{"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":"2025-10-21T16:09:20","modified_gmt":"2025-10-21T07:09:20","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,\\, \\cdots,\\, a_n)\\)\uc774 \ud568\uc218 \\(f: \\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\\[\\frac{\\partial f}{\\partial x_i}(a) = \\lim_{h \\to 0} \\frac{f(a_1,\\, \\cdots,\\, a_i + h,\\, \\cdots,\\, a_n) – f(a)}{h}\\]
\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\uc774 \uc544\ub2d0 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4,
\n\\[f(x,\\, y) =
\n\\begin{cases} \\frac{xy}{x^2 + y^2} & \\text{if }\\, (x,\\, y) \\neq (0,\\, 0), \\\\[8pt]
\n0 & \\text{if }\\, (x,\\, y) = (0,\\, 0)
\n\\end{cases}\\]
\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 \ubd88\uc5f0\uc18d\uc774\ub2e4.<\/p>\n

\ud568\uc218 \\(f: \\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: \\mathbb{R}^n \\to \\mathbb{R}^m\\)\uc774 \uc874\uc7ac\ud558\uc5ec
\n\\[\\lim_{h \\to 0} \\frac{\\|f(a + h) – f(a) – L(h)\\|}{\\|h\\|} = 0\\]
\n\uc778 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc774\ub54c \uc120\ud615\ubcc0\ud658 \\(L\\)\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\ud560 \uc870\uac74\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ud560 \uc218\ub3c4 \uc788\ub2e4.
\n\\[f(a + h) = f(a) + Df(a)(h) + o(\\|h\\|).\\]
\n\uc5ec\uae30\uc11c \\(o(\\|h\\|)\\)\ub294 \\(\\|h\\| \\to 0\\)\uc77c \ub54c \\(o(\\|h\\|)\/\\|h\\| \\to 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud56d\uc774\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<\/p>\n

\uc99d\uba85\uc740 \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uacbd\uc6b0\uc640 \uc720\uc0ac\ud558\ub2e4. \\(f\\)\uac00 \\(a\\)\uc5d0\uc11c \uc804\ubbf8\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \\(f(a+h)=f(a)+Df(a)h+r(h)\\), \\(\\|r(h)\\|\/\\|h\\|\\to 0\\)\uc774\ub2e4. \ub530\ub77c\uc11c
\n\\(\\|f(a+h)-f(a)\\|\\le (\\|Df(a)\\|+\\varepsilon(h))\\|h\\|\\to 0\\)\uc774\ubbc0\ub85c, \\(f\\)\ub294 \\(a\\)\uc5d0 \uc5f0\uc18d\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\lVert Df(a)\\rVert\\)\ub294 \uc720\ud074\ub9ac\ub4dc \ub178\ub984\uc5d0 \ub300\ud55c \uc5f0\uc0b0\uc790 \ub178\ub984\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

\ud568\uc218 \\(f: \\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. \uc989 \\(f = (f_1,\\, \\ldots,\\, f_m)\\)\uc77c \ub54c
\n\\[Df(a) = \\begin{pmatrix}
\n\\frac{\\partial f_1}{\\partial x_1}(a) & \\cdots & \\frac{\\partial f_1}{\\partial x_n}(a) \\\\
\n\\vdots & \\ddots & \\vdots \\\\
\n\\frac{\\partial f_m}{\\partial x_1}(a) & \\cdots & \\frac{\\partial f_m}{\\partial x_n}(a)
\n\\end{pmatrix}\\]
\n\uc774\ub2e4. \\(a\\)\uc5d0\uc11c \ubaa8\ub4e0 \ubcc0\uc218\uc5d0 \ub300\ud55c \\(f\\)\uc758 \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\ub974\uace0 \\(Df(a)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud2b9\ud788 \\(a\\)\uc5d0\uc11c \\(f\\)\uac00 \uc804\ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c \uc704 \ud589\ub82c\uc744 \uc804\ubbf8\ubd84 \ud589\ub82c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\ubaa8\ub4e0 \ud3b8\ubbf8\ubd84\uc774 \uc874\uc7ac\ud574\ub3c4 \uc804\ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\uc744 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4
\n\\[f(x,\\,y)=
\n\\begin{cases}
\n\\frac{x^3 -xy^2}{x^2 +y^2} & \\quad\\text{if }\\, (x,\\,y)\\neq (0,\\,0) ,\\\\[6pt]
\n0 & \\quad\\text{if }\\, (x,\\,y) = (0,\\,0)
\n\\end{cases}\\]
\n\uc774\ub77c\uace0 \ud558\uba74 \ub450 \ud3b8\ubbf8\ubd84\uacc4\uc218 \\(f_x (0,\\,0)\\)\uacfc \\(f_y (0,\\,0)\\)\uc774 \ubaa8\ub450 \uc874\uc7ac\ud558\uc9c0\ub9cc, \\(f\\)\ub294 \\((0,\\,0)\\)\uc5d0\uc11c \ubbf8\ubd84 \ubd88\uac00\ub2a5\ud558\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: \\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 \uac1c\uc694<\/span>
\n\uac04\ub2e8\ud788 \\(f: \\mathbb{R}^2 \\to \\mathbb{R}\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ub4f1\uc2dd
\n\\[\\begin{aligned}
\nf(a + h) – f(a) &= f(a_1 + h_1,\\, a_2 + h_2) – f(a_1,\\, a_2) \\\\[6pt]
\n&= [f(a_1 + h_1,\\, a_2 + h_2) – f(a_1,\\, a_2 + h_2)] \\\\[6pt]
\n&\\quad + [f(a_1,\\, a_2 + h_2) – f(a_1,\\, a_2)]
\n\\end{aligned}\\]
\n\uc5d0\uc11c \uac01 \ud56d\uc5d0 \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uace0, \ud3b8\ubbf8\ubd84\uc758 \uc5f0\uc18d\uc131\uc744 \uc0ac\uc6a9\ud558\uba74 \ubc14\ub77c\ub294 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

\n

\ubb38\uc81c 9.1.<\/span>
\n\ud568\uc218 \\(f:\\mathbb{R}^2\\rightarrow\\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\\[f(x,\\,y) = ({a_{11}}x+{a_{12}}y,\\,{a_{21}}x+{a_{22}}y,\\,{a_{31}}x+{a_{32}}y).\\]<\/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)=\\left( \\cos xy ,\\, e^y – \\ln y \\right)\\)\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\\[f(x,\\,y)=
    \n\\begin{cases}
    \n\\frac{y^2}{x^2+y^2} & \\quad\\text{if }\\, (x,\\,y)\\neq (0,\\,0),\\\\[6pt]
    \n0 & \\quad\\text{if }\\, (x,\\,y)= (0,\\,0).
    \n\\end{cases}\\]<\/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:D\\rightarrow\\mathbb{R}^m\\)\uacfc \\(g:D\\rightarrow\\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 \\(D(f \\cdot g)(a)
      \n= g(a)^{\\top} Df(a) + f(a)^{\\top} Dg(a)\\)\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: \\mathbb{R}^n \\to \\mathbb{R}^m\\)\uc774 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(g: \\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\\[D(g \\circ f)(a) = Dg(f(a)) \\cdot Df(a).\\]<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \ub2e4\uc74c \ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130 \uace7\ubc14\ub85c \uc5bb\ub294\ub2e4.
      \n\\[\\begin{align*}
      \ng(f(a + h)) – g(f(a)) &= Dg(f(a))[f(a + h) – f(a)] + o(\\|f(a + h) – f(a)\\|) \\\\[6pt]
      \n&= Dg(f(a))[Df(a)(h) + o(\\|h\\|)] + o(\\|Df(a)(h) + o(\\|h\\|)\\|) \\\\[6pt]
      \n&= Dg(f(a)) \\cdot Df(a)(h) + o(\\|h\\|).\\tag*{\\(\\blacksquare\\)}
      \n\\end{align*}\\]<\/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\\[\\frac{\\partial w}{\\partial r} = \\frac{\\partial w}{\\partial x}\\frac{\\partial x}{\\partial r} + \\frac{\\partial w}{\\partial y}\\frac{\\partial y}{\\partial r} + \\frac{\\partial w}{\\partial z}\\frac{\\partial z}{\\partial r},\\quad
      \n\\frac{\\partial w}{\\partial s} = \\frac{\\partial w}{\\partial x}\\frac{\\partial x}{\\partial s} + \\frac{\\partial w}{\\partial y}\\frac{\\partial y}{\\partial s} + \\frac{\\partial w}{\\partial z}\\frac{\\partial z}{\\partial s}.
      \n\\]<\/p>\n<\/div>\n

      \uc810 \\(a\\)\uc5d0\uc11c \ud568\uc218 \\(f: \\mathbb{R}^n \\to \\mathbb{R}^m\\)\uc758 \ubbf8\ubd84 \\(Df(a)\\)\ub97c \uc0dd\uac01\ud558\uc790. \\(m = 1\\)\uc778 \uacbd\uc6b0 \uc774 \ud589\ub82c\uc740 \\(n\\)\ucc28\uc6d0 \ud589\ubca1\ud130\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774 \ubca1\ud130\ub97c \uae30\uc6b8\uae30 \ubca1\ud130<\/span>(gradient vector)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\nabla f\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989
      \n\\[\\nabla f = \\left(\\frac{\\partial f}{\\partial x_1},\\, \\ldots,\\, \\frac{\\partial f}{\\partial x_n}\\right).\\]<\/p>\n

      \uae30\uc6b8\uae30 \ubca1\ud130 \\(\\nabla f\\)\ub294 \\(x\\)\uac00 \\(a\\)\uc5d0\uc11c \ucd9c\ubc1c\ud558\uc5ec \uc6c0\uc9c1\uc77c \ub54c \ud568\uc22b\uac12 \\(f(x)\\)\uac00 \uac00\uc7a5 \ube60\ub974\uac8c \uc99d\uac00\ud558\ub294 \ubc29\ud5a5\uc744 \ub098\ud0c0\ub0b8\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\\ne 0\\)\uc778 \uc810\uc5d0\uc11c \\(\\nabla(f\/g) = (g\\nabla f – f\\nabla g)\/(g^2)\\).<\/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 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
        \n\\[D_v f(a) = \\lim_{t \\to 0} \\frac{f(a + tv) – f(a)}{t}.\\]
        \n\uc774 \ubc29\ud5a5\ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.
        \n\\[D_v f(a) = \\nabla f(a) \\cdot v\\tag{9.1}\\]
        \n\ubc29\ud5a5\ub3c4\ud568\uc218\ub294 \\(v\\) \ubc29\ud5a5\uc73c\ub85c\uc758 \\(f\\)\uc758 \ubcc0\ud654\uc728\uc774\ub2e4.<\/p>\n

