\\(\\mathbb{R}^3\\)\uc5d0\uc11c\uc758 \uc720\ud5a5\uc120\uc801\ubd84\uc740 1\ud615\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec
\n\\[\\int_C \\mathbb{F} \\cdot \\mathbb{T} \\,dx = \\int_C P \\,dx + Q\\,dy + R\\,dz\\]
\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \\(C\\)\ub294 \\(\\mathbb{R}^3\\)\uc5d0 \ud3ec\ud568\ub418\ub294 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc774\ub2e4. \uc989 \\(C\\)\ub294 \\(\\mathbb{R}^3\\)\uc5d0 \ud3ec\ud568\ub418\ub294 1\ucc28\uc6d0 \uc9d1\ud569\uc774\ub2e4. \ud55c\ud3b8 \\(\\mathbb{R}^3\\)\uc5d0\uc11c\uc758 \uc720\ud5a5\uba74\uc801\ubd84\uc740 2\ud615\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec
\n\\[\\iint_S \\mathbb{F} \\cdot \\mathbb{n} \\,d\\sigma = \\iint_S P\\,dy\\,dz + Q\\,dz\\,dx + R\\,dx\\,dy\\]
\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \\(S\\)\ub294 \\(\\mathbb{R}^3\\)\uc5d0 \ud3ec\ud568\ub418\ub294 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uba74\uc774\ub2e4. \uc989 \\(S\\)\ub294 \\(\\mathbb{R}^3\\)\uc5d0 \ud3ec\ud568\ub418\ub294 2\ucc28\uc6d0 \uc9d1\ud569\uc774\ub2e4.<\/p>\n
\uc774\ub7ec\ud55c \uac1c\ub150\uc744 \ud655\uc7a5\ud558\uc5ec \\(r\\)\uac00 \\(n\\)\ubcf4\ub2e4 \uc791\uc740 \uc790\uc5f0\uc218\uc77c \ub54c \\(\\mathbb{R}^n\\)\uc5d0 \ud3ec\ud568\ub418\ub294 \\(r\\)\ucc28\uc6d0 \uc9d1\ud569\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\uc73c\uba70, \uadf8 \uc9d1\ud569 \uc704\uc5d0\uc11c \uc815\uc758\ub418\ub294 \\(r\\)\ud615\uc2dd\uc758 \uc801\ubd84\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n
\ubbf8\ubd84\ud615\uc2dd\uc758 \uc77c\ubc18\uc801\uc778 \uc815\uc758\ub97c \ub3c4\uc785\ud558\uae30 \uc804\uc5d0 \uba3c\uc800 \uad6c\uccb4\uc801\uc778 \uc608\ub85c\uc11c \\(\\mathbb{R}^4\\)\uc5d0\uc11c\uc758 \ubbf8\ubd84\ud615\uc2dd\uc744 \uc0b4\ud3b4\ubcf4\uc790. \\(V\\)\uac00 \\(\\mathbb{R}^4\\)\uc758 \uc5f4\ub9b0\ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(f\\)\uc5d0 \ucca8\uc790\uac00 \ubd99\uc740 \ud568\uc218\ub294 \ubaa8\ub450 \\(V\\)\ub85c\ubd80\ud130 \\(\\mathbb{R}\\)\ub85c\uc758 \ud568\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud558\uc790.<\/p>\n
\u2018\\(r\\)\ucc28 \ubbf8\ubd84\ud615\uc2dd\u2019\uc774\ub77c\ub294 \uc6a9\uc5b4\uc5d0\uc11c \\(r\\)\ub294 \ubbf8\ubd84\uc18c\uac00 \uacf1\ud574\uc9c4 \ucc28\uc218\ub97c \uc758\ubbf8\ud55c\ub2e4. \ub530\ub77c\uc11c 0\ucc28 \ubbf8\ubd84\ud615\uc2dd\uc740 \ubbf8\ubd84\uc18c\uac00 \ud558\ub098\ub3c4 \uacf1\ud574\uc838 \uc788\uc9c0 \uc54a\uc740 \uac83\uc73c\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\uc73c\ubbc0\ub85c \\(V\\)\ub85c\ubd80\ud130 \\(\\mathbb{R}\\)\ub85c\uc758 \ud568\uc218\ub97c 0\ucc28 \ubbf8\ubd84\ud615\uc2dd\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n
\uc774\uc81c \uc774\ub7ec\ud55c \ubbf8\ubd84\ud615\uc2dd\uc758 \uac1c\ub150\uc744 \\(\\mathbb{R}^n\\)\uc5d0\uc11c\uc758 \\(r\\)\ucc28 \ubbf8\ubd84\ud615\uc2dd\uc73c\ub85c \ud655\uc7a5\ud558\uc5ec \uc815\uc758\ud558\uc790.<\/p>\n
\uc815\uc758 1. (\ubbf8\ubd84\ud615\uc2dd)<\/span><\/p>\n \\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uace0 \\(r\\)\uac00 \uc815\uc218\uc774\uba70 \\(0\\le r\\le n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(V\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n \n\ucc38\uace0 1.<\/span> \\(V\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(r=n\\)\uc77c \ub54c \\(i\\le i_1 < \\cdots < i_r \\le n\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ucca8\uc218\uc5f4\uc740 \\(i_k = k\\)\ub85c \uc720\uc77c\ud558\ubbc0\ub85c \\(V\\)\uc5d0\uc11c\uc758 \\(n\\)\ud615\uc2dd\uc740 \ubaa8\ub450\n\\[\\omega = f\\,dx_1 \\,dx_2 \\cdots dx_n\\]\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \\(f\\)\ub294 0\ud615\uc2dd, \uc989 \\(V\\)\ub85c\ubd80\ud130 \\(\\mathbb{R}\\)\ub85c\uc758 \ud568\uc218\uc774\ub2e4. \ub530\ub77c\uc11c \\(n\\)\ucc28\uc6d0 \uc5f4\ub9b0\uc9d1\ud569 \\(V\\)\uc5d0\uc11c\uc758 \\(n\\)\ud615\uc2dd\uc740 \ubaa8\ub450 \ubd84\ud574 \uac00\ub2a5\ud558\ub2e4. \ud55c\ud3b8 \\(V\\)\uc5d0\uc11c\uc758 1\ud615\uc2dd\uc740 \ubaa8\ub450\n\\[\\omega = \\sum_{j=1}^n f_j \\,dx_j\\]\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \\(f_j\\)\ub294 \ubaa8\ub450 0\ud615\uc2dd\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218 \\(z = f(x,\\,y)\\)\uc758 \uc804\ubbf8\ubd84\uc18c \\(dz = f_x \\,dx + f_y \\,dy\\)\ub294 \ub300\ud45c\uc801\uc778 1\ud615\uc2dd\uc774\ub2e4.<\/p>\n \\(\\mathbb{R}^n\\)\uc5d0\uc11c\uc758 \\((n-1)\\)\ud615\uc2dd\uc740 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uba74 \\(S = (\\phi ,\\,E )\\) \uc704\uc5d0\uc11c 2\ucc28 \ubbf8\ubd84\ud615\uc2dd\uc744 \uc801\ubd84\ud560 \ub54c \uc57c\ucf54\ube44\uc548\uc744 \uc0ac\uc6a9\ud558\ub294 \uac83\ucc98\ub7fc \\(n\\)\ucc28\uc6d0 \ub2e4\uc591\uccb4\ub77c\uace0 \ubd88\ub9ac\ub294 \uc601\uc5ed\uc5d0\uc11c \\(n\\)\ud615\uc2dd\uc758 \uc801\ubd84\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc774\uac83\uc744 \uc704\ud558\uc5ec \uba3c\uc800 \ubbf8\ubd84\ud615\uc2dd\uc758 \uc5ec\ub7ec \uac00\uc9c0 \uc5f0\uc0b0\uc744 \uc815\uc758\ud558\uace0 \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n \ub450 \ubbf8\ubd84\ud615\uc2dd\uc758 \ud569\uacfc \ucc28\ub294 \uacc4\uc218\ud568\uc218\uc758 \ud569\uacfc \ucc28\ub85c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc989 \\(V\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc774\uace0 0\ud615\uc2dd\uacfc \\(r\\)\ud615\uc2dd\uc758 \uacf1\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4. \ud55c\ud3b8 \ud589\ub82c\uc758 \ub450 \uc5f4\uc774 \uc11c\ub85c \uac19\uc744 \ub54c \ud589\ub82c\uc2dd\uc758 \uac12\uc740 \\(0\\)\uc774\ubbc0\ub85c, \uc11c\ub85c \uac19\uc740 \ub450 1\ud615\uc2dd\uc758 \uacf1\uc758 \uacb0\uacfc\ub294 \\(0\\)\uc774\ub2e4. \uc989 \\(dx\\,dx =0\\)\uc774\ub2e4. \uc774\ub7ec\ud55c \ubc95\uce59\uc744 \uac70\ub4ed\uc81c\uacf1\ubc95\uce59 \ub610\ub294 \uba71\ub4f1\ubc95\uce59<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n \uc774\ub7ec\ud55c \ub0b4\uc6a9\uc744 \ubc14\ud0d5\uc73c\ub85c \ubbf8\ubd84\ud615\uc2dd\uc758 \uacf1\uc744 \uc815\uc758\ud558\uc790. \uba3c\uc800 \ubd84\ud574 \uac00\ub2a5\ud55c \ub450 1\ud615\uc2dd\uc758 \uacf1\uc5d0 \ub300\ud558\uc5ec \ubc18\uad50\ud658\ubc95\uce59\uacfc \uba71\ub4f1\ubc95\uce59\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4. \uc989 \\(i\\ne j\\)\uc77c \ub54c \uc608\ub97c \ub4e4\uc5b4 \\(x^2 \\,dx\\)\uc640 \\(y \\,dy + z\\,dz\\)\uc758 \uacf1\uc740 \ubc18\uad50\ud658\ubc95\uce59\uc5d0 \uc758\ud558\uc5ec \\(i \\ne j\\)\uc77c \ub54c \\(dx_i \\,dx_j = -dx_j \\,dx_i\\)\uc774\ub2e4. \uc774\ub7ec\ud55c \uc774\uc720\ub85c \ubbf8\ubd84\ud615\uc2dd\uc744 \ub098\ud0c0\ub0bc \ub54c \uc2e4\uc218\uc758 \uacf1\uacfc \ubbf8\ubd84\uc18c\uc758 \uacf1\uc744 \uad6c\ubd84\ud558\uae30 \uc704\ud558\uc5ec \ubbf8\ubd84\uc18c \uc0ac\uc774\uc5d0 \uc410\uae30\uacf1\uc744 \ucca9\uac00\ud558\uc5ec \\(dx_i \\wedge dx_j\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \ucc38\uace0\ub85c \uc410\uae30\uacf1\uc744 \uc0ac\uc6a9\ud558\uba74 \n\uc608\uc81c 1.<\/span> \ud480\uc774.<\/span> \\(r\\)\uac1c\uc758 1\ud615\uc2dd\uc744 \uacf1\ud558\uba74 \n\uc815\ub9ac 1. (\ubbf8\ubd84\ud615\uc2dd\uc758 \uc5f0\uc0b0\uc758 \uc131\uc9c8)<\/span><\/p>\n \\(V\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0 \\(f\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 0\ud615\uc2dd\uc774\uba70 \\(\\omega\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(r\\)\ud615\uc2dd\uc774\uace0 \\(r\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(s\\)\ud615\uc2dd\uc774\uba70 \\(\\theta\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(t\\)\ud615\uc2dd\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n \uc99d\uba85<\/p>\n [1]\uacfc [3]\uc740 \uc5f0\uc0b0\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uba85\ud558\uac8c \uc131\ub9bd\ud55c\ub2e4. \uc5ec\uae30\uc11c\ub294 [2]\ub97c \uc99d\uba85\ud558\uc790.<\/p>\n \\(\\omega\\)\uc640 \\(\\eta\\)\uac00 \ubd84\ud574 \uac00\ub2a5\ud558\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uc989 \ubbf8\ubd84\uc801\ubd84\ud559\uc5d0\uc11c \ubcc0\uc218\uac00 \ub450 \uac1c\uc778 \ud568\uc218 \\(z = f(x,\\,y)\\)\uc758 \ubbf8\ubd84\uc18c\uac00 \ub2e4\uc74c \uc815\ub9ac\ub294 \\(n\\)\uac1c\uc758 \\(n\\)\ud615\uc2dd \ub2e4\ud56d\uc2dd\uc758 \uacf1\uc774 \\(n\\times n\\) \ud589\ub82c\uacfc \uc5b4\ub5a0\ud55c \uad00\uacc4\uac00 \uc788\ub294\uc9c0 \uc124\uba85\ud55c\ub2e4.<\/p>\n \n\uc815\ub9ac 2. (\ubbf8\ubd84\ud615\uc2dd \ub2e4\ud56d\uc2dd\uc758 \uacf1)<\/span><\/p>\n \\(V\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0 \\(\\omega_1 ,\\) \\(\\omega _2 ,\\) \\(\\cdots ,\\) \\(\\omega _n\\)\uc774 \\(V\\)\uc5d0\uc11c\uc758 1\ud615\uc2dd\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(A = \\left[a_{ij}\\right]_{n\\times n}\\)\uc774 \uc2e4\uc218 \uc131\ubd84\uc744 \uac16\ub294 \ud589\ub82c\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc99d\uba85<\/p>\n \\(n\\)\uc5d0 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc801\uc6a9\ud558\uc790. \\(n=1\\)\uc77c \ub54c\uc5d0\ub294 \uc815\ub9ac\uc758 \ub4f1\uc2dd\uc774 \uc790\uba85\ud558\uac8c \uc131\ub9bd\ud55c\ub2e4. \uc774\uc81c \\(n \\ge 2\\)\uc774\uace0, \\(n\\)\uc744 \\(n=1\\)\ub85c \ubc14\uafb8\uc5c8\uc744 \ub54c \uc815\ub9ac\uc758 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uc815\ub9ac 1\uacfc \uba71\ub4f1\ubc95\uce59, \uadf8\ub9ac\uace0 \ubc18\uad50\ud658\ubc95\uce59\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4. \ubbf8\ubd84\ud615\uc2dd\uc740 \uc678\ubbf8\ubd84\uc744 \ud1b5\ud574 \ub354 \ub192\uc740 \ucc28\uc218\uc758 \ubbf8\ubd84\ud615\uc2dd\uc73c\ub85c \ubcc0\ud615\ud560 \uc218 \uc788\ub2e4. \uc678\ubbf8\ubd84\uc740 \uc77c\ubc18\ud654\ub41c \uc2a4\ud1a0\ud06c\uc2a4 \uc815\ub9ac\ub97c \uac04\ub2e8\ud558\uac8c \ud45c\ud604\ud560 \ub54c \uc720\uc6a9\ud558\uac8c \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n \uc815\uc758 2. (\uc678\ubbf8\ubd84)<\/span><\/p>\n \\(V\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc774\uace0 \\(\\omega\\)\uac00 \\(V\\)\uc5d0\uc11c \\(C^1\\)\uae09 \\(r\\)\ud615\uc2dd\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n \uc608\uc81c 2.<\/span> \ud480\uc774.<\/span> \ubcf4\uae30 3.<\/span> \ubcf4\uae30 4.<\/span> \uc815\ub9ac 3. (\uc678\ubbf8\ubd84\uc758 \uc131\uc9c8)<\/span><\/p>\n \\(V\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0 \\(\\alpha\\)\uac00 \uc2a4\uce7c\ub77c\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(\\omega\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(C^1\\)\uae09 \\(r\\)\ud615\uc2dd\uc774\uace0 \\(\\eta\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(C^1\\)\uae09 \\(s\\)\ud615\uc2dd\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n \uc99d\uba85<\/p>\n \uc678\ubbf8\ubd84\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uba74 \\(\\omega\\)\uc640 \\(\\eta\\)\uac00 \ubd84\ud574 \uac00\ub2a5\ud55c \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \uc989 \uc774\uc81c [3]\uc744 \uc99d\uba85\ud558\uc790. \uba3c\uc800 \\(r=0\\)\uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud558\uc790. \\(f\\)\uac00 \uc2e4\uc22b\uac12 \ud568\uc218\uc774\uace0 \\(\\omega = f\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\)\uac00 \\(\\mathbb{R}^3\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0 \\(f : V \\,\\to\\,\\mathbb{R}\\)\uac00 \\(C^2\\)\uae09 \ud568\uc218\uc774\uba70 \\(\\mathrm{x} \\in V\\)\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc815\ub9ac 4. (\uc774\uacc4\uc678\ubbf8\ubd84\uc758 \uc131\uc9c8)<\/span><\/p>\n \\(V\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0 \\(\\omega\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(C^2\\)\uae09 \ubbf8\ubd84\ud615\uc2dd\uc77c \ub54c \\(d^2 \\,\\omega =0\\)\uc774\ub2e4.