{"id":9497,"date":"2025-10-20T19:00:04","date_gmt":"2025-10-20T10:00:04","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9497"},"modified":"2025-10-21T16:09:34","modified_gmt":"2025-10-21T07:09:34","slug":"ch10-multiple-integral","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\/","title":{"rendered":"\uc911\uc801\ubd84"},"content":{"rendered":"
<\/p>\n

<\/p>\n

\uc774 \uc7a5\uc5d0\uc11c\ub294 \ub9ac\ub9cc \ub2e4\uc911\uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed\uc5d0\uc11c\uc758 \uc801\ubd84\ubd80\ud130 \uc2dc\uc791\ud558\uc5ec \uc77c\ubc18 \uc601\uc5ed\uc73c\ub85c \ud655\uc7a5\ud558\uace0, \ud478\ube44\ub2c8 \uc815\ub9ac\uc640 \ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac\ub97c \ud1b5\ud574 \uc801\ubd84\uc744 \uacc4\uc0b0\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\ub2e4\uc911\uc801\ubd84\uc758 \uc815\uc758<\/h3>\n

\\(\\mathbb{R}^n\\)\uc5d0\uc11c \\(R = [a_1,\\, b_1] \\times \\cdots \\times [a_n,\\, b_n]\\) \ud615\ud0dc\uc758 \uc9d1\ud569\uc744 \uc9c1\uc0ac\uac01\ud615 \uc9d1\ud569<\/span>(rectangle) \ub610\ub294 \uac04\ub2e8\ud788 \uc9c1\uc0ac\uac01\ud615\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc9c1\uc0ac\uac01\ud615\uc758 \ubd80\ud53c<\/span>(volume)\ub97c \\(|R| = \\prod_{i=1}^{n} (b_i – a_i)\\)\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

\uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc758 \ubd84\ud560<\/span>(partition) \\(P\\)\ub294 \uac01 \uc88c\ud45c\ucd95\uc5d0 \ub300\ud55c \ubd84\ud560\ub4e4\uc758 \uacf1\uc774\ub2e4. \uc989, \uac01 \uad6c\uac04 \\([a_i,\\, b_i]\\)\ub97c \\[a_i = x_{i,0} < x_{i,1} < \\cdots < x_{i,m_i} = b_i\\]\ub85c \ubd84\ud560\ud558\uba74, \\(R\\)\uc740 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\ub4e4\n\\[R_{j_1,\\ldots,j_n} = [x_{1,j_1 -1} ,\\, x_{1,j_1}] \\times [x_{2,j_2 -1} ,\\, x_{2,j_2}] \\times \\cdots \\times [x_{n,j_n -1} ,\\, x_{n,j_n}]\\]\n\uc73c\ub85c \ub098\ub25c\ub2e4. \uc11c\ub85c \ub2e4\ub978 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\uc740 \uc11c\ub85c \ub0b4\ubd80\ub97c \uacf5\uc720\ud558\uc9c0 \uc54a\ub294\ub2e4. \ub610\ud55c \ubaa8\ub4e0 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\uc744 \ud569\uc9d1\ud569\ud558\uba74 \uc6d0\ub798\uc758 \uc9c1\uc0ac\uac01\ud615\uacfc \uac19\ub2e4.<\/p>\n

\uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(f: R \\to \\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec, \\(R\\)\uc758 \uac01 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615 \\(S\\)\uc5d0\uc11c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
\n\\[m_S = \\inf\\{f(x) \\mid x \\in S\\} ,\\quad M_S = \\sup\\{f(x) \\mid x \\in S\\} .\\]<\/p>\n

\uc774\ub54c \ub9ac\ub9cc \uc0c1\ud569<\/span>\uacfc \ub9ac\ub9cc \ud558\ud569<\/span>\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[U(f,\\, P) = \\sum_{S} M_S |S|, \\quad L(f,\\, P) = \\sum_{S} m_S |S|.\\]<\/p>\n

\uc77c\ubcc0\uc218 \ud568\uc218\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c, \uc0c1\uc801\ubd84<\/span>\uacfc \ud558\uc801\ubd84<\/span>\uc744 \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[\\begin{aligned}
\n\\overline{\\int_R} f(x) \\,dx&= \\inf \\left\\{ U(f,\\, P) \\mid P \\text{ is a partition of }R\\right\\}, \\\\[6pt]
\n\\underline{\\int_R} f(x) \\,dx &= \\sup \\left\\{ L(f,\\, P)\\mid P \\text{ is a partition of }R\\right\\}.
\n\\end{aligned}\\]
\n \ud568\uc218 \\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5<\/span>\ud558\ub2e4(Riemann integrable)\ub294 \uac83\uc740 \\(R\\)\uc5d0\uc11c \\(f\\)\uac00 \uc720\uacc4\uc774\uace0 \\(f\\)\uc758 \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc774 \uac19\uc740 \uac83\uc774\ub2e4. \uc774 \uacf5\ud1b5\uac12\uc744
\n\\[\\int_R f(x) dx\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774 \uac12\uc744 \uac04\ub2e8\ud788 \\(\\int_R f\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n

