{"id":6709,"date":"2021-07-21T00:01:18","date_gmt":"2021-07-20T15:01:18","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6709"},"modified":"2021-09-22T19:18:13","modified_gmt":"2021-09-22T10:18:13","slug":"vector-valued-functions","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/vector-valued-functions\/","title":{"rendered":"\ubca1\ud130\ud568\uc218\uc758 \uadf9\ud55c"},"content":{"rendered":"<div style=\"display: none; visibility: hidden;\">\n<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<p>\\[<br \/>\n\\newcommand{\\vecf}{{\\mathbf{f}}}<br \/>\n\\newcommand{\\vecL}{{\\mathbf{L}}}<br \/>\n\\newcommand{\\vecR}{{\\mathbb{R}}}<br \/>\n\\newcommand{\\imI}{\\boldsymbol{i}}<br \/>\n\\]\n<\/p><\/div>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 4\uc7a5 5\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\ubca1\ud130\ud568\uc218\uc758 \uadf9\ud55c\uc740 \ubca1\ud130\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ud06c\uac8c \ub2e4\ub974\uc9c0 \uc54a\ub2e4.<\/p>\n<p>\\(\\vecf : D \\rightarrow \\vecR ^d\\)\uac00 \ubca1\ud130\ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d<br \/>\n\\[\\lim_{x\\rightarrow c} \\lvert \\vecf (x) &#8211; \\vecL \\rvert = 0\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubca1\ud130 \\(\\vecL \\in \\vecR ^d\\)\uac00 \uc874\uc7ac\ud558\uba74<\/p>\n<p class=\"aligncenter\">\u201c\\(c\\)\uc5d0\uc11c \\(\\vecf (x)\\)\uac00 \\(\\vecL\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\u201d<\/p>\n<p>\ub610\ub294<\/p>\n<p class=\"aligncenter\">\u201c\\(x\\)\uac00 \\(c\\)\uc5d0 \ub2e4\uac00\uac08 \ub54c \\(\\vecf (x)\\)\uac00 \\(\\vecL\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\u201d<\/p>\n<p>\ub610\ub294<\/p>\n<p class=\"aligncenter\">\u201c\\(x\\)\uac00 \\(c\\)\uc5d0 \ub2e4\uac00\uac08 \ub54c \\(\\vecf (x)\\)\uac00 \\(\\vecL\\)\uc5d0 \ub2e4\uac00\uac04\ub2e4\u201d<\/p>\n<p>\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\ub54c \\(\\vecL\\)\uc744 \u2018\\(c\\)\uc5d0\uc11c \\(\\vecf\\)\uc758 <span class=\"defined\">\uadf9\ud55c<\/span>\u2019\uc774\ub77c\uace0 \ubd80\ub974\uba70, \uc774 \uc0c1\ud669\uc744 \uae30\ud638\ub85c<br \/>\n\\[\\lim_{x\\rightarrow c} \\vecf (x) = \\vecL\\]<br \/>\n\ub610\ub294<br \/>\n\\[\\vecf (x) \\rightarrow \\vecL \\quad\\text{as}\\quad x\\rightarrow c\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 4.5.1. (\ubca1\ud130\ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc131\ubd84\ubcc4 \uacc4\uc0b0)<\/span><\/p>\n<p>\\(f_1,\\) \\(f_2,\\) \\(f_3\\)\uc774 \\(D\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc218\uac12 \ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \\(\\vecf\\)\uac00 \\(D\\)\uc5d0\uc11c<br \/>\n\\[\\vecf (x) = (f_1 (x) ,\\, f_2 (x) ,\\, f_3 (x))\\]<br \/>\n\ub85c \uc815\uc758\ub41c \ubca1\ud130\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\vecf (x)\\)\uac00 \\(c\\)\uc5d0\uc11c \ubca1\ud130 \\((L_1 ,\\, L_2 ,\\, L_3 )\\)\uc5d0 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f_1 (x),\\) \\(f_2 (x) ,\\) \\(f_3 (x)\\)\uac00 \\(c\\)\uc5d0\uc11c \uac01\uac01 \\(L_1 ,\\) \\(L_2 ,\\) \\(L_3\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4. \\(\\vecf\\)\uc758 \uacf5\uc5ed\uc774 2\ucc28\uc6d0\uc774\uac70\ub098 4\ucc28\uc6d0 \uc774\uc0c1\uc77c \ub54c\uc5d0\ub3c4 \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.5.2.