{"id":2529,"date":"2019-03-05T14:33:38","date_gmt":"2019-03-05T05:33:38","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=2529"},"modified":"2025-10-20T15:21:13","modified_gmt":"2025-10-20T06:21:13","slug":"differentiation-of-abstract-functions","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/differentiation-of-abstract-functions\/","title":{"rendered":"\ucd94\uc0c1\uacf5\uac04\uc5d0\uc11c\uc758 \ubbf8\ubd84"},"content":{"rendered":"<p><!-- Differentiation of Abstract Functions --><\/p>\n<p>\uc774 \uae00\uc5d0\uc11c\ub294 \ucd94\uc0c1\uacf5\uac04\uc5d0\uc11c\uc758 \ubbf8\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<h3>\ub3c4\ud568\uc218\uc758 \uc815\uc758<\/h3>\n<p>\\(E\\)\ub97c \ub178\ub984\ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uace0, \\(K\\)\ub97c \ub2eb\ud78c\uad6c\uac04 \\([0,\\,1]\\)\uc774\ub77c\uace0 \ud558\uc790. \\(K\\)\uc5d0\uc11c \\(E\\)\ub85c\uc758 \ud568\uc218 \\(x = x(t)\\)\ub97c \uc0dd\uac01\ud558\uc790. (\uc120\ud615\uc778 \uacbd\uc6b0\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc77c\ubc18\uc801\uc778 \ud568\uc218\ub97c \uc0dd\uac01\ud558\uc790.) \uc55e\uc73c\ub85c \uc774\ub7ec\ud55c \ud568\uc218\ub97c \uad6c\uac04 \\([0,\\,1]\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c <span class=\"defined\">\ucd94\uc0c1\ud654\ub41c \ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub97c \uac83\uc774\ub2e4.<\/p>\n<p>\uc774\ub7ec\ud55c \ud568\uc218\uc5d0 \ub300\ud558\uc5ec, \ud574\uc11d\ud559\uc758 \uae30\ubcf8\uc801\uc778 \uc5f0\uc0b0\uc744 \uc815\uc758\ud558\uace0 \uadf8 \uc131\uc9c8\uc744 \uc720\ub3c4\ud558\uc790.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1.<\/span><br \/>\n\\(x\\in E\\)\uc774\uace0 \\(t\\in [0,\\,1]\\)\uc778 \ud568\uc218 \\(x = x(t)\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc774 \ud568\uc218\uc758 <span class=\"defined\">\ub3c4\ud568\uc218 \\(x &#8216; (t)\\)<\/span>\ub97c<br \/>\n\\[x &#8216; (t) := \\frac{d}{dt} x(t) := \\lim_{\\varDelta t \\to 0} \\frac{1}{\\varDelta t}[x(t+\\varDelta t) &#8211; x(t)]\\tag{1}\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \ubb3c\ub860, (1)\uc758 \uc6b0\ubcc0\uc758 \uadf9\ud55c\uc774 \uc874\uc7ac\ud560 \ub54c\ub9cc \uc815\uc758\ub41c\ub2e4.<\/p>\n<\/div>\n<p>\ub3c4\ud568\uc218\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[x &#8216; (t) = \\frac{1}{\\varDelta t}[x(t+\\varDelta t) &#8211; x(t)] + \\alpha (t,\\, \\varDelta t).\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\varDelta t \\to 0\\)\uc77c \ub54c \\(\\alpha (t,\\,\\varDelta t) \\to 0\\)\uc774\ub2e4.<br \/>\n\ub530\ub77c\uc11c \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[x(t+\\varDelta t) &#8211; x(t) = x &#8216; (t) \\varDelta t &#8211; \\alpha (t,\\,\\varDelta t) \\varDelta t .\\tag{2}\\]<br \/>\n\\(\\varDelta t \\to 0\\)\uc77c \ub54c, \ub4f1\uc2dd (2)\uc758 \uc6b0\ubcc0\uc740 \\(0\\)\uc73c\ub85c \uc218\ub834\ud55c\ub2e4. \ub530\ub77c\uc11c \\(x(t)\\)\uac00 \\(t\\)\uc5d0 \uad00\ud55c \ub3c4\ud568\uc218\ub97c \uac00\uc9c0\uba74, \\(x(t)\\)\ub294 \uc810 \\(t\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<p>\ub3c4\ud568\uc218\uc758 \ub300\ud45c\uc801\uc778 \uc608\ub85c\uc11c \ubca1\ud130\ud568\uc218 \\(x_n (t)\\)\uc758 \ub3c4\ud568\uc218\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \\(t\\)\ub97c \uc2dc\uac04\uc73c\ub85c \ud574\uc11d\ud558\uba74 \\(x_n &#8216; (t)\\)\ub294 \uc18d\ub3c4 \ubca1\ud130\uc774\ub2e4. \uc989, \uc774\uc0b0\uc5ed\ud559\uc5d0\uc11c \uc5f0\uc18d\uc5ed\ud559\uc73c\ub85c\uc758 \uc804\ud658\uc740 \\(n\\)\ucc28\uc6d0 \ubca1\ud130 \\(x_n (t)\\)\uc5d0\uc11c \uc2dc\uac04 \\((t)\\)\uc5d0 \uc758\uc874\ud558\ub294 \ud2b9\uc815 \ud568\uc218\uacf5\uac04\uc758 \uc6d0\uc18c \\(x(t)\\)\ub85c\uc758 \uc774\ud589\uc5d0 \ub300\uc751\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(x &#8216; (t)\\)\ub294 \uc18d\ub3c4\uc774\ub2e4.<\/p>\n<p>1\ucc28\uc6d0 \ubb38\uc81c(\ub048, \ub9c9\ub300 \ub4f1)\uc5d0\uc11c\ub294 \\(x(t)\\)\uc640 \\(x &#8216; (t)\\)\ub97c \uacf5\uac04 \\(C[0,\\,1]\\)\uc758 \uc6d0\uc18c\ub85c \uc0dd\uac01\ud55c\ub2e4.<\/p>\n<p>\ubbf8\ubd84\uc758 \uc815\uc758\uc640 \uadf9\ud55c\uc758 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud558\uba74 \ub2e4\uc74c \uc131\uc9c8\uc744 \ub04c\uc5b4\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\([x(t) + y(t)] &#8216; = x &#8216; (t) + y &#8216; (t).\\)<\/li>\n<li>\ubaa8\ub4e0 \uc0c1\uc218 \\(\\lambda\\)\uc5d0 \ub300\ud558\uc5ec \\([\\lambda x(t) ] &#8216; = \\lambda x &#8216; (t)\\)\uc774\ub2e4.<\/li>\n<li>\uc6d0\uc18c \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \uc6d0\uc18c \\(y\\in E\\)\uc640\uc758 \uc88c\uce21 \uacf1\uc148(\ub610\ub294 \uc6b0\uce21 \uacf1\uc148)\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \uc774\uac83\uc774 \uc5f0\uc18d\uc774\uace0 \ub367\uc148\uc5d0 \uad00\ud574 \ubd84\ubc30\ubc95\uce59\uc744 \ub9cc\uc871\ud558\uba70 \uc2a4\uce7c\ub77c \uacf1\uc148\uacfc \uac00\ud658\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[[yx(t)] &#8216; = yx &#8216; (t) . \\tag{3}\\]<br \/>\n\uc989, \uc0c1\uc218\uacf1\uc740 \ubbf8\ubd84 \uae30\ud638 \ubc16\uc73c\ub85c \uaebc\ub0bc \uc218 \uc788\ub2e4.<\/li>\n<\/ol>\n<p>\uc131\uc9c8 [1]\uacfc [2]\ub294 \uba85\ubc31\ud558\ub2e4. \uc131\uc9c8 [3]\uc740 \uc88c\uce21 \uacf1\uc148\uc758 \uc5f0\uc18d\uc131\uacfc \ubd84\ubc30\ubc95\uce59\uc73c\ub85c\ubd80\ud130 \ub530\ub77c \ub098\uc628\ub2e4. \uc989<br \/>\n\\[\\begin{align}<br \/>\nyx &#8216; (t)<br \/>\n&#038;= y \\lim_{\\varDelta t \\to 0} \\frac{1}{\\varDelta t} [x(t+ \\varDelta t) &#8211; x(t)] \\\\[6pt]<br \/>\n&#038;= \\lim_{\\varDelta t \\to 0} y\\left\\{ \\frac{1}{\\varDelta t} [ x(t+ \\varDelta t) &#8211; x(t)] \\right\\} \\\\[6pt]<br \/>\n&#038;= \\lim_{\\varDelta t \\to 0} \\frac{1}{\\varDelta t}[yx(t+\\varDelta t) &#8211; yx(t) ] \\\\[6pt]<br \/>\n&#038;= \\frac{d}{dt} [yx(t)].