\\(R\\)\uc774 \\(\\mathbb{R}^2\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc778 \uacbd\uc6b0 \uc774 \uc801\ubd84\uc744 \\(\\iint_R f(x,\\, y) dA\\)\ub85c \ub098\ud0c0\ub0b4\uace0, \\(R\\)\uc774 \\(\\mathbb{R}^3\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc778 \uacbd\uc6b0 \uc774 \uc801\ubd84\uc744 \\(\\iiint_R f(x,\\, y,\\, z) dV\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\uc9c1\uc0ac\uac01\ud615\uc774 \uc544\ub2c8\uace0 \uc720\uacc4\uc778 \uc9d1\ud569 \\(E \\subseteq \\mathbb{R}^n\\) \uc704\uc5d0\uc11c \uc801\ubd84\uc744 \uc815\uc758\ud558\uae30 \uc704\ud574 \ud2b9\uc131\ud568\uc218<\/span>(characteristic function)\ub97c \ub3c4\uc785\ud55c\ub2e4.
\n\\[\\chi_E(x) = \\begin{cases} 1 & \\text{if }\\, x \\in E, \\\\ 0 & \\text{if }\\, x \\notin E. \\end{cases}\\]<\/p>\n

\ud568\uc218 \\(f: E \\to \\mathbb{R}\\)\uc774 \uc720\uacc4\uc77c \ub54c, \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[\\int_E f(x) dx = \\int_R f(x) \\, \\chi_E (x) dx .\\]
\n\uc774\uc640 \uac19\uc774 \uc815\uc758\ud55c \uc801\ubd84\uc758 \uac12\uc740 \\(R\\)\uc758 \uc120\ud0dd\uacfc \uc0c1\uad00\uc5c6\uc774 \uc77c\uc815\ud558\ub2e4.<\/p>\n

\uc801\ubd84 \uac00\ub2a5\uc131\uacfc \uc801\ubd84\uc758 \uc131\uc9c8<\/h3>\n

\uc9d1\ud569 \\(E\\)\uac00 \uc870\ub974\ub2e8 \uac00\uce21<\/span>(Jordan measurable)\uc774\ub77c\ub294 \uac83\uc740 \\(\\chi_E\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud55c \uac83\uc744 \ub73b\ud55c\ub2e4. \uc774\ub54c \\[|E| = \\int_R \\chi_E\\]\ub97c \\(E\\)\uc758 \uc870\ub974\ub2e8 \uce21\ub3c4<\/span>(Jordan measure) \ub610\ub294 \ubd80\ud53c \ub610\ub294 \uccb4\uc801\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\n

\uc815\ub9ac 10.1.<\/span><\/p>\n

\uc9d1\ud569 \\(E\\)\uac00 \ucef4\ud329\ud2b8\uc774\uace0 \uc870\ub974\ub2e8 \uac00\uce21\uc9d1\ud569\uc774\uba70 \ud568\uc218 \\(f\\)\uac00 \\(E\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \\(f\\)\ub294 \\(E\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85 \uac1c\uc694<\/span>
\n\ucef4\ud329\ud2b8 \uc9d1\ud569\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218\uac00 \uade0\ub4f1\uc5f0\uc18d\uc774\ub77c\ub294 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574 \ucda9\ubd84\ud788 \uc138\ubc00\ud55c \ubd84\ud560\uc744 \uc7a1\uc73c\uba74 \uc0c1\ud569\uacfc \ud558\ud569\uc758 \ucc28\uc774\uac00 \\(\\varepsilon\\) \ubbf8\ub9cc\uc774 \ub418\ub3c4\ub85d \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/span><\/p>\n<\/div>\n

\uc2e4\ud568\uc218\uc758 \uc801\ubd84\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \ub2e4\ubcc0\uc218\ud568\uc218\uc758 \uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

    \n
  • \uc120\ud615\uc131<\/strong>: \\(f\\)\uc640 \\(g\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \\(\\alpha\\)\uc640 \\(\\beta\\)\uac00 \uc2e4\uc218\uc77c \ub54c,\\[\\int_E (\\alpha f + \\beta g) = \\alpha \\int_E f + \\beta \\int_E g .\\]<\/li>\n
  • \ub2e8\uc870\uc131<\/strong>: \\(f \\leq g\\)\uc774\uba74 \\[\\int_E f \\leq \\int_E g.\\]<\/li>\n
  • \uc601\uc5ed\uc758 \uac00\ubc95\uc131<\/strong>: \\(E = E_1 \\cup E_2\\)\uc774\uace0 \\(E_1 \\cap E_2\\)\uc758 \uce21\ub3c4\uac00 0\uc774\uba74
    \n \\[\\int_E f = \\int_{E_1} f + \\int_{E_2} f .\\]<\/li>\n
  • \ud3c9\ud589\uc774\ub3d9 \ubd88\ubcc0\uc131<\/strong>: \\(a\\in\\mathbb{R}^n\\), \\(E\\subseteq\\mathbb{R}^n\\)\uc774\uace0 \\(E+a = \\left\\{ x+a \\mid x\\in E \\right\\}\\)\uc77c \ub54c
    \n\\[\\int_{E + a} f(x – a) dx = \\int_E f(x) dx.\\]<\/li>\n<\/ul>\n
    \n

    \ubb38\uc81c 10.1.<\/span>
    \n\\(E\\)\uac00 \\(\\mathbb{R}^2\\)\uc5d0\uc11c \uc138 \uc810 \\((0,\\,0)\\), \\((2,\\,3)\\), \\((4,\\,0)\\)\uc744 \uc787\ub294 \uc0bc\uac01\ud615\uacfc \uadf8 \uc548\ucabd\uc744 \ud3ec\ud568\ud558\ub294 \uc601\uc5ed\uc77c \ub54c, \ub9ac\ub9cc \ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(E\\) \uc704\uc5d0\uc11c \ud568\uc218 \\(f(x,\\,y)=x-2y\\)\uc758 \uc801\ubd84\uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 10.2.<\/span>
    \n\ub2e4\ubcc0\uc218\ud568\uc218\uc758 \uc801\ubd84\uc758 \uae30\ubcf8\uc131\uc9c8\uc744 \uc99d\uba85\ud558\uc2dc\uc624. \uc989 \uc801\ubd84\uc758 \uc120\ud615\uc131, \ub2e8\uc870\uc131, \uc601\uc5ed\uc758 \uac00\ubc95\uc131, \ud3c9\ud589\uc774\ub3d9 \ubd88\ubcc0\uc131\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \uc815\ub9ac 10.2. (\uc911\uc801\ubd84\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n

