{"id":4614,"date":"2020-05-23T00:01:01","date_gmt":"2020-05-22T15:01:01","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=4614"},"modified":"2020-05-23T00:56:44","modified_gmt":"2020-05-22T15:56:44","slug":"calculus-partial-derivatives-with-constrained-variables-examples","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-partial-derivatives-with-constrained-variables-examples\/","title":{"rendered":"\ubcc0\uc218\uc5d0 \uc81c\ud55c\uc870\uac74\uc774 \uc788\uc744 \ub54c\uc758 \ud3b8\ubbf8\ubd84 \uc608"},"content":{"rendered":"
\n

\uc608\uc81c 1.<\/span>
\n\ud568\uc218 \\(f(x,\\,y,\\,z)\\)\uac00 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0
\n\\[f(x,\\,y,\\,z)=0 \\tag{1.1}\\]
\n\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c
\n\\[\\left( \\frac{\\partial x}{\\partial g} \\right)_z \\left( \\frac{\\partial y}{\\partial z}\\right)_x \\left( \\frac{\\partial z}{\\partial x}\\right)_y =-1\\tag{1.2}\\]
\n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.
\n(Thomas’ Calculus 13ed 14.10. Exercise 9.)\n<\/p>\n<\/div>\n

\n

\ud480\uc774.<\/span>
\n\uba3c\uc800 \\(y,\\) \\(z\\)\ub97c \ub3c5\ub9bd\ubcc0\uc218\ub85c \ub450\uace0 (1.1)\uc758 \uc591\ubcc0\uc744 \\(y\\)\uc5d0 \uad00\ud558\uc5ec \ubbf8\ubd84\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\begin{align}
\n&\\frac{\\partial f}{\\partial x} \\frac{\\partial x}{\\partial y} + \\frac{\\partial f}{\\partial y} \\frac{\\partial y}{\\partial y} + \\frac{\\partial f}{\\partial z} \\frac{\\partial z}{\\partial y} =0 \\\\[6pt]
\n\\Rightarrow \\quad &
\n\\frac{\\partial f}{\\partial x} \\frac{\\partial x}{\\partial y} = – \\frac{\\partial f}{\\partial y} \\\\[6pt]
\n\\Rightarrow \\quad & \\left( \\frac{\\partial x}{\\partial y} \\right)_z = – \\frac{\\partial f \/ \\partial y}{\\partial f\/\\partial x} . \\tag{1.3}
\n\\end{align}\\]<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \\(x,\\) \\(y\\)\ub97c \ub3c5\ub9bd\ubcc0\uc218\ub85c \ub450\uace0 (1.1)\uc758 \uc591\ubcc0\uc744 \\(y\\)\uc5d0 \uad00\ud558\uc5ec \ubbf8\ubd84\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\begin{align}
\n&\\frac{\\partial f}{\\partial x} \\frac{\\partial x}{\\partial x} + \\frac{\\partial f}{\\partial y} \\frac{\\partial y}{\\partial x} + \\frac{\\partial f}{\\partial z} \\frac{\\partial z}{\\partial x} =0 \\\\[6pt]
\n\\Rightarrow \\quad &
\n\\frac{\\partial f}{\\partial z} \\frac{\\partial z}{\\partial x} = – \\frac{\\partial f}{\\partial x} \\\\[6pt]
\n\\Rightarrow \\quad & \\left( \\frac{\\partial z}{\\partial x} \\right)_y = – \\frac{\\partial f \/ \\partial x}{\\partial f\/\\partial z} . \\tag{1.4}
\n\\end{align}\\]<\/p>\n

\ub05d\uc73c\ub85c \\(z,\\) \\(x\\)\ub97c \ub3c5\ub9bd\ubcc0\uc218\ub85c \ub450\uace0 (1.1)\uc758 \uc591\ubcc0\uc744 \\(z\\)\uc5d0 \uad00\ud558\uc5ec \ubbf8\ubd84\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\begin{align}
\n&\\frac{\\partial f}{\\partial x} \\frac{\\partial x}{\\partial z} + \\frac{\\partial f}{\\partial y} \\frac{\\partial y}{\\partial z} + \\frac{\\partial f}{\\partial z} \\frac{\\partial z}{\\partial z} =0 \\\\[6pt]
\n\\Rightarrow \\quad &
\n\\frac{\\partial f}{\\partial y} \\frac{\\partial y}{\\partial z} = – \\frac{\\partial f}{\\partial z} \\\\[6pt]
\n\\Rightarrow \\quad & \\left( \\frac{\\partial y}{\\partial z} \\right)_x = – \\frac{\\partial f \/ \\partial z}{\\partial f\/\\partial y} . \\tag{1.5}
\n\\end{align}\\]<\/p>\n

