{"id":9495,"date":"2025-10-20T18:59:43","date_gmt":"2025-10-20T09:59:43","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9495"},"modified":"2026-09-27T16:14:33","modified_gmt":"2026-09-27T07:14:33","slug":"ch09-differentiation-of-functions-of-several-variables","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\/","title":{"rendered":"\ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84"},"content":{"rendered":"
<\/p>\n

<\/p>\n

\uc774 \uc7a5\uc5d0\uc11c\ub294 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04 \\(\\mathbb{R}^n\\)\uc5d0\uc11c \\(\\mathbb{R}^m\\)\uc73c\ub85c\uc758 \ud568\uc218\uc758 \ubbf8\ubd84\uc744 \ub2e4\ub8ec\ub2e4. \ud3b8\ubbf8\ubd84\uacfc \uc804\ubbf8\ubd84\uc758 \uac1c\ub150\uc744 \uc815\uc758\ud558\uace0, \uc5f0\uc1c4\ubc95\uce59, \ud3c9\uade0\uac12 \uc815\ub9ac, \uc74c\ud568\uc218 \uc815\ub9ac \ub4f1 \uc911\uc694\ud55c \uacb0\uacfc\ub4e4\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\ud3b8\ubbf8\ubd84\uacfc \uc804\ubbf8\ubd84<\/h3>\n

\uc810 \\(a=(a_1,\\ldots,a_n)\\)\uc774 \ud568\uc218 \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}\\)\uc758 \uc815\uc758\uc5ed\uc758 \ub0b4\uc810\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \uc810 \\(a\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 \\(x_i\\)\uc5d0 \ub300\ud55c \ud3b8\ubbf8\ubd84<\/span>(partial derivative)\uc744
\n\\[
\n\\frac{\\partial f}{\\partial x_i}(a)
\n=
\n\\lim_{h\\to0}
\n\\frac{f(a_1,\\ldots,a_i+h,\\ldots,a_n)-f(a)}{h}
\n\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc704 \ud3b8\ubbf8\ubd84\uc744 \\(f_{x_i}(a)\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n

\ubaa8\ub4e0 \ubcc0\uc218\uc5d0 \ub300\ud55c \ud3b8\ubbf8\ubd84\uc774 \uc874\uc7ac\ud574\ub3c4 \ud568\uc218\uac00 \uc5f0\uc18d\uc77c \ud544\uc694\ub294 \uc5c6\ub2e4. \uc608\ub97c \ub4e4\uc5b4,
\n\\[
\nf(x,y)=
\n\\begin{cases}
\n\\dfrac{xy}{x^2+y^2} & \\text{if }\\;(x,y)\\ne(0,0),\\\\[8pt]
\n0 & \\text{if }\\;(x,y)=(0,0)
\n\\end{cases}
\n\\]
\n\uc774\ub77c\uace0 \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\ub294 \\((0,0)\\)\uc5d0\uc11c \ub450 \ud3b8\ubbf8\ubd84\uacc4\uc218\uac00 \ubaa8\ub450 \\(0\\)\uc774\uc9c0\ub9cc, \\(f(t,t)=1\/2\\)\uc774\ubbc0\ub85c \uc6d0\uc810\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2c8\ub2e4.<\/p>\n

\ud568\uc218 \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}^m\\)\uc774 \uc810 \\(a\\)\uc5d0\uc11c \uc804\ubbf8\ubd84<\/span> \uac00\ub2a5\ud558\ub2e4(differentiable), \ub610\ub294 \ud504\ub808\uc170 \ubbf8\ubd84<\/span> \uac00\ub2a5\ud558\ub2e4\ub294 \uac83\uc740 \uc120\ud615\ubcc0\ud658 \\(L\\colon\\mathbb{R}^n\\to\\mathbb{R}^m\\)\uc774 \uc874\uc7ac\ud558\uc5ec
\n\\[
\n\\lim_{h\\to0}
\n\\frac{\\|f(a+h)-f(a)-L(h)\\|}{\\|h\\|}
\n=0
\n\\]
\n\uc778 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc774\ub7ec\ud55c \\(L\\)\uc740 \uc720\uc77c\ud558\ub2e4. \uc2e4\uc81c\ub85c \\(L_1,L_2\\)\uac00 \ubaa8\ub450 \uc704 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\uace0 \\(v\\in\\mathbb{R}^n\\)\uc774\uba74 \\(h=tv\\)\ub97c \ub300\uc785\ud558\uc5ec \\(t\\to0\\)\uc73c\ub85c \ubcf4\ub0bc \ub54c
\n\\[
\n\\|(L_1-L_2)v\\|
\n\\le
\n\\frac{\\|f(a+tv)-f(a)-L_1(tv)\\|}{|t|}
\n+
\n\\frac{\\|f(a+tv)-f(a)-L_2(tv)\\|}{|t|}
\n\\to0
\n\\]
\n\uc774\ubbc0\ub85c \\(L_1v=L_2v\\)\uc774\ub2e4. \uc774 \uc720\uc77c\ud55c \uc120\ud615\ubcc0\ud658\uc744 \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 \ubbf8\ubd84<\/span>(differential) \ub610\ub294 \ub3c4\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \\(Df(a)\\) \ub610\ub294 \\(f'(a)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\ud568\uc218 \\(f\\)\uac00 \\(a\\)\uc5d0\uc11c \uc804\ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\ub294 \uc870\uac74\uc740
\n\\[
\nf(a+h)=f(a)+Df(a)(h)+r(h),
\n\\qquad
\n\\frac{\\|r(h)\\|}{\\|h\\|}\\to0
\n\\]
\n\uc73c\ub85c \uc4f8 \uc218\ub3c4 \uc788\ub2e4. \ub9c8\uc9c0\ub9c9 \ud56d\uc744 \uac04\ub2e8\ud788 \\(o(\\|h\\|)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\uc120\ud615\ubcc0\ud658 \\(T\\colon\\mathbb{R}^n\\to\\mathbb{R}^m\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\n\\|T\\|=\\sup_{\\|v\\|=1}\\|T(v)\\|
\n\\]
\n\ub97c \\(T\\)\uc758 \uc5f0\uc0b0\uc790 \ub178\ub984<\/span>(operator norm)\uc774\ub77c\uace0 \ud55c\ub2e4. \uc720\ud55c\ucc28\uc6d0\uc5d0\uc11c\ub294 \uc774 \uc0c1\ud55c\uc774 \uc720\ud55c\ud558\uba70, \ubaa8\ub4e0 \\(v\\)\uc5d0 \ub300\ud558\uc5ec \\(\\|T(v)\\|\\le\\|T\\|\\|v\\|\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 9.1. (\ubbf8\ubd84 \uac00\ub2a5\uc131\uacfc \uc5f0\uc18d\uc131\uc758 \uad00\uacc4)<\/span><\/p>\n