<\/p>\n<\/div>\n \uc99d\uba85<\/p>\n \\(\\omega\\)\uac00 \ubd84\ud574 \uac00\ub2a5\ud55c \ubbf8\ubd84\ud615\uc2dd\uc774\uace0 \ub2e4\uc74c\uc73c\ub85c \\(r=1\\)\uc774\uace0 \\(\\omega = f\\,dx_k\\)\ub77c\uace0 \ud558\uc790. \\[d^2 \\,x_k = d(1\\,dx_k )=0\\]\uc774\ubbc0\ub85c \\(r=0\\)\uc77c \ub54c\uc758 \uacb0\uacfc\ub97c \uc774\uc6a9\ud558\uba74 \ub05d\uc73c\ub85c \\(r > 1\\)\uc774\uace0, \\(0 \\le s < r\\)\uc778 \uc784\uc758\uc758 \\(s\\)\ud615\uc2dd\uc758 \uc774\uacc4\uc678\ubbf8\ubd84\uc774 \\(0\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74\n\\[d\\omega = d\\left( f\\,dx_{i_1} \\cdots dx_{i_{r-1}} \\right) dx_{i_r}\\]\n\uc774\ubbc0\ub85c \uc678\ubbf8\ubd84\uc758 \uacf1\uc758 \uc131\uc9c8\uacfc \uadc0\ub0a9\uc801 \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.\n\\[\\begin{align}\nd^2\\,\\omega &= d^2 \\left( f\\,dx_{i_1} \\cdots dx_{i_{r-1}} \\right) dx_{i_r} \\\\[6pt]\n& \\quad + (-1)^r \\,d\\left(f\\,dx_{i_1}\\cdots dx_{i_{r-1}} \\right) d^2 \\,dx_{i_r} =0\n\\tag*{\\(\\blacksquare\\)}\n\\end{align}\\]\n<\/p>\n<\/div>\n \ub2e4\uc74c \uc815\uc758\ub294 \\(C^1\\)\uae09 \ud568\uc218 \\(\\phi : \\mathbb{R}^n \\,\\to\\,\\mathbb{R}^m\\)\uc744 \uc774\uc6a9\ud558\uc5ec \\(\\mathbb{R}^m\\)\uc758 \ubbf8\ubd84\ud615\uc2dd\uc744 \\(\\mathbb{R}^n\\)\uc758 \ubbf8\ubd84\ud615\uc2dd\uc73c\ub85c \ubcc0\ud658\ud558\ub294 \ubc29\ubc95\uc744 \uc124\uba85\ud55c\ub2e4.<\/p>\n \uc815\uc758 3. (\ubbf8\ubd84\ubcc0\ud658; differential transform)<\/span><\/p>\n \\(U\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0 \\(V\\)\uac00 \\(\\mathbb{R}^m\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uba70 \\(\\phi : U \\,\\to\\,V\\)\uac00 \\(U\\)\uc5d0\uc11c \\(C^1\\)\uae09\uc774\uace0 \uc9c0\uae08\ubd80\ud130 \uc774 \uae00\uc774 \ub05d\ub0a0 \ub54c\uae4c\uc9c0 \\(U\\)\ub294 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0 \\(V\\)\ub294 \\(\\mathbb{R}^m\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uba70 \\(\\phi\\)\ub294 \\(U\\)\ub85c\ubd80\ud130 \\(V\\)\ub85c\uc758 \ud568\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n \ucc38\uace0 2.<\/span> \uc99d\uba85<\/p>\n \\(V\\)\uc5d0\uc11c\uc758 \uc784\uc758\uc758 0\ud615\uc2dd \\(f\\)\uc5d0 \ub300\ud558\uc5ec \ucc38\uace0 3.<\/span> \uc99d\uba85<\/p>\n \ubbf8\ubd84\ud615\uc2dd\uc758 \ud569\uc740 \uacc4\uc218\uc758 \ud569\uc73c\ub85c \uc815\uc758\ub418\ubbc0\ub85c \\(\\omega\\)\uc640 \\(\\eta\\)\uac00 \ubaa8\ub450 \ubd84\ud574 \uac00\ub2a5\ud55c \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \ucc38\uace0 4.<\/span> \uc99d\uba85<\/p>\n \\(\\omega\\)\uc640 \\(\\eta\\)\uac00 \ubaa8\ub450 \ubd84\ud574 \uac00\ub2a5\ud55c \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \ucc38\uace0 5.<\/span> \uc99d\uba85<\/p>\n \\(\\omega\\)\uac00 \ubd84\ud574 \uac00\ub2a5\ud55c \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \\(r\\)\uc5d0 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc801\uc6a9\ud558\uc790.<\/p>\n \\(r=0\\)\uc774\uace0 \\(\\omega = f\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubbf8\ubd84\ubcc0\ud658\uc758 \uc815\uc758\uc640 \uc5f0\uc1c4\ubc95\uce59\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4. \ub05d\uc73c\ub85c \\(r > 1\\)\uc774\uace0 \\(0 \\le s < r\\)\uc778 \uc784\uc758\uc758 \uc815\uc218 \\(s\\)\uc5d0 \ub300\ud558\uc5ec \\(s\\)\ud615\uc2dd\uc774 \uc815\ub9ac\uc758 \ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \uac00\uc815\ud558\uc790. \\(\\omega\\)\uac00 \ubd84\ud574 \uac00\ub2a5\ud55c \\(r\\)\ud615\uc2dd\uc774\uace0 \\(\\omega = \\theta \\eta\\)\uc774\uba70 \\(\\theta\\)\ub294 1\ud615\uc2dd\uc774\uace0 \\(\\eta\\)\ub294 \\((r-1)\\)\ud615\uc2dd\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74\n\\[d\\omega = (d\\theta ) \\eta - \\theta \\,d\\eta\\]\n\uc774\ubbc0\ub85c \uadc0\ub0a9\uc801 \uac00\uc815\uacfc \uc678\ubbf8\ubd84\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.\n\\[\\begin{align}\nd^* (d\\omega ) &= \\phi^* (d\\theta )\\phi^* (\\eta ) - \\phi^* (\\theta ) \\phi^* (d\\eta ) \\\\[6pt]\n&= d(\\phi^* \\theta )\\phi^* (\\eta ) - \\phi^* (\\theta ) d(\\phi^* \\eta ) \\\\[6pt]\n&= d((\\phi^* \\theta )(\\phi^* \\eta )) \\\\[6pt]\n&= d(\\phi^* (\\theta \\eta )) \\\\[6pt]\n&= d(\\phi^* (\\omega )).\n\\tag*{\\(\\blacksquare\\)}\n\\end{align}\\]\n<\/p>\n<\/div>\n \ub2e4\uc74c \uc815\ub9ac\ub294 \ubbf8\ubd84\ubcc0\ud658\uc774 \ubbf8\ubd84\uc801\ubd84\ud559\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \uc720\ud5a5\uc120\uc801\ubd84\uacfc \uc720\ud5a5\uba74\uc801\ubd84\uc758 \uc815\uc758\uc5d0\uc11c \uc81c\uc2dc\ud55c \ud53c\uc801\ubd84\ud568\uc218\uc758 \ud615\ud0dc\ub97c \uc77c\ubc18\ud654\ud560 \ub54c \uc0ac\uc6a9\ub41c\ub2e4\ub294 \uac83\uc744 \uc124\uba85\ud55c\ub2e4.