    \\(f\\)\uac00 \uc870\ub974\ub2e8 \uac00\uce21\uc774\uace0 \uc5f0\uacb0\ub41c \ucef4\ud329\ud2b8 \uc9d1\ud569 \\(E\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(c \\in E\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.
    \n\\[\\int_E f = f(c) |E|\\]<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(f\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c \ucd5c\ub313\uac12 \\(M\\)\uacfc \ucd5c\uc19f\uac12 \\(m\\)\uc744 \uac00\uc9c4\ub2e4. \uc774\ub54c
    \n\\[m |E| \\leq \\int_E f \\leq M |E|\\]
    \n\uc774\ubbc0\ub85c, \uc0ac\uc787\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(f(c)=\\frac{1}{\\lvert E \\rvert} \\int_E f(x)dx \\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(c\\in E\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

    \ubc18\ubcf5\uc801\ubd84\uacfc \ud478\ube44\ub2c8 \uc815\ub9ac<\/h3>\n

    \uc9c1\uc0ac\uac01\ud615 \\(R = [a,\\, b] \\times [c,\\, d]\\)\uc5d0\uc11c \ud568\uc218 \\(f(x,\\, y)\\)\uc758 \uc801\ubd84\uc744 \uacc4\uc0b0\ud558\ub294 \uc720\uc6a9\ud55c \ubc29\ubc95\uc740 \ubc18\ubcf5\uc801\ubd84\uc774\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 10.3. (\ud478\ube44\ub2c8 \uc815\ub9ac – \uc9c1\uc0ac\uac01\ud615\uc758 \uacbd\uc6b0)<\/span><\/p>\n

    \\(f\\)\uac00 \uc9c1\uc0ac\uac01\ud615 \\(R = [a,\\, b] \\times [c,\\, d]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
    \n\\[\\iint_R f(x,\\, y) dA = \\int_a^b \\left(\\int_c^d f(x,\\, y) dy\\right) dx = \\int_c^d \\left(\\int_a^b f(x,\\, y) dx\\right) dy.\\tag{10.1}\\]
    \n\uc704 \ub4f1\uc2dd\uc5d0\uc11c \ub450 \ubc88\uc9f8 \uc801\ubd84\uacfc \uc138 \ubc88\uc9f8 \uc801\ubd84\uc744 \ubc18\ubcf5\uc801\ubd84<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \uc9c1\uc0ac\uac01\ud615 \\(R = [a,\\, b] \\times [c,\\, d]\\)\uc5d0\uc11c \uc5f0\uc18d\ud568\uc218 \\(f\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc790.
    \n\\[\\iint_R f(x,\\, y) dA = \\int_a^b \\left(\\int_c^d f(x,\\, y) dy\\right) dx\\]
    \n\uba3c\uc800 \\(f(x,\\, y) = g(x)h(y)\\) \ud615\ud0dc\uc758 \ud568\uc218\ub97c \uc0dd\uac01\ud558\uc790. \uc5ec\uae30\uc11c \\(g: [a,\\, b] \\to \\mathbb{R}\\)\uacfc \\(h: [c,\\, d] \\to \\mathbb{R}\\)\uc740 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4. \uc77c\ubcc0\uc218 \uc801\ubd84\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec, \ubd84\ud560 \\(P=P_x \\times P_y\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
    \n\\[\\begin{aligned}
    \n\\iint_R g(x)h(y) dA &= \\int_R g(x)h(y) dA \\\\[6pt]
    \n&= \\lim_{\\|P\\| \\to 0} \\sum_{i,j} g(x_i^*)h(y_j^*) \\Delta x_i \\Delta y_j \\\\[6pt]
    \n&= \\lim_{\\|P_x\\| \\to 0} \\sum_{i} g(x_i^*) \\Delta x_i \\cdot \\lim_{\\|P_y\\| \\to 0} \\sum_{j} h(y_j^*) \\Delta y_j \\\\[6pt]
    \n&= \\int_a^b g(x) dx \\cdot \\int_c^d h(y) dy
    \n\\end{aligned}\\]
    \n\ud55c\ud3b8, \ubc18\ubcf5\uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
    \n\\[\\begin{aligned}
    \n\\int_a^b \\left(\\int_c^d g(x)h(y) dy\\right) dx &= \\int_a^b g(x) \\left(\\int_c^d h(y) dy\\right) dx \\\\[6pt]
    \n&= \\int_a^b g(x) dx \\cdot \\int_c^d h(y) dy
    \n\\end{aligned}\\]
    \n\ub530\ub77c\uc11c \ubd84\ub9ac\uac00\ub2a5 \ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub294 \ubc14\ub77c\ub294 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \\(R\\)\uc758 \ubd84\ud560 \\(P = P_x \\times P_y\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc5ec\uae30\uc11c \\(P_x\\)\uc640 \\(P_y\\)\ub294 \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\uc740 \ubd84\ud560\uc774\ub2e4.
    \n\\[\\begin{aligned}
    \nP_x &: a = x_0 < x_1 < \\cdots < x_m = b , \\\\[6pt]\nP_y &: c = y_0 < y_1 < \\cdots < y_n = d .\n\\end{aligned}\\]\n\ubd80\ubd84\uc9c1\uc0ac\uac01\ud615 \\(R_{ij} = [x_{i-1},\\, x_i] \\times [y_{j-1},\\, y_j]\\)\uc5d0\uc11c \ud568\uc22b\uac12\uc774 \uc0c1\uc218 \\(c_{ij}\\)\uc778 \uacc4\ub2e8\ud568\uc218\n\\[s(x,\\, y) = \\sum_{i=1}^{m} \\sum_{j=1}^{n} c_{ij} \\chi_{R_{ij}}(x,\\, y)\\]\n\ub97c \uc0dd\uac01\ud558\uc790. \uc5ec\uae30\uc11c \\(\\chi_{R_{ij}}\\)\ub294 \\(R_{ij}\\)\uc758 \ud2b9\uc131\ud568\uc218\uc774\ub2e4. \uc774 \uacc4\ub2e8\ud568\uc218\ub294 \ubd84\ub9ac\uac00\ub2a5 \ud568\uc218\ub4e4\uc758 \uc77c\ucc28\uacb0\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4.\n\\[s(x,\\, y) = \\sum_{i,j} c_{ij} \\chi_{[x_{i-1},x_i]}(x) \\cdot \\chi_{[y_{j-1},y_j]}(y).\\]\n\uc801\ubd84\uc758 \uc120\ud615\uc131\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc911\uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\\[\\begin{aligned}\n\\iint_R s(x,\\, y) dA &= \\sum_{i,j} c_{ij} \\iint_R \\chi_{R_{ij}}(x,\\, y) dA \\\\[6pt]\n&= \\sum_{i,j} c_{ij} (x_i - x_{i-1})(y_j - y_{j-1}).\n\\end{aligned}\\]\n\ubc18\ubcf5\uc801\ubd84\uc758 \uacc4\uc0b0 \uacb0\uacfc\ub3c4 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\\[\\begin{aligned}\n\\int_a^b \\left(\\int_c^d s(x,\\, y) dy\\right) dx &= \\sum_{i=1}^{m} \\int_{x_{i-1}}^{x_i}\\left( \\sum_{j=1}^{n} c_{ij}\\left( y_j - y_{j-1}\\right) \\right) dx \\\\[6pt]\n&= \\sum_{i=1}^{m} \\sum_{j=1}^{n} c_{ij} (x_i - x_{i-1}) (y_j - y_{j-1}).\n\\end{aligned}\\]<\/p>\n

