{"id":4650,"date":"2020-05-28T23:15:11","date_gmt":"2020-05-28T14:15:11","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=4650"},"modified":"2020-06-02T15:52:11","modified_gmt":"2020-06-02T06:52:11","slug":"calculus-tetrahedron-with-minimum-surface-area","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-tetrahedron-with-minimum-surface-area\/","title":{"rendered":"\uc0ac\uba74\uccb4\uc758 \uac89\ub113\uc774\uac00 \ucd5c\uc18c\uac00 \ub418\ub3c4\ub85d \ud558\ub294 \uaf2d\uc9d3\uc810\uc758 \uc704\uce58"},"content":{"rendered":"

\ub77c\uadf8\ub791\uc8fc\uc758 \ubc29\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \ub3c4\ud615\uacfc \uad00\ub828\ub41c \ubb38\uc81c\ub97c \ud574\uacb0\ud558\ub294 \uc608\ub97c \uc0b4\ud3b4 \ubcf4\uc790.<\/p>\n

\n

\ubb38\uc81c.<\/span>
\n\\(\\mathbb{R}^3\\)\uc5d0 \ub193\uc740 \uc0ac\uba74\uccb4 \\(\\mathrm{P-ABC}\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc0bc\uac01\ud615 \\(\\mathrm{ABC}\\)\uac00 \\(xy\\) \ud3c9\uba74\uc5d0 \uace0\uc815\ub418\uc5b4 \uc788\uace0, \uc810 \\(\\mathrm{P}\\)\ub294 \\(z > 0\\)\uc778 \uc704\ucabd \ubc18\uacf5\uac04\uc5d0 \ub193\uc5ec \uc788\uc73c\uba70 \uc0ac\uba74\uccb4 \\(\\mathrm{P-ABC}\\)\uc758 \ubd80\ud53c\uac00 \uc77c\uc815\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc810 \\(\\mathrm{P}\\)\ub85c\ubd80\ud130 \\(xy\\) \ud3c9\uba74\uc5d0 \ub0b4\ub9b0 \uc218\uc120\uc758 \ubc1c\uc744 \\(\\mathrm{Q}\\)\ub77c \ud558\uc790. \uc774\ub54c \uc0ac\uba74\uccb4 \\(\\mathrm{P-ABC}\\)\uc758 \uac89\ub113\uc774\uac00 \ucd5c\uc18c\uac00 \ub418\ub3c4\ub85d \ud558\ub294 \uc810 \\(\\mathrm{Q}\\)\uc758 \uc704\uce58\ub97c \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n

\n

\n\ud480\uc774.<\/span>
\n\uc810 \\(\\mathrm{Q}\\)\uac00 \uc0bc\uac01\ud615 \\(\\mathrm{ABC}\\)\uc758 \ub0b4\ubd80\uc5d0 \uc788\ub294 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud558\uc790. \uadf8\ub9ac\uace0 \uc810 \\(\\mathrm{Q}\\)\ub85c\ubd80\ud130 \uc138 \ubcc0 \\(\\mathrm{BC},\\) \\(\\mathrm{CA},\\) \\(\\mathrm{AB}\\)\uae4c\uc9c0\uc758 \uac70\ub9ac\ub97c \uac01\uac01 \\(x,\\) \\(y,\\) \\(z\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc810 \\(\\mathrm{Q}\\)\uac00 \uc0bc\uac01\ud615 \\(\\mathrm{ABC}\\)\uc758 \ub0b4\ubd80\uc5d0 \uc788\uc9c0 \uc54a\ub2e4\uba74 \\(x,\\) \\(y,\\) \\(z\\) \uc911 \ud558\ub098 \uc774\uc0c1\uc758 \uac12\uc744 \\(0\\) \uc774\ud558\uac00 \ub418\ub3c4\ub85d \ud568\uc73c\ub85c\uc368 \uc810 \\(\\mathrm{Q}\\)\uc758 \uc704\uce58\ub97c \\(x,\\) \\(y,\\) \\(z\\)\uc5d0 \ub300\uc751\uc2dc\ud0ac \uc218 \uc788\ub2e4.<\/p>\n