<\/p>\n \uc815\ub9ac 4. (\ubbf8\ubd84\ubcc0\ud658\uc758 \uae30\ubcf8\uc815\ub9ac)<\/span><\/p>\n \\(m \\ge n\\)\uc774\uace0 \\(U\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uba70 \\(V\\)\uac00 \\(\\mathbb{R}^m\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(\\phi : U \\,\\to\\,V\\)\uac00 \\(U\\)\uc5d0\uc11c \\(C^1\\)\uae09\uc774\uace0 \uc99d\uba85<\/p>\n \\(\\omega\\)\uac00 \ubd84\ud574 \uac00\ub2a5\ud55c \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \\(n=1\\)\uc774\uace0 \\(\\omega = f\\,dx_j\\)\uc774\uba74 \\(\\mathbb{R}^3\\)\uc5d0\uc11c\uc758 \uc720\ud5a5\uc120\uc801\ubd84\uc740 1\ud615\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \\(\\int_C \\mathbb{F} \\cdot \\mathbb{T} \\,dx = \\int_C P \\,dx + Q\\,dy + R\\,dz\\) \uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \\(C\\)\ub294 \\(\\mathbb{R}^3\\)\uc5d0 \ud3ec\ud568\ub418\ub294 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc774\ub2e4. \uc989 \\(C\\)\ub294 \\(\\mathbb{R}^3\\)\uc5d0 \ud3ec\ud568\ub418\ub294 1\ucc28\uc6d0 \uc9d1\ud569\uc774\ub2e4. \ud55c\ud3b8 \\(\\mathbb{R}^3\\)\uc5d0\uc11c\uc758 \uc720\ud5a5\uba74\uc801\ubd84\uc740 2\ud615\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \\(\\iint_S \\mathbb{F} \\cdot \\mathbb{n} \\,d\\sigma = \\iint_S P\\,dy\\,dz + Q\\,dz\\,dx + R\\,dx\\,dy\\) \uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \\(S\\)\ub294 \\(\\mathbb{R}^3\\)\uc5d0 \ud3ec\ud568\ub418\ub294 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uba74\uc774\ub2e4. \uc989 \\(S\\)\ub294…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[62],"tags":[],"class_list":["post-2358","post","type-post","status-publish","format-standard","hentry","category-differential-geometry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2358","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=2358"}],"version-history":[{"count":56,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2358\/revisions"}],"predecessor-version":[{"id":4863,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2358\/revisions\/4863"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2358"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2358"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\n\\[\\omega = \\sum_{1\\le i_1 < i_2 < \\cdots < i_r \\le n} f_{i_1 \\, i_2 \\cdots i_r} dx_{i_1} \\, dx_{i_2} \\, \\dots dx_{i_r} \\tag{1}\\]\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(f_{i_1 \\, i_2 \\, \\cdots i_r}\\)\ub97c \\(\\omega\\)\uc758 \uacc4\uc218\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/li>\n
\n\\[\\omega = \\sum_{1\\le i_1 < i_2 < \\cdots < i_r \\le n} f_{i_1 \\, i_2 \\cdots i_r} dx_{i_1} \\, dx_{i_2} \\, \\dots dx_{i_r}\\]\n\uc640\n\\[\\eta = \\sum_{1\\le i_1 < i_2 < \\cdots < i_r \\le n} g_{i_1 \\, i_2 \\cdots i_r} dx_{i_1} \\, dx_{i_2} \\, \\dots dx_{i_r}\\]\n\uac00 \uc11c\ub85c \uac19\ub2e4<\/span>\ub294 \uac83\uc740 \\(1\\le i_1 < \\cdots < i_r \\le n\\)\uc778 \ubaa8\ub4e0 \\(i_r\\)\uc5d0 \ub300\ud558\uc5ec \\(f_{i_1 \\cdots i_r} = g_{i_1 \\cdots i_r}\\)\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/li>\n
\n\uc815\uc758 1\uc758 \ubbf8\ubd84\ud615\uc2dd\ub4e4\uc740 \uadf8 \uc790\uccb4\ub85c \uc2e4\uc22b\uac12\uc744 \uac16\ub294 \uac83\uc774 \uc544\ub2c8\uba70 \uc801\ubd84 \uae30\ud638\uc640 \uc801\ubd84 \uc601\uc5ed\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c \uac12\uc744 \uac16\uac8c \ub41c\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \ubbf8\ubd84\ud615\uc2dd\uc744 \u2018\uc801\ubd84 \uc601\uc5ed\uc744 \uc2e4\uc22b\uac12\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\u2019\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.\n<\/p>\n<\/div>\n
\n\\[\\omega = \\sum_{j=1}^n f_j \\,dx_1 \\cdots \\hat{dx_j} \\cdots dx_n\\]
\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \\(\\hat{dx_j}\\)\ub294 \ubbf8\ubd84\uc18c \\(dx_j\\)\uac00 \ube60\uc84c\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(\\mathbb{R}^3\\)\uc5d0\uc11c\uc758 2\ud615\uc2dd\uc740
\n\\[\\omega = f_1 \\,dy\\,dz + f_2 \\,dx\\,dz + f_3\\,dx\\,dy \\tag{2}\\]
\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n\ubbf8\ubd84\ud615\uc2dd\uc758 \uc5f0\uc0b0<\/h3>\n
\n\\[\\omega = \\sum_{1\\le i_1 < i_2 < \\cdots < i_r \\le n} f_{i_1 \\, i_2 \\cdots i_r} dx_{i_1} \\, dx_{i_2} \\, \\dots dx_{i_r} , \\\\[8pt]\n\\eta = \\sum_{1\\le i_1 < i_2 < \\cdots < i_r \\le n} g_{i_1 \\, i_2 \\cdots i_r} dx_{i_1} \\, dx_{i_2} \\, \\dots dx_{i_r}\n\\]\n\uac00 \\(V\\)\uc5d0\uc11c\uc758 \ubbf8\ubd84\ud615\uc2dd\uc77c \ub54c\n\\[\\begin{align}\n\\omega +\\eta &= \\sum_{1\\le i_1 < i_2 < \\cdots < i_r \\le n} \\left(f_{i_1 \\, i_2 \\cdots i_r} + g_{i_1 \\, i_2 \\cdots i_r} \\right) dx_{i_1} \\, dx_{i_2} \\, \\dots dx_{i_r} ,\\\\[8pt]\n\\omega -\\eta &= \\sum_{1\\le i_1 < i_2 < \\cdots < i_r \\le n} \\left(f_{i_1 \\, i_2 \\cdots i_r} - g_{i_1 \\, i_2 \\cdots i_r} \\right) dx_{i_1} \\, dx_{i_2} \\, \\dots dx_{i_r}\n\\end{align}\\]\n\uc774\ub2e4. \uba85\ubc31\ud788 \ubbf8\ubd84\ud615\uc2dd\uc758 \ud569\uc5d0 \ub300\ud558\uc5ec \uacb0\ud569\ubc95\uce59\uacfc \uad50\ud658\ubc95\uce59\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n
\n\\[g \\left( \\sum f_{i_1 \\cdots i_r} \\,dx_{i_1} \\cdots dx_{i_r} \\right)
\n=
\n\\left( \\sum gf_{i_1 \\cdots i_r} \\,dx_{i_1} \\cdots dx_{i_r} \\right) \\]
\n\\(\\omega ,\\) \\(\\eta\\)\uac00 \\(r\\)\ud615\uc2dd\uc774\uace0 \\(f,\\) \\(g\\)\uac00 0\ud615\uc2dd\uc77c \ub54c \uba85\ubc31\ud788 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[f(\\omega + \\eta) = f\\omega + f\\eta ,\\\\[8pt]
\n(f+g)\\omega = f\\omega + g\\omega .\\]