<br \/>\n\\end{align}\\]<br \/>\n\ub9c8\ucc2c\uac00\uc9c0\ub85c, \uc6b0\uce21 \uacf1\uc148\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[[x(t) y] &#8216; = x &#8216; (t) y \\tag{3a}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><br \/>\n\\(x\\in E\\)\uc778 \\(x = x(t)\\)\ub77c\uace0 \ud558\uace0, \\(A\\)\ub97c \\((E \\to E_1 )\\)\uc778 \uc5f0\uc0b0\uc790\ub77c\uace0 \ud558\uc790.<br \/>\n\\[[ Ax(t) ] &#8216; = Ax &#8216; (t) . \\tag{4}\\]<br \/>\n\ub9cc\uc57d \\(A = A(t) \\in (E \\to E_1 )\\)\uc774\uace0 \\(x\\in E\\)\uc774\uba74,<br \/>\n\\[[A(t)x] &#8216; = A &#8216; (t) x . \\tag{4a}\\]<br \/>\n\ud2b9\ud788, \uc120\ud615\ubc94\ud568\uc218 \\(f\\in \\overline{E}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n\\left\\{ f[x(t)] \\right\\} &#8216; &#038;= f[x &#8216; (t)] ,\\tag{5} \\\\[6pt]<br \/>\n\\left\\{ f(t)(x) \\right\\} &#8216; &#038;= f &#8216; (t)(x) .\\tag{5a}<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<h3>\uace0\uacc4 \ub3c4\ud568\uc218<\/h3>\n<p>\uc774\uc81c \uace0\uacc4 \ub3c4\ud568\uc218\ub97c \uc815\uc758\ud558\uc790. \uc2e4\ud568\uc218\uc758 \uacbd\uc6b0\ucc98\ub7fc, \ucd94\uc0c1\ud654\ub41c \ud568\uc218 \\(x=x(t)\\)\uc758 \\(n\\)\uacc4 \ub3c4\ud568\uc218\uc5d0 \ub300\ud55c \ub450 \uac00\uc9c0 \uc815\uc758\ub97c \uc81c\uc2dc\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[ {\\overline{\\varDelta}}_{\\varDelta t} ^n x(t)<br \/>\n=\\sum_{k=0}^n (-1)^{n-k} \\binom{n}{k} x(t+k \\varDelta t)\\]<br \/>\n\uc740 \uc810 \\(t\\)\uc5d0\uc11c \\(x(t)\\)\uc758 \\(n\\)\uacc4 \ucc28\ubd84\uc774\ub2e4. \ub610\ud55c<br \/>\n\\[\\varDelta _{\\varDelta t}^n x(t)<br \/>\n={\\overline{\\varDelta}}_{\\varDelta t}^n x \\left( t- \\frac{n}{2} \\varDelta t \\right)\\]<br \/>\n\ub97c <span class=\"defined\">\\(n\\)\uacc4 \uc911\uc2ec\ucc28\ubd84<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc774\uc81c \uc810 \\(t\\)\uc5d0\uc11c \ud568\uc218 \\(x(t)\\)\uc758 <span class=\"defined\">\\(n\\)\uacc4 \ucc28\ubd84\ub3c4\ud568\uc218<\/span>\ub97c \ub2e4\uc74c \uc2dd\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[x^{[n]}(t) = \\lim_{\\varDelta t \\to 0} \\frac{1}{(\\varDelta t)^n} \\varDelta_{\\varDelta t}^{n} x(t). \\tag{6}\\]<br \/>\n\ubb3c\ub860 \uc6b0\ubcc0\uc758 \uadf9\ud55c\uc774 \uc874\uc7ac\ud560 \ub54c\ub9cc \uc815\uc758\ub41c\ub2e4.<\/p>\n<p>\ub9cc\uc57d (6)\uc5d0\uc11c\uc758 \uadf9\ud55c\uc774 \ubaa8\ub4e0 \uc810 \\(t\\)\uc5d0\uc11c \uade0\ub4f1\ud558\uac8c \uc218\ub834\ud558\uba74, \\(x^{[n]}(t)\\)\ub97c <span class=\"defined\">\uade0\ub4f1 \\(n\\)\uacc4 \ucc28\ubd84\ub3c4\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(n\\)\uacc4 \ub3c4\ud568\uc218 \\(x^{(n)}(t)\\)\ub294 \\(n\\)\ubc88 \ubbf8\ubd84\ud55c \ud568\uc218\ub85c \uc815\uc758\ub41c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nx &#8216; (t)_0 &#038;= \\frac{d}{dt} x(t),\\\\[6pt]<br \/>\nx &#8216; &#8216; (t)_0 &#038;= \\frac{d}{dt} [x &#8216; (t)_0 ] ,\\\\[6pt]<br \/>\n&#038;\\,\\,\\,\\vdots \\\\[6pt]<br \/>\nx^{(n)} (t)_0 &#038;= \\frac{d}{dt} [x^{(n-1)}(t)_0 ].<br \/>\n\\end{align}\\]\n<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span><br \/>\n\uc810 \\(t\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \uc5f0\uc18d\uc778 \\(n\\)\uacc4 \ub3c4\ud568\uc218 \\(x^{(n)}(t)_0\\)\uac00 \uc874\uc7ac\ud558\uba74, \uc774 \uadfc\ubc29\uc5d0\uc11c \uade0\ub4f1 \\(n\\)\uacc4 \ucc28\ubd84\ub3c4\ud568\uc218 \\(x^{[n]}(t)\\)\ub3c4 \uc874\uc7ac\ud558\uba70,<br \/>\n\\[x^{[n]}(t) = x^{(n)}(t)_0\\]<br \/>\n\uc774\ub2e4.<br \/>\n\uac70\uafb8\ub85c, \uc810 \\(t\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc778 \uade0\ub4f1 \ucc28\ubd84\ub3c4\ud568\uc218 \\(x^{[n]}(t)\\)\uac00 \uc874\uc7ac\ud558\uba74, \uc774 \uadfc\ubc29\uc5d0\uc11c \\(n\\)\uacc4 \ub3c4\ud568\uc218 \\(x^{(n)} (t)_0\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\uc774\ub7ec\ud55c \uba85\uc81c\ub4e4\uc740 \uce58\uc5ed\uc774 \uc2e4\uc218\ub098 \ubcf5\uc18c\uc218\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ud568\uc218\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud55c\ub2e4. \ucd94\uc0c1\ud654\ub41c \ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub3c4 \uc774 \uc131\uc9c8\uc774 \uc131\ub9bd\ud568\uc744 \ud655\uc778\ud558\ub824\uba74 \ud568\uc218\ud574\uc11d\ud559\uc5d0\uc11c \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \ubc29\ubc95\uc744 \ud65c\uc6a9\ud574\uc57c \ud55c\ub2e4. \uccab \ubc88\uc9f8 \uba85\uc81c\ub97c \ud655\uc778\ud574 \ubcf4\uc790.<\/p>\n<p>\uc784\uc758\uc758 \uc120\ud615\ubc94\ud568\uc218 \\(f\\in \\overline{E}\\)\uc5d0 \ub300\ud558\uc5ec,<br \/>\n\\[\\varphi (t) = f[x(t)]\\]<br \/>\n\ub294 \uc815\uc758\uc5ed\uacfc \uce58\uc5ed\uc774 \uc218\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ud568\uc218\uc774\ub2e4. (5)\uc5d0 \uc758\ud558\uc5ec,<br \/>\n\\[\\begin{align}<br \/>\nf[x &#8216; (t)_0 ] &#038;= \\left\\{ f [ x(t) ] \\right\\} &#8216; = \\varphi &#8216; (t), \\\\[6pt]<br \/>\nf[x &#8216; &#8216; (t)_0 ] &#038;= \\left\\{ f [ x &#8216; (t)_0 ] \\right\\} &#8216; = \\left\\{ \\varphi &#8216; (t) \\right\\} &#8216; = \\varphi &#8216; &#8216; (t) ,\\\\[6pt]<br \/>\n&#038;\\,\\,\\,\\vdots\\\\[6pt]<br \/>\nf[x^{(n)}(t)_0 ] &#038;= \\left\\{ f[x^{(n-1)}(t)_0 ] \\right\\} &#8216; = \\left\\{ \\varphi^{n-1} (t) \\right\\} &#8216; = \\varphi^{(n)} (t) .