    \uc774\uc81c \uc801\ubd84\ud568\uc218\uc758 \uc5f0\uc18d\uc131\uc744 \ubcf4\uc774\uc790. \uc989 \\(F(x) = \\int_c^d f(x,\\, y) dy\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc790.<\/p>\n

    \\(x_0 \\in [a,\\, b]\\)\uc640 \\(\\varepsilon > 0\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(f\\)\ub294 \ucef4\ud329\ud2b8 \uc9d1\ud569 \\(R\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ubbc0\ub85c, \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(\\|(x_1,\\, y_1) – (x_2,\\, y_2)\\| < \\delta\\)\uc77c \ub54c\ub9c8\ub2e4\n\\[|f(x_1,\\, y_1) - f(x_2,\\, y_2)| < \\frac{\\varepsilon}{d - c}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \\(|x - x_0| < \\delta\\)\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[\\begin{aligned}\n|F(x) - F(x_0)| &= \\left|\\int_c^d f(x,\\, y) dy - \\int_c^d f(x_0,\\, y) dy\\right| \\\\[6pt]\n&= \\left|\\int_c^d [f(x,\\, y) - f(x_0,\\, y)] dy\\right| \\\\[6pt]\n&\\leq \\int_c^d |f(x,\\, y) - f(x_0,\\, y)| dy \\\\[6pt]\n&< \\int_c^d \\frac{\\varepsilon}{d - c} dy = \\varepsilon .\n\\end{aligned}\\]\n\uadf8\ub7ec\ubbc0\ub85c \\(F\\)\ub294 \uc5f0\uc18d\uc774\ub2e4.<\/p>\n