\n0\ud615\uc2dd\uc774 \uc544\ub2cc \ubbf8\ubd84\ud615\uc2dd\uc758 \uacf1, \uc989 \\(r > 0,\\) \\(s > 0\\)\uc77c \ub54c \\(r\\)\ud615\uc2dd\uacfc \\(s\\)\ud615\uc2dd\uc758 \uacf1\uc744 \uc815\uc758\ud558\ub294 \uac83\uc740 \ube44\uad50\uc801 \uae4c\ub2e4\ub86d\ub2e4. \uba3c\uc800 \uac04\ub2e8\ud55c \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(S = (\\phi ,\\,E )\\)\uac00 \\(\\mathbb{R}^3\\)\uc5d0 \ud3ec\ud568\ub418\uc5b4 \uc788\uace0 \ubc29\ud5a5\uc744 \uac00\uc9c4 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uba74\uc774\uba70
\n\\[x = \\phi_1 (u,\\,v) ,\\, y=\\phi_2 (u,\\,v) ,\\, z=\\phi_3 (u,\\,v)\\]
\n\uc77c \ub54c, 2\ucc28 \ubbf8\ubd84\uc18c\ub294
\n\\[\\begin{gather}
\ndy \\,dz =\\frac{\\partial (y,\\,z)}{\\partial (u,\\,v)} d(u,\\,v) ,\\\\[6pt]
\ndz \\,dx =\\frac{\\partial (z,\\,x)}{\\partial (u,\\,v)} d(u,\\,v) ,\\\\[6pt]
\ndx \\,dy =\\frac{\\partial (x,\\,y)}{\\partial (u,\\,v)} d(u,\\,v)
\n\\end{gather}\\]
\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(dx \\,dy\\)\ub294 \ub450 1\ud615\uc2dd\uc758 \uacf1\ucc98\ub7fc \ubcf4\uc778\ub2e4. \uadf8\ub7f0\ub370 \uc774 \uacf1\uc5d0 \ub300\ud574\uc11c\ub294 \uad50\ud658\ubc95\uce59\uc774 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc65c\ub0d0\ud558\uba74 \ud589\ub82c\uc758 \ub450 \uc5f4\uc744 \uc11c\ub85c \ubc14\uafb8\uba74 \ud589\ub82c\uc2dd\uc758 \ubd80\ud638\uac00 \ubc18\ub300\uac00 \ub418\uae30 \ub54c\ubb38\uc5d0, \ub450 1\ud615\uc2dd\uc758 \uacf1\uc5d0\uc11c \uacf1\ud558\ub294 \uc21c\uc11c\ub97c \ubc14\uafb8\uba74 \ubd80\ud638\uac00 \ubc14\ub010\ub2e4. \uc989 \\(dx\\,dy = -dy\\,dx\\)\uc774\ub2e4. \uc774\ub7ec\ud55c \ubc95\uce59\uc744 \ubc18\uad50\ud658\ubc95\uce59<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n
\n\\[dx_j \\,dx_k = -dx_k \\,dx_j\\]
\n\uadf8\ub9ac\uace0
\n\\[dx_j \\,dx_j = 0\\]
\n\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4. \ub2e4\uc74c\uc73c\ub3c4 \ub450 \ubbf8\ubd84\ud615\uc2dd\uc744 \uacf1\ud560 \ub54c \ubd84\ubc30\ubc95\uce59\uc744 \uc774\uc6a9\ud558\uc5ec \uacf1\ud55c \ub4a4 \ubc18\uad50\ud658\ubc95\uce59\uacfc \uba71\ub4f1\ubc95\uce59\uc744 \uc774\uc6a9\ud558\uc5ec \uc815\uc758 1\uc5d0 \ub9de\ub3c4\ub85d \ubcc0\ud615\ud560 \uc218 \uc788\ub2e4\uace0 \uc57d\uc18d\ud55c\ub2e4.<\/p>\n
\n\\[(x^2 \\,dx)(y\\,dy + z\\,dz ) = x^2 y \\,dx\\,dy + x^2 z \\,dx\\,dz\\]
\n\uc774\uba70, \\(\\sin x \\,dz\\)\uc640 \\(x^2 \\,dx + xy \\,dy + \\ln z \\,dz\\)\uc758 \uacf1\uc740
\n\\[(\\sin x\\,dz)(x^2\\,dx + xy\\,dy + \\ln z\\,dz) = -xy \\sin x \\,dy\\,dz – x^2 \\sin x \\,dx\\,dz\\]
\n\uc774\ub2e4. \ud2b9\ud788
\n\\[\\omega = \\sum_{j=1}^N \\omega _j ,\\quad \\eta= \\sum_{k=1}^L \\eta_k\\]
\n\uac00 \uac01\uac01 \ubbf8\ubd84\ud615\uc2dd\ub4e4\uc758 \ud569\uc77c \ub54c
\n\\[\\omega \\eta = \\sum_{j=1}^N \\sum_{k=1}^L \\omega_j \\,\\eta_k\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ubc18\uad50\ud658\ubc95\uce59\uacfc \uba71\ub4f1\ubc95\uce59\uc774 \ub2e4\uc18c \uc774\uc0c1\ud574 \ubcf4\uc77c \uc218\ub3c4 \uc788\ub2e4. \ud558\uc9c0\ub9cc \ubbf8\ubd84\ud615\uc2dd\uc758 \ubbf8\ubd84\uc18c \\(dx\\,dy\\)\uac00 \ubc18\ubcf5\uc801\ubd84\uc758 \uac1c\ub150\uc5d0\uc11c \uac00\uc838\uc628 \uac83\uc774 \uc544\ub2c8\ub77c \uc720\ud5a5\uc801\ubd84\uc758 \uac1c\ub150\uc5d0\uc11c \uac00\uc838\uc628 \uac83\uc784\uc744 \uc0dd\uac01\ud574 \ubcf4\uba74 \uc774\ub7ec\ud55c \ubc95\uce59\uc740 \uc720\ud5a5\uc801\ubd84\uc758 \uc131\uc9c8\ub85c\ubd80\ud130 \uc5bb\uc5b4\uc9c4 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uacb0\uacfc\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \ubc18\uad50\ud658\ubc95\uce59\uc740 \uc720\ud5a5\uc801\ubd84\uc5d0\uc11c \uc801\ubd84 \uc601\uc5ed\uc758 \ubc29\ud5a5\uc774 \ubc18\ub300\uac00 \ub420 \uacbd\uc6b0 \uc801\ubd84\uac12\uc758 \ubd80\ud638\uac00 \ubc18\ub300\uac00 \ub41c\ub2e4\ub294 \uc131\uc9c8\ub85c\ubd80\ud130 \uc5bb\uc5b4\uc9c4 \uac83\uc774\ub2e4.<\/p>\n
\n\\[dx_1 \\, dx_2 \\cdots dx_r = \\bigwedge_{j=1}^{r} dx_j\\]
\n\uc640 \uac19\uc774 \uac04\ub2e8\ud788 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n
\n\\(\\omega = x^2 \\,dx \\,dz + xy \\,dy\\,dz ,\\) \\(\\eta = 2y \\,dx \\,dz\\)\uc77c \ub54c \\(\\omega + \\eta ,\\) \\(\\omega – \\eta,\\) \\(\\omega \\eta\\)\ub97c \uac01\uac01 \uad6c\ud558\uc2dc\uc624.<\/p>\n
\n\ubbf8\ubd84\ud615\uc2dd\uc758 \uc5f0\uc0b0\uc758 \uc815\uc758\uc640 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\omega + \\eta &= (x^2 + 2y )dx\\,dz + xy \\,dy\\,dz ,\\\\[8pt]
\n\\omega – \\eta &= (x^2 – 2y ) dx\\,dz + xy\\,dy\\,dz ,\\\\[8pt]
\n\\omega \\eta &= (x^2 \\,dx\\,dz + xy\\,dy\\,dz)(2y\\,dx\\,dz) \\\\[6pt]
\n&= 2x^2 y \\,dx\\,dz\\,dx\\,dz + 2xy^2 \\,dy\\,dz\\,dx\\,dz \\\\[6pt]
\n&= -2x^2 y \\,dx\\,dx\\,dz\\,dz – 2xy^2 \\,dy\\,dx\\,dz\\,dz =0. \\tag*{\\(\\square\\)}
\n\\end{align}\\]\n<\/p>\n<\/div>\n
\n\\[\\sum f_{i_1 \\cdots i_r} \\,dx_{i_1} \\cdots dx_{i_r}\\]
\n\uc758 \uaf34\uc774 \ub41c\ub2e4. \ubc18\uad50\ud658\ubc95\uce59\uacfc \uba71\ub4f1\ubc95\uce59\uc744 \uc774\uc6a9\ud558\uba74 \uc704\uc640 \uac19\uc740 \ud45c\ud604\uc5d0\uc11c \\(i_j\\)\uac00 \uc99d\uac00\uc218\uc5f4\uc774 \ub418\ub3c4\ub85d \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uc989 \ubbf8\ubd84\ud615\uc2dd\uc758 \ud45c\ud604\uc5d0\uc11c \ucca8\uc218\uac00 \uc99d\uac00\uc218\uc5f4\uc774 \uc544\ub2cc \uacbd\uc6b0\uc5d0\ub3c4 \ubcc0\ud615\ud558\uc5ec \ucca8\uc218\uac00 \uc99d\uac00\uc218\uc5f4\uc774 \ub418\ub3c4\ub85d \ud560 \uc218 \uc788\uc73c\ubbc0\ub85c, \uc9c0\uae08\ubd80\ud130\ub294 \ubbf8\ubd84\ud615\uc2dd\uc758 \uc815\uc758\uc5d0\uc11c \ucca8\uc218\uac00 \uc99d\uac00\uc218\uc5f4\uc774\ub77c\ub294 \uc870\uac74