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\vecf : \\vecR \\rightarrow \\vecR ^3\\)\uac00<br \/>\n\\[\\vecf (x) = (2x+3 ,\\, x-4 ,\\, 1-x)\\]<br \/>\n\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(-2\\)\uc5d0\uc11c \\(\\vecf\\)\uc758 \uadf9\ud55c\uc740<br \/>\n\\[\\lim_{x\\rightarrow -2} \\vecf (x) = \\left( \\lim_{x\\rightarrow -2}(2x+3) ,\\, \\lim_{x\\rightarrow -2}(x-4) ,\\, \\lim_{x\\rightarrow -2}(1-x)  \\right) = (-1,\\,-6,\\,3)\\]<br \/>\n\uc774\ub2e4.<\/li>\n<li style=\"margin-top: 0.7em;\">\\(\\vecf : \\vecR ^2 \\rightarrow \\vecR ^2\\)\uac00<br \/>\n\\[\\vecf (x,\\,y) = (f_1 (x,\\,y) ,\\, f_2 ( x,\\,y))\\]<br \/>\n\ub85c \uc8fc\uc5b4\uc838 \uc788\uace0,<br \/>\n\\[f_1 (x,\\,y) = x+2y , \\quad f_2 ( x,\\,y) = -x+3y\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\((2,\\,1)\\)\uc5d0\uc11c \\(\\vecf\\)\uc758 \uadf9\ud55c\uc740<br \/>\n\\[\\lim_{(x,\\,y)\\rightarrow (2,\\,1)} \\vecf (x,\\,y) =<br \/>\n\\left( \\lim_{(x,\\,y)\\rightarrow (2,\\,1)} f_1 (x,\\,y) ,\\, \\lim_{(x,\\,y)\\rightarrow (2,\\,1)} f_2 ( x,\\,y)  \\right) = (4,\\,1)\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\\(f\\)\uac00 \ubcf5\uc18c\ud568\uc218\uc774\uace0<br \/>\n\\[f(x+y \\imI ) = u (x,\\,y) + v(x,\\,y) \\imI\\]<br \/>\n\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \\(u\\)\uc640 \\(v\\)\ub294 \uac01\uac01 \\(\\vecR ^2\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc778 \\(D\\)\ub97c \uc815\uc758\uc5ed\uc73c\ub85c \uac16\uace0 \\(\\vecR ^2\\)\ub97c \uacf5\uc5ed\uc73c\ub85c \uac16\ub294 \ud568\uc218\uc774\ub2e4. \\(a,\\) \\(b,\\) \\(A,\\) \\(B\\)\uac00 \uc2e4\uc218\uc774\uace0 \\(z_0 = a+b \\imI\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\lim_{z\\rightarrow z_0} f(z) = A+B \\imI\\]<br \/>\n\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740<br \/>\n\\[\\lim_{(x,\\,y) \\rightarrow (a,\\,b)} u(x,\\,y) = A \\quad\\text{and}\\quad \\lim_{(x,\\,y)\\rightarrow (a,\\,b)} v(x,\\,y) = B\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.5.2.<\/span><br \/>\n\ub2e4\uc74c \uc608\uc5d0\uc11c \\(x\\)\uc640 \\(y\\)\ub294 \uc2e4\uc218 \ubcc0\uc218\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\ub9cc\uc57d \\(u(x,\\,y \\imI ) = x-2y\\)\uc774\uba74<br \/>\n\\[\\lim_{z\\rightarrow 1-2 \\imI} u(z) = \\lim_{(x,\\,y) \\rightarrow (1,\\,-2)} u(x+y \\imI ) = 5\\]<br \/>\n\uc774\ub2e4.