        \n

        \ubb38\uc81c 9.6.<\/span>
        \n\ubc29\ud5a5\ub3c4\ud568\uc218 \uacf5\uc2dd (9.1)\uc744 \uc99d\uba85\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 \ub2e4\uc74c\uacfc \uac19\uc740 \ud568\uc218\ub85c \ud45c\ud604\ub41c\ub2e4\uace0 \ud558\uc790.
        \n\\[r(t) = (x(t),\\,y(t),\\,z(t)) ,\\,\\, t\\in I.\\]
        \n\uc774\ub54c \uc810 \\(t_0\\)\uc5d0\uc11c \\(r\\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
        \n\\[r'(t_0 ) = \\left( \\frac{d}{dt}x(t_0) ,\\,\\, \\frac{d}{dt}y(t_0) ,\\,\\, \\frac{d}{dt}z(t_0) \\right).\\]
        \n\uace1\uc120 \\(C\\) \uc704\uc758 \uc810\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 \\(w=f(r(t))\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
        \n\\[\\frac{dw}{dt} = \\frac{\\partial w}{\\partial x}\\frac{dx}{dt} + \\frac{\\partial w}{\\partial y} \\frac{dy}{dt} + \\frac{\\partial w}{\\partial z}\\frac{dz}{dt}. \\]
        \n\uc774\uac83\uc744 \uae30\uc6b8\uae30 \uc5f0\uc0b0\uc790\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
        \n\\[\\frac{d}{dt} f(r(t)) = \\nabla f(r(t)) \\cdot r'(t).\\]<\/p>\n

        \n

        \ubb38\uc81c 9.7.<\/span>
        \n\ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218 \\(F:\\mathbb{R}^2 \\rightarrow\\mathbb{R}\\)\uacfc \uc0c1\uc218 \\(c\\)\uc5d0 \ub300\ud558\uc5ec, \ub4f1\uc704\uace1\uc120<\/span> \\(F(x,\\,y)=c\\)\ub97c \uc0dd\uac01\ud558\uc790. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

          \n
        1. \uace1\uc120 \\(F(x,\\,y)=c\\)\uac00 \ub9e4\uac1c\ubcc0\uc218 \\(t\\)\uc5d0 \ub300\ud55c \ud568\uc218 \\(r(t) = (f(t),\\,h(t))\\), \\(a\\le t\\le b\\)\ub85c \ud45c\ud604\ub41c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ub4f1\uc704\uace1\uc120 \uc704\uc758 \uc810\uc5d0\uc11c \\(\\nabla F\\)\uc640 \\(dr\/dt\\)\uac00 \uc11c\ub85c \uc218\uc9c1\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
        2. \ub4f1\uc704\uace1\uc120 \\(F(x,\\,y)=c\\) \uc704\uc758 \uc810 \\((a ,\\, b)\\)\uc5d0\uc11c \uc774 \uace1\uc120\uc5d0 \uc811\ud558\ub294 \uc9c1\uc120\uc758 \ubc29\uc815\uc2dd\uc744 \uc720\ub3c4\ud558\uc2dc\uc624.
          \n\\[ \\frac{\\partial }{\\partial x}F(a,\\,b) (x-a) + \\frac{\\partial}{\\partial y}F(a,\\,b)(y-b)=0.\\]<\/li>\n<\/ol>\n<\/div>\n

          \ud568\uc218 \\(f:\\mathbb{R}^n\\rightarrow\\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 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.
          \n\\[f_{x_i \\,x_j} \\quad\\text{\ub610\ub294}\\quad \\frac{\\partial^2}{\\partial x_j \\,\\partial x_i} f .\\]
          \n\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 9.4. (\ud074\ub808\ub85c \uc815\ub9ac)<\/span><\/p>\n

          \\(f: \\mathbb{R}^n \\to \\mathbb{R}\\)\uc758 \uc774\uacc4\ud3b8\ubbf8\ubd84 \\(\\frac{\\partial^2 f}{\\partial x_i \\partial x_j}\\)\uc640 \\(\\frac{\\partial^2 f}{\\partial x_j \\partial x_i}\\)\uac00 \\(c\\) \uadfc\ubc29\uc5d0\uc11c \uc874\uc7ac\ud558\uace0 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74
          \n\\[\\frac{\\partial^2 f}{\\partial x_i \\,\\partial x_j}(c) = \\frac{\\partial^2 f}{\\partial x_j \\,\\partial x_i}(c).\\]<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/p>\n

          \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(n = 2\\)\uc774\uace0 \\(i = 1\\), \\(j = 2\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud55c\ub2e4. \uc989, \\(f: \\mathbb{R}^2 \\to \\mathbb{R}\\)\uc5d0 \ub300\ud574
          \n\\[\\frac{\\partial^2 f}{\\partial x \\,\\partial y}(a,\\, b) = \\frac{\\partial^2 f}{\\partial y \\,\\partial x}(a,\\, b)\\]
          \n\uc784\uc744 \ubcf4\uc774\uc790. \uc5ec\uae30\uc11c \\(c = (a,\\, b)\\)\uc774\ub2e4.<\/p>\n

          \uc808\ub313\uac12\uc774 \ucda9\ubd84\ud788 \uc791\uace0 \\(0\\)\uc544\ub2cc \\(h\\), \\(k\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \ucc28\ubd84\uc744 \uc815\uc758\ud55c\ub2e4.
          \n\\[\\Delta(h,\\, k) = f(a+h,\\, b+k) – f(a+h,\\, b) – f(a,\\, b+k) + f(a,\\, b).\\]
          \n\\(g(x) = f(x,\\, b+k) – f(x,\\, b)\\)\ub77c\uace0 \uc815\uc758\ud558\uba74
          \n\\[\\Delta(h,\\, k) = g(a+h) – g(a)\\]
          \n\uc774\ubbc0\ub85c, \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574, \uc801\ub2f9\ud55c \\(\\xi \\in (a,\\, a+h)\\)\uac00 \uc874\uc7ac\ud558\uc5ec
          \n\\[\\Delta(h,\\, k) = h \\cdot g'(\\xi) = h \\cdot \\left[\\frac{\\partial f}{\\partial x}(\\xi,\\, b+k) – \\frac{\\partial f}{\\partial x}(\\xi,\\, b)\\right]\\]
          \n\uc774\ub2e4. \ub2e4\uc2dc \\(\\frac{\\partial f}{\\partial x}(\\xi,\\, y)\\)\uc5d0 \\(y\\)\uc5d0 \ub300\ud55c \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74, \uc801\ub2f9\ud55c \\(\\eta_1 \\in (b,\\, b+k)\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.
          \n\\[\\Delta(h,\\, k) = hk \\cdot \\frac{\\partial^2 f}{\\partial y \\partial x}(\\xi,\\, \\eta_1).\\]<\/p>\n

          \uc774\ubc88\uc5d0\ub294 \\(\\phi(y) = f(a+h,\\, y) – f(a,\\, y)\\)\ub77c\uace0 \uc815\uc758\ud558\uba74
          \n\\[\\Delta(h,\\, k) = \\phi(b+k) – \\phi(b)\\]
          \n\uc774\ubbc0\ub85c, \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574, \uc801\ub2f9\ud55c \\(\\eta \\in (b,\\, b+k)\\)\uac00 \uc874\uc7ac\ud558\uc5ec
          \n\\[\\Delta(h,\\, k) = k \\cdot \\phi'(\\eta) = k \\cdot \\left[\\frac{\\partial f}{\\partial y}(a+h,\\, \\eta) – \\frac{\\partial f}{\\partial y}(a,\\, \\eta)\\right]\\]
          \n\uc774\ub2e4. \ub2e4\uc2dc \\(\\frac{\\partial f}{\\partial y}(x,\\, \\eta)\\)\uc5d0 \\(x\\)\uc5d0 \ub300\ud55c \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74, \uc801\ub2f9\ud55c \\(\\xi_1 \\in (a,\\, a+h)\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.
          \n\\[\\Delta(h,\\, k) = hk \\cdot \\frac{\\partial^2 f}{\\partial x \\partial y}(\\xi_1,\\, \\eta).\\]<\/p>\n

          \ub450 \uacb0\uacfc\ub97c \ube44\uad50\ud558\uba74
          \n\\[hk \\cdot \\frac{\\partial^2 f}{\\partial y \\partial x}(\\xi,\\, \\eta_1) = hk \\cdot \\frac{\\partial^2 f}{\\partial x \\partial y}(\\xi_1,\\, \\eta)\\]
          \n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(h \\neq 0\\), \\(k \\neq 0\\)\uc774\ubbc0\ub85c
          \n\\[\\frac{\\partial^2 f}{\\partial y \\partial x}(\\xi,\\, \\eta_1) = \\frac{\\partial^2 f}{\\partial x \\partial y}(\\xi_1,\\, \\eta)\\]
          \n\uc774\ub2e4.
          \n\\(h \\to 0\\), \\(k \\to 0\\)\uc77c \ub54c \\(\\xi \\to a\\), \\(\\xi_1 \\to a\\)\uc774\uace0 \\(\\eta \\to b\\), \\(\\eta_1 \\to b\\)\uc774\ubbc0\ub85c \uc774\uacc4\ud3b8\ub3c4\ud568\uc218\uc758 \uc5f0\uc18d\uc131\uc5d0 \uc758\ud574
          \n\\[\\frac{\\partial^2 f}{\\partial y \\partial x}(\\xi,\\, \\eta_1) \\to \\frac{\\partial^2 f}{\\partial y \\partial x}(a,\\, b),\\]
          \n\\[\\frac{\\partial^2 f}{\\partial x \\partial y}(\\xi_1,\\, \\eta) \\to \\frac{\\partial^2 f}{\\partial x \\partial y}(a,\\, b)\\]
          \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)=\\frac{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:H\\rightarrow\\mathbb{R}\\)\uc774 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c
            \n\\[F(y)=\\int_a^b f(x,\\,y)dx\\]
            \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:H\\rightarrow\\mathbb{R}\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uac01 \\(y\\in [c,\\,d]\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\)\ub97c \ubcc0\uc218\ub85c \ud558\ub294 \ud568\uc218 \\(f(x,\\,y)\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0, \uac01 \\(x\\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(y\\)\ub97c \ubcc0\uc218\ub85c \ud558\ub294 \ud568\uc218 \\(f(x,\\,y)\\)\uac00 \\([c,\\,d]\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70, \\(x\\), \\(y\\)\ub97c \ubaa8\ub450 \ubcc0\uc218\ub85c \ud558\ub294 \ud3b8\ub3c4\ud568\uc218 \\(f_y (x,\\,y)\\)\uac00 \\(H\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \uc784\uc758\uc758 \\(y\\in [c,\\,d]\\)\uc5d0 \ub300\ud558\uc5ec
            \n\\[\\frac{d}{dy}\\int_a^b f(x,\\,y)dx = \\int_a^b \\frac{\\partial f}{\\partial y}(x,\\,y)dx\\]
            \n\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \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 9.5. (\ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n