\uc804\ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\ub294 \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/span>
\n\\(f(a+h)=f(a)+Df(a)(h)+r(h)\\)\uc774\uace0 \\(\\|r(h)\\|\/\\|h\\|\\to0\\)\uc774\ubbc0\ub85c
\n\\[
\n\\|f(a+h)-f(a)\\|
\n\\le
\n\\|Df(a)\\|\\,\\|h\\|+\\|r(h)\\|
\n\\to0.
\n\\]
\n\ub530\ub77c\uc11c \\(f\\)\ub294 \\(a\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

\ud568\uc218 \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}^m\\)\uc774 \\(a\\)\uc5d0\uc11c \uc804\ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 \ubaa8\ub4e0 \ud3b8\ubbf8\ubd84\uc774 \uc874\uc7ac\ud558\uace0, \\(Df(a)\\)\uc758 \ud589\ub82c \ud45c\ud604\uc740 \ud3b8\ubbf8\ubd84\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4\ub2e4. \uc2e4\uc81c\ub85c \\(e_i\\)\ub97c \\(i\\)\ubc88\uc9f8 \ud45c\uc900\uae30\uc800\ubca1\ud130\ub77c\uace0 \ud558\uba74
\n\\[
\nDf(a)e_i
\n=
\n\\lim_{t\\to0}\\frac{f(a+te_i)-f(a)}{t},
\n\\]
\n\uc774\ubbc0\ub85c \\(Df(a)e_i\\)\ub294 \uac01 \uc131\ubd84\uc758 \\(x_i\\)\uc5d0 \ub300\ud55c \ud3b8\ubbf8\ubd84\uc744 \ubaa8\uc740 \uc5f4\ubca1\ud130\uc774\ub2e4. \ub530\ub77c\uc11c \\(f=(f_1,\\ldots,f_m)\\)\uc77c \ub54c
\n\\[
\nDf(a)=
\n\\begin{pmatrix}
\n\\dfrac{\\partial f_1}{\\partial x_1}(a) & \\cdots & \\dfrac{\\partial f_1}{\\partial x_n}(a)\\\\
\n\\vdots & \\ddots & \\vdots\\\\
\n\\dfrac{\\partial f_m}{\\partial x_1}(a) & \\cdots & \\dfrac{\\partial f_m}{\\partial x_n}(a)
\n\\end{pmatrix}.
\n\\]
\n\\(a\\)\uc5d0\uc11c \ubaa8\ub4e0 \uc77c\uacc4\ud3b8\ubbf8\ubd84\uacc4\uc218\uac00 \uc874\uc7ac\ud560 \ub54c \uc704 \ud589\ub82c\uc744 \uc57c\ucf54\ube44 \ud589\ub82c<\/span>(Jacobian matrix)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\ud788 \\(a\\)\uc5d0\uc11c \\(f\\)\uac00 \uc804\ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c \uc774 \ud589\ub82c\uc774 \uc120\ud615\ubcc0\ud658 \\(Df(a)\\)\ub97c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\ubaa8\ub4e0 \ud3b8\ubbf8\ubd84\uc774 \uc874\uc7ac\ud574\ub3c4 \uc804\ubbf8\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ub294 \uc5c6\ub2e4. \uc608\ub97c \ub4e4\uc5b4
\n\\[
\nf(x,y)=
\n\\begin{cases}
\n\\dfrac{x^3-xy^2}{x^2+y^2} & \\text{if }\\;(x,y)\\ne(0,0),\\\\[6pt]
\n0 & \\text{if }\\;(x,y)=(0,0)
\n\\end{cases}
\n\\]
\n\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f_x(0,0)=1\\), \\(f_y(0,0)=0\\)\uc774\ubbc0\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uba74 \\(Df(0,0)(h_1,h_2)=h_1\\)\uc774\uc5b4\uc57c \ud55c\ub2e4. \uadf8\ub7ec\ub098 \\(f(t,t)=0\\)\uc774\ubbc0\ub85c
\n\\[
\n\\frac{|f(t,t)-Df(0,0)(t,t)|}{\\sqrt2|t|}
\n=
\n\\frac1{\\sqrt2}
\n\\]
\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(f\\)\ub294 \\((0,0)\\)\uc5d0\uc11c \uc804\ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\ub2e4.<\/p>\n