<br \/>\n\\end{align}\\]<br \/>\n\ub610\ud55c<br \/>\n\\[\\begin{align}<br \/>\nf \\left[ \\frac{1}{(\\varDelta t)_n} \\varDelta_{\\varDelta t}^{n} x(t) \\right]<br \/>\n&#038;= \\frac{1}{(\\varDelta t)^n} \\sum_{k=0}^{n} (-1)^{n-k} \\binom{n}{k} \\varphi \\left( t+ \\left( k- \\frac{n}{2} \\right) \\varDelta t \\right) \\\\[6pt]<br \/>\n&#038;= \\frac{1}{(\\varDelta t)^n} \\varDelta_{\\varDelta t}^n \\varphi (t) \\\\[6pt]<br \/>\n&#038;= \\varphi^{(n)} (t+ \\theta \\varDelta t ) \\\\[6pt]<br \/>\n&#038;=f[x^{(n)} (t+\\theta \\varDelta t)]<br \/>\n\\end{align}\\]<br \/>\n\uc774\uace0, \uc5ec\uae30\uc11c \\(-\\frac{1}{2} \\le \\theta \\le \\frac{1}{2}\\)\uc774\ub2e4. \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \\(x^{(n)} (t)_0\\)\ub294 \uc810 \\(t\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c,<br \/>\n\\[\\lVert x^{(n)} (t+ \\theta \\varDelta t)_0 &#8211; x^{(n)} (t)_0 \\rVert \\le \\epsilon_{\\varDelta t}\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\varDelta t \\to 0\\)\uc77c \ub54c \\(\\epsilon_{\\varDelta t} \\to 0\\)\uc774\uace0, \uc774\ub294 \uc810 \\(t\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \uade0\uc77c\ud558\ub2e4.<\/p>\n<p>\uc774\ub85c\ubd80\ud130<br \/>\n\\[\\left\\lvert f\\left[ \\frac{1}{(\\varDelta t)^n} \\varDelta_{\\varDelta t}^{n} x(t) \\right] &#8211; f[x^{(n)} (t)_0 ] \\right\\rvert<br \/>\n\\le \\epsilon_{\\varDelta t} \\lVert f \\rVert \\tag{7}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.<br \/>\n\ubd80\ub4f1\uc2dd (7)\uc740 \uc784\uc758\uc758 \\(f\\in \\overline{E}\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud558\ubbc0\ub85c,<br \/>\n\\[\\left\\lVert \\frac{1}{(\\varDelta t)^n} \\varDelta_{\\varDelta t}^{n} x(t) &#8211; x^{(n)} (t)_0 \\right\\rVert \\le \\epsilon_{\\varDelta t}\\]<br \/>\n\uc774\uace0, \ub530\ub77c\uc11c<br \/>\n\\[x^{(n)}(t)_0 = \\lim_{\\varDelta t \\to 0} \\frac{1}{(\\varDelta )^n} \\varDelta_{\\varDelta t}^{n} x(t)\\]<br \/>\n\uc774\ub2e4.<br \/>\n\uc774 \uacbd\uc6b0, \uc218\ub834\uc740 \ubaa8\ub4e0 \uc810 \\(t\\)\uc5d0 \ub300\ud55c \uade0\ub4f1\uc218\ub834\uc774\ub2e4.<\/p>\n<p>\uc774\uac83\uc73c\ub85c \uc815\ub9ac\uc758 \uccab \ubc88\uc9f8 \uba85\uc81c\uac00 \uc99d\uba85\ub418\uc5c8\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<h3>\ud3b8\ub3c4\ud568\uc218<\/h3>\n<p>\ucd94\uc0c1\ud654\ub41c \ud568\uc218\uc758 <span class=\"defined\">\ud3b8\ub3c4\ud568\uc218<\/span> \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uc790. \uc774\ub97c \uc704\ud558\uc5ec, \uce58\uc5ed\uc774 \ub178\ub984\ubca1\ud130\uacf5\uac04 \\(E\\)\uc5d0 \ub193\uc5ec \uc788\ub294 \\(n\\)\uac1c \uc2e4\ubcc0\uc218 \\(t_1 ,\\) \\(t_2 ,\\) \\(\\cdots ,\\) \\(t_n\\)\uc758 \ud568\uc218\ub97c \uc0dd\uac01\ud55c\ub2e4.<br \/>\n\\[y = f(t_1 ,\\, t_2 ,\\, \\cdots,\\, t_n ) \\in E .\\]<br \/>\n\\(t_1 ,\\) \\(t_2 ,\\) \\(\\cdots ,\\) \\(t_n\\)\uc744 \\(n\\)\ucc28\uc6d0 \ubca1\ud130<br \/>\n\\[T = \\sum_{i=1}^{n} t_i e_i\\]<br \/>\n\uc758 \uc131\ubd84\uc73c\ub85c \ud574\uc11d\ud560 \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \\(e_i\\)\ub294 \uc815\uaddc\uc9c1\uad50 \uae30\uc800\ubca1\ud130, \uc989 \uc11c\ub85c \uc9c1\uad50\ud558\ub294 \\(n\\)\ucc28\uc6d0 \ub2e8\uc704\ubca1\ud130\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \uc810 \\(T_0 = \\sum_{i=1}^{n} t_i^{(0)} e_i\\)\uc5d0\uc11c <span class=\"defined\">\\(n\\)\uacc4 \ud3b8\ucc28\ubd84\ub3c4\ud568\uc218<\/span><br \/>\n\\[\\frac{\\partial ^n}{\\partial t_1 \\,\\partial t_2 \\cdots \\partial t_n} f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n )\\]<br \/>\n\ub97c \uc815\uc758\ud558\uc790.<\/p>\n<p>\uc774\ub97c \uc704\ud558\uc5ec, \\(n\\)\uacc4 \ud3b8\ucc28\ubd84\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\uc131\ud55c\ub2e4.<br \/>\n\\[\\varDelta_{t_1 , \\cdots, t_n}^n f(T_0 ) =<br \/>\n\\sum_{i_1 ,\\cdots, i_n} (-1)^{n-k} f[T_0 + \\varDelta t(e_{i_1} + \\cdots + e_{i_k} )].\\]<br \/>\n\uc5ec\uae30\uc11c \\(0\\le i_1 < i_2 < \\cdots < i_k \\le n\\)\uc774\uace0, \\((i_1 ,\\, i_2 ,\\, \\cdots ,\\, i_k )\\)\ub294 \\((1,\\,2,\\,\\cdots,\\,n )\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub2e4. \ud569\uc740 \uc774\ub7ec\ud55c \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc5d0 \ub300\ud558\uc5ec \ucde8\ud55c\ub2e4. \uacf5\uc9d1\ud569\uc5d0 \ub300\ud574\uc11c\ub294 \\(k=0\\)\uc774\uace0 \\(Y_0\\)\ub97c \uc778\uc218\ub85c \ub454\ub2e4. \uc810 \\(T_0\\)\uc5d0\uc11c\uc758 \\(n\\)\uacc4 \uc911\uc2ec \ud3b8\ucc28\ubd84\n\\[\\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(T_0 )\\]\n\uc740 \uc810\n\\[T_0 ' = T_0 - \\frac{1}{2} \\varDelta t \\sum_{i=1}^{n} e_i\\]\n\uc5d0\uc11c\uc758 \\(n\\)\uacc4 \ud3b8\ucc28\ubd84\uc774\ub2e4.\n\ub530\ub77c\uc11c\n\\[\\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^{n} f(T_0 )\n= {\\overline{\\varDelta}}_{t_1 ,\\,\\cdots,\\,t_n ;\\, \\varDelta t}^n\nf\\left( T_0 - \\frac{1}{2} \\varDelta t \\sum_{i=1}^{n} e_i \\right).\\]\n\uadf8\ub7ec\uba74\n\\[\\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\,t_n ;\\, \\varDelta t}^n f(T_0 )\\]\n\uc758 \\(\\varDelta t \\to 0\\)\uc77c \ub54c\uc758 \uadf9\ud55c\uc744 \uc810 \\(T_0 = (t_1^{(0)} ,\\, t_2^{(0)} ,\\, \\cdots ,\\, t_n^{(0)})\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 <span class=\"defined\">\\(n\\)\uacc4 \ud3b8\ucc28\ubd84\ub3c4\ud568\uc218<\/span><br \/>\n\\[\\frac{\\partial ^n}{\\partial t_1 \\, \\partial t_2 \\cdots \\partial t_n} f(t_1^{(0)} ,\\, t_2^{(0)} ,\\, \\cdots ,\\, t_n^{(0)})\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud55c\ub2e4. \ubb3c\ub860, \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\ub294 \uacbd\uc6b0\uc5d0 \uc815\uc758\ub41c\ub2e4.