{"id":214,"date":"2019-08-28T19:53:58","date_gmt":"2019-08-28T19:53:58","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=214"},"modified":"2020-06-18T17:37:26","modified_gmt":"2020-06-18T17:37:26","slug":"multi-variable-functions-surfaces-and-contours","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/multivariable-calculus\/multi-variable-functions-surfaces-and-contours\/","title":{"rendered":"Multi-Variable Functions, Surfaces, and Contours"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Multivariable Functions, Surfaces, and Contours &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n\t\tThe graphs of surfaces in 3-space can get very intricate and complex!\n\t\tIn this tutorial, we investigate some tools that can be used to help\n\t\tvisualize the <b>graph<\/b> of a function $f(x,y)$, defined as the graph\n\t\tof the equation $z=f(x,y)$.\n\t\n<\/p>\n\n\n\n<p>\n\t\t\tTry plotting\n\t\t\t$$\n\t\t\tz=\\sin (xy)!\n\t\t\t$$\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nLet $f(x,y) = x^2 + \\frac{y^2}{4}$.  Before actually graphing $z\n= x^2 + \\frac{y^2}{4}$, let&#8217;s see if we can visualize the surface that\nwill result.\n\n<\/p>\n\n\n\n<p>\nIf we set $y=0$, we find that the intersection of the surface with\nthe $xz$-plane is the parabola $z=x^2$.\n\n<\/p>\n\n\n\n<p>\nSimilarly, setting $x=0$, the intersection of the surface with the\n$yz$-plane is the parabola $z=\\frac{y^2}{4}$.\n\n<\/p>\n\n\n\n<p>\nCan you see what the surface, called an elliptic paraboloid, will look\nlike?\n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"243\" height=\"197\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/multi1.gif?resize=243%2C197&#038;ssl=1\" alt=\"The graph of z=x^2+y^2\/4\" class=\"wp-image-401\"\/><\/figure><\/div>\n\n\n\n<p>\nBy setting $x=0$ or $y=0$ in $z=f(x,y)$, we are really looking at the\nintersection of the surface $z=f(x,y)$ with the plane $x=0$ or $y=0$,\nrespectively.  If we take the intersection of a surface $z=f(x,y)$\nwith any plane, the resulting curve is called the <b>cross section<\/b>\nor <b>trace<\/b> of the surface in the plane.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nLet $f(x,y) = 5 &#8211; \\sqrt{x^2 +y^2}$.  What can we determine about the\nsurface given by $z = 5 &#8211; \\sqrt{x^2 +y^2}$?  Notice that $z \\leq 5$.\nIf we set $z=5$, $x^2 + y^2 =0$ and we get a single point $x=0, \\quad\ny=0$ in the plane $z=5$.\n\n<\/p>\n\n\n\n<p>\nIf we set $z=4$, $x^2+y^2 =1$, giving a circle of radius 1.\n\n<\/p>\n\n\n\n<p>\nIf $z=0$, $x^2 + y^2 =5$, a circle of radius $\\sqrt{5}$.\n\n<\/p>\n\n\n\n<p>\nIf $z=-4$, $x^2 + y^2 = 9$, a circle of radius 3.\n\n<\/p>\n\n\n\n<p>\nIs this another paraboloid?  Notice that the trace in the plane $y=0$\nis the pair of lines $z=5-x$ and $z=5+x$.  \n\n<\/p>\n\n\n\n<p>\nSimilarly, the trace in the plane $x=0$ is the pair of lines $z=5-y$\nand $z=5+y$.  The surface is a right circular cone.\n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"266\" height=\"197\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/multi2.gif?resize=266%2C197&#038;ssl=1\" alt=\"The graph of z = 5 - sqrt(x^2+y^2)\" class=\"wp-image-400\"\/><\/figure><\/div>\n\n\n\n<p>\nWhen we take the intersection of the surface $z=f(x,y)$ with the\nhorizontal plane $z=k$, as we did several times in the previous\nexample, the projection of the resulting curve onto the $xy$-plane is\ncalled the <b>level curve of height $k$<\/b>.  Along this curve, $f$ is\nconstant with value $k$.\n\n<\/p>\n\n\n\n<p>\n\t\tA collection of level curves of a surface, labeled with their heights,\n\t\tis called a <b>contour map<\/b>.\n\t\n<\/p>\n\n\n\n<p>\n\t\t\tA contour map is just a topographic map of the surface.\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nLet $f(x,y) = \\sqrt{9-x^2-y^2}$.  Notice here that $f(x,y) \\ge 0$.  We\nwill examine the level curves of $z=f(x,y)$.\n\n<\/p>\n\n\n\n<p>\nSetting $z=k, \\quad k \\ge 0$, squaring both sides of the equation and\nrearranging terms, we find that the level curves of $z=f(x,y)$ are\ncircles given by $x^2 + y^2 = 9 &#8211; k^2$.\n\n<\/p>\n\n\n\n<p>\nExamination of traces with $x=c$ or $y=c$ shows them to be portions of\ncircles.  Thus, $z=f(x,y)$ is a hemisphere here.\n\n<\/p>\n\n\n\n<p>\nSquaring $z=\\sqrt{9-x^2-y^2}$ from the previous example and\nrearranging terms, we obtain $x^2 + y^2 + z^2 = 9$, the equation of a\nsphere.  It is useful to be able to recognize some common quadric\nsurfaces such as this.\n\n<\/p>\n\n\n\n<center>\n<h4>Note<\/h4>\n<\/center>\n\n\n\n<p>\nFor a function $f(x,y,z)$ of <i>three<\/i> variables, $f(x,y,z) = k$\nis called the <b>level surface with constant $k$<\/b>.  The function\n$f(x,y,z)$ is constant over the level surface.\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h4>Key Concept<\/h4>\n<p>\n<\/p><\/center>\n\n\n\n<p>\nLet $z=f(x,y)$.\n\n<\/p>\n\n\n\n<p>\nThe projection onto the $xy$-plane of the intersection of the surface\n$z=f(x,y)$ with the horizontal plane $z=k$ is called the <b>level curve\nof height $k$<\/b>.  A collection of level curves, called a <b>contour\nmap<\/b> is a useful tool in visualizing the graph of a function $f(x,y)$.\n\n<!------------------------>\n\n\n<br>\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p> [<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ3110\/\">I&#8217;m ready to take the quiz.<\/a>] [<a href=\"#top\">I need to review more.<\/a>]<br> <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Multivariable Functions, Surfaces, and Contours &#8211; HMC Calculus Tutorial The graphs of surfaces in 3-space can get very intricate and complex! In this tutorial, we investigate some tools that can be used to help visualize the graph of a function $f(x,y)$, defined as the graph of the equation $z=f(x,y)$. Try plotting $$ z=\\sin (xy)! $$&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":59,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-214","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/214","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=214"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/214\/revisions"}],"predecessor-version":[{"id":1240,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/214\/revisions\/1240"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/59"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=214"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}