\ub2e4\ubcc0\uc218\ud568\uc218 \\(f\\)\uac00 \uc810 \\(a\\)\uc5d0\uc11c \uc804\ubbf8\ubd84 \uac00\ub2a5\ud558\uae30 \uc704\ud55c \ucda9\ubd84\uc870\uac74\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

\n

\uc815\ub9ac 9.2. (\uc804\ubbf8\ubd84 \uac00\ub2a5 \uc870\uac74)<\/span><\/p>\n

\ud568\uc218 \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}^m\\)\uc758 \ubaa8\ub4e0 \ud3b8\ubbf8\ubd84\uc774 \uc810 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \uc874\uc7ac\ud558\uace0 \\(a\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \\(f\\)\ub294 \\(a\\)\uc5d0\uc11c \uc804\ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/span>
\n\\(f=(f_1,\\ldots,f_m)\\)\uc774\ub77c\uace0 \ud558\uace0 \\(h=(h_1,\\ldots,h_n)\\)\uc774\ub77c \ud558\uc790. \\(j=0,1,\\ldots,n\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\nx^{(j)}=a+h_1e_1+\\cdots+h_je_j
\n\\]
\n\ub85c \ub450\uba74 \\(x^{(0)}=a\\), \\(x^{(n)}=a+h\\)\uc774\ub2e4. \\(h\\)\uac00 \ucda9\ubd84\ud788 \uc791\uc73c\uba74 \uc774 \uc810\ub4e4\uc744 \uc787\ub294 \ubaa8\ub4e0 \uc88c\ud45c\ubc29\ud5a5 \uc120\ubd84\uc774 \uc8fc\uc5b4\uc9c4 \uadfc\ubc29 \uc548\uc5d0 \ub193\uc778\ub2e4.<\/p>\n

\uac01 \uc131\ubd84 \\(f_\\ell\\)\uc5d0 \ub300\ud558\uc5ec \uc77c\ubcc0\uc218 \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc88c\ud45c\ubcc4\ub85c \uc801\uc6a9\ud558\uba74, \\(h_j\\ne0\\)\uc77c \ub54c \\(x^{(j-1)}\\)\uc640 \\(x^{(j)}\\) \uc0ac\uc774\uc758 \uc5b4\ub5a4 \uc810 \\(\\xi_{\\ell j}\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[
\nf_\\ell(x^{(j)})-f_\\ell(x^{(j-1)})
\n=
\n\\frac{\\partial f_\\ell}{\\partial x_j}(\\xi_{\\ell j})h_j
\n\\]
\n\uac00 \ub41c\ub2e4. \\(h_j=0\\)\uc778 \ud56d\uc740 \\(0\\)\uc774\ubbc0\ub85c \ub530\ub85c \uace0\ub824\ud560 \ud544\uc694\uac00 \uc5c6\ub2e4. \ub530\ub77c\uc11c
\n\\[
\n\\begin{aligned}
\n&f_\\ell(a+h)-f_\\ell(a)
\n-\\sum_{j=1}^n
\n\\frac{\\partial f_\\ell}{\\partial x_j}(a)h_j\\\\
\n&\\quad=
\n\\sum_{j=1}^n
\n\\left(
\n\\frac{\\partial f_\\ell}{\\partial x_j}(\\xi_{\\ell j})
\n–
\n\\frac{\\partial f_\\ell}{\\partial x_j}(a)
\n\\right)h_j.
\n\\end{aligned}
\n\\]
\n\ubaa8\ub4e0 \\(\\xi_{\\ell j}\\to a\\)\uc774\uace0 \ud3b8\ubbf8\ubd84\ub4e4\uc774 \\(a\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \\(h\\)\uac00 \ucda9\ubd84\ud788 \uc791\uc73c\uba74 \uc6b0\ubcc0\uc758 \uc808\ub313\uac12\uc740
\n\\[
\n\\varepsilon\\sum_{j=1}^n|h_j|
\n\\le
\n\\varepsilon\\sqrt n\\,\\|h\\|
\n\\]
\n\ubcf4\ub2e4 \uc791\ub2e4. \uc131\ubd84\uc758 \uc218\uac00 \uc720\ud55c\ud558\ubbc0\ub85c \ubca1\ud130 \uc804\uccb4\uc758 \ub098\uba38\uc9c0\ub3c4 \\(o(\\|h\\|)\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(f\\)\ub294 \\(a\\)\uc5d0\uc11c \uc804\ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \uadf8 \ubbf8\ubd84\uc740 \uc57c\ucf54\ube44 \ud589\ub82c\ub85c \uc8fc\uc5b4\uc9c4\ub2e4.<\/span><\/p>\n<\/div>\n

