_0^{\\pi\/2} \\cos^2\\phi\\sin^2\\theta \\cdot \\sin\\theta\\, d\\theta\\, d\\phi = \\frac{2\\pi}{3}.\\]<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 8.<\/span><\/p>\n<p>\\(\\mathbb{R}^3\\)\uc5d0\uc11c 3-\ud615\uc2dd \\(\\omega = xyz\\, dx \\wedge dy \\wedge dz\\)\ub97c \uc9c1\uc721\uba74\uccb4 \\(V = [0,\\,1] \\times [0,\\,1] \\times [0,\\,1]\\) \uc704\uc5d0\uc11c \uc801\ubd84\ud558\uc790. 3-\ud615\uc2dd \\(dx \\wedge dy \\wedge dz\\)\ub294 \ubd80\ud53c\uc18c\ub97c \ub098\ud0c0\ub0b4\ubbc0\ub85c<br \/>\n\\[\\iiint_V \\omega = \\iiint_V xyz\\, dx\\, dy\\, dz = \\int_0^1 \\int_0^1 \\int_0^1 xyz\\, dx\\, dy\\, dz.\\]<br \/>\n\uc774\uac83\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \ubc18\ubcf5\uc801\ubd84\uc73c\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[\\int_0^1 x\\, dx \\cdot \\int_0^1 y\\, dy \\cdot \\int_0^1 z\\, dz = \\frac{1}{2} \\cdot \\frac{1}{2} \\cdot \\frac{1}{2} = \\frac{1}{8}.\\]<\/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>2-\ud615\uc2dd \\(\\omega = y\\, dz \\wedge dx + z\\, dx \\wedge dy\\)\ub97c \uad6c\uba74 \\(S: x^2 + y^2 + z^2 = 1\\) \uc704\uc5d0\uc11c \uc801\ubd84\ud558\uc2dc\uc624.<\/li>\n<li>2-\ud615\uc2dd \\(\\omega = x\\, dy \\wedge dz\\)\ub97c \uc6d0\uae30\ub465 \ud45c\uba74 \\(S: x^2 + y^2 = 1\\), \\(0 \\leq z \\leq 2\\) \uc704\uc5d0\uc11c \uc801\ubd84\ud558\uc2dc\uc624. (\ub2e8, \ubc95\ubca1\ud130\ub294 \ubc14\uae65\ucabd\uc744 \ud5a5\ud55c\ub2e4.)<\/li>\n<\/ol>\n<\/div>\n<p>\\(\\mathbb{R}^n\\)\uc758 \uc77c\ubc18\uc801\uc778 \uc601\uc5ed\uc5d0 \uc801\ubd84\uc744 \uc815\uc758\ud558\uae30 \uc704\ud574 <span class=\"defined\">\ub2e4\uc591\uccb4<\/span>(manifold)\uc758 \uac1c\ub150\uc774 \ud544\uc694\ud558\ub2e4. \\(k\\)\ucc28\uc6d0 <span class=\"defined\">\ubbf8\ubd84 \uac00\ub2a5 \ub2e4\uc591\uccb4<\/span>\ub294 \uad6d\uc18c\uc801\uc73c\ub85c \\(\\mathbb{R}^k\\)\uc640 \uac19\uc544 \ubcf4\uc774\ub294 \uacf5\uac04\uc774\ub2e4.<\/p>\n<p>\uad6c\uccb4\uc801\uc73c\ub85c, \uc704\uc0c1\uacf5\uac04 \\(M\\)\uc774 \\(k\\)\ucc28\uc6d0 \ubbf8\ubd84 \uac00\ub2a5 \ub2e4\uc591\uccb4\ub77c\ub294 \uac83\uc740 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc774\ub2e4. \uac01 \uc810 \\(p \\in M\\)\uc5d0 \ub300\ud574 \\(p\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc5f4\ub9b0\uc9d1\ud569 \\(U \\subseteq M\\)\uacfc \\(\\mathbb{R}^k\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569 \\(V\\), \uadf8\ub9ac\uace0 \uc704\uc0c1\ub3d9\ud615\uc0ac\uc0c1 \\(\\phi: U \\to V\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub7ec\ud55c \uc30d \\((U,\\, \\phi)\\)\ub97c <span class=\"defined\">\uc88c\ud45c \ud328\uce58<\/span>(coordinate patch) \ub610\ub294 <span class=\"defined\">\ucc28\ud2b8<\/span>(chart)\ub77c\uace0 \ubd80\ub974\uace0, \\(\\phi\\)\ub97c <span class=\"defined\">\uc88c\ud45c \uc0ac\uc0c1<\/span>(coordinate map)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc88c\ud45c \ud328\uce58\ub4e4\uc758 \ubaa8\uc784\uc744 <span class=\"defined\">\uc544\ud2c0\ub77c\uc2a4<\/span>(atlas)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ub450 \uc88c\ud45c \ud328\uce58 \\((U_1,\\, \\phi_1)\\)\uacfc \\((U_2,\\, \\phi_2)\\)\uac00 \uacb9\uce58\ub294 \ubd80\ubd84 \\(U_1 \\cap U_2 \\neq \\varnothing\\)\uc5d0\uc11c <span class=\"defined\">\uc804\ud658 \uc0ac\uc0c1<\/span>(transition map) \\[\\phi_2 \\circ \\phi_1^{-1}: \\phi_1(U_1 \\cap U_2) \\to \\phi_2(U_1 \\cap U_2)\\]\uac00 \\(C^\\infty\\) \ubbf8\ubd84\ub3d9\ud615\uc0ac\uc0c1\uc77c \ub54c \ub2e4\uc591\uccb4\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ub9d0\ud55c\ub2e4. \uc774 \uc870\uac74\uc740 \uc11c\ub85c \ub2e4\ub978 \uc88c\ud45c\uacc4 \uc0ac\uc774\uc758 \ubcc0\ud658\uc774 \ub9e4\ub044\ub7fd\uac8c \uc774\ub8e8\uc5b4\uc9d0\uc744 \ubcf4\uc7a5\ud55c\ub2e4.<\/p>\n<p><span class=\"defined\">\ubc29\ud5a5\uc744 \uac00\uc9c4 \ub2e4\uc591\uccb4<\/span>(oriented manifold)\ub294 \uc77c\uad00\ub41c \uc88c\ud45c\uacc4\uc758 \uc120\ud0dd\uc774 \uac00\ub2a5\ud55c \ub2e4\uc591\uccb4\uc774\ub2e4. \uad6c\uccb4\uc801\uc73c\ub85c, \ubaa8\ub4e0 \uc804\ud658 \uc0ac\uc0c1\uc758 \uc57c\ucf54\ube44 \ud589\ub82c\uc2dd\uc774 \ud56d\uc0c1 \uc591\uc218\uac00 \ub418\ub3c4\ub85d \uc544\ud2c0\ub77c\uc2a4\ub97c \uc120\ud0dd\ud560 \uc218 \uc788\uc744 \ub54c \ub2e4\uc591\uccb4\uac00 \ubc29\ud5a5\uc744 \uac00\uc9c4\ub2e4\uace0 \ub9d0\ud55c\ub2e4. \uc774\uac83\uc740 &#8220;\uc624\ub978\uc190 \uc88c\ud45c\uacc4&#8221;\uc640 &#8220;\uc67c\uc190 \uc88c\ud45c\uacc4&#8221;\ub97c \uc77c\uad00\ub418\uac8c \uad6c\ubcc4\ud560 \uc218 \uc788\uc74c\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n<ul>\n<li>\uad6c\uba74 \\(S^2 = \\{x \\in \\mathbb{R}^3 \\mid \\|x\\| = 1\\}\\)\ub294 \ubc29\ud5a5\uc744 \uac00\uc9c4\ub2e4. \ubc14\uae65\ucabd \ubc95\ubca1\ud130\ub85c \ubc29\ud5a5\uc744 \uacb0\uc815\ud560 \uc218 \uc788\ub2e4.<\/li>\n<li>\ubafc\ube44\uc6b0\uc2a4 \ub760\ub294 \ubc29\ud5a5\uc744 \uac00\uc9c0\uc9c0 \uc54a\ub294\ub2e4. \ub760\ub97c \ud55c \ubc14\ud034 \ub3cc\uba74 \ubc29\ud5a5\uc774 \ubc18\ub300\uac00 \ub41c\ub2e4.<\/li>\n<li>\ud074\ub77c\uc778 \ubcd1\uc740 \ubc29\ud5a5\uc744 \uac00\uc9c0\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ul>\n<p>\ub2e4\uc591\uccb4 \\(M\\)\uc758 <span class=\"defined\">\uacbd\uacc4<\/span>(boundary) \\(\\partial M\\)\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4. \uc810 \\(p \\in M\\)\uc774 <span class=\"defined\">\uacbd\uacc4\uc810<\/span>\uc774\ub77c\ub294 \uac83\uc740, \\(p\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc88c\ud45c \ud328\uce58 \\((U,\\, \\phi)\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\phi(U)\\)\uac00 \uc0c1\ubc18\ud3c9\uba74 \\(H^k = \\{x \\in \\mathbb{R}^k \\mid x_k \\geq 0\\}\\)\uc758 \uc5f4\ub9b0\ubd80\ubd84\uc9d1\ud569\uc774\uace0, \\(\\phi(p)\\)\uac00 \\(x_k = 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc774\ub2e4. \uc774\ub7ec\ud55c \ubaa8\ub4e0 \uacbd\uacc4\uc810\ub4e4\uc758 \uc9d1\ud569\uc774 \ub2e4\uc591\uccb4\uc758 \uacbd\uacc4 \\(\\partial M\\)\uc774\ub2e4. \uacbd\uacc4\ub97c \uac00\uc9c4 \ub2e4\uc591\uccb4\ub97c <span class=\"defined\">\uacbd\uacc4\ub97c \uac00\uc9c4 \ub2e4\uc591\uccb4<\/span>(manifold with boundary)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ubbf8\ubd84\ud615\uc2dd\uc758 \uc801\ubd84\uc744 \uc77c\ubc18\uc801\uc778 \ub2e4\uc591\uccb4\ub85c \ud655\uc7a5\ud558\uae30 \uc704\ud574\uc11c\ub294 \uba87 \uac00\uc9c0 \ub3c4\uad6c\uac00 \ud544\uc694\ud558\ub2e4.<\/p>\n<p>\ud568\uc218 \\(f: M \\to \\mathbb{R}\\)\uc758 <span class=\"defined\">\uc9c0\uc9c0\uc9d1\ud569<\/span>(support)\uc740 \\[\\operatorname{supp}(f) = \\overline{\\{x \\in M \\mid f(x) \\neq 0\\}}\\]\uc73c\ub85c \uc815\uc758\ub41c\ub2e4. \uc989, \\(f\\)\uac00 0\uc774 \uc544\ub2cc \uc810\ub4e4\uc758 \ud3d0\ud3ec\uc774\ub2e4. \ud568\uc218\uac00 <span class=\"defined\">\ucef4\ud329\ud2b8 \uc9c0\uc9c0<\/span>(compact support)\ub97c \uac00\uc9c4\ub2e4\ub294 \uac83\uc740 \uadf8 \uc9c0\uc9c0\uc9d1\ud569\uc774 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc778 \uac83\uc774\ub2e4. \ucef4\ud329\ud2b8 \uc9c0\uc9c0\ub97c \uac00\uc9c4 \ud568\uc218\uc758 \uc911\uc694\ud55c \uc131\uc9c8\uc740 \ucda9\ubd84\ud788 \ud070 \uc601\uc5ed \ubc16\uc5d0\uc11c\ub294 0\uc774\ubbc0\ub85c \uc801\ubd84\uc774 \ud56d\uc0c1 \uc218\ub834\ud55c\ub2e4\ub294 \uac83\uc774\ub2e4.<\/p>\n<p><span class=\"defined\">\ub2e8\uc704\ubd84\ud560<\/span>(partition of unity)\uc740 \ub2e4\uc591\uccb4 \uc804\uccb4\uc5d0\uc11c \uc815\uc758\ub41c \ubb38\uc81c\ub97c \uad6d\uc18c\uc801\uc73c\ub85c \ud574\uacb0\ud558\uae30 \uc704\ud55c \ub3c4\uad6c\uc774\ub2e4. \ub2e4\uc591\uccb4 \\(M\\)\uc758 \uc5f4\ub9b0\ub36e\uac1c \\(\\{U_\\alpha\\}_{\\alpha \\in A}\\)\uc5d0 <span class=\"defined\">\uc885\uc18d\ub41c \ub2e8\uc704\ubd84\ud560<\/span>\uc774\ub780 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\ub4e4\uc758 \ubaa8\uc784 \\[\\{\\rho_\\alpha: M \\to [0,\\, 1]\\}_{\\alpha \\in A}\\]\uc774\ub2e4.<\/p>\n<ul>\n<li>\uac01 \\(\\rho_\\alpha\\)\ub294 \\(C^\\infty\\) \ud568\uc218\uc774\uace0 \\(\\operatorname{supp}(\\rho_\\alpha) \\subseteq U_\\alpha\\)\uc774\ub2e4.<\/li>\n<li>\uac01 \uc810 \\(x \\in M\\)\uc758 \uc5b4\ub5a4 \uadfc\ubc29\uc5d0\uc11c \uc720\ud55c \uac1c\ub97c \uc81c\uc678\ud55c \ubaa8\ub4e0 \\(\\rho_\\alpha\\)\uac00 0\uc774\ub2e4. (\uc774\uac83\uc744 &#8216;\uad6d\uc18c\uc720\ud55c\uc131&#8217;\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.)