    \\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f\\)\ub294 \\(R\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574, \ucda9\ubd84\ud788 \uc138\ubc00\ud55c \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud558\uba74 \uac01 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615 \\(R_{ij}\\)\uc5d0\uc11c \\(f\\)\uc758 \uc9c4\ub3d9\uc774 \\(\\varepsilon\/|R|\\) \ubbf8\ub9cc\uc774 \ub418\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4. \uacc4\ub2e8\ud568\uc218 \\(s^-\\)\uc640 \\(s^+\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790:
    \n\\[\\begin{aligned}
    \ns^-(x,\\, y) &= \\sum_{i,j} m_{ij} \\chi_{R_{ij}}(x,\\, y), \\\\[6pt]
    \ns^+(x,\\, y) &= \\sum_{i,j} M_{ij} \\chi_{R_{ij}}(x,\\, y).
    \n\\end{aligned}\\]
    \n\uc5ec\uae30\uc11c \\(m_{ij} = \\inf\\{f(x,\\, y) \\mid (x,\\, y) \\in R_{ij}\\}\\)\uc774\uace0 \\(M_{ij} = \\sup\\{f(x,\\, y) \\mid (x,\\, y) \\in R_{ij}\\}\\)\uc774\ub2e4. \uadf8\ub7ec\uba74 \\(s^- \\leq f \\leq s^+\\)\uc774\uace0
    \n\\[\\iint_R (s^+ – s^-) dA < \\varepsilon\\]\n\uc774\ub2e4. \uacc4\ub2e8\ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub294 \ud478\ube44\ub2c8 \uc815\ub9ac\uac00 \uc131\ub9bd\ud558\ubbc0\ub85c \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.\n\\[\\iint_R s^{\\pm} dA = \\int_a^b \\left(\\int_c^d s^{\\pm}(x,\\, y) dy\\right) dx.\\]\n\uc801\ubd84\uc758 \ub2e8\uc870\uc131\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.\n\\[\\begin{gathered}\n\\iint_R s^- dA \\leq \\iint_R f dA \\leq \\iint_R s^+ dA ,\\\\[3pt]\n\\int_a^b \\left(\\int_c^d s^-(x,\\, y) dy\\right) dx \\leq \\int_a^b \\left(\\int_c^d f(x,\\, y) dy\\right) dx \\leq \\int_a^b \\left(\\int_c^d s^+(x,\\, y) dy\\right) dx.\n\\end{gathered}\\]\n\uc5ec\uae30\uc5d0 \\(\\varepsilon \\to 0\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.\n\\[\\begin{aligned}\n\\iint_R s^{\\pm} dA &\\to \\iint_R f dA ,\\\\[6pt]\n\\int_a^b \\left(\\int_c^d s^{\\pm}(x,\\, y) dy\\right) dx &\\to \\int_a^b \\left(\\int_c^d f(x,\\, y) dy\\right) dx .\n\\end{aligned}\\]\n\ub530\ub77c\uc11c\n\\[\\iint_R f(x,\\, y) dA = \\int_a^b \\left(\\int_c^d f(x,\\, y) dy\\right) dx\\tag{10.2}\\]\n\uc774\ub2e4. \\(x\\)\uc640 \\(y\\)\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uba74 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc801\ubd84 \uc21c\uc11c\ub97c \ubc14\uafbc \uacbd\uc6b0\ub3c4 \uac19\uc740 \uac12\uc744 \uac00\uc9d0\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.\n\\[\\iint_R f(x,\\, y) dA = \\int_c^d \\left(\\int_a^b f(x,\\, y) dx\\right) dy.\\tag{10.3}\\]\n\ub450 \ub4f1\uc2dd (10.2)\uc640 (10.3)\ub97c \uacb0\ud569\ud558\uba74 \ubc14\ub77c\ub294 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

    \ud478\ube44\ub2c8 \uc815\ub9ac\ub97c \uc9c1\uc0ac\uac01\ud615\uc774 \uc544\ub2cc \uc601\uc5ed\uc73c\ub85c \ud655\uc7a5\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 10.4. (\ud478\ube44\ub2c8 \uc815\ub9ac – \uc9c1\uc0ac\uac01\ud615\uc774 \uc544\ub2cc \uc601\uc5ed)<\/span><\/p>\n

    \\(E \\subseteq \\mathbb{R}^2\\)\uac00 \uc870\ub974\ub2e8 \uac00\uce21\uc774\uace0 \\(f\\)\uac00 \\(E\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

      \n
    1. \\(E = \\{(x,\\, y) \\mid a \\leq x \\leq b,\\, \\phi(x) \\leq y \\leq \\psi(x)\\}\\) \ud615\ud0dc\uc77c \ub54c
      \n \\[\\iint_E f(x,\\, y) dA = \\int_a^b \\int_{\\phi(x)}^{\\psi(x)} f(x,\\, y) dy\\, dx.\\]<\/li>\n
    2. \\(E = \\{(x,\\, y) \\mid c \\leq y \\leq d,\\, \\alpha(y) \\leq x \\leq \\beta(y)\\}\\) \ud615\ud0dc\uc77c \ub54c
      \n \\[\\iint_E f(x,\\, y) dA = \\int_c^d \\int_{\\alpha(y)}^{\\beta(y)} f(x,\\, y) dx\\, dy.\\]<\/li>\n<\/ol>\n<\/div>\n
      \n

      \uc99d\uba85<\/p>\n

      \\(E = \\{(x,\\, y) \\mid a \\leq x \\leq b,\\, \\phi(x) \\leq y \\leq \\psi(x)\\}\\) \ud615\ud0dc\uc758 \uc601\uc5ed\uc5d0 \ub300\ud574 \uc99d\uba85\ud558\uc790.
      \n\\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc9c1\uc0ac\uac01\ud615 \\(R = [a,\\, b] \\times [c,\\, d]\\)\ub97c \ud0dd\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(c \\leq \\phi(x) \\leq \\psi(x) \\leq d\\)\uc774\ub2e4. \ud655\uc7a5\ub41c \ud568\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
      \n\\[F(x,\\, y) = f(x,\\, y) \\cdot \\chi_E(x,\\, y) = \\begin{cases}
      \nf(x,\\, y) & \\text{if } (x,\\, y) \\in E , \\\\
      \n0 & \\text{if } (x,\\, y) \\notin E .
      \n\\end{cases} \\]
      \n\\(E\\)\uac00 \uc870\ub974\ub2e8 \uac00\uce21\uc774\ubbc0\ub85c \\(\\chi_E\\)\ub294 \uac70\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0, \\(f\\)\uac00 \uc5f0\uc18d\uc774\ubbc0\ub85c \\(F\\)\ub294 \uac70\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \ub530\ub77c\uc11c
      \n\\[\\int_E f = \\iint_R F(x,\\, y) dA = \\int_a^b \\left(\\int_c^d F(x,\\, y) dy\\right) dx\\]
      \n\uc774\ub2e4.
      \n\uac01\uac01\uc758 \\(x \\in [a,\\, b]\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
      \n\\[\\int_c^d F(x,\\, y) dy = \\int_c^d f(x,\\, y) \\chi_E(x,\\, y) dy.\\]
      \n\\(E\\)\uc758 \uc815\uc758\uc5d0 \uc758\ud574 \\(\\chi_E(x,\\, y) = 1\\)\uc778 \uac83\uc740 \\(\\phi(x) \\leq y \\leq \\psi(x)\\)\uc77c \ub54c\ubfd0\uc774\ubbc0\ub85c \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.
      \n\\[\\int_c^d F(x,\\, y) dy = \\int_{\\phi(x)}^{\\psi(x)} f(x,\\, y) dy\\]
      \n\ub530\ub77c\uc11c
      \n\\[\\int_E f = \\int_a^b \\left(\\int_{\\phi(x)}^{\\psi(x)} f(x,\\, y) dy\\right) dx.\\]<\/p>\n