<\/li>\n<li style=\"margin-top: 0.7em;\">\ud568\uc218 \\(f : \\mathbb{C} \\rightarrow \\mathbb{C}\\)\uac00<br \/>\n\\[f(x+y\\imI ) = u(x+y\\imI ) + v(x+y\\imI ) \\imI \\]<br \/>\n\ub85c \uc815\uc758\ub418\uc5b4 \uc788\uace0,<br \/>\n\\[u(x+y\\imI ) = x-2y ,\\quad v(x+y\\imI ) =3x+4y\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(1-2\\imI\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ud55c\uc740<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{z\\rightarrow 1-2\\imI} f(z) &#038;= \\lim _ {(x,\\,y) \\rightarrow (1,\\,-2)} \\left\\{ u(x+y\\imI ) + v(x+y\\imI )\\imI \\right\\} \\\\[4pt]<br \/>\n&#038;= \\lim_{(x,\\,y)\\rightarrow (1,\\,-2)} u(x+y\\imI ) + \\lim_{(x,\\,y)\\rightarrow (1,\\,-2)} v(x+y\\imI )\\imI \\\\[4pt]<br \/>\n&#038;= 5- \\imI<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4.<\/li>\n<li style=\"margin-top: 0.7em;\">\ud568\uc218 \\(g : \\mathbb{C} \\rightarrow \\mathbb{C}\\)\uac00<br \/>\n\\[g(x+y\\imI ) = (x^2 + y^2 ) &#8211; (x+y) \\imI\\]<br \/>\n\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(3+2\\imI\\)\uc5d0\uc11c \\(g\\)\uc758 \uadf9\ud55c\uc740<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{z \\rightarrow 3+2\\imI} g(z) &#038;= \\lim_{(x,\\,y)\\rightarrow (3,\\,2)} (x^2 + y^2 ) &#8211; \\lim_{(x,\\,y)\\rightarrow (3,\\,2)} (x+y) \\imI = 13-5 \\imI<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n\n\n\n<p>.<\/p>\n\n\n\n\n\n<p>.<\/p>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/functions-of-several-variables\">\ub2e4\ubcc0\uc218\ud568\uc218\uc758 \uadf9\ud55c<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/definition-of-a-derivative\">\ubbf8\ubd84\uc758 \uc815\uc758<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\\( \\newcommand{\\vecf}{{\\mathbf{f}}} \\newcommand{\\vecL}{{\\mathbf{L}}} \\newcommand{\\vecR}{{\\mathbb{R}}} \\newcommand{\\imI}{\\boldsymbol{i}} \\) \uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 4\uc7a5 5\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \ubca1\ud130\ud568\uc218\uc758 \uadf9\ud55c\uc740 \ubca1\ud130\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ud06c\uac8c \ub2e4\ub974\uc9c0 \uc54a\ub2e4. \\(\\vecf : D \\rightarrow \\vecR ^d\\)\uac00 \ubca1\ud130\ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\lim_{x\\rightarrow c} \\lvert \\vecf (x) &#8211; \\vecL \\rvert = 0\\) \uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubca1\ud130 \\(\\vecL \\in \\vecR ^d\\)\uac00 \uc874\uc7ac\ud558\uba74 \u201c\\(c\\)\uc5d0\uc11c \\(\\vecf (x)\\)\uac00 \\(\\vecL\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\u201d \ub610\ub294 \u201c\\(x\\)\uac00 \\(c\\)\uc5d0 \ub2e4\uac00\uac08 \ub54c&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":405,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6709","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6709","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=6709"}],"version-history":[{"count":18,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6709\/revisions"}],"predecessor-version":[{"id":7933,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6709\/revisions\/7933"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6709"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}