            \\(f: \\mathbb{R}^n \\to \\mathbb{R}\\)\uc774 \uc5f4\ub9b0 \ubcfc\ub85d\uc9d1\ud569 \\(U\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74, \uc784\uc758\uc758 \\(a,\\, b \\in U\\)\uc5d0 \ub300\ud574 \uc5b4\ub5a4 \\(c \\in [a,\\, b]\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(f(b) – f(a) = \\nabla f(c) \\cdot (b – a)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc5ec\uae30\uc11c \\([a,\\,b]\\)\ub294 \ub450 \uc810 \\(a\\), \\(b\\)\ub97c \uc787\ub294 \uc120\ubd84\uc774\ub2e4.<\/p>\n<\/div>\n

            \n

            \uc99d\uba85<\/p>\n

            \ud568\uc218 \\(\\phi(t) = f(a + t(b – a))\\)\uc5d0 \uc77c\ubcc0\uc218 \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74 \ub41c\ub2e4. \\(\\phi'(t) = \\nabla f(a + t(b – a)) \\cdot (b – a)\\)\uc774\ubbc0\ub85c \uc815\ub9ac\uc758 \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: \\mathbb{R}^n \\to \\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec \uc0c1\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(\\|\\nabla f\\| \\leq 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\uc5d0\uc11c \\(\\nabla f = 0\\)\uc774\uba74 \\(f\\)\uac00 \uc0c1\uc218\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

            \ud568\uc218 \\(f\\)\uac00 \uc5f0\uc18d\uc778 \\(k\\)\uacc4\ub3c4\ud568\uc218\ub97c \uac00\uc9c8 \ub54c \\(f \\in C^k\\)\ub77c\uace0 \uc4f4\ub2e4. \\(f \\in C^\\infty\\)\uc778 \ud568\uc218\ub97c \ub9e4\ub044\ub7ec\uc6b4 \ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\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 = (\\alpha_1,\\, \\ldots,\\, \\alpha_n) \\in \\mathbb{N}^n\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n

              \n
            • \\(|\\alpha| = \\alpha_1 + \\cdots + \\alpha_n\\)<\/li>\n
            • \\(\\alpha! = \\alpha_1! \\cdots \\alpha_n!\\)<\/li>\n
            • \\(x^\\alpha = x_1^{\\alpha_1} \\cdots x_n^{\\alpha_n}\\)<\/li>\n
            • \\(D^\\alpha = \\frac{\\partial^{|\\alpha|}}{\\partial x_1^{\\alpha_1} \\cdots \\partial x_n^{\\alpha_n}}\\)<\/li>\n<\/ul>\n
              \n

              \uc815\ub9ac 9.6. (\ud14c\uc77c\ub7ec \uc815\ub9ac)<\/span><\/p>\n

              \ud568\uc218 \\(f: \\mathbb{R}^n \\to \\mathbb{R}\\)\uc774 \uc120\ubd84 \\([a,\\,h]\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc5f4\ub9b0 \uc601\uc5ed\uc5d0\uc11c \\(C^{k+1}\\)\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
              \n\\[f(a + h) = \\sum_{|\\alpha| \\leq k} \\frac{1}{\\alpha!} D^\\alpha f(a) h^\\alpha + R_k(h).\\]
              \n\uc5ec\uae30\uc11c \\(\\alpha = (\\alpha_1,\\, \\ldots,\\, \\alpha_n)\\)\uc740 \ub2e4\uc911\uc9c0\ud45c\uc774\uba70, \ub098\uba38\uc9c0\ub294 \\(R_k(h) = o(\\|h\\|^k)\\)\uc774\ub2e4.<\/p>\n<\/div>\n

              \n

              \uc99d\uba85<\/p>\n

              \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud55c\ub2e4.
              \n\\(\\phi(t) = f(a + th)\\)\ub77c\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \\(t \\in [0,\\, 1]\\)\uc774\uace0 \\(h\\)\ub294 \uace0\uc815\ub41c \ubca1\ud130\uc774\ub2e4. \uc5f0\uc1c4\ubc95\uce59\uc744 \uc0ac\uc6a9\ud558\uba74 \\(\\phi\\)\uc758 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
              \n\\[\\phi'(t) = \\sum_{i=1}^{n} \\frac{\\partial f}{\\partial x_i}(a + th) h_i = \\nabla f(a + th) \\cdot h .\\]
              \n\uc77c\ubc18\uc801\uc73c\ub85c \\(\\phi\\)\uc758 \\(m\\)\uacc4 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
              \n\\[\\phi^{(m)}(t) = \\sum_{|\\alpha| = m} \\frac{m!}{\\alpha!} D^\\alpha f(a + th) h^\\alpha.\\]
              \n\\(\\phi\\)\uc5d0 \uc77c\ubcc0\uc218 \ud14c\uc77c\ub7ec \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74, \uc801\ub2f9\ud55c \\(\\theta \\in (0,\\, 1)\\)\uc5d0 \ub300\ud558\uc5ec
              \n\\[\\phi(1) = \\sum_{m=0}^{k} \\frac{\\phi^{(m)}(0)}{m!} + \\frac{\\phi^{(k+1)}(\\theta)}{(k+1)!}\\]
              \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \\(\\phi(1) = f(a + h)\\), \\(\\phi(0) = f(a)\\)\uc774\ubbc0\ub85c \uc55e\uc758 \uacb0\uacfc\ub97c \ud65c\uc6a9\ud558\uba74
              \n\\[\\begin{aligned}
              \nf(a + h) &= \\sum_{m=0}^{k} \\sum_{|\\alpha| = m} \\frac{1}{\\alpha!} D^\\alpha f(a) h^\\alpha + R_k(h) ,\\\\[6pt]
              \nR_k(h) &= \\sum_{|\\alpha| = k+1} \\frac{1}{\\alpha!} D^\\alpha f(a + \\theta h) h^\\alpha
              \n\\end{aligned}\\]
              \n\uc774\ub2e4. \uc774\uc81c \ub098\uba38\uc9c0\ud56d\uc758 \ud06c\uae30\ub97c \ucd94\uc815\ud558\uc790. \\(D^\\alpha f\\)\uac00 \\(a\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(\\lVert h \\rVert \\rightarrow 0\\)\uc77c \ub54c
              \n\\[|D^\\alpha f(a + \\theta h) – D^\\alpha f(a)| \\to 0\\]
              \n\uc774\ub2e4. \\(|h^\\alpha| \\leq \\|h\\|^{|\\alpha|}\\)\uc774\ubbc0\ub85c
              \n\\[|R_k(h)| \\leq C \\|h\\|^{k+1}\\]
              \n\uc778 \uc0c1\uc218 \\(C\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \ub530\ub77c\uc11c \\(|R_k(h)|\\le C\\|h\\|^{k+1}=o(\\|h\\|^{k})\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 9.13.<\/span>
              \n\uc810 \\(a\\)\uc640 \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc744 \ub54c, \\(a\\)\ub97c \uc911\uc2ec\uc73c\ub85c \ud558\ub294 \\(f\\)\uc758 \\(3\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

                \n
              1. \\(f(x,\\,y)=\\sqrt{x}+\\sqrt{y}\\), \\(a=(1,\\,4)\\).<\/li>\n
              2. \\(f(x,\\,y)=e^{xy}\\), \\(a=(0,\\,0)\\).<\/li>\n<\/ol>\n<\/div>\n

                \uc810 \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 2\ucc28 \ud14c\uc77c\ub7ec \uc804\uac1c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
                \n\\[f(a + h) = f(a) + \\nabla f(a) \\cdot h + \\frac{1}{2} h^T H_f(a) h + o(\\|h\\|^2).\\]
                \n\uc5ec\uae30\uc11c \\(H_f(a)\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\ub294 \ud5e4\uc138 \ud589\ub82c<\/span>(Hessian matrix)\uc774\ub2e4.
                \n\\[H_f(a) = \\left(\\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(a)\\right)_{i,j}.\\tag{9.2}\\]
                \n\ud074\ub808\ub85c\uc758 \uc815\ub9ac(\uc815\ub9ac 9.4)\uc5d0 \uc758\ud558\uc5ec \ud5e4\uc138 \ud589\ub82c\uc740 \ub300\uce6d\ud589\ub82c\uc774\ub2e4.<\/p>\n

                2\ucc28 \ud14c\uc77c\ub7ec \uc804\uac1c\ub97c \uc0ac\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uadf9\uac12\uc5d0 \ub300\ud55c \uc774\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95<\/span>\uc744 \uc720\ub3c4\ud560 \uc218 \uc788\ub2e4.<\/p>\n

                \ud568\uc218 \\(f: \\mathbb{R}^n \\to \\mathbb{R}\\)\uc774 \\(a\\)\uc5d0\uc11c \uadf9\uac12\uc744 \uac00\uc9c0\uace0 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(\\nabla f(a) = 0\\)\uc774\ub2e4. \uc774\ub7ec\ud55c \uc810 \\(a\\)\ub97c \uc784\uacc4\uc810<\/span>(critical point)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

                \n

                \uc815\ub9ac 9.7. (\uc774\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

                \ud568\uc218 \\(f\\)\uac00 \\(a\\)\uc5d0\uc11c \ub450 \ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0, \\(\\nabla f(a)=0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

                  \n
                1. \\(H_f(a)\\)\uac00 \uc591\uc758 \uc815\ubd80\ud638\uc774\uba74 \\(f\\)\ub294 \\(a\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n
                2. \\(H_f(a)\\)\uac00 \uc74c\uc758 \uc815\ubd80\ud638\uc774\uba74 \\(f\\)\ub294 \\(a\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n
                3. \\(H_f(a)\\)\uac00 \ubd80\uc815\ubd80\ud638\uc774\uba74 \\(f\\)\ub294 \\(a\\)\uc5d0\uc11c \uc548\uc7a5\uc810\uc744 \uac00\uc9c4\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
                  \n