      \\(E = \\{(x,\\, y) \\mid c \\leq y \\leq d,\\, \\alpha(y) \\leq x \\leq \\beta(y)\\}\\) \ud615\ud0dc\uc758 \uc601\uc5ed\uc5d0 \ub300\ud574\uc11c\ub3c4 \ub3d9\uc77c\ud55c \ubc29\ubc95\uc73c\ub85c
      \n\\[\\int_E f = \\int_c^d \\left(\\int_{\\alpha(y)}^{\\beta(y)} f(x,\\, y) dx\\right) dy\\]
      \n\ub97c \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

      \uc77c\ubc18\uc801\uc73c\ub85c \uc601\uc5ed \\(E\\)\uac00 \uc720\ud55c \uac1c\uc758 \ud45c\uc900 \uc601\uc5ed(\ud568\uc218 \uadf8\ub798\ud504\ub85c \uc815\uc758\ub41c \uc601\uc5ed)\uc758 \ud569\uc9d1\ud569\uc73c\ub85c \ud45c\ud604\ub420 \ub54c, \uac01 \ubd80\ubd84\uc5d0 \ud478\ube44\ub2c8 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uace0 \uacb0\uacfc\ub97c \ud569\ud558\uba74 \ub41c\ub2e4. \uc774\ub54c \uacb9\uce58\ub294 \ubd80\ubd84\uc758 \uce21\ub3c4\uac00 0\uc774\uc5b4\uc57c \ud55c\ub2e4.<\/p>\n

      \n

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

      \ud568\uc218 \\(f\\)\uac00 \uc6d0\ud310 \\(D = \\{(x,\\, y) \\mid x^2 + y^2 \\leq r^2\\}\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \\(D\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n

        \n
      • \\(y\\)\uc5d0 \ub300\ud558\uc5ec \uba3c\uc800 \uc801\ubd84\ud558\ub294 \uacbd\uc6b0 \\(D = \\left\\{(x,\\, y) \\mid -r \\leq x \\leq r,\\, -\\sqrt{r^2 – x^2} \\leq y \\leq \\sqrt{r^2 – x^2}\\right\\}\\)\uc774\ubbc0\ub85c
        \n\\[\\iint_D f(x,\\, y) dA = \\int_{-r}^r \\int_{-\\sqrt{r^2-x^2}}^{\\sqrt{r^2-x^2}} f(x,\\, y) dy\\, dx.\\]<\/li>\n
      • \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uba3c\uc800 \uc801\ubd84\ud558\ub294 \uacbd\uc6b0 \\(D = \\left\\{(x,\\, y) \\mid -r \\leq y \\leq r,\\, -\\sqrt{r^2 – y^2} \\leq x \\leq \\sqrt{r^2 – y^2}\\right\\}\\)\uc774\ubbc0\ub85c
        \n\\[\\iint_D f(x,\\, y) dA = \\int_{-r}^r \\int_{-\\sqrt{r^2-y^2}}^{\\sqrt{r^2-y^2}} f(x,\\, y) dx\\, dy.\\]<\/li>\n<\/ul>\n<\/div>\n

        \uace0\ucc28\uc6d0\uc758 \uacbd\uc6b0\ub3c4 \uc720\uc0ac\ud558\uac8c \ubc18\ubcf5\uc801\ubd84\uc73c\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \ud2b9\ud788 \\(\\mathbb{R}^3\\)\uc5d0\uc11c \ub2e4\uc74c\uacfc \uac19\uc774 \uc911\uc801\ubd84\uc744 \uc138 \ubc88 \ubc18\ubcf5\ud558\ub294 \uc801\ubd84\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.
        \n\\[\\iiint_E f(x,\\, y,\\, z) dV = \\int \\int \\int f(x,\\, y,\\, z) dx\\, dy\\, dz.\\]
        \n\uc5ec\uae30\uc11c \uc801\ubd84 \uc21c\uc11c\ub294 \uc601\uc5ed\uc758 \ud615\ud0dc\uc5d0 \ub530\ub77c \uc120\ud0dd\ud55c\ub2e4.<\/p>\n

        \n

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

        \uc0ac\uba74\uccb4 \\(T = \\left\\{(x,\\, y,\\, z) \\mid x,\\, y,\\, z \\geq 0,\\, x + y + z \\leq 1\\right\\}\\)\uc758 \ubd80\ud53c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
        \n\\[\\begin{aligned}
        \n|T| &= \\int_0^1 \\int_0^{1-x} \\int_0^{1-x-y} dz\\, dy\\, dx \\\\[6pt]
        \n&= \\int_0^1 \\int_0^{1-x} (1 – x – y) dy\\, dx \\\\[6pt]
        \n&= \\int_0^1 \\frac{(1-x)^2}{2} dx = \\frac{1}{6}.
        \n\\end{aligned}\\]<\/p>\n<\/div>\n