<\/p>\n<p>\uc774\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c, <span class=\"defined\">\\(n\\)\uacc4 \ud3b8\ub3c4\ud568\uc218<\/span>\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc774\uac83\uc740 \ud568\uc218 \\(f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n )\\)\uc744 \\(t_{k_n} ,\\) \\(t_{k_{n-1}} ,\\) \uadf8\ub9ac\uace0 \ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(t_{k_1}\\)\uc5d0 \uad00\ud558\uc5ec \uc5f0\uc18d\uc801\uc73c\ub85c \ubbf8\ubd84\ud55c \uacb0\uacfc\uc774\ub2e4. \uc5ec\uae30\uc11c \\(k_1 ,\\) \\(k_2 ,\\) \\(\\cdots ,\\) \\(k_n\\)\uc740 \uc9c0\ud45c \\(1,\\) \\(2,\\) \\(\\cdots,\\) \\(n\\)\uc758 \uc784\uc758\uc758 \uc21c\uc5f4\uc774\ub2e4. \ubb3c\ub860, \ub3c4\ud568\uc218<br \/>\n\\[\\begin{align}<br \/>\n\\frac{\\partial}{\\partial t_{k_n}} &#038; f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n ) ,<br \/>\n\\\\[6pt]<br \/>\n\\frac{\\partial}{\\partial t_{k_{n-1}}} &#038; \\left( \\frac{\\partial}{\\partial t_{k_n}} f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n ) \\right) ,\\\\[6pt]<br \/>\n&#038;\\vdots<br \/>\n \\\\[6pt]<br \/>\n\\frac{\\partial}{\\partial t_{k_i}} &#038; \\left\\{<br \/>\n\\frac{\\partial}{\\partial t_{k_{i+1}}}  \\left( \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n ) \\right)<br \/>\n\\right\\}\\\\[6pt]<br \/>\n&#038;\\vdots<br \/>\n\\end{align}\\]<br \/>\n\uac00 \\(T_0\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \uc5f0\uc18d\uc801\uc73c\ub85c \uc874\uc7ac\ud55c\ub2e4\uace0 \uac00\uc815\ud574\uc57c \ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2.<\/span><br \/>\n\uc810 \\(T_0 = (t_1 ^{(0)} ,\\, t_2 ^{(0)} ,\\, \\cdots ,\\, t_n^{(0)})\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 \\(n\\)\uacc4 \ud3b8\ub3c4\ud568\uc218\uac00 \uc874\uc7ac\ud558\uace0, \uc774 \ub3c4\ud568\uc218\uac00 \\(T_0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \\(T_0\\)\uc5d0\uc11c \\(n\\)\uacc4 \ud3b8\ucc28\ubd84\ub3c4\ud568\uc218\ub3c4 \uc874\uc7ac\ud558\uba70, \ub450 \ub3c4\ud568\uc218\ub294 \uc77c\uce58\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof-eng\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(L\\)\uc744 \\(\\overline{E}\\)\uc758 \uc784\uc758\uc758 \uc120\ud615\ubc94\ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[L[f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n )] = L f(t_i )\\]<br \/>\n\ub294 \uce58\uc5ed\uc774 \uc2e4\uc218 \ub610\ub294 \ubcf5\uc18c\uc218\ub85c \uc774\ub8e8\uc5b4\uc9c4 \\(t_1 ,\\) \\(t_2 ,\\) \\(\\cdots ,\\) \\(t_n\\)\uc758 \ud568\uc218\uc774\ub2e4. \uc9c0\ud45c \\(1,\\) \\(2,\\) \\(\\cdots ,\\) \\(n\\)\uc758 \ubaa8\ub4e0 \uc21c\uc5f4 \\(k_1 ,\\) \\(k_2 ,\\) \\(\\cdots ,\\) \\(k_n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n L f(t_i^{(0)} )<br \/>\n= \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} L f(t_i^{(0)} + \\theta_i \\varDelta t ) \\right\\}\\]<br \/>\n\uc774\uace0,<br \/>\n\\[\\begin{align}<br \/>\nL \\left[ \\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i ^{(0)}) \\right]<br \/>\n&#038;= \\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n L f(t_i^{(0)} ) \\\\[6pt]<br \/>\n&#038;= \\frac{\\partial}{\\partial t_{k_1}} \\left\\{<br \/>\n\\cdots \\frac{\\partial}{\\partial t_{k_n}} L f(t_i ^{(0)} + \\theta_i \\varDelta t ) \\right\\} \\\\[6pt]<br \/>\n&#038;= L \\left[ \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} + \\theta_i \\varDelta t ) \\right\\} \\right]<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uc120\ud615\ubc94\ud568\uc218 \\(L\\)\uc740 \ub3c4\ud568\uc218 \uae30\ud638 \ubc16\uc73c\ub85c \uaebc\ub0bc \uc218 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p>\uac00\uc815\uc5d0 \uc758\ud558\uc5ec, \\(f(t_i )\\)\uc758 \\(n\\)\uacc4 \ud3b8\ub3c4\ud568\uc218\ub294 \uc810 \\((t_i ^{(0)} )\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \\(\\epsilon_{\\varDelta t} > 0\\)\uc5d0 \ub300\ud558\uc5ec,<br \/>\n\\[\\left\\lVert \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} + \\theta_i \\varDelta t ) \\right\\} &#8211; \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} )\\right\\} \\right\\rVert \\le \\epsilon_{\\varDelta t}\\]<br \/>\n\uac00 \ucda9\ubd84\ud788 \uc791\uc740 \\(\\lvert \\varDelta t \\rvert\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[\\begin{align}<br \/>\n\\,&#038;<br \/>\n\\left\\lvert L \\left[ \\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i^{(0)}) \\right] &#8211; L \\left[ \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} ) \\right\\} \\right] \\right\\rvert \\\\[6pt]<br \/>\n=&#038; \\left\\lvert L \\left[ \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} + \\theta_i \\varDelta t ) \\right\\} \\right] &#8211; L \\left[ \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} ) \\right\\} \\right] \\right\\rvert \\\\[6pt]<br \/>\n\\le &#038; \\lVert L \\rVert<br \/>\n\\left\\lVert \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} + \\theta_i \\varDelta t ) \\right\\} &#8211; \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} ) \\right\\} \\right\\rVert\\\\[6pt]<br \/>\n\\le &#038; \\lVert L \\rVert \\epsilon_{\\varDelta t} .<br \/>\n\\end{align}\\]<br \/>\n\uc774 \ubd80\ub4f1\uc2dd\uc740 \ubaa8\ub4e0 \\(L \\in \\overline{E}\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud558\ubbc0\ub85c,<br \/>\n\\[\\left\\lVert \\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i ^{(0)}) &#8211; \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} ) \\right\\} \\right\\rVert \\le \\epsilon_{\\varDelta t}\\]<br \/>\n\uc774\ub2e4.<br \/>\n\ub530\ub77c\uc11c<br \/>\n\\[\\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i ^{(0)})<br \/>\n\\,\\to\\,<br \/>\n\\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} ) \\right\\}\\]<br \/>\n\uac00 \\(\\varDelta t \\to 0\\)\uc77c \ub54c \uc131\ub9bd\ud558\uace0, \uc2dd<br \/>\n\\[\\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i^{(0)})\\]<br \/>\n\uc758 \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uba70 \uadf8\uac83\uc774 \\(n\\)\uacc4 \ud3b8\ub3c4\ud568\uc218\uc640 \uac19\uc74c\uc744 \uc99d\uba85\ud558\uc600\ub2e4.<br \/>\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac.<\/span><br \/>\n\uc9c0\ud45c \\(1,\\) \\(2,\\) \\(\\cdots,\\) \\(n\\)\uc758 \uc11c\ub85c \ub2e4\ub978 \uc21c\uc5f4\uc5d0 \ub300\uc751\ud558\ub294 \ub450 \\(n\\)\uacc4 \ud3b8\ub3c4\ud568\uc218\ub294, \ub458 \ub2e4 \uc5f0\uc18d\uc778 \uc810\uc5d0\uc11c \uc77c\uce58\ud55c\ub2e4. \uc774\uac83\uc740 \\(n\\)\uacc4 \ud3b8\ub3c4\ud568\uc218\uac00 \ubbf8\ubd84\uc758 \uc21c\uc11c\uc5d0 \uc758\uc874\ud558\uc9c0 \uc54a\ub294\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p>\uc774 \uacbd\uc6b0 \ub4f1\uc2dd<br \/>\n\\[\\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i) \\right\\}<br \/>\n=<br \/>\n\\lim_{\\varDelta t \\to 0} \\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i )\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 \uc6b0\ubcc0\uc740 \uc21c\uc5f4 \\(k_1 ,\\) \\(\\cdots ,\\) \\(k_n\\)\uc5d0 \uc758\uc874\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<p>\uc774\ud6c4\uc5d0 \ud2b9\uc815 \uc601\uc5ed\uc5d0\uc11c \\(n\\)\uacc4 \ud3b8\ub3c4\ud568\uc218\uc5d0 \ub300\ud574 \uc5b8\uae09\ud560 \ub54c, \uadf8\uac83\uc774 \uc774 \uc601\uc5ed\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \uac00\uc815\ud560 \uac83\uc774\ub2e4. \ub530\ub77c\uc11c \ub450 \uc815\uc758\uc5d0 \uc758\ud55c \uacb0\uacfc\ub294 \uac19\uc740 \ud568\uc218\uac00 \ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774 \ub3c4\ud568\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc740 \uae30\ud638\ub85c \ud45c\uc2dc\ud558\uc790.<br \/>\n\\[\\frac{\\partial ^n}{\\partial t_1 \\,\\partial t_2 \\cdots \\partial t_n} f(t_i ).\\]<br \/>\n\\(h_i\\)\uac00 \uacf5\uac04 \\(E\\)\uc758 \uc784\uc758\uc758 \uc6d0\uc18c\uc774\uace0,<br \/>\n\\[y = f \\left( \\sum_{i=1}^{n} t_i h_i \\right) \\in E \\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uba74, \\(y\\)\ub294 \\(n\\)\uac1c\uc758 \ub9e4\uac1c\ubcc0\uc218 \\(t_1 ,\\) \\(t_2 ,\\) \\(\\cdots ,\\) \\(t_n\\)\uc758 \ud568\uc218\uc774\ub2e4. \ud568\uc218<br \/>\n\\[f \\left( \\sum_{i=1}^n t_i h_i \\right)\\]<br \/>\n\ub294 \uc784\uc758\uc758 \\(h_1 ,\\) \\(\\cdots ,\\) \\(h_n\\)\uc5d0 \ub300\ud558\uc5ec \\(n\\)\uacc4 \ud3b8\ub3c4\ud568\uc218<br \/>\n\\[\\left[ \\frac{\\partial ^n}{\\partial t_1 \\, \\partial t_2 \\cdots \\partial t_n} f\\left( \\sum_{i=1}^{n} t_i h_i \\right) \\right]_{t_1 = \\cdots = t_n =0 }\\]<br \/>\n\ub97c \uac16\ub294\ub2e4.<br \/>\n\ub9cc\uc57d \\(h_1 = \\cdots = h_n\\)\uc774\uba74,<br \/>\n\\[\\left[ \\frac{\\partial ^n}{\\partial t_1 \\, \\partial t_2 \\cdots \\partial t_n} f \\left( x+\\sum_{i=1}^{n} t_i h_i \\right) \\right]_{t_1 = \\cdots = t_n =0}<br \/>\n= \\frac{d^n}{dt^n} f(x+th)_{t=0}\\]<br \/>\n\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.<\/p>\n<h3>\ucc38\uace0\ubb38\ud5cc<\/h3>\n<ul class=\"bracket\">\n<li>\ub958\uc2a4\ud14c\ub974\ub2c8\ud06c(L. A. Liusternik)\uc640 \uc18c\ubcfc\ub808\ud504(V. J. Sobolev), \u300e\ud568\uc218\ud574\uc11d\ud559(Elements of Functional Analysis)\u300f (\uc564\uc11c\ub2c8 \ub77c\ubc14\ub808(Anthony E. Labarre) \ubc88\uc5ed), \ud504\ub808\ub354\ub9ad \uc6c5\uac00\ub974 \ucd9c\ud310\uc0ac(Frederick Ungar Publishing Company, New York), 168-173\ucabd.\n<\/li>\n<\/ul>\n<p><!--\n\n\n\n<p>In this post we will study differentiation in abstract spaces.<\/p>\n\n\n\n\n\n<h3>Definition of Derivatives<\/h3>\n\n\n\n\n\n<p>Let \\(E\\) be a normed linear space and \\(K\\) the closed interval \\([0,\\,1]\\) of the real number line. We consider an operator \\(x = x(t),\\) which need not be linear and maps \\(K\\) into \\(E.\\) In the following, we will call such an operator an <span class=\"defined\">abstract function<\/span> on the interval \\([0,\\,1].\\)<\/p>\n\n\n\n\n\n<p>For these functions, we shall define and deduce properties of the fundamental operations of analysis.<\/p>\n\n\n\n\n\n<div class=\"definition\">\n\n\n<p><span class=\"definition\">Definition 1.<\/span>\nWe consider the function \\(x = x(t)\\) with \\(x\\in E\\) and \\(t\\in [0,\\,1].\\) We define the <span class=\"defined\">derivative \\(x ' (t)\\)<\/span> by\n\\[x ' (t) := \\frac{d}{dt} x(t) := \\lim_{\\varDelta t \\to 0} \\frac{1}{\\varDelta t}[x(t+\\varDelta t) - x(t)],\\tag{1}\\]\nin case the limit in the right-hand side exists.<\/p>\n\n\n<\/div>\n\n\n\n\n\n<p>It follows that\n\\[x ' (t) = \\frac{1}{\\varDelta t}[x(t+\\varDelta t) - x(t)] + \\alpha (t,\\, \\varDelta t),\\]\nwhere \\(\\alpha (t,\\,\\varDelta t) \\to 0\\) as \\(\\varDelta t \\to 0 ,\\)\nTherefore\n\\[x(t+\\varDelta t) - x(t) = x ' (t) \\varDelta t - \\alpha (t,\\,\\varDelta t) \\varDelta t .\\tag{2}\\]\nAs \\(\\varDelta t \\to 0,\\) the right side of the equation (2) tends to \\(0.\\) Hence if \\(x(t)\\) has a derivative with respect to \\(t,\\) then \\(x(t)\\) is continuous at the point \\(t.\\)<\/p>\n\n\n\n\n\n<p>As an example we mention differentiation of a vector function \\(x_n (t).\\) If we interpret \\(t\\) as time, then \\(x_n ' (t)\\) is the velocity vector.<\/p>\n\n\n\n\n\n<p>To the translation  from the mechanics of a system of points to the mechanics of a continuum there corresponds the transition from an \\(n\\)-dimensional vector \\(x_n (t)\\) to a time \\((t)\\) dependent element \\(x(t)\\) of a certain function space. Then \\(x ' (t)\\) is the velocity.<\/p>\n\n\n\n\n\n<p>In one-dimensional problems(string, rod, etc.) we consider \\(x(t)\\) and \\(x ' (t)\\) as elements of the space \\(C[0,\\,1].\\)<\/p>\n\n\n\n\n\n<p>We easily recognize the following properties of differentiation:<\/p>\n\n\n\n\n<ol class=\"bracket\">\n\n\n<li>\\([x(t) + y(t)] ' = x ' (t) + y ' (t).\\)<\/li>\n\n\n\n\n<li>\\([\\lambda x(t) ] ' = \\lambda x ' (t)\\) for all numbers \\(\\lambda .\\)<\/li>\n\n\n\n\n<li>Let there be defined for the elements \\(x\\in E\\) a left-sided (right-sided) multiplication with elements \\(y\\in E.\\) Suppose also that it is continuous and distributive with respect to addition and commutative with scalar multiplication. Then\n\\[[yx(t)] ' = yx ' (t) , \\tag{3}\\]\ni.e., constant factors can be removed from under the differentiation symbol.