                  \uc99d\uba85<\/p>\n

                  2\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uc790.
                  \n\\(\\nabla f(a) = 0\\)\uc774\ubbc0\ub85c \ud14c\uc77c\ub7ec \uc815\ub9ac\uc5d0 \uc758\ud574
                  \n\\[f(a + h) – f(a) = \\frac{1}{2} h^T H_f(a) h + o(\\|h\\|^2)\\]
                  \n\uc774\ub2e4. \uc774\uc81c \\(H_f (a)\\)\uc758 \ud2b9\uc131\uc5d0 \ub530\ub77c \uacbd\uc6b0\ub97c \ub098\ub204\uc5b4 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

                  \\(H_f(a)\\)\uac00 \uc591\uc758 \uc815\ubd80\ud638\uc778 \uacbd\uc6b0, \ud5e4\uc138 \ud589\ub82c \\(H_f(a)\\)\uac00 \ub300\uce6d\uc774\ubbc0\ub85c \ubaa8\ub4e0 \uace0\uc733\uac12\uc774 \uc591\uc218\uc774\ub2e4. \\(H_f (a)\\)\uc758 \ucd5c\uc18c\uace0\uc733\uac12\uc744 \\(\\lambda_{\\min} > 0\\)\uc774\ub77c\uace0 \ud558\uba74
                  \n\\[h^T H_f(a) h \\geq \\lambda_{\\min} \\|h\\|^2\\]
                  \n\uc774\ub2e4. \ucda9\ubd84\ud788 \uc791\uc740 \\(\\delta > 0\\)\uc5d0 \ub300\ud574 \\(\\|h\\| < \\delta\\)\uc77c \ub54c\n\\[\\left|\\frac{o(\\|h\\|^2)}{\\|h\\|^2}\\right| < \\frac{\\lambda_{\\min}}{4}\\]\n\uc774\ubbc0\ub85c\n\\[f(a + h) - f(a) \\geq \\frac{1}{2}\\lambda_{\\min}\\|h\\|^2 - \\frac{\\lambda_{\\min}}{4}\\|h\\|^2 = \\frac{\\lambda_{\\min}}{4}\\|h\\|^2 > 0\\]
                  \n\uc774\ub2e4. \ub530\ub77c\uc11c \\(0 < \\|h\\| < \\delta\\)\uc778 \ubaa8\ub4e0 \\(h\\)\uc5d0 \ub300\ud574 \\(f(a + h) > f(a)\\)\uc774\ubbc0\ub85c \\(a\\)\ub294 \uadf9\uc19f\uac12\uc774\ub2e4.<\/p>\n

                  \ub2e4\uc74c\uc73c\ub85c \\(H_f(a)\\)\uac00 \uc74c\uc758 \uc815\ubd80\ud638\uc778 \uacbd\uc6b0, \\(-f\\)\uc5d0 \uc55e\uc758 \uacb0\uacfc\ub97c \uc801\uc6a9\ud558\uba74 \ub41c\ub2e4. \ucd5c\ub300\uace0\uc733\uac12\uc744 \\(\\lambda_{\\max} < 0\\)\uc774\ub77c \ud558\uba74 \uc720\uc0ac\ud55c \ub17c\uc99d\uc73c\ub85c \\(f(a + h) < f(a)\\)\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

                  \ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(H_f(a)\\)\uac00 \ubd80\uc815\ubd80\ud638\uc778 \uacbd\uc6b0, \uc591\uc758 \uace0\uc733\uac12 \\(\\lambda_+ > 0\\)\uacfc \uc74c\uc758 \uace0\uc733\uac12 \\(\\lambda_- < 0\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \ub300\uc751\ud558\ub294 \ub2e8\uc704\uace0\uc720\ubca1\ud130\ub97c \uac01\uac01 \\(v_+\\), \\(v_-\\)\ub77c \ud558\uc790. \\(h = tv_+\\) \ubc29\ud5a5\uc73c\ub85c\ub294 \ucda9\ubd84\ud788 \uc791\uc740 \\(t > 0\\)\uc5d0 \ub300\ud574
                  \n\\[f(a + tv_+) – f(a) = \\frac{t^2}{2}\\lambda_+ + o(t^2) > 0\\]
                  \n\uc774\uace0, \\(h = tv_-\\) \ubc29\ud5a5\uc73c\ub85c\ub294 \ucda9\ubd84\ud788 \uc791\uc740 \\(t > 0\\)\uc5d0 \ub300\ud574
                  \n\\[f(a + tv_-) – f(a) = \\frac{t^2}{2}\\lambda_- + o(t^2) < 0\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(a\\)\uc758 \uc784\uc758\uc758 \uadfc\ubc29\uc5d0\uc11c \\(f(x) > f(a)\\)\uc778 \uc810 \\(x\\)\uc640 \\(f(x) < f(a)\\)\uc778 \uc810 \\(x\\)\uac00 \ubaa8\ub450 \uc874\uc7ac\ud558\ubbc0\ub85c, \\(f\\)\ub294 \\(a\\)\uc5d0\uc11c \uadf9\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

                  \n

                  \ubb38\uc81c 9.14.<\/span>
                  \n\ub2e4\uc74c\uacfc \uac19\uc740 \ud568\uc218 \\(f\\)\uc758 \uadf9\uac12\uc744 \ubaa8\ub450 \uad6c\ud558\uc2dc\uc624.<\/p>\n

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

                    \ubb38\uc81c 9.15.<\/span>
                    \n\ud568\uc218 \\(f\\)\uc640 \uc9d1\ud569 \\(H\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(H\\) \uc704\uc5d0\uc11c \\(f\\)\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

                      \n
                    1. \\(f(x,\\,y) = x^2 + 2x – y^2\\), \\(H = \\left\\{ (x,\\,y) \\mid x^2 + 4y^2 \\le 4 \\right\\}\\).<\/li>\n
                    2. \\(f(x,\\,y) = x^2 + 2xy + 3y^2\\), \\(H\\)\ub294 \uc138 \uc810 \\((1,\\,0)\\), \\((1,\\,2)\\), \\((3,\\,0)\\)\uc744 \uc787\ub294 \uc0bc\uac01\ud615\uc758 \uacbd\uacc4\uc640 \ub0b4\ubd80.<\/li>\n
                    3. \\(f(x,\\,y) = x^3 + 3xy – y^3\\), \\(H = [-1,\\,1]^2\\).<\/li>\n<\/ol>\n<\/div>\n

                      \uc774\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95\uc5d0\uc11c \ud5e4\uc138 \ud589\ub82c\uc774 \ubc18\uc815\ubd80\ud638(positive semidefinite \ub610\ub294 negative semidefinite)\uc778 \uacbd\uc6b0\ub294 \ud310\uc815\uc774 \ubd88\uac00\ub2a5\ud558\ub2e4. \ub2e4\uc74c \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

                        \n
                      • \\(f(x,\\, y) = x^4 + y^4\\)\ub294 \uc6d0\uc810\uc5d0\uc11c \\(H_f(0,\\, 0) = 0\\)\uc774\uc9c0\ub9cc \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n
                      • \\(f(x,\\, y) = x^4 – y^4\\)\ub294 \uc6d0\uc810\uc5d0\uc11c \\(H_f(0,\\, 0) = 0\\)\uc774\uace0, \uc810 \\((0,\\,0,\\,0)\\)\uc740 \uace1\uba74 \\(z=f(x,\\,y)\\)\uc758 \uc548\uc7a5\uc810\uc774\ub2e4.<\/li>\n
                      • \\(f(x,\\, y) = x^3 + y^3\\)\uc740 \uc6d0\uc810\uc5d0\uc11c \\(H_f(0,\\, 0) = 0\\)\uc774\uace0, \uc810 \\((0,\\,0,\\,0)\\)\uc740 \uace1\uba74 \\(z=f(x,\\,y)\\)\uc758 \uc548\uc7a5\uc810\uc774\ub2e4.<\/li>\n<\/ul>\n

                        \uc74c\ud568\uc218 \uc815\ub9ac\uc640 \uc5ed\ud568\uc218 \uc815\ub9ac<\/h3>\n

                        \\(F(x,\\,y) = x^2 + y^2 -25\\)\ub77c\uace0 \ud558\uba74 \uc6d0\uc758 \ubc29\uc815\uc2dd \\(F(x,\\,y)=0\\)\uc5d0\uc11c \\(y\\)\ub294 \\(x\\)\uc758 \ud568\uc218\uac00 \uc544\ub2c8\ub2e4. \uadf8\ub7ec\ub098 \uc6d0 \uc704\uc758 \uc810 \\((3,\\,-4)\\)\ub97c \ud3ec\ud568\ud558\ub294 \\(y < 0\\)\uc778 \ubc94\uc704\ub97c \ucde8\ud558\uba74, \\(F(x,\\,y)=0\\)\ub294 \\(y=-\\sqrt{25-x^2}\\)\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc73c\uba70, \\(y\\)\ub294 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uac00 \ub41c\ub2e4.<\/p>\n

                        \uc774\ub7ec\ud55c \uacb0\uacfc\ub97c \ub354 \ub192\uc740 \ucc28\uc6d0\uc73c\ub85c \uc77c\ubc18\ud654\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

                        \n

                        \uc815\ub9ac 9.8. (\uc74c\ud568\uc218 \uc815\ub9ac)<\/span><\/p>\n

                        \ud568\uc218 \\(F: \\mathbb{R}^{n+m} \\to \\mathbb{R}^m\\)\uc774 \uc810 \\((a,\\, b) \\in \\mathbb{R}^n \\times \\mathbb{R}^m\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \\(C^1\\)\uc774\uace0, \\(F(a,\\, b) = 0\\)\uc774\uba70, \\(y\\)\uc5d0 \ub300\ud55c \uc57c\ucf54\ube44 \ud589\ub82c
                        \n\\[\\frac{\\partial F}{\\partial y}(a,\\, b) = \\left(\\frac{\\partial F_i}{\\partial y_j}(a,\\, b)\\right)\\]
                        \n\uac00 \uac00\uc5ed\uc774\uba74, \\(a\\)\uc758 \ud55c \uadfc\ubc29 \\(U\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 \\(g: \\mathbb{R}^n \\to \\mathbb{R}^m\\)\uc774 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n