\n

\ubb38\uc81c 9.1.<\/span>
\n\ud568\uc218 \\(f\\colon\\mathbb{R}^2\\to\\mathbb{R}^3\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(f\\)\uc758 \uc804\ubbf8\ubd84 \ud589\ub82c\uc744 \uad6c\ud558\uc2dc\uc624.
\n\\[
\nf(x,y)=
\n(a_{11}x+a_{12}y,\\,
\na_{21}x+a_{22}y,\\,
\na_{31}x+a_{32}y).
\n\\]<\/p>\n<\/div>\n

\n

\ubb38\uc81c 9.2.<\/span>
\n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

    \n
  1. \ud568\uc218 \\(f(x,y)=(\\cos xy,\\,e^y-\\ln y)\\)\uac00 \uc810 \\((1,1)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624.<\/li>\n
  2. \ub2e4\uc74c \ud568\uc218\uac00 \\((0,0)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624.
    \n\\[
    \nf(x,y)=
    \n\\begin{cases}
    \n\\dfrac{y^2}{x^2+y^2} & \\text{if }\\;(x,y)\\ne(0,0),\\\\[8pt]
    \n0 & \\text{if }\\;(x,y)=(0,0).
    \n\\end{cases}
    \n\\]<\/li>\n<\/ol>\n<\/div>\n
    \n

    \ubb38\uc81c 9.3.<\/span>
    \n\\(k\\in\\mathbb{R}\\), \\(D\\subseteq\\mathbb{R}^n\\), \\(a\\in D^o\\)\uc774\uace0 \ub450 \ud568\uc218 \\(f\\colon D\\to\\mathbb{R}^m\\)\uacfc \\(g\\colon D\\to\\mathbb{R}^m\\)\uc774 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n

      \n
    1. \ub450 \ud568\uc218\uc758 \ud569 \\(f+g\\)\ub294 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(D(f+g)(a)=Df(a)+Dg(a)\\)\uc774\ub2e4.<\/li>\n
    2. \ud568\uc218\uc758 \uc2e4\uc218\ubc30 \\(kf\\)\ub294 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(D(kf)(a)=kDf(a)\\)\uc774\ub2e4.<\/li>\n
    3. \ub450 \ud568\uc218\uc758 \ub0b4\uc801 \\(f\\cdot g\\)\ub294 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0
      \n\\[
      \nD(f\\cdot g)(a)
      \n=
      \ng(a)^{\\top}Df(a)+f(a)^{\\top}Dg(a)
      \n\\]
      \n\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

      \ubbf8\ubd84\uc758 \uacc4\uc0b0<\/h3>\n

      \uc2e4\ud568\uc218\uc5d0\uc11c \ud569\uc131\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc774 \uc874\uc7ac\ud558\ub4ef \ub2e4\ubcc0\uc218\ud568\uc218\uc5d0\uc11c\ub3c4 \ud569\uc131\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 9.3. (\uc5f0\uc1c4\ubc95\uce59)<\/span><\/p>\n

      \ud568\uc218 \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}^m\\)\uc774 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(g\\colon\\mathbb{R}^m\\to\\mathbb{R}^p\\)\uac00 \\(f(a)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74, \\(g\\circ f\\)\ub294 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
      \n\\[
      \nD(g\\circ f)(a)=Dg(f(a))Df(a).
      \n\\]<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/span>
      \n\\(u(h)=f(a+h)-f(a)\\)\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c
      \n\\[
      \nu(h)=Df(a)h+r_f(h),
      \n\\qquad
      \nr_f(h)=o(\\|h\\|),
      \n\\]
      \n\uc774\uace0 \ud2b9\ud788 \\(\\|u(h)\\|=O(\\|h\\|)\\)\uc774\ub2e4. \ub610\ud55c \\(g\\)\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131\uc5d0 \uc758\ud574
      \n\\[
      \ng(f(a)+u)-g(f(a))
      \n=
      \nDg(f(a))u+r_g(u),
      \n\\qquad
      \nr_g(u)=o(\\|u\\|).
      \n\\]
      \n\ub530\ub77c\uc11c
      \n\\[
      \n\\begin{aligned}
      \ng(f(a+h))-g(f(a))
      \n&=Dg(f(a))Df(a)h
      \n+Dg(f(a))r_f(h)
      \n+r_g(u(h))\\\\
      \n&=Dg(f(a))Df(a)h+o(\\|h\\|).
      \n\\end{aligned}
      \n\\]
      \n\ub9c8\uc9c0\ub9c9 \ub4f1\uc2dd\uc5d0\uc11c\ub294 \\(r_g(u(h))=o(\\|u(h)\\|)=o(\\|h\\|)\\)\uc744 \uc0ac\uc6a9\ud588\ub2e4.<\/span><\/p>\n<\/div>\n

      \n

      \ubb38\uc81c 9.4.<\/span>
      \n\\(w=f(x,y,z)\\), \\(x=x(r,s)\\), \\(y=y(r,s)\\), \\(z=z(r,s)\\)\uac00 \ubaa8\ub450 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc77c \ub54c, \uc815\ub9ac 9.3\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \uacf5\uc2dd\uc744 \uc720\ub3c4\ud558\uc2dc\uc624.
      \n\\[
      \n\\frac{\\partial w}{\\partial r}
      \n=
      \n\\frac{\\partial w}{\\partial x}\\frac{\\partial x}{\\partial r}
      \n+
      \n\\frac{\\partial w}{\\partial y}\\frac{\\partial y}{\\partial r}
      \n+
      \n\\frac{\\partial w}{\\partial z}\\frac{\\partial z}{\\partial r},
      \n\\quad
      \n\\frac{\\partial w}{\\partial s}
      \n=
      \n\\frac{\\partial w}{\\partial x}\\frac{\\partial x}{\\partial s}
      \n+
      \n\\frac{\\partial w}{\\partial y}\\frac{\\partial y}{\\partial s}
      \n+
      \n\\frac{\\partial w}{\\partial z}\\frac{\\partial z}{\\partial s}.
      \n\\]<\/p>\n<\/div>\n