<\/li>\n\n\n<\/ol>\n\n\n\n\n\n<p>Properties [1] and [2] are clear. Property [3] follows from the continuity and distributivity of left-sided multiplication, namely\n\\[\\begin{align}\nyx ' (t)\n&= y \\lim_{\\varDelta t \\to 0} \\frac{1}{\\varDelta t} [x(t+ \\varDelta t) - x(t)] \\\\[6pt]\n&= \\lim_{\\varDelta t \\to 0} y\\left\\{ \\frac{1}{\\varDelta t} [ x(t+ \\varDelta t) - x(t)] \\right\\} \\\\[6pt]\n&= \\lim_{\\varDelta t \\to 0} \\frac{1}{\\varDelta t}[yx(t+\\varDelta t) - yx(t) ] \\\\[6pt]\n&= \\frac{d}{dt} [yx(t)].\n\\end{align}\\]\nAnalogously, we obtain\n\\[[x(t) y] ' = x ' (t) y \\tag{3a}\\]\nfor right-sided multiplication.<\/p>\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">Example 1.<\/span>\nLet \\(x = x(t)\\) with \\(x\\in E\\) and let \\(A\\) be an operator from \\((E \\to E_1 ).\\)\n\\[[ Ax(t) ] ' = Ax ' (t) . \\tag{4}\\]\nIf \\(A = A(t) \\in (E \\to E_1 )\\) and \\(x\\in E ,\\) then\n\\[[A(t)x] ' = A ' (t) x . \\tag{4a}\\]\nIn particular, for a linear functional \\(f\\in \\overline{E} :\\)\n\\[\\begin{align}\n\\left\\{ f[x(t)] \\right\\} ' &= f[x ' (t)] ,\\tag{5} \\\\[6pt]\n\\left\\{ f(t)(x) \\right\\} ' &= f ' (t)(x) .\\tag{5a}\n\\end{align}\\]\n<\/p>\n\n\n<\/div>\n\n\n\n\n\n<h3>Derivatives of Higher Order<\/h3>\n\n\n\n\n\n<p>We will now define derivatives of higher order. As in the case of ordinary functions, we can give two definitions of the \\(n\\)th derivative of an abstract function \\(x=x(t).\\)\n\\[ {\\overline{\\varDelta}}_{\\varDelta t} ^n x(t)\n=\\sum_{k=0}^n (-1)^{n-k} \\binom{n}{k} x(t+k \\varDelta t),\\]\nis the \\(n\\)th difference of \\(x(t)\\) at the point \\(t.\\) Further, we call\n\\[\\varDelta _{\\varDelta t}^n x(t)\n={\\overline{\\varDelta}}_{\\varDelta t}^n x \\left( t- \\frac{n}{2} \\varDelta t \\right)\\]\nthe <span class=\"defined\">\\(n\\)th central difference<\/span>.<\/p>\n\n\n\n\n<p>The expression\n\\[x^{[n]}(t) = \\lim_{\\varDelta t \\to 0} \\frac{1}{(\\varDelta t)^n} \\varDelta_{\\varDelta t}^{n} x(t) \\tag{6}\\]\nshall be called -- under the hypothesis that this limit exists -- the <span class=\"defined\">\\(n\\)th difference derivative<\/span> of the function \\(x(t)\\) at the point \\(t.\\)<\/p>\n\n\n\n\n\n<p>If the limit process in (6) is uniform in the neighborhood of every point \\(t,\\) then \\(x^{[n]}(t)\\) is called the <span class=\"defined\">uniform \\(n\\)th difference-derivative<\/span>.<\/p>\n\n\n\n\n\n<p>The \\(n\\)th derivative \\(x^{(n)}(t)\\) is defined by differentiating successively \\(n\\) times:\n\\[\\begin{align}\nx ' (t)_0 &= \\frac{d}{dt} x(t),\\\\[6pt]\nx ' ' (t)_0 &= \\frac{d}{dt} [x ' (t)_0 ] ,\\\\[6pt]\n&\\,\\,\\,\\vdots \\\\[6pt]\nx^{(n)} (t)_0 &= \\frac{d}{dt} [x^{(n-1)}(t)_0 ].\n\\end{align}\\]\n<\/p>\n\n\n\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">Theorem 1.<\/span>\nIf the continuous \\(n\\)th derivative \\(x^{(n)}(t)_0\\) exists in a neighborhood of a point \\(t,\\) then in this neighborhood the uniform \\(n\\)th difference-derivative \\(x^{[n]}(t)\\) exists too, and\n\\[x^{[n]}(t) = x^{(n)}(t)_0 .\\]\nConversely, if in a neighborhood of the point \\(t\\) the uniformly continuous, uniform difference-derivative \\(x^{[n]}(t)\\) exists, then there exists in this neighborhood the \\(n\\)th derivative \\(x^{(n)} (t)_0 .\\)<\/p>\n\n\n<\/div>\n\n\n\n\n\n<p>These statements hold for functions whose ranges consist of numbers. The transition to abstract functions is accomplished by a method which is often applied in functional analysis. We shall carry this out for the first statement.<\/p>\n\n\n\n\n\n<p>For an arbitrary linear functional \\(f\\in \\overline{E} ,\\)\n\\[\\varphi (t) = f[x(t)]\\]\nis a function whose domain of definition and range consist of numbers. Because of (5), we obtain\n\\[\\begin{align}\nf[x ' (t)_0 ] &= \\left\\{ f [ x(t) ] \\right\\} ' = \\varphi ' (t), \\\\[6pt]\nf[x ' ' (t)_0 ] &= \\left\\{ f [ x ' (t)_0 ] \\right\\} ' = \\left\\{ \\varphi ' (t) \\right\\} ' = \\varphi ' ' (t) ,\\\\[6pt]\n&\\,\\,\\,\\vdots\\\\[6pt]\nf[x^{(n)}(t)_0 ] &= \\left\\{ f[x^{(n-1)}(t)_0 ] \\right\\} ' = \\left\\{ \\varphi^{n-1} (t) \\right\\} ' = \\varphi^{(n)} (t) .\n\\end{align}\\]\nFurthermore\n\\[\\begin{align}\nf \\left[ \\frac{1}{(\\varDelta t)_n} \\varDelta_{\\varDelta t}^{n} x(t) \\right]\n&= \\frac{1}{(\\varDelta t)^n} \\sum_{k=0}^{n} (-1)^{n-k} \\binom{n}{k} \\varphi \\left( t+ \\left( k- \\frac{n}{2} \\right) \\varDelta t \\right) \\\\[6pt]\n&= \\frac{1}{(\\varDelta t)^n} \\varDelta_{\\varDelta t}^n \\varphi (t) \\\\[6pt]\n&= \\varphi^{(n)} (t+ \\theta \\varDelta t ) \\\\[6pt]\n&=f[x^{(n)} (t+\\theta \\varDelta t)]\n\\end{align}\\]\nwhere \\(-\\frac{1}{2} \\le \\theta \\le \\frac{1}{2} .\\) Since \\(x^{(n)} (t)_0 ,\\) according to the hypothesis, is continuous in a neighborhood of the point \\(t,\\)\n\\[\\lVert x^{(n)} (t+ \\theta \\varDelta t)_0 - x^{(n)} (t)_0 \\rVert \\le \\epsilon_{\\varDelta t} ,\\]\nwhere \\(\\epsilon_{\\varDelta t} \\to 0\\) as \\(\\varDelta t \\to 0\\) uniformly in a neighborhood of the point \\(t.\\)<\/p>\n\n\n\n\n\n<p>From this we have\n\\[\\left\\lvert f\\left[ \\frac{1}{(\\varDelta t)^n} \\varDelta_{\\varDelta t}^{n} x(t) \\right] - f[x^{(n)} (t)_0 ] \\right\\rvert \n\\le \\epsilon_{\\varDelta t} \\lVert f \\rVert . \\tag{7}\\]\nThe inequality (7) holds for arbitrary \\(f\\in \\overline{E} ,\\) therefore\n\\[\\left\\lVert \\frac{1}{(\\varDelta t)^n} \\varDelta_{\\varDelta t}^{n} x(t) - x^{(n)} (t)_0 \\right\\rVert \\le \\epsilon_{\\varDelta t}\\]\nand consequently\n\\[x^{(n)}(t)_0 = \\lim_{\\varDelta t \\to 0} \\frac{1}{(\\varDelta )^n} \\varDelta_{\\varDelta t}^{n} x(t) .\\]\nIn this case, the convergence is uniform in a neighborhood of every point \\(t.\\)<\/p>\n\n\n\n\n<p>This proves the first statement of the theorem.