                          \n
                        • \\(g(a) = b\\).<\/li>\n
                        • \uc784\uc758\uc758 \\(x\\in U\\)\uc5d0 \ub300\ud558\uc5ec \\(F(x,\\, g(x)) = 0\\)\uc774\ub2e4.<\/li>\n
                        • \\(g\\)\ub294 \\(C^1\\)\uc774\uace0 \\(\\frac{\\partial g}{\\partial x}(x)
                          \n= -\\Big[\\frac{\\partial F}{\\partial y}\\big(x,\\, g(x)\\big)\\Big]^{-1}
                          \n \\frac{\\partial F}{\\partial x}\\big(x,\\, g(x)\\big)\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<\/div>\n
                          \n

                          \uc99d\uba85<\/p>\n

                          \ud589\ub82c \\(A = \\frac{\\partial F}{\\partial y}(a,\\, b)\\)\uac00 \uac00\uc5ed\uc774\ubbc0\ub85c, \ubc29\uc815\uc2dd \\(F(x,\\, y) = 0\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc4f8 \uc218 \uc788\ub2e4.
                          \n\\[y = y – A^{-1} F(x,\\, y).\\]
                          \n\ub530\ub77c\uc11c \ud568\uc218 \\(\\Phi: \\mathbb{R}^n \\times \\mathbb{R}^m \\to \\mathbb{R}^m\\)\uc744
                          \n\\[\\Phi(x,\\, y) = y – A^{-1} F(x,\\, y)\\]
                          \n\ub77c\uace0 \uc815\uc758\ud558\uba74, \\(F(x,\\, y) = 0\\)\uc740 \\(y = \\Phi(x,\\, y)\\)\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n

                          \\(F(a,\\, b) = 0\\)\uc774\ubbc0\ub85c \\(\\Phi(a,\\, b) = b\\)\uc774\ub2e4. \ub610\ud55c
                          \n\\[\\frac{\\partial \\Phi}{\\partial y}(x,\\, y) = I – A^{-1} \\frac{\\partial F}{\\partial y}(x,\\, y)\\]
                          \n\uc774\ub2e4. \ud2b9\ud788 \\((a,\\, b)\\)\uc5d0\uc11c
                          \n\\[\\frac{\\partial \\Phi}{\\partial y}(a,\\, b) = I – A^{-1} A = 0\\]
                          \n\uc774\ub2e4. \\(\\frac{\\partial \\Phi}{\\partial y}(a,\\, b) = 0\\)\uc774\uace0 \\(\\Phi\\)\uc758 \ud3b8\ub3c4\ud568\uc218\uac00 \uc5f0\uc18d\uc774\ubbc0\ub85c, \ucda9\ubd84\ud788 \uc791\uc740 \\(\\delta > 0\\)\uc640 \\(\\varepsilon > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(\\|x – a\\| < \\delta\\), \\(\\|y - b\\| < \\varepsilon\\)\uc77c \ub54c\n\\[\\left\\|\\frac{\\partial \\Phi}{\\partial y}(x,\\, y)\\right\\| < \\frac{1}{2}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c, \uace0\uc815\ub41c \\(x\\)\uc5d0 \ub300\ud558\uc5ec, \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(\\|y_1 - b\\|,\\, \\|y_2 - b\\| < \\varepsilon\\)\uc77c \ub54c\n\\[\\|\\Phi(x,\\, y_1) - \\Phi(x,\\, y_2)\\| \\leq \\frac{1}{2} \\|y_1 - y_2\\|\\]\n\uc774\ub2e4. \uc989, \\(\\Phi_x(y) = \\Phi(x,\\, y)\\)\ub294 \\(y\\)\uc5d0 \ub300\ud55c \ucd95\uc18c\uc0ac\uc0c1\uc774\ub2e4.<\/p>\n

                          \\(\\delta\\)\ub97c \ub354 \uc791\uac8c \uc7a1\uc544\uc11c, \\(\\|x – a\\| < \\delta\\)\uc77c \ub54c\n\\[\\|\\Phi(x,\\, b) - b\\| = \\|b - A^{-1}F(x,\\, b) - b\\| = \\|A^{-1}F(x,\\, b)\\| < \\frac{\\varepsilon}{2}\\]\n\uc774 \ub418\ub3c4\ub85d \ud558\uc790. \uc0bc\uac01\ubd80\ub4f1\uc2dd\uacfc \ucd95\uc18c\uc0ac\uc0c1\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec, \\(\\|y - b\\| \\leq \\varepsilon\\)\uc774\uba74\n\\[\\begin{aligned}\n\\|\\Phi(x,\\, y) - b\\| &\\leq \\|\\Phi(x,\\, y) - \\Phi(x,\\, b)\\| + \\|\\Phi(x,\\, b) - b\\| \\\\[6pt]\n&\\leq \\frac{1}{2}\\|y - b\\| + \\frac{\\varepsilon}{2} \\leq \\varepsilon\n\\end{aligned}\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\Phi_x\\)\ub294 \\(\\overline{B}(b,\\, \\varepsilon)\\)\uc744 \uc790\uae30 \uc790\uc2e0\uc73c\ub85c \ubcf4\ub0b8\ub2e4.<\/p>\n

                          \uac01 \\(\\|x – a\\| < \\delta\\)\uc5d0 \ub300\ud574, \\(\\Phi_x: \\overline{B}(b,\\, \\varepsilon) \\to \\overline{B}(b,\\, \\varepsilon)\\)\uc740 \ucd95\uc18c\uc0ac\uc0c1\uc774\ubbc0\ub85c, \uace0\uc815\uc810 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc720\uc77c\ud55c \uace0\uc815\uc810 \\(g(x)\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc989, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[g(x) = \\Phi(x,\\, g(x)) \\quad \\Longleftrightarrow \\quad F(x,\\, g(x)) = 0.\\]\n\uc774\uc81c \\(g\\)\uac00 \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc790. \\(x_1,\\, x_2 \\in B(a,\\, \\delta)\\)\uc5d0 \ub300\ud574, \\(y_i = g(x_i)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74\n\\[\\begin{aligned}\n\\|y_1 - y_2\\| &= \\|\\Phi(x_1,\\, y_1) - \\Phi(x_2,\\, y_2)\\| \\\\[6pt]\n&\\leq \\|\\Phi(x_1,\\, y_1) - \\Phi(x_2,\\, y_1)\\| + \\|\\Phi(x_2,\\, y_1) - \\Phi(x_2,\\, y_2)\\| \\\\[6pt]\n&\\leq \\|\\Phi(x_1,\\, y_1) - \\Phi(x_2,\\, y_1)\\| + \\frac{1}{2}\\|y_1 - y_2\\|\n\\end{aligned}\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c\n\\[\\|y_1 - y_2\\| \\leq 2\\|\\Phi(x_1,\\, y_1) - \\Phi(x_2,\\, y_1)\\|\\]\n\uc774\ub2e4. \\(\\Phi\\)\uac00 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc5f0\uc18d\uc774\ubbc0\ub85c \\(g\\)\ub3c4 \uc5f0\uc18d\uc774\ub2e4.<\/p>\n

                          \ub2e4\uc74c\uc73c\ub85c \\(g\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc790. \uc5f0\uc1c4\ubc95\uce59\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(F(x,\\, g(x)) = 0\\)\uc758 \uc804\ubbf8\ubd84\uc744 \uad6c\ud558\uba74
                          \n\\[\\frac{\\partial F}{\\partial x}(x,\\, g(x)) + \\frac{\\partial F}{\\partial y}(x,\\, g(x)) \\cdot \\frac{\\partial g}{\\partial x}(x) = 0\\]
                          \n\uc774\ub2e4. \\(\\frac{\\partial F}{\\partial y}(x,\\, g(x))\\)\uac00 \uac00\uc5ed\uc774\ubbc0\ub85c
                          \n\\[\\frac{\\partial g}{\\partial x}(x) = -\\left[\\frac{\\partial F}{\\partial y}(x,\\, g(x))\\right]^{-1} \\frac{\\partial F}{\\partial x}(x,\\, g(x))\\]
                          \n\uc774\ub2e4. \uc6b0\ubcc0\uc758 \ud3b8\ub3c4\ud568\uc218\ub4e4\uc774 \uc5f0\uc18d\uc774\uace0 \\(g\\)\uac00 \uc5f0\uc18d\uc774\ubbc0\ub85c \\(\\frac{\\partial g}{\\partial x}\\)\ub3c4 \uc5f0\uc18d\uc774\ub2e4. \ub530\ub77c\uc11c \\(g \\in C^1\\)\uc774\ub2e4.<\/p>\n

                          \ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(g\\)\uc758 \uc720\uc77c\uc131\uc744 \ubcf4\uc774\uc790. \\(h: U’ \\to \\mathbb{R}^m\\)\uc774 \\(F(x,\\, h(x)) = 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub2e4\ub978 \ud568\uc218\ub77c\uace0 \ud558\uc790. \\(U\\)\uc640 \\(U’\\)\uc758 \uad50\uc9d1\ud569\uc744 \ucda9\ubd84\ud788 \uc791\uac8c \uc7a1\uc73c\uba74, \uac01 \\(x\\)\uc5d0 \ub300\ud574 \\(h(x)\\)\ub294 \\(\\Phi_x\\)\uc758 \uace0\uc815\uc810\uc774\ub2e4. \uace0\uc815\uc810\uc758 \uc720\uc77c\uc131\uc5d0 \uc758\ud574 \\(g(x) = h(x)\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

                          \uc74c\ud568\uc218 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud560 \ub54c \uc5fc\ub450\uc5d0 \ub458 \uc810\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

                            \n
                          1. \uc74c\ud568\uc218 \uc815\ub9ac\ub294 \uad6d\uc18c\uc801 \uacb0\uacfc\uc774\ub2e4. \uc989 \uc804\uc5ed\uc801\uc73c\ub85c\ub294 \uc5ec\ub7ec \uac1c\uc758 \ud574\uac00 \uc874\uc7ac\ud560 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(x^2 + y^2 = 25\\)\uc5d0\uc11c \\(y\\)\ub97c \\(x\\)\uc758 \ud568\uc218\ub85c \ub098\ud0c0\ub0b4\uba74 \uad6d\uc18c\uc801\uc73c\ub85c\ub9cc \uac00\ub2a5\ud558\ub2e4.<\/li>\n
                          2. \\(\\frac{\\partial F}{\\partial y}\\)\uac00 \uac00\uc5ed\uc774 \uc544\ub2c8\uba74 \uc74c\ud568\uc218\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uac70\ub098 \uc720\uc77c\ud558\uc9c0 \uc54a\uc744 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(F(x,\\, y) = x^2 – y^2\\)\uc77c \ub54c \\((0,\\, 0)\\)\uc5d0\uc11c\ub294 \\(\\frac{\\partial F}{\\partial y} = 0\\)\uc774\uace0, \uc2e4\uc81c\ub85c \\(x = 0\\) \uadfc\ucc98\uc5d0\uc11c \\(y\\)\ub97c \\(x\\)\uc758 \ud568\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc5c6\ub2e4.<\/li>\n
                          3. \\(F\\)\uac00 \\(C^k\\)\uc774\uba74 \\(g\\)\ub3c4 \\(C^k\\)\uc774\ub2e4.<\/li>\n<\/ol>\n
                            \n