      \uc810 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c \uc2e4\ud568\uc218 \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc774\ub54c \\(Df(a)\\)\uc758 \ud589\ub82c\uc740 \ud55c \ud589\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c0\uba70, \uadf8 \uc804\uce58\ubca1\ud130
      \n\\[
      \n\\nabla f(a)
      \n=
      \n\\left(
      \n\\frac{\\partial f}{\\partial x_1}(a),
      \n\\ldots,
      \n\\frac{\\partial f}{\\partial x_n}(a)
      \n\\right)
      \n\\]
      \n\ub97c \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 \uae30\uc6b8\uae30 \ubca1\ud130<\/span>(gradient vector)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

      \n

      \ubb38\uc81c 9.5.<\/span>
      \n\\(f\\)\uc640 \\(g\\)\uac00 \\(D\\subseteq\\mathbb{R}^n\\)\uc73c\ub85c\ubd80\ud130 \\(\\mathbb{R}\\)\ub85c\uc758 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc77c \ub54c \ub2e4\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n

        \n
      1. \\(\\nabla(f+g)=\\nabla f+\\nabla g\\).<\/li>\n
      2. \\(\\nabla(f-g)=\\nabla f-\\nabla g\\).<\/li>\n
      3. \\(k\\)\uac00 \uc2e4\uc218\uc778 \uc0c1\uc218\uc77c \ub54c \\(\\nabla(kf)=k\\nabla f\\).<\/li>\n
      4. \\(\\nabla(fg)=f\\nabla g+g\\nabla f\\).<\/li>\n
      5. \\(g\\ne0\\)\uc778 \uc810\uc5d0\uc11c
        \n\\[
        \n\\nabla(f\/g)
        \n=
        \n\\frac{g\\nabla f-f\\nabla g}{g^2}.
        \n\\]<\/li>\n<\/ol>\n<\/div>\n

        \\(v\\)\uac00 \ub2e8\uc704\ubca1\ud130\uc77c \ub54c, \\(v\\) \ubc29\ud5a5\uc73c\ub85c\uc758 \\(f\\)\uc758 \ubc29\ud5a5\ub3c4\ud568\uc218<\/span>(directional derivative)\ub97c
        \n\\[
        \nD_vf(a)
        \n=
        \n\\lim_{t\\to0}
        \n\\frac{f(a+tv)-f(a)}{t}
        \n\\]
        \n\ub85c \uc815\uc758\ud55c\ub2e4. \ubaa8\ub4e0 \ubc29\ud5a5\ub3c4\ud568\uc218\uac00 \uc874\uc7ac\ud558\ub354\ub77c\ub3c4 \uc804\ubbf8\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ub294 \uc5c6\uc9c0\ub9cc, \uc804\ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \ubc29\ud5a5\ub3c4\ud568\uc218\ub294 \uae30\uc6b8\uae30\ub85c \uacc4\uc0b0\ub41c\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 9.4. (\ubc29\ud5a5\ub3c4\ud568\uc218\uc640 \uae30\uc6b8\uae30)<\/span><\/p>\n

        \ud568\uc218 \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}\\)\uac00 \\(a\\)\uc5d0\uc11c \uc804\ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(v\\)\uac00 \ub2e8\uc704\ubca1\ud130\uc774\uba74
        \n\\[
        \nD_vf(a)=\\nabla f(a)\\cdot v.\\tag{9.1}
        \n\\]
        \n\ud2b9\ud788 \\(\\nabla f(a)\\ne0\\)\uc774\uba74 \ub2e8\uc704\ubca1\ud130 \uac00\uc6b4\ub370 \ubc29\ud5a5\ub3c4\ud568\uc218\uac00 \uac00\uc7a5 \ud070 \ubc29\ud5a5\uc740
        \n\\[
        \nv=\\frac{\\nabla f(a)}{\\|\\nabla f(a)\\|}
        \n\\]
        \n\uc774\uace0, \uac00\uc7a5 \uc791\uc740 \ubc29\ud5a5\uc740 \uadf8 \ubc18\ub300\ubc29\ud5a5\uc774\ub2e4. \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc740 \uac01\uac01 \\(\\|\\nabla f(a)\\|\\)\uc640 \\(-\\|\\nabla f(a)\\|\\)\uc774\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/span>
        \n\\(h=tv\\)\ub97c \uc804\ubbf8\ubd84\uc758 \uc815\uc758\uc5d0 \ub300\uc785\ud558\uba74
        \n\\[
        \nf(a+tv)-f(a)
        \n=
        \nt\\,Df(a)v+o(|t|).
        \n\\]
        \n\uc591\ubcc0\uc744 \\(t\\)\ub85c \ub098\ub204\uace0 \\(t\\to0\\)\uc73c\ub85c \ubcf4\ub0b4\uba74
        \n\\[
        \nD_vf(a)
        \n=
        \nDf(a)v
        \n=
        \n\\nabla f(a)\\cdot v
        \n\\]
        \n\uc774\ub2e4. \ub098\uba38\uc9c0\ub294 \ucf54\uc2dc\u2013\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd
        \n\\[
        \n|\\nabla f(a)\\cdot v|
        \n\\le
        \n\\|\\nabla f(a)\\|\\|v\\|
        \n\\]
        \n\uacfc \ub4f1\ud638 \uc870\uac74\uc5d0\uc11c \ub530\ub978\ub2e4.<\/span><\/p>\n<\/div>\n