<\/li>\n<li>\ubaa8\ub4e0 \\(x \\in M\\)\uc5d0 \ub300\ud574 \\(\\sum_{\\alpha \\in A} \\rho_\\alpha(x) = 1\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc9c1\uad00\uc801\uc73c\ub85c, \ub2e8\uc704\ubd84\ud560\uc740 \ub2e8\uc704 \\(1\\)\uc744 \uc5ec\ub7ec \uc870\uac01\uc73c\ub85c \ub098\ub204\uc5b4 \uac01 \uc88c\ud45c \ud328\uce58\uc5d0 \ud560\ub2f9\ud558\ub294 \ubc29\ubc95\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \uc6d0 \\(S^1\\)\uc744 \ub450 \uac1c\uc758 \uc5f4\ub9b0 \ud638\ub85c \ub36e\uc5c8\ub2e4\uba74, \uac01 \ud638\uc5d0\uc11c 1\uc5d0 \uac00\uae5d\uace0 \uacbd\uacc4\ub85c \uac08\uc218\ub85d 0\uc5d0 \uac00\uae4c\uc6cc\uc9c0\ub294 \ud568\uc218\ub4e4\uc744 \ub9cc\ub4e4\uc5b4 \uadf8 \ud569\uc774 \ud56d\uc0c1 1\uc774 \ub418\ub3c4\ub85d \uc870\uc815\ud560 \uc218 \uc788\ub2e4. \ud328\ub7ec\ucef4\ud329\ud2b8 \uacf5\uac04(\ud2b9\ud788 \ucef4\ud329\ud2b8 \ub2e4\uc591\uccb4)\uc5d0\uc11c\ub294 \uc784\uc758\uc758 \uc5f4\ub9b0\ub36e\uac1c\uc5d0 \ub300\ud574 \uc885\uc18d\ub41c \ub2e8\uc704\ubd84\ud560\uc774 \uc874\uc7ac\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc774 \uc54c\ub824\uc838 \uc788\ub2e4.<\/p>\n<p>\uc774\ub7ec\ud55c \ub3c4\uad6c\ub4e4\uc744 \uc0ac\uc6a9\ud558\uba74 \uc804\uccb4 \ub2e4\uc591\uccb4\uc5d0\uc11c\uc758 \uc801\ubd84\uc744 \uac01 \uc88c\ud45c \ud328\uce58\uc5d0\uc11c\uc758 \uc801\ubd84\uc73c\ub85c \ubd84\ud574\ud560 \uc218 \uc788\ub2e4. \ud568\uc218 \\(f\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\[f = \\sum_\\alpha \\rho_\\alpha f\\]\ub85c \uc4f8 \uc218 \uc788\uace0, \uac01 \\(\\rho_\\alpha f\\)\ub294 \\(U_\\alpha\\)\uc5d0\uc11c\ub9cc 0\uc774 \uc544\ub2c8\ubbc0\ub85c \uad6d\uc18c \uc88c\ud45c\uc5d0\uc11c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 11.6. (\uc77c\ubc18\ud654\ub41c \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac)<\/span><\/p>\n<p>\\(M\\)\uc774 \ubc29\ud5a5\uc744 \uac00\uc9c4 \\(k\\)\ucc28\uc6d0 \ubbf8\ubd84 \uac00\ub2a5 \ucef4\ud329\ud2b8 \ub2e4\uc591\uccb4\uc774\uace0, \\(\\partial M\\)\uc774 \uadf8 \uacbd\uacc4\uc774\uba70, \\(\\omega\\)\uac00 \\(M\\)\uc5d0\uc11c \uc815\uc758\ub41c \\(C^1\\) \\((k-1)\\)-\ud615\uc2dd\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\int_{\\partial M} \\omega = \\int_M d\\omega .\\tag{11.20}\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\partial M\\)\uc758 \ubc29\ud5a5\uc740 \\(M\\)\uc758 \ubc29\ud5a5\uc73c\ub85c\ubd80\ud130 \uc720\ub3c4\ub41c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\uc99d\uba85\uc740 \ub2e8\uacc4\uc801\uc73c\ub85c \uc77c\ubc18\ud654\ud558\uc5ec \uc9c4\ud589\ud55c\ub2e4. \\(M\\)\uc758 \ucef4\ud329\ud2b8\uc131\uc740 \uc720\ud55c \uac1c\uc758 \uc88c\ud45c \ud328\uce58\ub85c \ub36e\uc744 \uc218 \uc788\uc74c\uc744 \ubcf4\uc7a5\ud558\uace0, \uc801\ubd84\uc774 \uc798 \uc815\uc758\ub418\uba70, \ub2e8\uc704 \ubd84\ud560\uc774 \uc720\ud55c\ud569\uc73c\ub85c \ud45c\ud604\ub428\uc744 \ubcf4\uc7a5\ud55c\ub2e4.