\uc774\uc81c (1.3), (1.4), (1.5)\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\left( \\frac{\\partial x}{\\partial y} \\right)_z \\left( \\frac{\\partial y}{\\partial z} \\right)_x \\left( \\frac{\\partial z}{\\partial x}\\right)_y
\n= – \\frac{\\partial f \/ \\partial y}{\\partial f\/\\partial x} \\frac{\\partial f \/ \\partial z}{\\partial f\/\\partial y} \\frac{\\partial f \/ \\partial x}{\\partial f\/\\partial z} = -1. \\tag*{\\(\\square\\)}\\]\n<\/p>\n<\/div>\n

\n

\uc608\uc81c 2.<\/span>
\n\ub450 \ud568\uc218 \\(f(x,\\,y,\\,z,\\,w),\\) \\(g(x,\\,y,\\,z,\\,w)\\)\uac00 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0, \ub450 \ub4f1\uc2dd
\n\\[f(x,\\,y,\\,z,\\,w)=0,\\,\\,g(x,\\,y,\\,z,\\,w)=0 \\tag{2.1}\\]
\n\uc5d0 \uc758\ud558\uc5ec \\(z,\\) \\(w\\)\uac00 \\(x,\\) \\(y\\)\uc758 \ud568\uc218\ub85c\uc11c \uc74c\uc801\uc73c\ub85c \ub41c\ub2e4(defined implicitly)\uace0 \ud558\uc790. \ub9cc\uc57d
\n\\[\\frac{\\partial f}{\\partial z} \\frac{\\partial g}{\\partial w} – \\frac{\\partial f}{\\partial w} \\frac{\\partial g}{\\partial z} \\ne 0 \\tag{2.2}\\]
\n\uc774\uba74
\n\\[\\left( \\frac{\\partial z}{\\partial x} \\right)_y = – \\frac{ \\frac{\\partial f}{\\partial x} \\frac{\\partial g}{\\partial w} – \\frac{\\partial f}{\\partial w} \\frac{\\partial g}{\\partial x} }{ \\frac{\\partial f}{\\partial z} \\frac{\\partial g}{\\partial w} – \\frac{\\partial f}{\\partial w} \\frac{\\partial g}{\\partial x} }\\tag{2.3}\\]
\n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.
\n(Thomas’ Calculus 13ed 14.10. Exercise 12.)\n<\/p>\n<\/div>\n