                            \ubcf4\uae30 1.<\/span><\/p>\n

                            \ubc29\uc815\uc2dd \\(x^3 + y^3 + xy – 3 = 0\\)\uc774 \uc810 \\((1,\\, 1)\\) \uadfc\ucc98\uc5d0\uc11c \\(y = g(x)\\) \ud615\ud0dc\ub85c \uc720\uc77c\ud558\uac8c \ud480 \uc218 \uc788\ub294\uc9c0 \ud655\uc778\ud574\ubcf4\uc790.<\/p>\n

                            \\(F(x,\\, y) = x^3 + y^3 + xy – 3\\)\uc774\ub77c \ud558\uba74
                            \n\\[\\begin{aligned}
                            \nF(1,\\, 1) &= 1 + 1 + 1 – 3 = 0 ,\\\\[6pt]
                            \n\\frac{\\partial F}{\\partial y}(x,\\, y) &= 3y^2 + x , & \\frac{\\partial F}{\\partial x}(x,\\,y) &= 3x^2 +y ,\\\\[6pt]
                            \n\\frac{\\partial F}{\\partial y}(1,\\, 1) &= 4 \\neq 0 , & \\frac{\\partial F}{\\partial x}(1,\\,1) &= 4
                            \n\\end{aligned}\\]
                            \n\uc774\ub2e4. \ub530\ub77c\uc11c \uc74c\ud568\uc218 \uc815\ub9ac\uc5d0 \uc758\ud574 \\((1,\\, 1)\\) \uadfc\ucc98\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ubc29\uc815\uc2dd\uc744 \\(y = g(x)\\)\ub85c \uc720\uc77c\ud558\uac8c \ub098\ud0c0\ub0bc \uc218 \uc788\uc73c\uba70,
                            \n\\[g'(1) = -\\frac{\\partial F\/\\partial x(1,\\, 1)}{\\partial F\/\\partial y(1,\\, 1)} = -\\frac{4}{4} = -1 .\\]<\/p>\n<\/div>\n

                            \n

                            \uc815\ub9ac 9.9. (\uc5ed\ud568\uc218 \uc815\ub9ac)<\/span><\/p>\n

                            \ud568\uc218 \\(f: \\mathbb{R}^n \\to \\mathbb{R}^n\\)\uc774 \uc810 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \\(C^1\\)\uc774\uace0 \\(Df(a)\\)\uac00 \uac00\uc5ed\uc774\uba74, \\(a\\)\uc758 \uadfc\ubc29 \\(U\\)\uc640 \\(f(a)\\)\uc758 \uadfc\ubc29 \\(V\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(f: U \\to V\\)\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\uace0, \uc5ed\ud568\uc218 \\(f^{-1}: V \\to U\\)\ub3c4 \\(C^1\\)\uc774\uba70
                            \n\\[D(f^{-1})(y) = [Df(f^{-1}(y))]^{-1}\\]
                            \n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<\/div>\n

                            \n

                            \uc99d\uba85<\/p>\n

                            \ud568\uc218 \\(F: \\mathbb{R}^n \\times \\mathbb{R}^n \\to \\mathbb{R}^n\\)\uc744
                            \n\\[F(x,\\, y) = f(x) – y\\]
                            \n\ub77c\uace0 \uc815\uc758\ud55c\ub2e4. \uadf8\ub7ec\uba74 \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.
                            \n\\[\\begin{aligned}
                            \nF(a,\\, f(a)) &= f(a) – f(a) = 0 ,\\\\[6pt]
                            \n\\frac{\\partial F}{\\partial x}(x,\\, y) &= Df(x) ,\\\\[6pt]
                            \n\\frac{\\partial F}{\\partial y}(x,\\, y) &= -I .
                            \n\\end{aligned}\\]
                            \n\\(\\frac{\\partial F}{\\partial x}(a,\\, f(a)) = Df(a)\\)\uac00 \uac00\uc5ed\uc774\ubbc0\ub85c, \uc74c\ud568\uc218 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(f(a)\\)\uc758 \uadfc\ubc29 \\(V’\\)\uacfc \\(a\\)\uc758 \uadfc\ubc29 \\(U’\\)\uc5d0\uc11c \uc720\uc77c\ud55c \\(C^1\\) \ud568\uc218 \\(g: V’ \\to U’\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n

                              \n
                            • \\(g(f(a)) = a\\)\uc774\ub2e4.<\/li>\n
                            • \ubaa8\ub4e0 \\(y \\in V’\\)\uc5d0 \ub300\ud574 \\(F(g(y),\\, y) = 0\\)\uc774\ub2e4. \uc989 \\(f(g(y)) = y\\)\uc774\ub2e4.<\/li>\n<\/ul>\n

                              \ub9c8\ucc2c\uac00\uc9c0\ub85c \ud568\uc218 \\(G\\)\ub97c \\(G(x,\\, y) = y – f(x)\\)\ub77c\uace0 \uc815\uc758\ud558\uba74, \\(\\frac{\\partial G}{\\partial y} = I\\)\uac00 \uac00\uc5ed\uc774\ubbc0\ub85c \uc74c\ud568\uc218 \uc815\ub9ac\uc5d0 \uc758\ud574 \ud568\uc218 \\(h: U” \\to V”\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n

                                \n
                              • \\(h(a) = f(a)\\)\uc774\ub2e4.<\/li>\n
                              • \ubaa8\ub4e0 \\(x \\in U”\\)\uc5d0 \ub300\ud574 \\(h(x) = f(x)\\)\uc774\ub2e4.<\/li>\n<\/ul>\n

                                \uc801\uc808\ud55c \uadfc\ubc29 \\(U = U’ \\cap U”\\)\uc640 \\(V = V’ \\cap f(U)\\)\ub97c \ud0dd\ud558\uba74, \\(f: U \\to V\\)\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\uace0 \\(g = f^{-1}\\)\uc774\ub2e4. \uc74c\ud568\uc218 \uc815\ub9ac\ub85c\ubd80\ud130 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
                                \n\\[\\begin{aligned}
                                \n\\frac{\\partial g}{\\partial y} &= -\\left(\\frac{\\partial F}{\\partial x}(g(y),\\, y)\\right)^{-1} \\frac{\\partial F}{\\partial y}(g(y),\\, y) \\\\[6pt]
                                \n&= -[Df(g(y))]^{-1} \\cdot (-I) = [Df(g(y))]^{-1}.
                                \n\\end{aligned}\\]
                                \n\ub530\ub77c\uc11c \\(D(f^{-1})(y) = [Df(f^{-1}(y))]^{-1}\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

                                \n

                                \ud568\uc218 \\(f: \\mathbb{R}^n \\to \\mathbb{R}^n\\)\uc774 \uc5f4\ub9b0\uc9d1\ud569 \\(\\Omega\\)\uc5d0\uc11c \\(C^1\\)\uc774\uace0 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(Df(x)\\)\uac00 \uac00\uc5ed\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

                                  \n
                                1. \\(f\\)\ub294 \uad6d\uc18c \ubbf8\ubd84\ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4.<\/li>\n
                                2. \\(f(\\Omega)\\)\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n
                                3. \\(f\\)\uac00 \uc77c\ub300\uc77c\uc774\uba74 \\(f: \\Omega \\to f(\\Omega)\\)\ub294 \ubbf8\ubd84\ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

                                  \uc5ed\ud568\uc218 \uc815\ub9ac\ub294 \ube44\uc120\ud615 \ud568\uc218\uac00 \uad6d\uc18c\uc801\uc73c\ub85c \uc120\ud615\ubcc0\ud658\ucc98\ub7fc \ud589\ub3d9\ud568\uc744 \ubcf4\uc5ec\uc900\ub2e4. \uc810 \\(a\\) \uadfc\ucc98\uc5d0\uc11c \\(f\\)\ub294 \uadfc\uc0ac\uc801\uc73c\ub85c
                                  \n\\[f(x) \\approx f(a) + Df(a)(x – a)\\]
                                  \n\uc774\uace0, \\(Df(a)\\)\uac00 \uac00\uc5ed\uc774\uba74 \uc774 \uc120\ud615\uadfc\uc0ac\uac00 \uad6d\uc18c\uc801\uc73c\ub85c \uc77c\ub300\uc77c\ub300\uc751\uc774 \ub41c\ub2e4.<\/p>\n

                                  \n

                                  \ubcf4\uae30 2.<\/span><\/p>\n

                                  \uadf9\uc88c\ud45c \ubcc0\ud658 \\(f(r,\\, \\theta) = (r\\cos\\theta,\\, r\\sin\\theta)\\)\ub97c \uc0dd\uac01\ud558\uc790. \\(f\\)\uc758 \uc57c\ucf54\ube44 \ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
                                  \n\\[Df(r,\\, \\theta) = \\left[
                                  \n\\begin{array}{rr} \\cos\\theta & -r\\sin\\theta \\\\[3pt] \\sin\\theta & r\\cos\\theta \\end{array}
                                  \n\\right].\\]
                                  \n\uc774 \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc774 \\(\\det(Df) = r\\)\uc774\ubbc0\ub85c, \\(r \\neq 0\\)\uc778 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc5ed\ud568\uc218 \uc815\ub9ac\uac00 \uc801\uc6a9\ub41c\ub2e4. \uc2e4\uc81c\ub85c \\(r > 0\\)\uc778 \uc601\uc5ed\uc5d0\uc11c \uadf9\uc88c\ud45c \ubcc0\ud658\uc740 \uad6d\uc18c\uc801\uc73c\ub85c \uac00\uc5ed\uc774\ub2e4.<\/p>\n

                                  \uc6d0\uc810\uc5d0\uc11c\ub294 \\(r = 0\\)\uc774\ubbc0\ub85c \uc5ed\ud568\uc218 \uc815\ub9ac\ub97c \uc801\uc6a9\ud560 \uc218 \uc5c6\uace0, \uc2e4\uc81c\ub85c \uc774 \uc810\uc5d0\uc11c \uadf9\uc88c\ud45c \ubcc0\ud658\uc740 \uac00\uc5ed\uc774 \uc544\ub2c8\ub2e4.<\/p>\n<\/div>\n