<\/p>\n<p>\uba3c\uc800 \\(M\\)\uc774 \\(\\mathbb{R}^k\\)\uc758 \uc0c1\ubc18\ud3c9\uba74 \\(H^k = \\{x \\in \\mathbb{R}^k \\mid x_k \\geq 0\\}\\)\uc758 \uc5f4\ub9b0\ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(\\omega\\)\uac00 \ucef4\ud329\ud2b8 \uc9c0\uc9c0\uc9d1\ud569\uc744 \uac00\uc9c4 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud558\uc790. \\((k-1)\\)-\ud615\uc2dd \\(\\omega\\)\ub97c \uc88c\ud45c\ub85c \ud45c\ud604\ud558\uba74<br \/>\n\\[\\omega = \\sum_{i=1}^k (-1)^{i-1} a_i\\, dx_1 \\wedge \\cdots \\wedge \\widehat{dx_i} \\wedge \\cdots \\wedge dx_k\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\widehat{dx_i}\\)\ub294 \\(dx_i\\)\ub97c \uc0dd\ub7b5\ud55c\ub2e4\ub294 \uc758\ubbf8\uc774\ub2e4.<\/p>\n<p>\\(\\omega\\)\uc758 \uc678\ubbf8\ubd84\uc744 \uacc4\uc0b0\ud558\uba74<br \/>\n\\[d\\omega = \\sum_{i=1}^k \\frac{\\partial a_i}{\\partial x_i} dx_1 \\wedge \\cdots \\wedge dx_k\\]<br \/>\n\uc774\ub2e4. \uc774\uc81c \\(K\\)\ub97c \\(\\omega\\)\uc758 \uc9c0\uc9c0\uc9d1\ud569\uc744 \ud3ec\ud568\ud558\ub294 \\(H^k\\)\uc758 \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c \ud558\uc790. \\(K\\)\uc5d0\uc11c \\(\\omega\\)\uc758 \uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n\\int_K d\\omega &#038;= \\sum_{i=1}^k \\int_K \\frac{\\partial a_i}{\\partial x_i} dx_1 \\wedge \\cdots \\wedge dx_k \\\\[6pt]<br \/>\n&#038;= \\sum_{i=1}^k (-1)^{k-i} \\int_K \\frac{\\partial a_i}{\\partial x_i} dx_1 \\cdots dx_k.<br \/>\n\\end{aligned}\\]<\/p>\n<p>\\(i < k\\)\uc778 \uacbd\uc6b0, \\(a_i\\)\uac00 \ucef4\ud329\ud2b8 \uc9c0\uc9c0\ub97c \uac00\uc9c0\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \uad6c\uac04 \\([-R,\\, R]\\)\uc5d0\uc11c \\(x_i\\) \ubc29\ud5a5\uc73c\ub85c \uc801\ubd84\ud558\uba74\n\\[\\int_{-R}^R \\frac{\\partial a_i}{\\partial x_i} dx_i = a_i\\Big|_{x_i=-R}^{x_i=R} = 0\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(i = k\\)\uc778 \ud56d\ub9cc \ub0a8\ub294\ub2e4.<\/p>\n<p>\\(i = k\\)\uc77c \ub54c, \\(x_k \\geq 0\\)\uc778 \uc601\uc5ed\uc5d0\uc11c \uc801\ubd84\ud558\ubbc0\ub85c<br \/>\n\\[\\int_0^\\infty \\frac{\\partial a_k}{\\partial x_k} dx_k = -a_k(x_1,\\,\\ldots,\\,x_{k-1},\\,0)\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[\\int_K d\\omega = (-1)^{k-k} \\int_{\\mathbb{R}^{k-1}} \\left(-a_k(x_1,\\,\\ldots,\\,x_{k-1},\\,0)\\right) dx_1 \\cdots dx_{k-1}\\]<br \/>\n\uc774\ub2e4. \ud55c\ud3b8 \uacbd\uacc4 \\(\\partial M \\cap K\\)\ub294 \\(\\{x \\in K \\mid x_k = 0\\}\\)\uc774\uace0, \uc774 \uacbd\uacc4\uc758 \ubc29\ud5a5\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc720\ub3c4\ub41c\ub2e4. \uc810 \\(p \\in \\partial M\\)\uc5d0\uc11c \\(M\\)\uc73c\ub85c \ud5a5\ud558\ub294 \ub0b4\ubd80 \ubc95\ubca1\ud130 \\(n\\)\uacfc \\(\\partial M\\)\uc758 \ubc29\ud5a5\ubca1\ud130\ub4e4 \\(v_1,\\,\\ldots,\\,v_{k-1}\\)\uc774 \\((n,\\, v_1,\\,\\ldots,\\,v_{k-1})\\) \uc21c\uc11c\ub85c \\(M\\)\uc758 \ubc29\ud5a5\uacfc \uc77c\uce58\ud558\ub3c4\ub85d \uc815\ud55c\ub2e4. \uc774 \ubc29\ud5a5\uc5d0\uc11c<br \/>\n\\[\\int_{\\partial K} \\omega = (-1)^{k-1} \\int_{\\mathbb{R}^{k-1}} a_k(x_1,\\,\\ldots,\\,x_{k-1},\\,0) dx_1 \\cdots dx_{k-1}\\]<br \/>\n\uc774\ubbc0\ub85c \uc6d0\ud558\ub294 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \uc88c\ud45c \ubcc0\ud658\uc5d0 \ub300\ud55c \ubd88\ubcc0\uc131\uc744 \ud655\uc778\ud55c\ub2e4. \ubbf8\ubd84\ud615\uc2dd\uc758 \uc801\ubd84\uacfc \uc678\ubbf8\ubd84\uc740 \ubaa8\ub450 \uc88c\ud45c \ubcc0\ud658\uc5d0 \ub300\ud574 \ubd88\ubcc0\uc774\ubbc0\ub85c, \ub2e4\ub978 \uc88c\ud45c\uacc4\uc5d0\uc11c\ub3c4 \uac19\uc740 \uacb0\uacfc\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uc77c\ubc18\uc801\uc778 \ub2e4\uc591\uccb4 \\(M\\)\uc5d0 \ub300\ud574\uc11c\ub294 \ub2e8\uc704 \ubd84\ud560\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \\(M\\)\uc744 \uc88c\ud45c\uadfc\ubc29 \\(\\{U_\\alpha\\}\\)\ub85c \ub36e\uace0 \uc885\uc18d\uc801 \ub2e8\uc704 \ubd84\ud560 \\(\\{\\rho_\\alpha\\}\\)\ub97c \uc120\ud0dd\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(\\omega = \\sum_\\alpha \\rho_\\alpha \\omega\\)\uc774\uace0, \uac01 \\(\\rho_\\alpha \\omega\\)\ub294 \\(U_\\alpha\\)\uc5d0\uc11c\ub9cc 0\uc774 \uc544\ub2c8\ubbc0\ub85c \uad6d\uc18c \uc88c\ud45c\uc5d0\uc11c \uc55e\uc758 \uacb0\uacfc\ub97c \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4. \uacbd\uacc4 \uadfc\ucc98\uc5d0\uc11c\ub294 \ub0b4\ubd80\uc640 \uacbd\uacc4 \ubd80\ubd84\uc744 \uc801\uc808\ud788 \ubd84\ub9ac\ud558\uba74, \uac01 \ubd80\ubd84\uc5d0\uc11c<br \/>\n\\[\\int_{\\partial M} \\rho_\\alpha \\omega = \\int_M d(\\rho_\\alpha \\omega)\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub97c \ubaa8\ub450 \ub354\ud558\uba74<br \/>\n\\[\\int_{\\partial M} \\omega = \\sum_\\alpha \\int_{\\partial M} \\rho_\\alpha \\omega = \\sum_\\alpha \\int_M d(\\rho_\\alpha \\omega) = \\int_M d\\omega\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>\uc77c\ubc18\uc801\uc778 \\(C^1\\) \\((k-1)\\)-\ud615\uc2dd\uc5d0 \ub300\ud574\uc11c\ub294 \ucef4\ud329\ud2b8 \uc9c0\uc9c0\ub97c \uac00\uc9c4 \ud615\uc2dd\uc73c\ub85c \uadfc\uc0ac\ud558\uc5ec \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc77c\ubc18\ud654\ub41c \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ud2b9\uc218\ud55c \uacbd\uc6b0\ub97c \ud3ec\ud568\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(k = 1\\), \\(M = [a,\\, b]\\)\uc778 \uacbd\uc6b0 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub294 \ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac\uc774\ub2e4.<br \/>\n\\[\\int_a^b f'(x)\\, dx = f(b) &#8211; f(a).