{"id":160,"date":"2021-07-30T17:22:57","date_gmt":"2021-07-30T17:22:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=160"},"modified":"2022-10-29T01:11:48","modified_gmt":"2022-10-29T01:11:48","slug":"summary-of-functions-of-several-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-functions-of-several-variables\/","title":{"raw":"Summary of Functions of Several Variables","rendered":"Summary of Functions of Several Variables"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The graph of a function of two variables is a surface in [latex]\\mathbb{R}^{3}[\/latex] and can be studied using level curves and vertical traces.<\/li>\r\n \t<li>A set of level curves is called a contour map.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Vertical trace\r\n<\/strong>[latex]f(a,y)=x[\/latex] for\u00a0[latex]x=a[\/latex] or\u00a0[latex]f(x,b)=z[\/latex] for\u00a0[latex]y=b[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Level surface of a function of three variables<\/strong>\r\n[latex]f(x,y,z)=c[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>contour map<\/dt>\r\n \t<dd>a plot of the various level curves of a given function [latex]f(x,y)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>function of two variables<\/dt>\r\n \t<dd>a function [latex]z=f(x,y)[\/latex] that maps each ordered pair [latex](x,y)[\/latex]<strong>\u00a0<\/strong>in a subset [latex]D[\/latex]\u00a0of [latex]\\mathbb{R}^{2}[\/latex]<strong>\u00a0<\/strong>to a unique real number [latex]z[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>graph of a function of two variables<\/dt>\r\n \t<dd>a set of ordered triples\u00a0[latex](x,y,z)[\/latex]\u00a0that satisfies the equation [latex]z=f(x,y)[\/latex]\u00a0plotted in three-dimensional Cartesian space<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>level curve of a function of two variables<\/dt>\r\n \t<dd>the set of points satisfying the equation [latex]f(x,y)=c[\/latex]<strong>\u00a0<\/strong>for some real number [latex]c[\/latex]\u00a0in the range of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>level surface of a function of three variables<\/dt>\r\n \t<dd>the set of points satisfying the equation [latex]f(x,y,z)=c[\/latex]<strong>\u00a0<\/strong>for some real number [latex]c[\/latex] in the range of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>surface<\/dt>\r\n \t<dd>the graph of a function of two variables, [latex]z=f(x,y)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vertical trace<\/dt>\r\n \t<dd>the set of ordered triples [latex](c,y,z)[\/latex]<strong>\u00a0<\/strong>that solves the equation [latex]f(c,y)=z[\/latex]\u00a0for a given constant [latex]x=c[\/latex]\u00a0or the set of ordered triples [latex](x,d,z)[\/latex]\u00a0that solves the equation [latex]f(x,d)=z[\/latex]\u00a0for a given constant [latex]y=d[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The graph of a function of two variables is a surface in [latex]\\mathbb{R}^{3}[\/latex] and can be studied using level curves and vertical traces.<\/li>\n<li>A set of level curves is called a contour map.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Vertical trace<br \/>\n<\/strong>[latex]f(a,y)=x[\/latex] for\u00a0[latex]x=a[\/latex] or\u00a0[latex]f(x,b)=z[\/latex] for\u00a0[latex]y=b[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Level surface of a function of three variables<\/strong><br \/>\n[latex]f(x,y,z)=c[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>contour map<\/dt>\n<dd>a plot of the various level curves of a given function [latex]f(x,y)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>function of two variables<\/dt>\n<dd>a function [latex]z=f(x,y)[\/latex] that maps each ordered pair [latex](x,y)[\/latex]<strong>\u00a0<\/strong>in a subset [latex]D[\/latex]\u00a0of [latex]\\mathbb{R}^{2}[\/latex]<strong>\u00a0<\/strong>to a unique real number [latex]z[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>graph of a function of two variables<\/dt>\n<dd>a set of ordered triples\u00a0[latex](x,y,z)[\/latex]\u00a0that satisfies the equation [latex]z=f(x,y)[\/latex]\u00a0plotted in three-dimensional Cartesian space<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>level curve of a function of two variables<\/dt>\n<dd>the set of points satisfying the equation [latex]f(x,y)=c[\/latex]<strong>\u00a0<\/strong>for some real number [latex]c[\/latex]\u00a0in the range of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>level surface of a function of three variables<\/dt>\n<dd>the set of points satisfying the equation [latex]f(x,y,z)=c[\/latex]<strong>\u00a0<\/strong>for some real number [latex]c[\/latex] in the range of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>surface<\/dt>\n<dd>the graph of a function of two variables, [latex]z=f(x,y)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vertical trace<\/dt>\n<dd>the set of ordered triples [latex](c,y,z)[\/latex]<strong>\u00a0<\/strong>that solves the equation [latex]f(c,y)=z[\/latex]\u00a0for a given constant [latex]x=c[\/latex]\u00a0or the set of ordered triples [latex](x,d,z)[\/latex]\u00a0that solves the equation [latex]f(x,d)=z[\/latex]\u00a0for a given constant [latex]y=d[\/latex]<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-160\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-160","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/160","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/160\/revisions"}],"predecessor-version":[{"id":3762,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/160\/revisions\/3762"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/160\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=160"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=160"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=160"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}