        \n

        \ubb38\uc81c 10.3.<\/span>
        \n\ud568\uc218 \\(f\\)\uc640 \uc9d1\ud569 \\(E\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(E\\) \uc704\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n

          \n
        1. \\(f(x,\\,y)=\\frac{1}{1+x^2}\\), \\(E\\)\ub294 \ub450 \uc9c1\uc120 \\(x=1\\), \\(y=0\\)\uacfc \uace1\uc120 \\(y=x^3\\)\uc73c\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed.<\/li>\n
        2. \\(f(x,\\,y)=x+y\\), \\(E\\)\ub294 \uc138 \uc810 \\((0,\\,0)\\), \\((0,\\,1)\\), \\((2,\\,0)\\)\uc744 \uaf2d\uc9d3\uc810\uc73c\ub85c \ud558\ub294 \uc0bc\uac01\ud615 \uc601\uc5ed.<\/li>\n
        3. \\(f(x,\\,y,\\,z)=x\\), \\(E\\)\ub294 \uc138 \ubd80\ub4f1\uc2dd \\(0\\le z\\le 1-x^2\\), \\(0\\le y\\le x^2 + z^2\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\((x,\\,y,\\,z)\\)\uc758 \ubaa8\uc784.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \ubb38\uc81c 10.4.<\/span>
          \n\\(D\\)\uac00 \\(y = x^2\\)\uacfc \\(y = 2x\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed\uc77c \ub54c, \uc801\ubd84 \\(\\iint_D xy\\, dA\\)\ub97c \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

          \ubcc0\uc218\ubcc0\ud658<\/h3>\n

          \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84\uc5d0\uc11c \uce58\ud658\uc801\ubd84\uc5d0 \ud574\ub2f9\ud558\ub294 \uac83\uc774 \ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84\uc5d0\uc11c\ub294 \ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac\uc774\ub2e4.<\/p>\n

          \n

          \uc815\ub9ac 10.5. (\ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac)<\/span><\/p>\n

          \\(\\phi: U \\to V\\)\uac00 \uc5f4\ub9b0\uc9d1\ud569 \uc0ac\uc774\uc758 \uc77c\ub300\uc77c\ub300\uc751\uc774\uace0, \\(\\phi\\)\uc640 \\(\\phi^{-1}\\)\uc774 \ubaa8\ub450 \\(C^1\\)\uc774\uba70, \\(K \\subseteq U\\)\uac00 \ucef4\ud329\ud2b8 \uc870\ub974\ub2e8 \uac00\uce21\uc9d1\ud569\uc774\uace0, \\(f\\)\uac00 \\(\\phi(K)\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba70, \\(U\\)\uc5d0\uc11c \\(\\det D \\phi (x)\\ne 0\\)\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
          \n\\[\\int_{\\phi(K)} f(y) dy = \\int_K f(\\phi(x)) |\\det D\\phi(x)| dx.\\tag{10.4}\\]
          \n\uc5ec\uae30\uc11c \\(\\det D\\phi(x)\\)\ub97c \uc57c\ucf54\ube44 \ud589\ub82c\uc2dd<\/span>(Jacobian)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/p>\n

          \uba3c\uc800 \\(\\phi(x) = Ax + b\\)\uc778 \uc544\ud540 \ubcc0\ud658\uc744 \uc0dd\uac01\ud558\uc790. \uc5ec\uae30\uc11c \\(A\\)\ub294 \uac00\uc5ed\ud589\ub82c\ub85c \ud45c\ud604\ub418\ub294 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \ud3c9\ud589\uc774\ub3d9\uc740 \uc801\ubd84\uac12\uc744 \ubcc0\ud654\uc2dc\ud0a4\uc9c0 \uc54a\uc73c\ubbc0\ub85c \\(\\phi(x) = Ax\\)\uc778 \uc120\ud615\ubcc0\ud658\ub9cc \uace0\ub824\ud558\uba74 \ub41c\ub2e4.<\/p>\n