\\]<\/li>\n<li>\\(k = 2\\), \\(M \\subset \\mathbb{R}^2\\)\uc778 \uacbd\uc6b0 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub294 \uadf8\ub9b0\uc758 \uc815\ub9ac\uc774\ub2e4.<\/li>\n<li>\\(k = 2\\), \\(M \\subset \\mathbb{R}^3\\)\uc778 \uacbd\uc6b0 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub294 \\(3\\)\ucc28\uc6d0 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\uc774\ub2e4.<\/li>\n<li>\\(k = 3\\), \\(M \\subset \\mathbb{R}^3\\)\uc778 \uacbd\uc6b0 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub294 \ubc1c\uc0b0 \uc815\ub9ac\uc774\ub2e4.<\/li>\n<\/ul>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 11.14.<\/span><br \/>\n<span class=\"defined\">\uc644\uc804\ud615\uc2dd<\/span>(exact form)\uacfc <span class=\"defined\">\ub2eb\ud78c\ud615\uc2dd<\/span>(closed form)\uc758 \uc815\uc758\ub97c \uc870\uc0ac\ud558\uace0, \uc644\uc804\ud615\uc2dd\uacfc \ub2eb\ud78c\ud615\uc2dd\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\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 11.6)\ub97c \uc0ac\uc6a9\ud558\uc5ec \uadf8\ub9b0\uc758 \uc815\ub9ac(\uc815\ub9ac 11.3), \ubc1c\uc0b0 \uc815\ub9ac(\uc815\ub9ac 11.4), 3\ucc28\uc6d0 \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac(\uc815\ub9ac 11.5)\ub97c \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 \ud575\uc2ec\uc815\ub9ac \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<\/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>\uc774 \uc7a5\uc5d0\uc11c\ub294 \ubca1\ud130\ud574\uc11d\uc758 \uae30\ubcf8 \uac1c\ub150\uacfc \uadf8\ub9b0\uc758 \uc815\ub9ac, \ubc1c\uc0b0 \uc815\ub9ac, \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac \ub4f1 \uc911\uc694\ud55c \uc801\ubd84\uc815\ub9ac\ub4e4\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc774\ub7ec\ud55c \uc815\ub9ac\ub4e4\uc740 \ubb3c\ub9ac\ud559\uacfc \uacf5\ud559 \ub4f1 \uc751\uc6a9 \ubd84\uc57c\uc5d0\uc11c \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud55c\ub2e4. \ubca1\ud130\uc7a5\uc758 \uae30\ucd08 \\(\\mathbb{R}^n\\)\uc758 \uc601\uc5ed \\(D\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 \\(f: D \\to \\mathbb{R}\\)\uc744 \uc2a4\uce7c\ub77c\uc7a5(scalar field)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \\(F: 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. \uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uba87 \uac00\uc9c0 \uc5f0\uc0b0\uc744 \uc815\uc758\ud558\uc790. \uc2a4\uce7c\ub77c\uc7a5 \\(f\\)\uc758 \uae30\uc6b8\uae30(gradient)\ub97c&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":17,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9499\/revisions"}],"predecessor-version":[{"id":9616,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9499\/revisions\/9616"}],"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}]}}