\n\\[1 \\le i_1 < \\cdots < i_r \\le n\\]\n\uc744 \ube7c\uace0\n\\[i_j \\in \\left\\{ 1,\\,\\cdots,\\,n \\right\\}\\]\n\uc73c\ub85c \ub300\uccb4\ud558\ub3c4\ub85d \ud558\uc790. \uc774\ub85c\uc368 \ubbf8\ubd84\ud615\uc2dd\uc744 \ub098\ud0c0\ub0bc \ub54c \uc5d0 \ubbf8\ubd84\uc18c\ub4e4\uc758 \uc21c\uc11c\ub97c \uace0\ub824\ud560 \ud544\uc694\uac00 \uc5c6\uc73c\ubbc0\ub85c \ud45c\ud604\uc774 \ub354 \uc790\uc720\ub85c\uc6cc\uc9c4\ub2e4. \uc608\ub97c \ub4e4\uc5b4 2\ud615\uc2dd\uc740 \uc720\ud5a5\uba74\uc801\ubd84\uacfc \uae4a\uc774 \uc5f0\uad00\ub418\uae30 \ub54c\ubb38\uc5d0 (2)\ub294 \ubcf4\ud1b5\n\\[\\omega = P\\,dy\\,dz + Q\\,dz\\,dx + R \\,dx\\,dy\\]\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0b4\ub294 \uacbd\uc6b0\uac00 \ub9ce\ub2e4.<\/p>\n\n
\n\\[\\omega = f\\,dx_{i_1} \\cdots dx_{i_r} ,\\\\[6pt] \\eta = g\\,dx_{j_1} \\cdots dx_{j_s}\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\omega\\)\uc640 \\(\\eta\\)\uc758 \uacf1\uc740
\n\\[\\omega \\eta = fg \\,x_{i_1} \\cdots dx_{i_r} \\,dx_{j_1} \\cdots dx_{j_s}\\]
\n\ub85c\uc11c \\((r+s)\\)\ud615\uc2dd\uc774\ub2e4. \uc5ec\uae30\uc5d0 \ubc18\uad50\ud658\ubc95\uce59\uc744 \ubc18\ubcf5\ud558\uc5ec \uc801\uc6a9\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\omega \\eta &= fg\\,dx_{i_1} \\cdots dx_{i_r} \\,dx_{j_1} \\cdots dx_{j_s} \\\\[6pt]
\n&= (-1)^r fg \\,dx_{j_1} \\,x_{j_2} \\cdots dx_{i_r} \\,dx_{j_2} \\cdots dx_{j_s} \\\\[6pt]
\n&= \\cdots = (-1)^{rs} \\,gf\\,dx_{j_1} \\cdots dx_{j_s} \\,dx_{i_1} \\cdots dx_{i_r} \\\\[6pt]
\n&= (-1)^{rs} \\eta \\omega \\tag*{\\(\\blacksquare\\)}
\n\\end{align}\\]\n<\/p><\/div>\n
\n\\[dz = f_x \\,dx + f_y \\,dy\\]
\n\uc784\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub294\ub370, \uc774\ub7ec\ud55c \ubbf8\ubd84\uc18c \ub610\ud55c \ubbf8\ubd84\ud615\uc2dd\uc774\ub2e4. \uc774\ub85c\uc368 \\(dz\\,dx\\)\ub098 \\(dy\\,dz\\) \uac19\uc740 \ubbf8\ubd84\ud615\uc2dd\uc740 \ub450 \uac00\uc9c0 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ub428\uc744 \uc54c \uc218 \uc788\ub2e4. \uc989 \ubbf8\ubd84\ud615\uc2dd\uc740 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc758 \uc57c\ucf54\ube44\uc548\uc73c\ub85c \uc0dd\uac01\ud560 \uc218\ub3c4 \uc788\uc73c\uba70, \ub450 1\ud615\uc2dd\uc758 \uacf1\uc758 \uacb0\uacfc\ub85c \uc0dd\uac01\ud560 \uc218\ub3c4 \uc788\ub2e4. \ub458 \uc911 \uc5b4\ub290 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\ub354\ub77c\ub3c4 \ub3d9\uc77c\ud55c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4. \uc2e4\uc81c\ub85c \uace1\uba74 \\(z=f(x,\\,y)\\)\uc758 \uc790\uba85\ud55c \ub9e4\uac1c\ubcc0\uc218\ud654\ub97c \uc774\uc6a9\ud558\uc5ec \uc57c\ucf54\ube44\uc548\uc744 \uad6c\ud558\uba74
\n\\[dy \\,dz = -f_x \\,d(x,\\,y) ,\\quad fz\\,dx = -f_y \\,d(x,\\,y) \\tag{3}\\]
\n\ub97c \uc5bb\ub294\ub2e4. \ud55c\ud3b8 1\ud615\uc2dd
\n\\[dz = f_x\\,dx + f_y\\,dy\\]
\n\uc758 \uc591\ubcc0\uc758 \uc67c\ucabd\uc5d0 \\(dy\\)\ub97c \uacf1\ud558\uba74
\n\\[\\begin{align}
\ndy\\,dz &= dy(f_x \\,dx + f_y \\,dy ) \\\\[6pt]
\n&= f_x \\,dy\\,dx + f_y \\,dy\\,dy \\\\[6pt]
\n&= -f_x \\,dx\\,dy\\end{align}\\]
\n\ub97c \uc5bb\ub294\ub2e4. \ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c \ub4f1\uc2dd \\(dz\\,dx = -f_y \\,dx\\,dy\\)\ub3c4 \uc5bb\uc744 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \ub9cc\uc57d \\(d(x,\\,y)\\)\ub97c \\(dx\\,dy\\)\uc640 \ub3d9\uc77c\uc2dc\ud558\uba74 \ubbf8\ubd84\ud615\uc2dd\uc744 \uc815\uc758\ud558\ub294 \ub450 \ubc29\ubc95 \uc911 \uc5b4\ub290 \uac83\uc744 \uc0ac\uc6a9\ud558\ub4e0 \uc0c1\uad00 \uc5c6\uc774 (3)\uc744 \uc5bb\ub294\ub2e4.<\/p>\n
\n\\[\\left( \\sum_{j=1}^n \\,a_{1j} \\,\\omega_j \\right) \\left( \\sum_{j=1}^n \\,a_{2j} \\,\\omega_j \\right) = (\\det A )\\omega_1 \\,\\omega_2 \\cdots \\omega_n \\]\n<\/p>\n<\/div>\n
\n\\[\\begin{align}
\n\\left( \\sum_{j=1}^n \\,a_{1j} \\,\\omega_j \\right) \\cdots \\left( \\sum_{j=1}^n \\,a_{nj} \\,\\omega_j \\right)
\n&= \\left( a_{11} \\,\\omega_1 + \\sum_{j=2}^n \\,a_{1j}\\,\\omega_j \\right) \\cdots \\left(a_{n1} \\,\\omega_1 + \\sum_{j=2}^n \\,a_{nj} \\,\\omega_j \\right) \\\\[6pt]
\n&= a_{11}\\,\\omega_1 \\left( \\sum_{j=2}^n \\,a_{2j} \\,\\omega_j \\right) \\cdots \\left( \\sum_{j=2}^{n} \\,a_{nj}\\,\\omega_j \\right) \\\\[6pt]
\n&\\quad + \\left( \\sum_{j=2}^n \\,a_{1j}\\,\\omega_j \\right) a_{21} \\,\\omega_1 \\left( \\sum_{j=2}^n \\,a_{3j}\\,\\omega_j \\right) \\cdots \\left( \\sum_{j=2}^n \\,a_{nj} \\,\\omega_j \\right) \\\\[6pt]
\n&\\quad + \\cdots + \\left( \\sum_{j=2}^n \\,a_{1j} \\,\\omega_j \\right) \\cdots \\left( \\sum_{j=2}^n \\,a_{3j} \\,\\omega_j \\right) a_{n1} \\,\\omega_1 \\\\[6pt]
\n&\\quad + \\left( \\sum_{j=2}^n \\,a_{1j} \\,\\omega_j \\right) \\cdots \\left( \\sum_{j=2}^n \\,a_{nj} \\,\\omega_j \\right) \\\\[6pt]
\n&= a_{11} \\,\\det A_{11} (\\omega_1 \\, \\omega_2 \\cdots \\omega_n ) \\\\[8pt]
\n&\\quad + (-1)^1 a_{21} \\,\\det A_{21} (\\omega_1 \\,\\omega_2 \\cdots \\omega_n ) \\\\[8pt]
\n&\\quad + \\cdots + (-1)^{n-1} a_{n1} \\,\\det A_{n1} ( \\omega_1 \\,\\omega_2 \\cdots \\omega_n ) \\\\[8pt]
\n&\\quad + \\left( \\sum_{j=2}^n \\,a_{1j}\\,\\omega_j \\right) \\det A_{11} (\\omega_2 \\cdots \\omega_n ) \\\\[8pt]
\n&= (\\det A) \\omega_1 \\cdots \\omega_n + 0 \\\\[8pt]
\n&= (\\det A) \\omega_1 \\, \\omega_2 \\cdots \\omega_n .
\n\\tag*{\\(\\blacksquare\\)}
\n\\end{align}\\]\n<\/p>\n<\/div>\n\uc678\ubbf8\ubd84<\/h3>\n
\n
\n\\(\\omega ( x,\\,y,\\,z,\\,t) = xy\\,dx\\,dy + (x+z+t)dz\\,dt\\)\uc77c \ub54c \\(d\\omega \\)\uc640 \\(d^2 \\omega \\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n
\n\\[\\begin{align}
\nd\\omega &= (y\\,dx + x\\,dy) dx\\,dy + (dx+dz+dt)dz\\,dt \\\\[6pt]
\n&= dx\\,dz\\,dt ,\\\\[8pt]
\nd^2 \\omega &= (d1) dx\\,dy\\,dz =0 .