          \uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc5d0 \ub300\ud574, \\(A(R)\\)\uc758 \ubd80\ud53c\ub294 \\(|\\det A| \\cdot |R|\\)\uc774\ub2e4. \ub530\ub77c\uc11c
          \n\\[\\int_{A(R)} f(y) dy = |\\det A| \\int_R f(Ax) dx\\]
          \n\uc774\ub2e4. \uc774\ub97c \uc9c1\uc0ac\uac01\ud615\uc774 \uc544\ub2cc \uc9d1\ud569\uc73c\ub85c \ud655\uc7a5\ud558\uba74
          \n\\[\\int_{A(K)} f(y) dy = |\\det A| \\int_K f(Ax) dx\\]
          \n\uc774\ub2e4.
          \n\\(\\phi\\)\uac00 \\(C^1\\)\uc774\ubbc0\ub85c, \uc810 \\(x_0 \\in K\\) \uadfc\ucc98\uc5d0\uc11c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
          \n\\[\\phi(x_0 + h) = \\phi(x_0) + D\\phi(x_0)h + o(\\|h\\|).\\]
          \n\uc989 \ucda9\ubd84\ud788 \uc791\uc740 \uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc5d0 \ub300\ud558\uc5ec, \\(\\phi\\)\ub294 \uadfc\uc0ac\uc801\uc73c\ub85c \uc120\ud615\ubcc0\ud658 \\(x \\mapsto \\phi(x_0) + D\\phi(x_0)(x – x_0)\\)\ucc98\ub7fc \ud589\ub3d9\ud55c\ub2e4.
          \n\\(K\\)\ub97c \ucda9\ubd84\ud788 \uc791\uc740 \uc9c1\uc0ac\uac01\ud615\ub4e4 \\(\\{R_i\\}\\)\ub85c \ubd84\ud560\ud558\uc790. \uac01 \\(R_i\\)\uc5d0\uc11c \uc810 \\(x_i\\)\ub97c \uc120\ud0dd\ud55c\ub2e4. \\(R_i\\)\uac00 \ucda9\ubd84\ud788 \uc791\uc73c\uba74 \\(\\phi\\)\uc5d0 \uc758\ud55c \\(R_i\\)\uc758 \uc0c1(image)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uadfc\uc0ac\ud560 \uc218 \uc788\ub2e4.
          \n\\[\\begin{aligned}
          \n\\phi(R_i) &\\approx \\phi(x_i) + D\\phi(x_i)(R_i – x_i) , \\\\[6pt]
          \n|\\phi(R_i)| &\\approx |\\det D\\phi(x_i)| \\cdot |R_i| .
          \n\\end{aligned}\\]
          \n\uadf8\ub7ec\ubbc0\ub85c \ubd84\ud560\uc774 \ucda9\ubd84\ud788 \uc138\ubc00\ud560 \ub54c \\(\\phi(K)\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uadfc\uc0ac\ub41c\ub2e4.
          \n\\[\\begin{align*}
          \n\\int_{\\phi(K)} f(y) dy &\\approx \\sum_i f(\\phi(x_i)) |\\phi(R_i)| \\tag{10.5}\\\\[6pt]
          \n&\\approx \\sum_i f(\\phi(x_i)) |\\det D\\phi(x_i)| |R_i| \\tag{10.6}\\\\[6pt]
          \n&\\approx \\int_K f(\\phi(x)) |\\det D\\phi(x)| dx. \\tag{10.7}
          \n\\end{align*}\\]
          \n\\(\\phi\\)\uac00 \\(C^1\\)\uc774\uace0 \\(K\\)\uac00 \ucef4\ud329\ud2b8\uc774\ubbc0\ub85c \\(D\\phi\\)\ub294 \\(K\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\uace0, \\(|\\det D\\phi|\\)\ub3c4 \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ucda9\ubd84\ud788 \uc138\ubc00\ud55c \ubd84\ud560\uc744 \ud0dd\ud558\uba74 \uc120\ud615 \uadfc\uc0ac\uc758 \uc624\ucc28\uac00 \uade0\ub4f1\ud558\uac8c 0\uc73c\ub85c \uc218\ub834\ud55c\ub2e4. \uc989 \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec, \ucda9\ubd84\ud788 \uc138\ubc00\ud55c \ubd84\ud560\uc774 \uc874\uc7ac\ud558\uc5ec, \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.
          \n\\[\\left|\\frac{|\\phi(R_i)|}{|R_i|} – |\\det D\\phi(x_i)|\\right| < \\varepsilon .\\]\n\ubd84\ud560\uc758 \ub178\ub984\uc774 0\uc73c\ub85c \uac00\ub294 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 (10.5)\uc758 \uc6b0\ubcc0\uc758 \ub9ac\ub9cc\ud569\uc774 (10.5)\uc758 \uc88c\ubcc0\uc758 \uc801\ubd84\uc5d0 \uc218\ub834\ud558\uace0, (10.6)\uc758 \ub9ac\ub9cc\ud569\uc774 \uac01\uac01 (10.7)\uc758 \uc801\ubd84\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

          \uc704 \uc815\ub9ac\uc5d0\uc11c \uc57c\ucf54\ube44 \ud589\ub82c\uc2dd\uc758 \uc808\ub313\uac12 \\(|\\det D\\phi(x)|\\)\ub294 \ubcc0\ud658 \\(\\phi\\)\uac00 \uc810 \\(x\\) \uadfc\ucc98\uc5d0\uc11c \ubd80\ud53c\ub97c \ud655\ub300\ud558\uac70\ub098 \ucd95\uc18c\ud558\ub294 \ube44\uc728\uc744 \ub098\ud0c0\ub0b8\ub2e4. \uc5ec\uae30\uc11c \uc57c\ucf54\ube44 \ud589\ub82c\uc2dd\uc5d0 \uc808\ub313\uac12\uc744 \ucde8\ud558\ub294 \uc774\uc720\ub294 \\(\\det D\\phi < 0\\)\uc77c \ub54c\ub3c4 \ubcc0\ud658\ub41c \uc870\uac01\uc758 \ubd80\ud53c \uc591\uc218\uc5ec\uc57c \ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

          \ucc38\uace0\ub85c \uc704 \uc815\ub9ac\uc5d0\uc11c \uc870\uac74 “\\(C^1\\) \uc77c\ub300\uc77c\ub300\uc751”\uc740 \uc644\ud654\ud560 \uc218 \uc788\ub2e4. \uc989 \uce21\ub3c4 0\uc778 \uc9d1\ud569\uc744 \uc81c\uc678\ud558\uace0 \uc77c\ub300\uc77c\uc774\uba74 \ucda9\ubd84\ud558\ub2e4.<\/p>\n

          \uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uc88c\ud45c\ubcc0\ud658\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n