\n\"\"\n<\/div>\n

\ub610\ud55c \uc138 \ubcc0 \\(\\mathrm{BC},\\) \\(\\mathrm{CA},\\) \\(\\mathrm{AB}\\)\uc758 \uae38\uc774\ub97c \uac01\uac01 \\(a,\\) \\(b,\\) \\(c\\)\ub77c\uace0 \ud558\uace0, \uc0ac\uba74\uccb4\uc758 \ub192\uc774\uc778 \uc120\ubd84 \\(\\mathrm{PQ}\\)\uc758 \uae38\uc774\ub97c \\(h\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

\uc0bc\uac01\ud615 \\(\\mathrm{ABC}\\)\uc758 \ub113\uc774\ub97c \\(S\\)\ub77c\uace0 \ud558\uba74
\n\\[\\frac{1}{2} (ax+ by + cz) = S\\]
\n\uc774\uace0, \uc0ac\uba74\uccb4\uc758 \ubd80\ud53c\ub294
\n\\[\\frac{1}{3} Sh = \\frac{1}{6} (ax+by+cz)h = \\text{(constant)}\\]
\n\uc774\ub2e4. \uc0ac\uba74\uccb4\uc758 \ubd80\ud53c\uc640 \ubc11\uba74\uc758 \ub113\uc774\uac00 \uc77c\uc815\ud558\ubbc0\ub85c \uc0ac\uba74\uccb4\uc758 \ub192\uc774 \\(h\\) \ub610\ud55c \uc0c1\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc810 \\(\\mathrm{P}\\)\uc758 \uc704\uce58\ub294 \uc810 \\(\\mathrm{Q}\\)\uc758 \uc704\uce58\uc5d0 \uc758\ud558\uc5ec \uc644\uc804\ud788 \uacb0\uc815\ub41c\ub2e4.<\/p>\n

\ud55c\ud3b8 \uc0ac\uba74\uccb4\uc758 \uac89\ub113\uc774\ub294 \ubc11\uba74\uc758 \ub113\uc774\uc640 \uc606\uba74\uc758 \ub113\uc774\ub97c \ub354\ud558\uba74 \ub418\ub294\ub370 \ubc11\uba74\uc758 \ub113\uc774\uac00 \uc0c1\uc218\uc774\ubbc0\ub85c, \uc6b0\ub9ac\uac00 \uc6d0\ud558\ub294 \uacb0\uacfc\ub97c \uc5bb\uae30 \uc704\ud574\uc11c\ub294 \uc606\uba74\uc758 \ub113\uc774\uac00 \ucd5c\uc18c\uac00 \ub418\ub3c4\ub85d \ud558\uba74 \ub41c\ub2e4.
\n\\[\\begin{gather}
\n\\overline{\\mathrm{PD}} = \\sqrt{x^2 + h^2} , \\\\[6pt]
\n\\overline{\\mathrm{PE}} = \\sqrt{y^2 + h^2} , \\\\[6pt]
\n\\overline{\\mathrm{PF}} = \\sqrt{z^2 + h^2}
\n\\end{gather}\\]
\n\uc774\ubbc0\ub85c, \uc0ac\uba74\uccb4\uc758 \uc606\uba74\uc758 \ub113\uc774\ub294
\n\\[\\frac{1}{2} \\left( a\\sqrt{x^2 + h^2} + b\\sqrt{y^2 + h^2} + c\\sqrt{z^2 + h^2} \\right)\\]
\n\uc774\ub2e4.<\/p>\n