                                  \n

                                  \ubb38\uc81c 9.16.<\/span>
                                  \n\ub2e4\uc74c\uacfc \uac19\uc740 \ud568\uc218 \\(f\\)\uc5d0 \ub300\ud558\uc5ec \uc8fc\uc5b4\uc9c4 \uc810 \\((a,\\,b)\\)\uc758 \uc801\ub2f9\ud55c \uc5f4\ub9b0\uadfc\ubc29\uc5d0\uc11c \\(f^{-1}\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uace0 \\(D(f^{-1})(a,\\,b)\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n

                                    \n
                                  1. \\(f(u,\\,v) = (3u-v ,\\, 2u+5v)\\), \\((a,\\,b)\\in\\mathbb{R}^2\\).<\/li>\n
                                  2. \\(f(u,\\,v) = (u+v,\\,\\sin u + \\cos v)\\), \\((a,\\,b) = (0,\\,1)\\).<\/li>\n
                                  3. \\(f(u,\\,v) = (uv, u^2 + v^2)\\), \\((a,\\,b) = (2,\\,5)\\).<\/li>\n
                                  4. \\(f(u,\\,v) = (u^3 – v^2 ,\\, \\sin u – \\ln v)\\), \\((a,\\,b) = (-1,\\,0)\\).<\/li>\n<\/ol>\n<\/div>\n
                                    \n

                                    \ubb38\uc81c 9.17.<\/span>
                                    \n\ub2e4\uc74c \uac01 \ub4f1\uc2dd\uc5d0 \ub300\ud558\uc5ec \uc810 \\((0,\\,0,\\,0)\\)\uc758 \uc5f4\ub9b0\uadfc\ubc29 \\(V\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc8fc\uc5b4\uc9c4 \ub4f1\uc2dd\uc744 \\(V\\)\uc5d0\uc11c \\(z\\)\uc5d0 \ub300\ud558\uc5ec \ud480 \uc218 \uc788\ub294\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624. \ub610\ud55c \\(z\\)\uc5d0 \ub300\ud558\uc5ec \ud47c \uc2dd\uc774 \\((0,\\,0)\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624.<\/p>\n

                                      \n
                                    1. \\(xyz + \\sin(x+y+z)=0\\)<\/li>\n
                                    2. \\(x^2 + y^2 + z^2 + \\sqrt{\\sin(x^2 + y^2 )+3z+4} =2\\)<\/li>\n
                                    3. \\(xyz(2\\cos y – \\cos z)+(z\\cos x – x\\cos y)=0\\)<\/li>\n<\/ol>\n<\/div>\n
                                      \n

                                      \ubb38\uc81c 9.18.<\/span>
                                      \n\ud568\uc218 \\(z=F(x,\\,y)\\)\uac00 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(F_y (a,\\,b)\\neq 0\\)\uc774\uba70 \\(I\\)\uac00 \\(a\\)\uc758 \uc5f4\ub9b0\uadfc\ubc29\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \ud568\uc218 \\(f:I\\rightarrow\\mathbb{R}\\)\uc774 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f(a)=b\\)\uc774\uba70 \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(F(x,\\,f(x))=0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624.
                                      \n\\[\\frac{d}{dx}f(a) = -\\frac{ \\frac{\\partial}{\\partial x}F(a,\\,b) }{ \\frac{\\partial}{\\partial y}F(a,\\,b) }.\\]<\/p>\n<\/div>\n

                                      \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95<\/h3>\n

                                      \ud604\uc2e4\uc758 \ubb38\uc81c\ub97c \ubaa8\ub378\ub9c1\ud55c \ucd5c\uc801\ud654 \ubb38\uc81c\ub294 \uc81c\uc57d\uc870\uac74\uc744 \ub3d9\ubc18\ud558\ub294 \uacbd\uc6b0\uac00 \ub9ce\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \ub458\ub808\uc758 \uae38\uc774\uac00 \uace0\uc815\ub418\uc5b4 \uc788\uc744 \ub54c \ub113\uc774\uac00 \ucd5c\ub300\uc778 \uc9c1\uc0ac\uac01\ud615\uc744 \uad6c\ud558\uac70\ub098, \uac89\ub113\uc774\uac00 \uace0\uc815\ub418\uc5b4 \uc788\uc744 \ub54c \ubd80\ud53c\uac00 \ucd5c\ub300\uc778 \uc6d0\uae30\ub465\uc744 \uad6c\ud558\ub294 \ubb38\uc81c \ub4f1\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774\ucc98\ub7fc \uc81c\uc57d\uc870\uac74 \ud558\uc5d0\uc11c \uadf9\uac12\uc744 \uad6c\ud558\ub294 \ubc29\ubc95 \uc911 \ud558\ub098\uac00 \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95<\/span>(Lagrange multiplier method)\uc774\ub2e4.<\/p>\n

                                      \uba3c\uc800 \uae30\ud558\ud559\uc801 \uad00\uc810\uc5d0\uc11c \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc81c\uc57d\uc870\uac74 \\(g(x) = c\\) \ud558\uc5d0\uc11c \ud568\uc218 \\(f: \\mathbb{R}^n \\to \\mathbb{R}\\)\uc758 \uadf9\uac12\uc744 \uad6c\ud558\ub294 \uc0c1\ud669\uc744 \uc0dd\uac01\ud558\uc790. \uc81c\uc57d\uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810\ub4e4\uc758 \uc9d1\ud569\uc740 \\((n-1)\\)\ucc28\uc6d0 \uace1\uba74\uc744 \uc774\ub8ec\ub2e4. \ub9cc\uc57d \uc810 \\(a\\)\uac00 \uc774 \uace1\uba74 \uc704\uc5d0\uc11c \\(f\\)\uc758 \uadf9\uac12\uc774\ub77c\uba74, \\(f\\)\uc758 \ub4f1\uc704\uba74 \\(f(x) = f(a)\\)\uc640 \uc81c\uc57d\uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uace1\uba74 \\(g(x) = c\\)\uac00 \\(a\\)\uc5d0\uc11c \uc811\ud55c\ub2e4. \uc774\uac83\uc740 \ub450 \uace1\uba74\uc758 \ubc95\ubca1\ud130\uc778 \\(\\nabla f(a)\\)\uc640 \\(\\nabla g(a)\\)\uac00 \ud3c9\ud589\ud568\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n

                                      \n

                                      \uc815\ub9ac 9.10. (\ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95 – \uc81c\uc57d\uc870\uac74\uc774 \ud558\ub098\uc778 \uacbd\uc6b0)<\/span><\/p>\n

                                      \ud568\uc218 \\(f,\\, g: \\mathbb{R}^n \\to \\mathbb{R}\\)\uc774 \\(C^1\\)\uc774\uace0, \\(a\\)\uac00 \uc81c\uc57d\uc870\uac74 \\(g(x) = c\\) \ud558\uc5d0\uc11c \\(f\\)\uc758 \uadf9\uac12\uc774\uba70, \\(\\nabla g(a) \\neq 0\\)\uc774\uba74, \uc2e4\uc218 \\(\\lambda\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.
                                      \n\\[\\nabla f(a) = \\lambda \\nabla g(a).\\]<\/p>\n<\/div>\n

                                      \n

                                      \uc99d\uba85<\/p>\n

                                      \uc81c\uc57d\uc870\uac74 \\(g(x) = c\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810\ub4e4\uc758 \uc9d1\ud569\uc744 \\(S\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

                                      \\(\\nabla g(a) \\neq 0\\)\uc774\ubbc0\ub85c \uc74c\ud568\uc218 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(S\\)\ub294 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \\((n-1)\\)\ucc28\uc6d0 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uba74\uc774\ub2e4. \ub610\ud55c
                                      \n\\(S\\) \uc704\uc5d0\uc11c \\(a\\)\ub97c \uc9c0\ub098\ub294 \uc784\uc758\uc758 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120 \\(\\gamma(t)\\)\uc5d0 \ub300\ud574 \\(\\gamma(0) = a\\)\uc774\uace0 \\(\\gamma'(0)\\)\uc740 \\(S\\)\uc758 \uc811\ubca1\ud130\uc774\ub2e4. \\(f(\\gamma(t))\\)\uac00 \\(t = 0\\)\uc5d0\uc11c \uadf9\uac12\uc744 \uac00\uc9c0\ubbc0\ub85c
                                      \n\\[\\frac{d}{dt}f(\\gamma(t))\\bigg|_{t=0} = \\nabla f(a) \\cdot \\gamma'(0) = 0\\]
                                      \n\uc774\ub2e4. \ubaa8\ub4e0 \uc811\ubca1\ud130 \\(v\\)\uc5d0 \ub300\ud574 \\(\\nabla f(a) \\cdot v = 0\\)\uc774\ubbc0\ub85c, \\(\\nabla f(a)\\)\ub294 \uc811\uacf5\uac04\uc5d0 \uc218\uc9c1\uc774\ub2e4.<\/p>\n

                                      \\(\\nabla g(a)\\)\ub3c4 \uc811\uacf5\uac04\uc5d0 \uc218\uc9c1\uc774\ubbc0\ub85c, \\(\\nabla f(a) = \\lambda \\nabla g(a)\\)\uc778 \\(\\lambda\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

                                      \uc774 \uc815\ub9ac\uc5d0\uc11c \\(\\lambda\\)\ub97c \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218<\/span>(Lagrange multiplier)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc2e4\uc81c \uacc4\uc0b0\uc5d0\uc11c\ub294 \ub2e4\uc74c \uc5f0\ub9bd\ubc29\uc815\uc2dd\uc744 \ud47c\ub2e4.
                                      \n\\[\\begin{cases}
                                      \n\\nabla f(x) = \\lambda \\nabla g(x) \\\\
                                      \ng(x) = c
                                      \n\\end{cases}\\]<\/p>\n

                                      \uc774\uac83\uc740 \\(n+1\\)\uac1c\uc758 \ubc29\uc815\uc2dd\uacfc \\(n+1\\)\uac1c\uc758 \ubbf8\uc9c0\uc218 \\((x_1,\\, \\ldots,\\, x_n,\\, \\lambda)\\)\ub97c \uac00\uc9c4 \uc5f0\ub9bd\ubc29\uc815\uc2dd\uc774\ub2e4.<\/p>\n

                                      \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95\uc744 \uc81c\uc57d\uc870\uac74\uc774 \uc5ec\ub7ec \uac1c \uc788\ub294 \uacbd\uc6b0\ub85c \ud655\uc7a5\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