        \n

        \ubb38\uc81c 9.6.<\/span>
        \n\\(f(x,y)=x^2y+e^{xy}\\), \\(a=(1,0)\\), \\(v=(3\/5,4\/5)\\)\ub77c\uace0 \ud558\uc790. \\(D_vf(a)\\)\ub97c \uad6c\ud558\uace0, \\(a\\)\uc5d0\uc11c \ubc29\ud5a5\ub3c4\ud568\uc218\uac00 \uac00\uc7a5 \ud070 \ub2e8\uc704\ubc29\ud5a5\uacfc \uac00\uc7a5 \uc791\uc740 \ub2e8\uc704\ubc29\ud5a5 \ubc0f \uadf8 \uac12\uc744 \uac01\uac01 \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

        \\(x\\), \\(y\\), \\(z\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c \uc2e4\ud568\uc218\uc774\uace0 \uace1\uc120 \\(C\\)\uac00
        \n\\[
        \nr(t)=(x(t),y(t),z(t)),
        \n\\qquad
        \nt\\in I
        \n\\]
        \n\uc640 \uac19\uc740 \ud568\uc218\ub85c \ud45c\ud604\ub41c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \uc810 \\(t_0\\)\uc5d0\uc11c \\(r\\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c
        \n\\[
        \nr'(t_0)
        \n=
        \n\\left(
        \n\\frac{d}{dt}x(t_0),
        \n\\frac{d}{dt}y(t_0),
        \n\\frac{d}{dt}z(t_0)
        \n\\right)
        \n\\]
        \n\ub85c \uc815\uc758\ud55c\ub2e4. \uace1\uc120 \\(C\\) \uc704\uc758 \uc810\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 \\(w=f(r(t))\\)\uc5d0 \ub300\ud558\uc5ec
        \n\\[
        \n\\frac{dw}{dt}
        \n=
        \n\\frac{\\partial w}{\\partial x}\\frac{dx}{dt}
        \n+
        \n\\frac{\\partial w}{\\partial y}\\frac{dy}{dt}
        \n+
        \n\\frac{\\partial w}{\\partial z}\\frac{dz}{dt}
        \n\\]
        \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\uac83\uc744 \uae30\uc6b8\uae30 \uc5f0\uc0b0\uc790\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub098\ud0c0\ub0b4\uba74
        \n\\[
        \n\\frac{d}{dt}f(r(t))
        \n=
        \n\\nabla f(r(t))\\cdot r'(t)
        \n\\]
        \n\uc774\ub2e4.<\/p>\n

        \n

        \ubb38\uc81c 9.7.<\/span>
        \n\ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218 \\(F\\colon\\mathbb{R}^2\\to\\mathbb{R}\\)\uacfc \uc0c1\uc218 \\(c\\)\uc5d0 \ub300\ud558\uc5ec \ub4f1\uc704\uace1\uc120<\/span> \\(F(x,y)=c\\)\ub97c \uc0dd\uac01\ud558\uc790.<\/p>\n

          \n
        1. \uc774 \ub4f1\uc704\uace1\uc120\uc758 \uc77c\ubd80\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ub9e4\uac1c\ud654 \\(r(t)=(u(t),v(t))\\)\ub85c \uc8fc\uc5b4\uc9c0\uace0 \\(r'(t_0)\\ne0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(\\nabla F(r(t_0))\\)\uc640 \\(r'(t_0)\\)\uac00 \uc11c\ub85c \uc218\uc9c1\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
        2. \\((a,b)\\)\uac00 \ub4f1\uc704\uace1\uc120 \uc704\uc758 \uc810\uc774\uace0 \\(\\nabla F(a,b)\\ne0\\)\uc774\uba70, \uc774 \uc810\uc744 \uc9c0\ub098\ub294 \uc704\uc640 \uac19\uc740 \uc815\uce59 \ub9e4\uac1c\ud654\uac00 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc774
          \n\\[
          \n\\frac{\\partial F}{\\partial x}(a,b)(x-a)
          \n+
          \n\\frac{\\partial F}{\\partial y}(a,b)(y-b)
          \n=
          \n0
          \n\\]
          \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

          \ud568\uc218 \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}\\)\uc744 \ubcc0\uc218 \\(x_i\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud55c \ub4a4 \ub2e4\uc2dc \ubcc0\uc218 \\(x_j\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud55c \uc774\uacc4 \ud3b8\ub3c4\ud568\uc218\ub97c
          \n\\[
          \nf_{x_ix_j}
          \n\\quad\\text{\ub610\ub294}\\quad
          \n\\frac{\\partial^2}{\\partial x_j\\,\\partial x_i}f
          \n\\]
          \n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud568\uc218 \\(f\\)\uac00 \uc801\uc808\ud55c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(f\\)\uc758 \uc774\uacc4\ud3b8\ubbf8\ubd84\uc758 \ubbf8\ubd84 \uc21c\uc11c\ub97c \ubc14\uafb8\uc5b4\ub3c4 \ub3d9\uc77c\ud55c \ub3c4\ud568\uc218\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