\n<span class=\"qed\"><\/span><\/p>\n\n\n\n\n\n\n<h3>Partial Derivatives<\/h3>\n\n\n\n\n\n<p>We introduce the concept of the <span class=\"defined\">partial derivative<\/span> of an abstract function. To do this, we consider a function of \\(n\\) real variables \\(t_1 ,\\) \\(t_2 ,\\) \\(\\cdots ,\\) \\(t_n\\) with a range lying in a normed linear space \\(E:\\)\n\\[y = f(t_1 ,\\, t_2 ,\\, \\cdots,\\, t_n ) \\in E .\\]\nWe can interpret \\(t_1 ,\\) \\(t_2 ,\\) \\(\\cdots ,\\) \\(t_n\\) as components of the \\(n\\)-dimensional vector\n\\[T = \\sum_{i=1}^{n} t_i e_i ,\\]\nwhere the \\(e_i\\) are orthonormal basis vectors, i.e., \\(n\\)-dimensional mutually orthogonal unit vectors.<\/p>\n\n\n\n\n\n<p>We now define the <span class=\"defined\">\\(n\\)th partial difference-derivative<\/span>\n\\[\\frac{\\partial ^n}{\\partial t_1 \\,\\partial t_2 \\cdots \\partial t_n} f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n )\\]\nat the point \\(T_0 = \\sum_{i=1}^{n} t_i^{(0)} e_i .\\)<\/p>\n\n\n\n\n\n<p>To do this, we form the \\(n\\)th partial difference:\n\\[\\varDelta_{t_1 , \\cdots, t_n}^n f(T_0 ) =\n\\sum_{i_1 ,\\cdots, i_n} (-1)^{n-k} f[T_0 + \\varDelta t(e_{i_1} + \\cdots + e_{i_k} )].\\]\nHere \\(0\\le i_1 < i_2 < \\cdots < i_k \\le n(i_1 ,\\, i_2 ,\\, \\cdots ,\\, i_k )\\) is a subset of \\((1,\\,2,\\,\\cdots,\\,n ).\\) The summation extends over all such subsets. For the empty set we put \\(k=0\\) and \\(Y_0\\) as the argument. The \\(n\\)th central partial difference at the point \\(T_0\\)\n\\[\\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(T_0 )\\]\nis the \\(n\\)th partial difference at the point\n\\[T_0 ' = T_0 - \\frac{1}{2} \\varDelta t \\sum_{i=1}^{n} e_i .\\]\nTherefore\n\\[\\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^{n} f(T_0 )\n= {\\overline{\\varDelta}}_{t_1 ,\\,\\cdots,\\,t_n ;\\, \\varDelta t}^n\nf\\left( T_0 - \\frac{1}{2} \\varDelta t \\sum_{i=1}^{n} e_i \\right).\\]\nThen the limit as \\(\\varDelta t \\to 0\\) of\n\\[\\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\,t_n ;\\, \\varDelta t}^n f(T_0 ),\\]\nin case it exists, is called the <span class=\"defined\">\\(n\\)th partial difference derivative<\/span>\n\\[\\frac{\\partial ^n}{\\partial t_1 \\, \\partial t_2 \\cdots \\partial t_n} f(t_1^{(0)} ,\\, t_2^{(0)} ,\\, \\cdots ,\\, t_n^{(0)})\\]\nof the function \\(f\\) at the point \\(T_0 = (t_1^{(0)} ,\\, t_2^{(0)} ,\\, \\cdots ,\\, t_n^{(0)}).\\)<\/p>\n\n\n\n\n\n<p>Parallel to this, we can define the <span class=\"defined\">\\(n\\)th partial derivative<\/span>. It is the result of carrying out successive differentiations of the function \\(f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n )\\) with respect to \\(t_{k_n} ,\\) \\(t_{k_{n-1}} ,\\) and finally \\(t_{k_1} ,\\) where \\(k_1 ,\\) \\(k_2 ,\\) \\(\\cdots ,\\) \\(k_n\\) is an arbitrary permutation of the indices \\(1,\\) \\(2,\\) \\(\\cdots,\\) \\(n.\\) Of course, we have to assume that the derivative\n\\[\\begin{align}\n\\frac{\\partial}{\\partial t_{k_n}} & f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n ) ,\n\\\\[6pt]\n\\frac{\\partial}{\\partial t_{k_{n-1}}} & \\left( \\frac{\\partial}{\\partial t_{k_n}} f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n ) \\right) ,\\\\[6pt]\n&\\vdots\n \\\\[6pt]\n\\frac{\\partial}{\\partial t_{k_i}} & \\left\\{ \n\\frac{\\partial}{\\partial t_{k_{i+1}}}  \\left( \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n ) \\right)\n\\right\\}\\\\[6pt]\n&\\vdots\n\\end{align}\\]\nobtained successively exist in a neighborhood of \\(T_0 .\\)<\/p>\n\n\n\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">Theorem 2.<\/span>\nIf in a neighborhood of the point \\(T_0 = (t_1 ^{(0)} ,\\, t_2 ^{(0)} ,\\, \\cdots ,\\, t_n^{(0)})\\) there exists an \\(n\\)th partial derivative of the function \\(f\\) and if this derivative is continuous at \\(T_0 ,\\) then there also exists at \\(T_0\\) the \\(n\\)th partial difference-derivative and both derivatives coincide.\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof-eng\">\n\n\n<p class=\"proofname\">Proof<\/p>\n\n\n\n\n<p>Let \\(L\\) be an arbitrary linear functional of \\(\\overline{E}.\\) Then\n\\[L[f(t_1 ,\\, t_2 ,\\, \\cdots ,\\, t_n )] = L f(t_i )\\]\nis a function of \\(t_1 ,\\) \\(t_2 ,\\) \\(\\cdots ,\\) \\(t_n ,\\) whose range consists of numbers. We have\n\\[\\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n L f(t_i^{(0)} )\n= \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} L f(t_i^{(0)} + \\theta_i \\varDelta t ) \\right\\}\\]\nfor every permutation \\(k_1 ,\\) \\(k_2 ,\\) \\(\\cdots ,\\) \\(k_n\\) of the indices \\(1,\\) \\(2,\\) \\(\\cdots ,\\) \\(n\\) and\n\\[\\begin{align}\nL \\left[ \\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i ^{(0)}) \\right]\n&= \\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n L f(t_i^{(0)} ) \\\\[6pt]\n&= \\frac{\\partial}{\\partial t_{k_1}} \\left\\{\n\\cdots \\frac{\\partial}{\\partial t_{k_n}} L f(t_i ^{(0)} + \\theta_i \\varDelta t ) \\right\\} \\\\[6pt]\n&= L \\left[ \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} + \\theta_i \\varDelta t ) \\right\\} \\right] ,\n\\end{align}\\]\nbecause the linear functional \\(L\\) can be removed from under the derivative sign.<\/p>\n\n\n\n\n\n<p>According to the hypothesis, the \\(n\\)th partial derivative of \\(f(t_i )\\) is continuous at the point \\((t_i ^{(0)} ),\\) so that for an arbitrary \\(\\epsilon_{\\varDelta t} > 0 ,\\)\n\\[\\left\\lVert \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} + \\theta_i \\varDelta t ) \\right\\} - \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} )\\right\\} \\right\\rVert \\le \\epsilon_{\\varDelta t}\\]\nfor sufficiently small \\(\\lvert \\varDelta t \\rvert .