\n\\tag*{\\(\\square\\)}
\n\\end{align}\\]\n<\/p>\n<\/div>\n
\n\\(V\\)\uac00 \\(\\mathbb{R}^3\\)\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0 \ub450 \ud568\uc218 \\(f:V \\,\\to\\, \\mathbb{R}\\)\uc640 \\(\\mathbb{F} : V \\,\\to\\, \\mathbb{R}^3\\)\uac00 \ubaa8\ub450 \\(V\\)\uc5d0\uc11c \\(C^1\\)\uae09\uc774\uba70 \\(\\mathbb{F} = (P,\\,Q,\\,R)\\)\ub77c\uace0 \ud558\uc790. \uc774 \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n\n
\n\\(V\\)\uac00 \\(\\mathbb{R}^n\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc774\uace0 \\(\\omega\\)\uc640 \\(\\eta\\)\uac00 \\(V\\)\uc5d0\uc11c \\(C^1\\)\uae09\uc778 0\ud615\uc2dd\uc774\uba70 \\(\\alpha\\)\uac00 \uc2a4\uce7c\ub77c\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(d(\\alpha \\omega ),\\) \\(d(\\omega + \\eta ),\\) \\(d(\\omega \\eta )\\)\ub294 \ubaa8\ub450 \\(V\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 1\ud615\uc2dd\uc774\uba70, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\begin{align}
\nd(\\alpha \\omega ) &= \\alpha \\,d\\omega ,\\\\[6pt]
\nd(\\omega + \\eta ) &= d\\omega + d\\eta ,\\\\[6pt]
\nd(\\omega \\eta ) &= \\eta \\,d\\omega + \\omega \\,d\\eta.
\n\\tag*{\\(\\square\\)}
\n\\end{align}\\]\n<\/div>\n\n
\n\\[\\omega = f\\,dx_{i_1} \\cdots dx_{i_r} ,\\\\[6pt]
\n\\eta = g\\,dx_{j_1} \\cdots dx_{j_s}\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[\\begin{align}
\nd(\\alpha \\omega ) &= d(\\alpha f)dx_{i_1} \\cdots dx_{i_r} \\\\[6pt]
\n&=\\alpha \\,df\\,dx_{i_1} \\cdots dx_{i_r} \\\\[6pt]
\n&=\\alpha \\,d\\omega
\n\\end{align}\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4 \\(r=s\\)\uc774\uace0 \ubaa8\ub4e0 \\(\\nu\\)\uc5d0 \ub300\ud558\uc5ec \\(i_{\\nu} = j_{\\nu}\\)\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uc704\uc640 \ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c
\n\\[\\begin{align}
\nd(\\omega + \\eta ) &= d(f+g) dx_{i_1} \\cdots dx_{i_r}\\\\[6pt]
\n&=df\\,dx_{i_1} \\cdots dx_{i_r} + dg\\,dx_{i_1} \\cdots dx_{i_r} \\\\[6pt]
\n&= d\\omega + d\\eta
\n\\end{align}\\]
\n\ub97c \uc5bb\ub294\ub2e4. \uc774\ub85c\uc368 [1]\uacfc [2]\uac00 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n
\n\\[\\begin{align}
\nd(\\omega \\eta) &= d(fg) dx_{j_1} \\cdots dx_{j_s} \\\\[6pt]
\n&=(g\\,df + f\\,dg) dx_{j_1} \\cdots dx_{j_s} \\\\[6pt]
\n&= df\\,g\\,dx_{j_1} \\cdots dx_{j_s} + f\\,dg\\,dx_{j_1} \\cdots dx_{j_s} \\\\[6pt]
\n&= (d\\omega )\\eta + \\omega \\,d\\eta
\n\\end{align}\\]
\n\ub97c \uc5bb\ub294\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(r > 0\\)\uc774\uace0 \\(i_{\\nu} = j_{\\mu}\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(\\nu\\)\uc640 \\(\\mu\\)\uac00 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uba71\ub4f1\ubc95\uce59\uc5d0 \uc758\ud558\uc5ec
\n\\[\\omega \\eta = 0 = (d\\omega ) \\eta = \\omega \\,d\\eta\\]
\n\ub97c \uc5bb\ub294\ub2e4. \uadf8 \uc678\uc758 \uacbd\uc6b0, \uc989 \\(i_{\\nu}\\)\uc640 \\(j_{\\mu}\\) \uc911\uc5d0\uc11c \uc77c\uce58\ud558\ub294 \uac83\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \uacbd\uc6b0, \\(g\\)\ub294 0\ud615\uc2dd\uc774\uace0 \\(dg\\)\ub294 1\ud615\uc2dd\uc774\ubbc0\ub85c \uc815\ub9ac 1\uc758 [2]\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\nd(\\omega \\eta ) &= d \\left( fg \\,dx_{i_1} \\cdots dx_{i_r} \\,dx_{j_1} \\cdots dx_{j_s} \\right) \\\\[6pt]
\n&= \\left( g\\,df + f\\,dg ) dx_{i_1} \\cdots dx_{i_r} \\,dx_{j_1} \\cdots dx_{j_s} \\right) \\\\[6pt]
\n&= df \\,dx_{i_1} \\cdots dx_{i_r} \\,g\\,dx_{j_1} \\cdots dx_{j_s} +(-1)^{r\\cdot 1} \\,f\\,dx_{i_1} \\cdots dx_{i_r} \\,dg\\,dx_{j_1} \\cdots dx_{j_s} \\\\[6pt]
\n&= (d\\omega ) \\eta + (-1)^r \\,\\omega \\,d\\eta .
\n\\tag*{\\(\\blacksquare\\)}
\n\\end{align}\\]\n<\/p>\n<\/div>\n
\n\\[\\operatorname{curl} \\operatorname{grad} f(\\mathrm{x} ) =0\\]
\n\uc774\uac83\uc744 \uc77c\ubc18\ud654\ud558\uba74 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n
\n\\[\\omega = f\\,dx_{i_1} \\cdots dx_{i_r}\\]
\n\ub77c\uace0 \ud558\uc790. \\(r\\)\uc5d0 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc801\uc6a9\ud558\uc790. \\(r=0\\)\uc774\uace0 \\(\\omega = f\\)\uc77c \ub54c \uba71\ub4f1\ubc95\uce59\uacfc \ubc18\ub300\uce6d\ubc95\uce59, \uadf8\ub9ac\uace0 \ud074\ub808\ub85c(Clairaut)\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
\n\\[\\begin{align}
\nd^2\\,\\omega &= d\\left( \\sum_{j=1}^n \\frac{\\partial f}{\\partial x_j} dx_j \\right) \\\\[6pt]
\n&= \\sum_{j=1}^n \\sum_{k=1}^n \\frac{\\partial ^2 f}{\\partial x_k \\,\\partial x_j} dx_k\\,dx_j \\\\[6pt]
\n&= \\sum_{j < k} \\left( \\frac{\\partial ^2 f}{\\partial x_k\\, \\partial x_j} - \\frac{\\partial^2 f}{\\partial x_j \\,\\partial x_k} \\right) dx_k \\,dx_j =0\n\\end{align}\\]\n\uc744 \uc5bb\ub294\ub2e4.<\/p>\n
\n\\[\\begin{align}
\nd^2 \\,\\omega &= d(df \\,dx_k ) \\\\[6pt]
\n&=d^2 \\,f\\,dx_k – df\\,d^2\\,x_k =0
\n\\end{align}\\]
\n\uc744 \uc5bb\ub294\ub2e4.<\/p>\n\ubbf8\ubd84\ubcc0\ud658<\/h3>\n
\n\\[\\omega = \\sum f_{i_1 \\cdots i_r} \\,dx_{i_1} \\cdots dx_{i_r}\\]
\n\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(r\\)\ud615\uc2dd\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \uc784\uc758\uc758 0\ud615\uc2dd \\(f\\)\uc5d0 \ub300\ud558\uc5ec \\(\\phi^* (f) := f \\circ \\phi\\)\uc774\uace0
\n\\[\\phi^* (dx_i ) := d\\phi_i = \\sum_{j=1}^n \\frac{\\partial \\phi_i}{\\partial u_j} du_j \\quad (i=1,\\,2,\\,\\cdots,\\,m)\\]
\n\ub77c\uace0 \ud558\uc790. \uc774 \ub54c \\(\\phi\\)\uc5d0 \uc758\ud558\uc5ec \uc720\ub3c4\ub41c \\(\\omega\\)\uc758 \ubbf8\ubd84\ubcc0\ud658\uc744
\n\\[\\phi^* (\\omega ) := \\sum \\phi^* \\left( f_{i_1 \\cdots i_r} \\right) \\phi^* (dx_{i_1} ) \\cdots \\phi^* (dx_{i_r} )\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc774 \ub54c \\(\\phi^* (\\omega )\\)\ub294 \\(U\\)\uc5d0\uc11c\uc758 \\(r\\)\ud615\uc2dd\uc774 \ub41c\ub2e4.<\/p>\n<\/div>\n
\n\\(\\omega\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(C^1\\)\uae09 \\(r\\)\ud615\uc2dd\uc774\uace0 \\(\\phi\\)\uac00 \\(U\\)\uc5d0\uc11c \\(C^2\\)\uae09\uc774\uba74 \\(\\phi^* (\\omega )\\)\ub294 \\(U\\)\uc5d0\uc11c\uc758 \\(C^1\\)\uae09 \\(r\\)\ud615\uc2dd\uc774\ub2e4.<\/p>\n<\/div>\n
\n\\[(\\phi^* \\circ f)(\\mathrm{u}) = f(\\phi(\\mathrm{u}))\\]
\n\uac00 \\(U\\)\uc5d0\uc11c\uc758 \\(C^1\\)\uae09 0\ud615\uc2dd\uc774\uba70, \\(i=1,\\) \\(2,\\) \\(\\cdots,\\) \\(m\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\phi ^* (dx_i ) = \\sum_{j=1}^{n}\\frac{\\partial \\phi_i}{\\partial u_j} du_j\\]
\n\ub294 \\(U\\)\uc5d0\uc11c\uc758 \\(C^1\\)\uae09 1\ud615\uc2dd\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\omega\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(C^1\\)\uae09 \\(r\\)\ud615\uc2dd\uc77c \ub54c \\(\\phi^* (\\omega )\\)\ub294 \\(U\\)\uc5d0\uc11c\uc758 \\(C^1\\)\uae09 \\(r\\)\ud615\uc2dd\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n
\n\ubbf8\ubd84\ubcc0\ud658\uc740 \uc120\ud615 \ud568\uc218\uc774\ub2e4. \uc989 \\(\\omega\\)\uc640 \\(\\eta\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(r\\)\ud615\uc2dd\uc774\uace0 \\(\\phi\\)\uac00 \\(U\\)\uc5d0\uc11c \\(C^1\\)\uae09\uc774\uba74
\n\\[\\phi^* (\\omega + \\eta ) = \\phi^* (\\omega ) + \\phi^* (\\eta )\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n
\n\\[\\omega = f\\,dx_{i_1} \\cdots dx_{i_r},\\\\[6pt]
\n\\eta = g\\,dx_{i_1}\\cdots dx_{i_r}\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\phi^* (\\omega + \\eta ) &= \\phi^* ((f+g) x_{i_1} \\cdots dx_{i_r} ) \\\\[6pt]
\n&= (f\\circ \\phi + g \\circ \\phi )\\phi^* ( dx_{i_1} ) \\\\[6pt]
\n&= \\phi^* (\\omega ) + \\phi^* (\\eta ).
