{"id":582,"date":"2025-10-09T15:23:44","date_gmt":"2025-10-09T15:23:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=582"},"modified":"2025-10-15T14:15:37","modified_gmt":"2025-10-15T14:15:37","slug":"introduction-to-multivariable-functions-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-multivariable-functions-cheat-sheet\/","title":{"raw":"Introduction to Multivariable Functions: Cheat Sheet","rendered":"Introduction to Multivariable Functions: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Functions of Several Variables<\/strong>\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<strong>Limits and Continuity<\/strong>\r\n<ul>\r\n \t<li>To study limits and continuity for functions of two variables, we use a [latex]\\delta[\/latex]\u00a0disk centered around a given point.<\/li>\r\n \t<li>A function of several variables has a limit if for any point in a\u00a0[latex]\\delta[\/latex]\u00a0ball centered at a point [latex]P[\/latex],\u00a0the value of the function at that point is arbitrarily close to a fixed value (the limit value).<\/li>\r\n \t<li>The limit laws established for a function of one variable have natural extensions to functions of more than one variable.<\/li>\r\n \t<li>A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point.<\/li>\r\n<\/ul>\r\n<strong>Introduction to Partial Derivatives<\/strong>\r\n<ul id=\"fs-id1167794160198\" data-bullet-style=\"bullet\">\r\n \t<li>A partial derivative is a derivative involving a function of more than one independent variable.<\/li>\r\n \t<li>To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules.<\/li>\r\n \t<li>Higher-order partial derivatives can be calculated in the same way as higher-order derivatives.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul>\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 \t<li><strong>Partial derivative of [latex]f[\/latex]\u00a0with respect to\u00a0[latex]x[\/latex]\r\n<\/strong>[latex]\\frac{\\partial f}{\\partial x}=\\underset{h \\to{0}}{\\lim} \\frac{f(x+h,y)-f(x,y)}{h}[\/latex]<\/li>\r\n \t<li><strong>Partial derivative of [latex]f[\/latex]\u00a0with respect to\u00a0[latex]y[\/latex]\r\n<\/strong>[latex]\\frac{\\partial f}{\\partial y}=\\underset{k \\to{0}}{\\lim} \\frac{f(x,y+k)-f(x,y)}{k}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>boundary point<\/dt>\r\n \t<dd>a point\u00a0[latex]P_{0}[\/latex]\u00a0of [latex]R[\/latex]\u00a0is a boundary point if every [latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contains points both inside and outside [latex]R[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>closed set<\/dt>\r\n \t<dd>a set [latex]S[\/latex]<strong>\u00a0<\/strong>that contains all its boundary points<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>connected set<\/dt>\r\n \t<dd>an open set [latex]S[\/latex]\u00a0that cannot be represented as the union of two or more disjoint, nonempty open subsets<\/dd>\r\n<\/dl>\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>higher-order partial derivatives<\/dt>\r\n \t<dd>second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>interior point<\/dt>\r\n \t<dd>a point\u00a0[latex]P_{0}[\/latex]\u00a0of\u00a0[latex]R[\/latex]\u00a0is a boundary point if there is a\u00a0[latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contained completely in\u00a0[latex]R[\/latex]<\/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>mixed partial derivatives<\/dt>\r\n \t<dd>second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>open set<\/dt>\r\n \t<dd>a set\u00a0[latex]S[\/latex]\u00a0that contains none of its boundary points<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>partial derivative<\/dt>\r\n \t<dd>a derivative of a function of more than one independent variable in which all the variables but one are held constant<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>partial differential equation<\/dt>\r\n \t<dd>an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>region<\/dt>\r\n \t<dd>an open, connected, nonempty subset of [latex]\\mathbb{R}^{2}[\/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>\r\n<dl class=\"definition\">\r\n \t<dt><strong>[latex]\\delta[\/latex]<\/strong>\u00a0ball<\/dt>\r\n \t<dd>all points in [latex]\\mathbb{R}^{3}[\/latex]\u00a0lying at a distance of less than\u00a0[latex]\\delta[\/latex]\u00a0from [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt><strong>[latex]\\delta[\/latex] d<\/strong>isk<\/dt>\r\n \t<dd>an open disk of radius [latex]\\delta[\/latex]\u00a0centered at point\u00a0[latex](a,b)[\/latex]<\/dd>\r\n<\/dl>","rendered":"<h2>Essential Concepts<\/h2>\n<p><strong>Functions of Several Variables<\/strong><\/p>\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<p><strong>Limits and Continuity<\/strong><\/p>\n<ul>\n<li>To study limits and continuity for functions of two variables, we use a [latex]\\delta[\/latex]\u00a0disk centered around a given point.