\n

\ud480\uc774.<\/span>
\n(2.1)\uc758 \ub450 \ub4f1\uc2dd\uc758 \uc591\ubcc0\uc744 \\(x\\)\uc5d0 \uad00\ud558\uc5ec \ubbf8\ubd84\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\begin{gather}
\n\\frac{\\partial f}{\\partial x} \\frac{\\partial x}{\\partial x} + \\frac{\\partial f}{\\partial y} \\frac{\\partial y}{\\partial x} + \\frac{\\partial f}{\\partial z} \\frac{\\partial z}{\\partial x} + \\frac{\\partial f}{\\partial w} \\frac{\\partial w}{\\partial x} =0, \\tag{2.4} \\\\[6pt]
\n\\frac{\\partial g}{\\partial x} \\frac{\\partial x}{\\partial x} + \\frac{\\partial g}{\\partial y} \\frac{\\partial y}{\\partial x} + \\frac{\\partial g}{\\partial z} \\frac{\\partial z}{\\partial x} + \\frac{\\partial g}{\\partial w} \\frac{\\partial w}{\\partial x} =0. \\tag{2.5}
\n\\end{gather}\\]
\n\uc5ec\uae30\uc11c \\(\\partial x \/ \\partial x = 1\\)\uc774\uace0 \\(\\partial y \/ \\partial x = 0\\)\uc774\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{gather}
\n\\frac{\\partial f}{\\partial z} \\frac{\\partial z}{\\partial z} + \\frac{\\partial f}{\\partial w} \\frac{\\partial w}{\\partial x} = – \\frac{\\partial f}{\\partial x}, \\tag{2.6} \\\\[6pt]
\n\\frac{\\partial g}{\\partial z} \\frac{\\partial z}{\\partial z} + \\frac{\\partial g}{\\partial w} \\frac{\\partial w}{\\partial x} = – \\frac{\\partial g}{\\partial x}. \\tag{2.7}
\n\\end{gather}\\]
\n(2.6)\uc758 \uc591\ubcc0\uc5d0 \\(\\partial g \/ \\partial w\\)\ub97c \uacf1\ud558\uace0, (2.7)\uc758 \uc591\ubcc0\uc5d0 \\(\\partial f \/ \\partial w\\)\ub97c \uacf1\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{gather}
\n\\frac{\\partial f}{\\partial z} \\frac{\\partial g}{\\partial w} \\frac{\\partial z}{\\partial x} + \\frac{\\partial f}{\\partial w} \\frac{\\partial g}{\\partial w} \\frac{\\partial w}{\\partial x} = – \\frac{\\partial f}{\\partial x} \\frac{\\partial g}{\\partial w}, \\tag{2.8} \\\\[6pt]
\n\\frac{\\partial f}{\\partial w} \\frac{\\partial g}{\\partial z} \\frac{\\partial z}{\\partial x} + \\frac{\\partial f}{\\partial w} \\frac{\\partial g}{\\partial w} \\frac{\\partial w}{\\partial x} = – \\frac{\\partial f}{\\partial w} \\frac{\\partial g}{\\partial x}. \\tag{2.9} \\\\[6pt]
\n\\end{gather}\\]
\n(2.8)\uc5d0\uc11c (2.9)\ub97c \ubcc0\ub9c8\ub2e4 \ube7c\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\left( \\frac{\\partial f}{\\partial z} \\frac{\\partial g}{\\partial w} – \\frac{\\partial f}{\\partial w} \\frac{\\partial g}{\\partial z} \\right) \\frac{\\partial z}{\\partial x} = – \\left( \\frac{\\partial f}{\\partial x} \\frac{\\partial g}{\\partial w} \\frac{\\partial g}{\\partial x} \\right) \\tag{2.10}\\]
\n(2.2)\uc5d0 \uc758\ud558\uc5ec (2.10)\uc758 \uc88c\ubcc0\uc758 \uccab\uc9f8 \uc778\uc218\ub294 \\(0\\)\uc774 \uc544\ub2c8\ubbc0\ub85c, \uc591\ubcc0\uc744 \uadf8 \uc778\uc218\ub85c \ub098\ub20c \uc218 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[ \\frac{\\partial z}{\\partial x} = – \\frac{ \\frac{\\partial f}{\\partial x} \\frac{\\partial g}{\\partial w} – \\frac{\\partial f}{\\partial w} \\frac{\\partial g}{\\partial x} }{ \\frac{\\partial f}{\\partial z} \\frac{\\partial g}{\\partial w} – \\frac{\\partial f}{\\partial w} \\frac{\\partial g}{\\partial x} }\\tag{2.11}\\]
\n\uc774 \ub4f1\uc2dd\uc740 \\(x\\)\uc640 \\(y\\)\ub97c \ub3c5\ub9bd\ubcc0\uc218\ub85c \ub450\uace0 \uad6c\ud55c \uac83\uc774\ubbc0\ub85c (2.3)\uacfc \uac19\ub2e4.
\n<\/span><\/p>\n<\/div>\n

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

\uc608\uc81c 1. \ud568\uc218 \\(f(x,\\,y,\\,z)\\)\uac00 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f(x,\\,y,\\,z)=0 \\tag{1.1}\\) \uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c \\(\\left( \\frac{\\partial x}{\\partial g} \\right)_z \\left( \\frac{\\partial y}{\\partial z}\\right)_x \\left( \\frac{\\partial z}{\\partial x}\\right)_y =-1\\tag{1.2}\\) \uc784\uc744 \ubcf4\uc774\uc2dc\uc624. (Thomas’ Calculus 13ed 14.10. Exercise 9.) \ud480\uc774. \uba3c\uc800 \\(y,\\) \\(z\\)\ub97c \ub3c5\ub9bd\ubcc0\uc218\ub85c \ub450\uace0 (1.1)\uc758 \uc591\ubcc0\uc744 \\(y\\)\uc5d0 \uad00\ud558\uc5ec \ubbf8\ubd84\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4. \\(\\begin{align} &\\frac{\\partial f}{\\partial x} \\frac{\\partial x}{\\partial y} + \\frac{\\partial f}{\\partial y} \\frac{\\partial y}{\\partial y} + \\frac{\\partial…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[382,88,383,381],"class_list":["post-4614","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-constrained-variable","tag-partial-derivative","tag-383","tag-381"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4614","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=4614"}],"version-history":[{"count":15,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4614\/revisions"}],"predecessor-version":[{"id":4629,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4614\/revisions\/4629"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=4614"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=4614"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=4614"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}