          \n

          \uc815\ub9ac 9.5. (\ud074\ub808\ub85c \uc815\ub9ac)<\/span><\/p>\n

          \ud568\uc218 \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}\\)\uc758 \ub450 \ud63c\ud569 \uc774\uacc4\ud3b8\ubbf8\ubd84
          \n\\[
          \n\\frac{\\partial^2f}{\\partial x_i\\partial x_j},
          \n\\qquad
          \n\\frac{\\partial^2f}{\\partial x_j\\partial x_i}
          \n\\]
          \n\uac00 \uc810 \\(c\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \uc874\uc7ac\ud558\uace0 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74
          \n\\[
          \n\\frac{\\partial^2f}{\\partial x_i\\,\\partial x_j}(c)
          \n=
          \n\\frac{\\partial^2f}{\\partial x_j\\,\\partial x_i}(c)
          \n\\]
          \n\uc774\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/span>
          \n\ub2e4\ub978 \uc88c\ud45c\ub97c \uace0\uc815\ud558\uba74 \ub418\ubbc0\ub85c \\(n=2\\)\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud558\uba74 \ucda9\ubd84\ud558\ub2e4. \\(c=(a,b)\\)\ub77c\uace0 \ud558\uc790. \ucda9\ubd84\ud788 \uc791\uc740 \\(0\\ne h,k\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[
          \n\\Delta(h,k)
          \n=
          \nf(a+h,b+k)-f(a+h,b)-f(a,b+k)+f(a,b)
          \n\\]
          \n\ub85c \ub454\ub2e4.<\/p>\n

          \uba3c\uc800 \\(g(x)=f(x,b+k)-f(x,b)\\)\ub77c\uace0 \ud558\uba74 \uc77c\ubcc0\uc218 \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(a\\)\uc640 \\(a+h\\) \uc0ac\uc774\uc758 \uc5b4\ub5a4 \\(\\xi\\)\uac00 \uc874\uc7ac\ud558\uc5ec
          \n\\[
          \n\\Delta(h,k)
          \n=
          \nh\\bigl[f_x(\\xi,b+k)-f_x(\\xi,b)\\bigr].
          \n\\]
          \n\ub2e4\uc2dc \\(y\\)\uc5d0 \ub300\ud55c \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74 \\(b\\)\uc640 \\(b+k\\) \uc0ac\uc774\uc758 \uc5b4\ub5a4 \\(\\eta_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec
          \n\\[
          \n\\Delta(h,k)
          \n=
          \nhk\\,f_{xy}(\\xi,\\eta_1)
          \n\\]
          \n\uc774\ub2e4. \uac19\uc740 \ubc29\uc2dd\uc73c\ub85c \\(\\phi(y)=f(a+h,y)-f(a,y)\\)\uc5d0 \uba3c\uc800 \\(y\\)\uc5d0 \ub300\ud55c \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc801\uc6a9\ud55c \ub4a4 \\(x\\)\uc5d0 \ub300\ud55c \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74
          \n\\[
          \n\\Delta(h,k)
          \n=
          \nhk\\,f_{yx}(\\xi_1,\\eta)
          \n\\]
          \n\uc778 \\(\\xi_1,\\eta\\)\ub97c \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c
          \n\\[
          \nf_{xy}(\\xi,\\eta_1)
          \n=
          \nf_{yx}(\\xi_1,\\eta).
          \n\\]
          \n\\(h,k\\to0\\)\uc774\uba74 \\(\\xi,\\xi_1\\to a\\), \\(\\eta,\\eta_1\\to b\\)\uc774\ubbc0\ub85c \ub450 \ud63c\ud569 \ud3b8\ubbf8\ubd84\uc758 \\(c\\)\uc5d0\uc11c\uc758 \uc5f0\uc18d\uc131\uc5d0 \uc758\ud574
          \n\\[
          \nf_{xy}(a,b)
          \n=
          \nf_{yx}(a,b)
          \n\\]
          \n\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

          \n

          \ubb38\uc81c 9.8.<\/span>
          \n\ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc744 \ub54c, \\(f\\)\uc758 \uc774\uacc4\ud3b8\ub3c4\ud568\uc218\ub97c \ubaa8\ub450 \uad6c\ud558\uc2dc\uc624.<\/p>\n

            \n
          1. \\(f(x,y)=xe^y\\)<\/li>\n
          2. \\(f(x,y)=\\cos xy\\)<\/li>\n
          3. \\(f(x,y)=\\dfrac{x+y}{x^2+1}\\)<\/li>\n<\/ol>\n<\/div>\n
            \n

            \ubb38\uc81c 9.9.<\/span>
            \n\\(H=[a,b]\\times[c,d]\\)\uc774\uace0 \\(f\\colon H\\to\\mathbb{R}\\)\uc774 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c
            \n\\[
            \nF(y)=\\int_a^b f(x,y)\\,dx
            \n\\]
            \n\ub77c\uace0 \uc815\uc758\ub41c \ud568\uc218 \\(F\\)\uac00 \\([c,d]\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