                                      \n

                                      \uc815\ub9ac 9.11. (\ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95 – \uc81c\uc57d\uc870\uac74\uc774 \uc5ec\ub7ec \uac1c\uc778 \uacbd\uc6b0)<\/span><\/p>\n

                                      \ud568\uc218 \\(f,\\, g_1,\\, \\ldots,\\, g_k: \\mathbb{R}^n \\to \\mathbb{R}\\)\uc774 \\(C^1\\)\uc774\uace0, \\(a\\)\uac00 \uc81c\uc57d\uc870\uac74 \\(g_i(x) = c_i\\) (\\(i = 1,\\, \\ldots,\\, k\\)) \ud558\uc5d0\uc11c \\(f\\)\uc758 \uadf9\uac12\uc774\uba70, \ubca1\ud130 \\(\\nabla g_1(a),\\, \\ldots,\\, \\nabla g_k(a)\\)\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc774\uba74, \uc2e4\uc218 \\(\\lambda_1,\\, \\ldots,\\, \\lambda_k\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.
                                      \n\\[\\nabla f(a) = \\sum_{i=1}^{k} \\lambda_i \\nabla g_i(a).\\]<\/p>\n<\/div>\n

                                      \uc2e4\uc81c\ub85c \uacc4\uc0b0\ud560 \ub54c\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ub77c\uadf8\ub791\uc8fc \ud568\uc218<\/span>(Lagrangian)\ub97c \uc815\uc758\ud558\ub294 \uac83\uc774 \ud3b8\ub9ac\ud558\ub2e4.
                                      \n\\[L(x,\\, \\lambda) = f(x) – \\sum_{i=1}^{k} \\lambda_i (g_i(x) – c_i).\\]
                                      \n\uc774\ub54c \uc8fc\uc5b4\uc9c4 \ud568\uc218\uc758 \uadf9\uac12\uc774 \uc874\uc7ac\ud560 \ud544\uc694\uc870\uac74\uc740 \\(L\\)\uc758 \ubaa8\ub4e0 \ud3b8\ubbf8\ubd84\uc774 0\uc778 \uac83\uc774\ub2e4.
                                      \n\\[\\frac{\\partial L}{\\partial x_j} = 0 \\quad (j = 1,\\, \\ldots,\\, n), \\quad \\frac{\\partial L}{\\partial \\lambda_i} = 0 \\quad (i = 1,\\, \\ldots,\\, k).\\]<\/p>\n

                                      \n

                                      \ubcf4\uae30 3.<\/span><\/p>\n

                                      \\(a > b > 0\\)\uc77c \ub54c, \ud0c0\uc6d0 \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\) \uc704\uc758 \uc810 \uc911 \uc6d0\uc810\uc5d0\uc11c \uac00\uc7a5 \uba3c \uc810\uacfc \uac00\uc7a5 \uac00\uae4c\uc6b4 \uc810\uc744 \uad6c\ud574\ubcf4\uc790.<\/p>\n

                                      \ubaa9\uc801\ud568\uc218\ub294 \\(f(x,\\, y) = x^2 + y^2\\)\uc774\uace0, \uc81c\uc57d\uc870\uac74\uc740 \\(g(x,\\, y) = \\frac{x^2}{a^2} + \\frac{y^2}{b^2} – 1 = 0\\)\uc774\ub2e4.<\/p>\n

                                      \ub77c\uadf8\ub791\uc8fc \uc870\uac74 \\(\\nabla f = \\lambda \\nabla g\\)\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
                                      \n\\[(2x,\\, 2y) = \\lambda \\left(\\frac{2x}{a^2},\\, \\frac{2y}{b^2}\\right).\\]<\/p>\n

                                      \uc774 \uc2dd\uc73c\ub85c\ubd80\ud130 \\(x\\left(1 – \\frac{\\lambda}{a^2}\\right) = 0\\)\uacfc \\(y\\left(1 – \\frac{\\lambda}{b^2}\\right) = 0\\)\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

                                      \\(x = 0\\) \ub610\ub294 \\(y = 0\\)\uc778 \uacbd\uc6b0\ub97c \uac80\ud1a0\ud558\uba74, \ud568\uc218 \\(f\\)\ub294 \\((\\pm a,\\, 0)\\)\uc5d0\uc11c \ucd5c\ub313\uac12 \\(a^2\\)\uc744 \uac00\uc9c0\uba70, \\((0,\\, \\pm b)\\)\uc5d0\uc11c \ucd5c\uc19f\uac12 \\(b^2\\)\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n

                                      \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95\uc744 \ud65c\uc6a9\ud558\uc5ec \uc798 \uc54c\ub824\uc9c4 \ubd80\ub4f1\uc2dd\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<\/p>\n

                                      \n

                                      \ubcf4\uae30 4.<\/span><\/p>\n

                                      \uc0b0\uc220-\uae30\ud558 \ud3c9\uade0 \ubd80\ub4f1\uc2dd<\/strong><\/p>\n

                                      \uc591\uc218 \\(x_1,\\, \\ldots,\\, x_n\\)\uc5d0 \ub300\ud574
                                      \n\\[\\frac{x_1 + \\cdots + x_n}{n} \\geq \\sqrt[n]{x_1 \\cdots x_n}\\]
                                      \n\uc784\uc744 \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \ubcf4\uc77c \uc218 \uc788\ub2e4. \uc81c\uc57d\uc870\uac74 \\(x_1 \\cdots x_n = c^n\\) \ud558\uc5d0\uc11c
                                      \n\\[f(x_1,\\, \\ldots,\\, x_n) = x_1 + \\cdots + x_n\\]
                                      \n\uc758 \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\uba74, \\(x_1 = \\cdots = x_n = c\\)\uc77c \ub54c \\(f\\)\uac00 \uadf9\uac12\uc744 \uac00\uc9d0 \uc54c \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n

                                      \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95\uc740 \uadf9\uac12\uc758 \ud544\uc694\uc870\uac74\ub9cc\uc744 \uc81c\uacf5\ud55c\ub2e4\ub294 \uc810\uc744 \uc720\ub150\ud574\uc57c \ud55c\ub2e4. \uad6c\ud55c \uc810\uc5d0\uc11c \ud568\uc218\uac00 \uc2e4\uc81c\ub85c \uadf9\uac12\uc744 \uac16\ub294\uc9c0 \uc5ec\ubd80\ub294 \uc9c1\uc811 \ud655\uc778\ud574\uc57c \ud55c\ub2e4.<\/p>\n

                                      \n

                                      \ubb38\uc81c 9.19.<\/span>
                                      \n\uc81c\uc57d\uc870\uac74 \\(x^2 + y^2 + z^2 = 1\\) \ud558\uc5d0\uc11c \\(f(x,\\, y,\\, z) = x + 2y + 3z\\)\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                                      \n

                                      \ubb38\uc81c 9.20.<\/span>
                                      \n\uad6c \\(x^2 + y^2 + z^2 = 1\\) \uc704\uc5d0\uc11c \ud568\uc218 \\(f(x,\\, y,\\, z) = xyz\\)\uc758 \uadf9\uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                                      \n

                                      \ubb38\uc81c 9.21.<\/span>
                                      \n\uc8fc\uc5b4\uc9c4 \ud45c\uba74\uc801 \\(S\\)\ub97c \uac00\uc9c4 \uc9c1\uc721\uba74\uccb4 \uc911 \ubd80\ud53c\uac00 \ucd5c\ub300\uc778 \uac83\uc758 \ubaa8\uc11c\ub9ac\uc758 \uae38\uc774\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                                      \n

                                      \ubb38\uc81c 9.22.<\/span>
                                      \n\ud0c0\uc6d0\uccb4 \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} = 1\\)\uc5d0 \ub0b4\uc811\ud558\ub294 \uc9c1\uc721\uba74\uccb4\uc758 \ucd5c\ub300 \ubd80\ud53c\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                                      \n

                                      \ubb38\uc81c 9.23.<\/span>
                                      \n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

                                        \n
                                      1. \uc6d0 \\(x^2 + y^2=4\\) \uc704\uc5d0\uc11c \ud568\uc218 \\(f(x,\\,y)=x+y^2\\)\uc758 \uadf9\uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/li>\n
                                      2. \\(x^2 + y^2 + z^2 = 1\\)\uacfc \\(x+y+z=0\\)\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubc94\uc704\uc5d0\uc11c \ud568\uc218 \\(f(x,\\,y,\\,z)=xy\\)\uc758 \uadf9\uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/li>\n
                                      3. \\(3x^2 +y+4z^3=1\\)\uacfc \\(-x^3 +3z^4 +w=0\\)\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubc94\uc704\uc5d0\uc11c \ud568\uc218 \\(f(x,\\,y,\\,z,\\,w)=3x+y+w\\)\uc758 \uadf9\uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
                                        \n

                                        \ud574\uc11d\ud559 \ud575\uc2ec\uc815\ub9ac \ub178\ud2b8<\/a><\/p>\n

                                          \n
                                        1. \uc2e4\uc218\uacc4\uc758 \uc131\uc9c8<\/a><\/li>\n
                                        2. \uac70\ub9ac\uacf5\uac04<\/a><\/li>\n
                                        3. \uc218\uc5f4\uc758 \uadf9\ud55c<\/a><\/li>\n
                                        4. \ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131<\/a><\/li>\n
                                        5. \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n
                                        6. \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84<\/a><\/li>\n
                                        7. \ubb34\ud55c\uae09\uc218<\/a><\/li>\n
                                        8. \uc2e4\ud574\uc11d\uc801 \ud568\uc218<\/a><\/li>\n
                                        9. \ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n
                                        10. \uc911\uc801\ubd84<\/a><\/li>\n
                                        11. \ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                          <\/p>\n","protected":false},"excerpt":{"rendered":"

                                          \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. \ud3b8\ubbf8\ubd84\uacfc \uc804\ubbf8\ubd84 \uc810 \\(a = (a_1,\\, \\cdots,\\, a_n)\\)\uc774 \ud568\uc218 \\(f: \\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(partial derivative)\uc744 \\(\\frac{\\partial f}{\\partial x_i}(a) = \\lim_{h \\to 0} \\frac{f(a_1,\\, \\cdots,\\, a_i + h,\\, \\cdots,\\, a_n) – f(a)}{h}\\) \ub85c…<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9470,"menu_order":109,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9495","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9495","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9495"}],"version-history":[{"count":12,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9495\/revisions"}],"predecessor-version":[{"id":9614,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9495\/revisions\/9614"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9470"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9495"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}