\ud568\uc218 \\(f\\)\uc640 \\(g\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
\n\\[\\begin{gather}
\nf(x,\\,y,\\,z) = a\\sqrt{x^2 + h^2} + b\\sqrt{y^2 + h^2} + c\\sqrt{z^2 + h^2} ,\\\\[6pt]
\ng(x,\\,y,\\,z) = ax+by+cz.
\n\\end{gather}\\]
\n\uc774\uc81c \uc6b0\ub9ac\ub294 \uc81c\ud55c\uc870\uac74
\n\\[g(x,\\,y,\\,z) = \\text{(constant)}\\]
\n\uc544\ub798\uc5d0\uc11c \ud568\uc218 \\(f(z,\\,y,\\,z)\\)\uac00 \ucd5c\uc19f\uac12\uc744 \uac16\ub3c4\ub85d \ud558\ub294 \uc810 \\((x,\\,y,\\,z)\\)\ub97c \uad6c\ud574\uc57c \ud55c\ub2e4. \ub77c\uadf8\ub791\uc8fc\uc758 \ubc29\ubc95\uc744 \uc774\uc6a9\ud558\uc790. \uadf8\ub7ec\ud55c \uc810 \\((x,\\,y,\\,z)\\)\uac00 \uc874\uc7ac\ud55c\ub2e4\uba74 \uc0c1\uc218 \\(\\lambda\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c \ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ucf1c\uc57c \ud55c\ub2e4.
\n\\[\\begin{gather}
\n\\nabla f(x,\\,y,\\,z) = \\lambda \\nabla g(x,\\,y,\\,z)
\n\\end{gather}\\]
\n\uc774 \uc2dd\uc744 \ud480\uc5b4 \uc4f0\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[ \\frac{ax}{\\sqrt{x^2 + h^2}} \\textbf{i} + \\frac{by}{\\sqrt{y^2 + h^2}} \\textbf{j} + \\frac{cz}{\\sqrt{z^2 + h^2}} \\textbf{k} = \\lambda \\left(
\na\\textbf{i} + b\\textbf{j}+ c \\textbf{k} \\right)\\]
\n\uc774 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\ub824\uba74
\n\\[\\frac{x}{\\sqrt{x^2 + h^2}} = \\frac{y}{\\sqrt{y^2 + h^2}} = \\frac{z}{\\sqrt{z^2 + h^2}} \\]
\n\uc774\uc5b4\uc57c \ud558\uace0, \uc774\uac83\uc744 \ud480\uba74
\n\\[x = y = z\\]
\n\uc774\ub2e4. \uc989 \uc810 \\(\\mathrm{Q}\\)\uac00 \uc0bc\uac01\ud615 \\(\\mathrm{ABC}\\)\uc758 \ub0b4\uc2ec\uc77c \ub54c \uc0ac\uba74\uccb4\uc758 \uac89\ub113\uc774\ub294 \uadf9\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

\uc774\uc81c \\(x = y = z\\)\uc778 \uc810 \\((x,\\,y,\\,z)\\)\uc5d0\uc11c \\(f\\)\uac00 \ucd5c\uc19f\uac12\uc744 \uac00\uc9d0\uc744 \ubcf4\uc774\uc790. \uc77c\ub2e8 \ub77c\uadf8\ub791\uc8fc\uc758 \ubc29\ubc95\uc744 \ud1b5\ud574\uc11c \ubc1c\uacac\ub41c \uc810\uc740 \\(x=y=z\\)\uc778 \uc810 \\((x,\\,y,\\,z)\\) \ubfd0\uc774\ubbc0\ub85c, \ub2e4\ub978 \uc810\uc5d0\uc11c\ub294 \\(f\\)\uac00 \uadf9\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\((x,\\,y,\\,z)\\)\uac00 \\((0,\\,0,\\,0)\\)\uc73c\ub85c\ubd80\ud130 \ubb34\ud55c\ud788 \uba40\uc5b4\uc9c0\ub294 \uacbd\uc6b0\ub97c \uc870\uc0ac\ud558\uba74 \ub41c\ub2e4.<\/p>\n