\n\\tag*{\\(\\blacksquare\\)}
\n\\end{align}\\]\n<\/p>\n<\/div>\n
\n\ubbf8\ubd84\ubcc0\ud658\uc740 \ubbf8\ubd84\ud615\uc2dd\uc758 \uacf1\uc744 \ubcf4\uc874\ud55c\ub2e4. \uc989 \\(\\omega\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(r\\)\ud615\uc2dd\uc774\uace0 \\(\\eta\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(s\\)\ud615\uc2dd\uc774\uba70 \\(\\phi\\)\uac00 \\(U\\)\uc5d0\uc11c \\(C^1\\)\uae09\uc77c \ub54c\\[\\phi^* (\\omega \\eta ) = \\phi^* (\\omega ) \\,\\phi^* (\\eta )\\]\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n
\n\\[\\omega = f\\,dx_{i_1} \\cdots dx_{i_r},\\\\[6pt]
\n\\eta = g\\,dx_{i_1}\\cdots dx_{i_r}\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\phi^* (\\omega \\eta ) &=
\n\\phi^* ((fg) \\,dx_{i_1} \\cdots dx_{i_r} \\, dx_{j_1} \\cdots dx_{j_s} )\\\\[6pt]
\n&= (f\\circ \\phi )(g \\circ \\phi ) \\phi^* (dx_{i_1} ) \\cdots \\phi^* (dx_{i_r} ) \\phi^* (dx_{j_1} ) \\cdots \\phi^* (dx_{j_s}) \\\\[6pt]
\n&= \\phi^* (\\omega )\\, \\phi^* (\\eta ).
\n\\tag*{\\(\\blacksquare\\)}
\n\\end{align}\\]\n<\/p>\n<\/div>\n
\n\ubbf8\ubd84\ubcc0\ud658\uacfc \uc678\ubbf8\ubd84\uc740 \uc704\uce58\ub97c \uc11c\ub85c \uad50\ud658\ud560 \uc218 \uc788\ub2e4. \uc989 \\(\\omega\\)\uac00 \\(V\\)\uc5d0\uc11c\uc758 \\(C^1\\)\uae09 \\(r\\)\ud615\uc2dd\uc774\uace0 \\(\\phi\\)\uac00 \\(U\\)\uc5d0\uc11c \\(C^2\\)\uae09\uc774\uba74\\[\\phi^* (d\\omega ) = d(\\phi^* (\\omega ))\\]\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n
\n\\[\\begin{align}
\n\\phi^* (d\\omega ) &= \\phi^* \\left( \\sum_{k=1}^m \\frac{\\partial f}{\\partial x_k} dx_k \\right)\\\\[6pt]
\n&=\\sum_{k=1}^{m} \\phi^* \\left( \\frac{\\partial f}{\\partial x_k}\\right)\\phi^* (dx_k )\\\\[6pt]
\n&=\\sum_{k=1}^{m}\\left( \\frac{\\partial f}{\\partial x_k} \\circ \\phi \\right) \\sum_{j=1}^{n} \\frac{\\partial \\phi_k}{\\partial u_j} du_j \\\\[6pt]
\n&=\\sum_{j=1}^{n} \\frac{\\partial (f\\circ \\phi)}{\\partial u_j} du_j \\\\[6pt]
\n&=d(f\\circ\\phi ) = d(\\phi^* (\\omega )).
\n\\end{align}\\]
\n\ub2e4\uc74c\uc73c\ub85c \\(r=1\\)\uc774\uace0 \\(\\omega = f\\,dx_j\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ucc38\uace0 4\uc640 \\(r=0\\)\uc778 \uacbd\uc6b0\uc758 \ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\phi^* (d\\omega ) &= \\phi^* (df\\,dx_k ) \\\\[6pt]
\n&=\\phi^* (df) \\phi^* (dx_k )\\\\[6pt]
\n&= d(f\\circ \\phi ) d\\phi_k .
\n\\end{align}\\]
\n\ud55c\ud3b8
\n\\[\\phi^* (\\omega ) = (f \\circ \\phi )\\phi^* (dx_k ) = (f \\circ \\phi ) d\\phi_k\\]
\n\uc774\ubbc0\ub85c \uc815\ub9ac 3\uc758 [3]\uacfc \uba71\ub4f1\ubc95\uce59, \uc774\uacc4\uc678\ubbf8\ubd84\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\nd(\\phi^* (\\omega )) &= d(f\\circ \\phi ) d\\phi_k (f\\circ \\phi )d^2 \\,\\phi_k \\\\[6pt]
\n&= d(f\\circ \\phi)d\\phi _k .
\n\\end{align}\\]
\n\ub530\ub77c\uc11c \\(\\omega\\)\uac00 1\ud615\uc2dd\uc77c \ub54c\uc5d0\ub3c4 \uc815\ub9ac\uc758 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n
\n\\[\\omega = \\sum f_{i_1 \\cdots i_n} \\,dx_{i_1} \\cdots dx_{i_n}\\]
\n\uc774 \\(V\\)\uc5d0\uc11c\uc758 \\(n\\)\ud615\uc2dd\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\phi^* (\\omega) = \\sum \\left( f_{i_1 \\cdots i_n} \\circ \\phi \\right) \\frac{\\partial ( \\phi_{i_1} ,\\,\\cdots,\\,\\phi_{i_n})}{\\partial (u_1 ,\\,\\cdots,\\,u_n )} du_1 \\cdots du_n .\\tag{5}\\]\n<\/p>\n<\/div>\n
\n\\[\\phi^* (\\omega ) = \\phi^* (f) \\phi^* (dx_j ) = (f\\circ \\phi) \\phi \\prime \\,du\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \\(n > 1\\)\uc774\uace0 \\[\\omega = f\\,dx_{i_1} \\cdots dx_{i_n}\\]\uc778 \uacbd\uc6b0\uc5d0\ub294 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\phi^* (\\omega ) &= \\phi^* (f) \\phi^* (dx_{i_1} ) \\cdots \\phi^* (dx_{i_n} ) \\\\[6pt]
\n&= (f \\circ \\phi ) \\left( \\sum_{k=1}^n \\frac{\\partial \\phi_{i_1}}{\\partial u_k} du_k \\right) \\cdots \\left( \\sum_{k=1}^n \\frac{\\partial \\phi_{i_n}}{\\partial u_k} du_k \\right)\\\\[6pt]
\n&= (f \\circ \\phi ) \\frac{\\partial ( \\phi_{i_1} ,\\, \\cdots ,\\, \\phi_{i_n})}{\\partial (u_1 ,\\,\\cdots,\\,u_n )} du_1 \\cdots du_n .
\n\\tag*{\\(\\blacksquare\\)}
\n\\end{align}\\]\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"