            \n

            \ubb38\uc81c 9.10.<\/span>
            \n\\(H=[a,b]\\times[c,d]\\)\uc774\uace0 \ud568\uc218 \\(f\\colon H\\to\\mathbb{R}\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uac01 \\(y\\in[c,d]\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\mapsto f(x,y)\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0, \uac01 \\(x\\in[a,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(y\\mapsto f(x,y)\\)\uac00 \\((c,d)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70, \ud3b8\ub3c4\ud568\uc218 \\(f_y\\)\uac00 \\(H\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(y\\in(c,d)\\)\uc5d0 \ub300\ud558\uc5ec
            \n\\[
            \n\\frac{d}{dy}\\int_a^b f(x,y)\\,dx
            \n=
            \n\\int_a^b\\frac{\\partial f}{\\partial y}(x,y)\\,dx
            \n\\]
            \n\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \ub05d\uc810\uc5d0\uc11c\ub294 \ud574\ub2f9\ud558\ub294 \ud55c\ubc29\ud5a5 \ubbf8\ubd84\uc73c\ub85c \uac19\uc740 \uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774 \uacf5\uc2dd\uc744 \ud3b8\uc801\ubd84\uc758 \ubbf8\ubd84\uc5d0 \ub300\ud55c \ub77c\uc774\ud504\ub2c8\uce20 \uacf5\uc2dd<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

            \ud3c9\uade0\uac12 \uc815\ub9ac\uc640 \ud14c\uc77c\ub7ec \uc815\ub9ac<\/h3>\n
            \n

            \uc815\ub9ac 9.6. (\ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n

            \\(f\\colon U\\to\\mathbb{R}\\)\uac00 \uc5f4\ub9b0 \ubcfc\ub85d\uc9d1\ud569 \\(U\\subseteq\\mathbb{R}^n\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc11c\ub85c \ub2e4\ub978 \\(a,b\\in U\\)\uc5d0 \ub300\ud558\uc5ec \ub450 \uc810\uc744 \uc787\ub294 \uc5f4\ub9b0 \uc120\ubd84 \uc704\uc758 \uc5b4\ub5a4 \\(c\\)\uac00 \uc874\uc7ac\ud558\uc5ec
            \n\\[
            \nf(b)-f(a)
            \n=
            \n\\nabla f(c)\\cdot(b-a)
            \n\\]
            \n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<\/div>\n

            \n

            \uc99d\uba85<\/span>
            \n\\[
            \n\\phi(t)=f(a+t(b-a)),
            \n\\qquad
            \n0\\le t\\le1
            \n\\]
            \n\ub85c \ub450\uba74 \uc5f0\uc1c4\ubc95\uce59\uc5d0 \uc758\ud574
            \n\\[
            \n\\phi'(t)
            \n=
            \n\\nabla f(a+t(b-a))\\cdot(b-a).
            \n\\]
            \n\uc77c\ubcc0\uc218 \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \\(\\phi\\)\uc5d0 \uc801\uc6a9\ud558\uba74 \uc5b4\ub5a4 \\(\\theta\\in(0,1)\\)\uc5d0 \ub300\ud558\uc5ec \\(\\phi(1)-\\phi(0)=\\phi'(\\theta)\\)\uc774\ub2e4. \\(c=a+\\theta(b-a)\\)\ub85c \ub450\uba74 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

            \uc774\ub85c\ubd80\ud130 \ub2e4\uc74c\uacfc \uac19\uc740 \uc911\uc694\ud55c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

            \n

            \ubb38\uc81c 9.11.<\/span>
            \n\ud568\uc218 \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec \uc0c1\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(\\|\\nabla f\\|\\le M\\)\uc774\uba74 \\(f\\)\uac00 \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

            \n

            \ubb38\uc81c 9.12.<\/span>
            \n\uc5f0\uacb0\ub41c \uc5f4\ub9b0\uc9d1\ud569 \\(U\\subseteq\\mathbb{R}^n\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218 \\(f\\colon U\\to\\mathbb{R}\\)\uac00 \\(\\nabla f=0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(f\\)\uac00 \uc0c1\uc218\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \uc5f4\ub9b0 \uc5f0\uacb0\uc9d1\ud569\uc5d0\uc11c \uc784\uc758\uc758 \ub450 \uc810\uc744 \uc720\ud55c \uac1c\uc758 \uc120\ubd84\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uacbd\ub85c\ub85c \uc774\uc744 \uc218 \uc788\uc74c\uc744 \uba3c\uc800 \ubcf4\uc5ec\ub3c4 \uc88b\ub2e4.<\/p>\n<\/div>\n

            \uace0\uacc4\ubbf8\ubd84\uacfc \uad00\ub828\ub41c \ub0b4\uc6a9\uc744 \uae30\uc220\ud560 \ub54c \ub2e4\uc911\uc9c0\ud45c \ud45c\uae30\ubc95<\/span>(multi-index notation)\uc744 \uc0ac\uc6a9\ud558\uba74 \ud3b8\ub9ac\ud558\ub2e4.<\/p>\n

            \\(\\alpha_1,\\alpha_2,\\ldots,\\alpha_n\\)\uc774 \\(0\\) \uc774\uc0c1\uc778 \uc815\uc218\uc77c \ub54c,
            \n\\[
            \n\\alpha=(\\alpha_1,\\alpha_2,\\ldots,\\alpha_n)
            \n\\]
            \n\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n