\uc810 \\((x,\\,y,\\,z)\\)\uac00 \\((0,\\,0,\\,0)\\)\uc73c\ub85c\ubd80\ud130 \ubb34\ud55c\ud788 \uba40\uc5b4\uc9c0\ub294 \uac83\uc740 \uc810 \\(\\mathrm{Q}\\)\uac00 \\((0,\\,0)\\)\uc73c\ub85c\ubd80\ud130 \ubb34\ud55c\ud788 \uba40\uc5b4\uc9c0\ub294 \uac83\uacfc \uac19\ub2e4. \uadf8\ub7f0\ub370 \uc810 \\(\\mathrm{Q}\\)\uac00 \\((0,\\,0)\\)\uc73c\ub85c\ubd80\ud130 \ubb34\ud55c\ud788 \uba40\uc5b4\uc9c0\uba74 \uc0ac\uba74\uccb4\uc758 \uc606\uba74\uc744 \uc774\ub8e8\ub294 \uc138 \uc0bc\uac01\ud615 \uc911 \ud558\ub098 \uc774\uc0c1\uc758 \ub192\uc774\uac00 \ubb34\ud55c\ud788 \uae38\uc5b4\uc9c0\ubbc0\ub85c, \uc0ac\uba74\uccb4\uc758 \uac89\ub113\uc774 \ub610\ud55c \ubb34\ud55c\ud788 \ucee4\uc9c4\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\(x=y=z\\)\uc778 \uc810 \\((x,\\,y,\\,z)\\)\uc5d0\uc11c\uc758 \ud568\uc22b\uac12\ubcf4\ub2e4 \ub354 \uc791\uc740 \ud568\uc22b\uac12\uc744 \uac00\uc9c8 \uc218 \uc5c6\ub2e4.
\n<\/span>\n<\/p>\n<\/div>\n

\ud765\ubbf8\ub85c\uc6b4 \ubb38\uc81c\ub97c \uc18c\uac1c\ud574\uc900 \uc7a5\uc608\uc900 \ud559\uc0dd\uc5d0\uac8c \uac10\uc0ac\ub4dc\ub9bd\ub2c8\ub2e4.<\/p>\n

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

\ub77c\uadf8\ub791\uc8fc\uc758 \ubc29\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \ub3c4\ud615\uacfc \uad00\ub828\ub41c \ubb38\uc81c\ub97c \ud574\uacb0\ud558\ub294 \uc608\ub97c \uc0b4\ud3b4 \ubcf4\uc790. \ubb38\uc81c. \\(\\mathbb{R}^3\\)\uc5d0 \ub193\uc740 \uc0ac\uba74\uccb4 \\(\\mathrm{P-ABC}\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc0bc\uac01\ud615 \\(\\mathrm{ABC}\\)\uac00 \\(xy\\) \ud3c9\uba74\uc5d0 \uace0\uc815\ub418\uc5b4 \uc788\uace0, \uc810 \\(\\mathrm{P}\\)\ub294 \\(z > 0\\)\uc778 \uc704\ucabd \ubc18\uacf5\uac04\uc5d0 \ub193\uc5ec \uc788\uc73c\uba70 \uc0ac\uba74\uccb4 \\(\\mathrm{P-ABC}\\)\uc758 \ubd80\ud53c\uac00 \uc77c\uc815\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc810 \\(\\mathrm{P}\\)\ub85c\ubd80\ud130 \\(xy\\) \ud3c9\uba74\uc5d0 \ub0b4\ub9b0 \uc218\uc120\uc758 \ubc1c\uc744 \\(\\mathrm{Q}\\)\ub77c \ud558\uc790. \uc774\ub54c \uc0ac\uba74\uccb4 \\(\\mathrm{P-ABC}\\)\uc758 \uac89\ub113\uc774\uac00 \ucd5c\uc18c\uac00 \ub418\ub3c4\ub85d \ud558\ub294 \uc810 \\(\\mathrm{Q}\\)\uc758 \uc704\uce58\ub97c \uad6c\ud558\uc2dc\uc624. \ud480\uc774. \uc810 \\(\\mathrm{Q}\\)\uac00 \uc0bc\uac01\ud615 \\(\\mathrm{ABC}\\)\uc758 \ub0b4\ubd80\uc5d0 \uc788\ub294…<\/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":[384,386,387,385,388,389],"class_list":["post-4650","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-calculus","tag-lagrangian-method","tag-lagrangian-multiplier","tag-tetrahedron","tag-388","tag-389"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4650","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=4650"}],"version-history":[{"count":29,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4650\/revisions"}],"predecessor-version":[{"id":4733,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4650\/revisions\/4733"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=4650"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=4650"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=4650"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}