\\) Hence\n\\[\\begin{align}\n\\,&\n\\left\\lvert L \\left[ \\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i^{(0)}) \\right] - L \\left[ \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} ) \\right\\} \\right] \\right\\rvert \\\\[6pt]\n=& \\left\\lvert L \\left[ \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} + \\theta_i \\varDelta t ) \\right\\} \\right] - L \\left[ \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} ) \\right\\} \\right] \\right\\rvert \\\\[6pt]\n\\le & \\lVert L \\rVert\n\\left\\lVert \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} + \\theta_i \\varDelta t ) \\right\\} - \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} ) \\right\\} \\right\\rVert\\\\[6pt]\n\\le & \\lVert L \\rVert \\epsilon_{\\varDelta t} .\n\\end{align}\\]\nThis inequality is valid for every \\(L \\in \\overline{E},\\) therefore\n\\[\\left\\lVert \\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i ^{(0)}) - \\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} ) \\right\\} \\right\\rVert \\le \\epsilon_{\\varDelta t} .\\]\nIt follows that\n\\[\\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i ^{(0)})\n\\,\\to\\,\n\\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i ^{(0)} ) \\right\\}\\]\nas \\(\\varDelta t \\to 0,\\) and we have proved the existence of the limit of the expression\n\\[\\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i^{(0)})\\]\nand its equality with the \\(n\\)th partial derivative.\n<span class=\"qed\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n\n<div class=\"box\">\n\n\n<p><span class=\"theorem\">Corollary.<\/span>\nTwo \\(n\\)th partial derivatives which correspond to different permutations of the indices \\(1,\\) \\(2,\\) \\(\\cdots,\\) \\(n,\\) coincide at the points where they both are continuous. This means that the \\(n\\)th partial derivative does not depend on the order of the differentiations.\n<\/p>\n\n<\/div>\n\n\n\n\n\n<p>The equality\n\\[\\frac{\\partial}{\\partial t_{k_1}} \\left\\{ \\cdots \\frac{\\partial}{\\partial t_{k_n}} f(t_i) \\right\\}\n=\n\\lim_{\\varDelta t \\to 0} \\frac{1}{(\\varDelta t)^n} \\varDelta_{t_1 ,\\, \\cdots ,\\, t_n ;\\, \\varDelta t}^n f(t_i )\\]\nholds in this case. The right hand side, however, does not depend on the permutation \\(k_1 ,\\) \\(\\cdots ,\\) \\(k_n .\\)<\/p>\n\n\n\n\n\n<p>In the sequel, if we speak about the \\(n\\)th partial derivative in a certain region, we shall assume also that it is continuous in this region. Therefore, both definitions give the same result and we shall designate this derivative by the symbol\n\\[\\frac{\\partial ^n}{\\partial t_1 \\,\\partial t_2 \\cdots \\partial t_n} f(t_i ) .\\]\nIf \\(h_i\\) are arbitrary elements of the space \\(E,\\) and if\n\\[y = f \\left( \\sum_{i=1}^{n} t_i h_i \\right) \\in E \\]\nholds, then \\(y\\) is a function of the \\(n\\) parameters \\(t_1 ,\\) \\(t_2 ,\\) \\(\\cdots ,\\) \\(t_n .\\) The function\n\\[f \\left( \\sum_{i=1}^n t_i h_i \\right)\\]\nhas, for arbitrary \\(h_1 ,\\) \\(\\cdots ,\\) \\(h_n ,\\) an \\(n\\)th partial derivative\n\\[\\left[ \\frac{\\partial ^n}{\\partial t_1 \\, \\partial t_2 \\cdots \\partial t_n} f\\left( \\sum_{i=1}^{n} t_i h_i \\right) \\right]_{t_1 = \\cdots = t_n =0 } .\\]\nIf \\(h_1 = \\cdots = h_n ,\\) then it is not difficult to show that\n\\[\\left[ \\frac{\\partial ^n}{\\partial t_1 \\, \\partial t_2 \\cdots \\partial t_n} f \\left( x+\\sum_{i=1}^{n} t_i h_i \\right) \\right]_{t_1 = \\cdots = t_n =0}\n= \\frac{d^n}{dt^n} f(x+th)_{t=0}\\]\nholds.<\/p>\n\n\n\n\n\n<h3>Reference<\/h3>\n\n\n\n\n<ul class=\"bracket\">\n\n\n<li>L. A. Liusternik ad V. J. Sobolev, \u300eElements of Functional Analysis\u300f (Translated by Anthony E. Labarre), Frederick Ungar Publishing Company(New York), 168-173.\n<\/li>\n\n\n<\/ul>\n\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc5d0\uc11c\ub294 \ucd94\uc0c1\uacf5\uac04\uc5d0\uc11c\uc758 \ubbf8\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790. \ub3c4\ud568\uc218\uc758 \uc815\uc758 \\(E\\)\ub97c \ub178\ub984\ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uace0, \\(K\\)\ub97c \ub2eb\ud78c\uad6c\uac04 \\([0,\\,1]\\)\uc774\ub77c\uace0 \ud558\uc790. \\(K\\)\uc5d0\uc11c \\(E\\)\ub85c\uc758 \ud568\uc218 \\(x = x(t)\\)\ub97c \uc0dd\uac01\ud558\uc790. (\uc120\ud615\uc778 \uacbd\uc6b0\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc77c\ubc18\uc801\uc778 \ud568\uc218\ub97c \uc0dd\uac01\ud558\uc790.) \uc55e\uc73c\ub85c \uc774\ub7ec\ud55c \ud568\uc218\ub97c \uad6c\uac04 \\([0,\\,1]\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ucd94\uc0c1\ud654\ub41c \ud568\uc218\ub77c\uace0 \ubd80\ub97c \uac83\uc774\ub2e4. \uc774\ub7ec\ud55c \ud568\uc218\uc5d0 \ub300\ud558\uc5ec, \ud574\uc11d\ud559\uc758 \uae30\ubcf8\uc801\uc778 \uc5f0\uc0b0\uc744 \uc815\uc758\ud558\uace0 \uadf8 \uc131\uc9c8\uc744 \uc720\ub3c4\ud558\uc790. \uc815\uc758 1. \\(x\\in E\\)\uc774\uace0 \\(t\\in [0,\\,1]\\)\uc778 \ud568\uc218 \\(x = x(t)\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc774 \ud568\uc218\uc758 \ub3c4\ud568\uc218 \\(x &#8216;&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"categories":[70],"tags":[82,83,85,84,86,88,89,87],"class_list":["post-2529","post","type-post","status-publish","format-standard","hentry","category-functional-analysis","tag-abstract-function","tag-abstract-space","tag-central-difference","tag-derivative","tag-difference-derivative","tag-partial-derivative","tag-partial-difference-derivative","tag-uniform-difference-derivative"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2529","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=2529"}],"version-history":[{"count":58,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2529\/revisions"}],"predecessor-version":[{"id":9463,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2529\/revisions\/9463"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2529"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2529"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2529"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}