<\/li>\n<li>A function of several variables has a limit if for any point in a\u00a0[latex]\\delta[\/latex]\u00a0ball centered at a point [latex]P[\/latex],\u00a0the value of the function at that point is arbitrarily close to a fixed value (the limit value).<\/li>\n<li>The limit laws established for a function of one variable have natural extensions to functions of more than one variable.<\/li>\n<li>A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point.<\/li>\n<\/ul>\n<p><strong>Introduction to Partial Derivatives<\/strong><\/p>\n<ul id=\"fs-id1167794160198\" data-bullet-style=\"bullet\">\n<li>A partial derivative is a derivative involving a function of more than one independent variable.<\/li>\n<li>To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules.<\/li>\n<li>Higher-order partial derivatives can be calculated in the same way as higher-order derivatives.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul>\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<li><strong>Partial derivative of [latex]f[\/latex]\u00a0with respect to\u00a0[latex]x[\/latex]<br \/>\n<\/strong>[latex]\\frac{\\partial f}{\\partial x}=\\underset{h \\to{0}}{\\lim} \\frac{f(x+h,y)-f(x,y)}{h}[\/latex]<\/li>\n<li><strong>Partial derivative of [latex]f[\/latex]\u00a0with respect to\u00a0[latex]y[\/latex]<br \/>\n<\/strong>[latex]\\frac{\\partial f}{\\partial y}=\\underset{k \\to{0}}{\\lim} \\frac{f(x,y+k)-f(x,y)}{k}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>boundary point<\/dt>\n<dd>a point\u00a0[latex]P_{0}[\/latex]\u00a0of [latex]R[\/latex]\u00a0is a boundary point if every [latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contains points both inside and outside [latex]R[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>closed set<\/dt>\n<dd>a set [latex]S[\/latex]<strong>\u00a0<\/strong>that contains all its boundary points<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>connected set<\/dt>\n<dd>an open set [latex]S[\/latex]\u00a0that cannot be represented as the union of two or more disjoint, nonempty open subsets<\/dd>\n<\/dl>\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>higher-order partial derivatives<\/dt>\n<dd>second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>interior point<\/dt>\n<dd>a point\u00a0[latex]P_{0}[\/latex]\u00a0of\u00a0[latex]R[\/latex]\u00a0is a boundary point if there is a\u00a0[latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contained completely in\u00a0[latex]R[\/latex]<\/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>mixed partial derivatives<\/dt>\n<dd>second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>open set<\/dt>\n<dd>a set\u00a0[latex]S[\/latex]\u00a0that contains none of its boundary points<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>partial derivative<\/dt>\n<dd>a derivative of a function of more than one independent variable in which all the variables but one are held constant<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>partial differential equation<\/dt>\n<dd>an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>region<\/dt>\n<dd>an open, connected, nonempty subset of [latex]\\mathbb{R}^{2}[\/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<dl class=\"definition\">\n<dt><strong>[latex]\\delta[\/latex]<\/strong>\u00a0ball<\/dt>\n<dd>all points in [latex]\\mathbb{R}^{3}[\/latex]\u00a0lying at a distance of less than\u00a0[latex]\\delta[\/latex]\u00a0from [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>[latex]\\delta[\/latex] d<\/strong>isk<\/dt>\n<dd>an open disk of radius [latex]\\delta[\/latex]\u00a0centered at point\u00a0[latex](a,b)[\/latex]<\/dd>\n<\/dl>\n","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-582","chapter","type-chapter","status-publish","hentry"],"part":426,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/582","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/582\/revisions"}],"predecessor-version":[{"id":1105,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/582\/revisions\/1105"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/426"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/582\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=582"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=582"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=582"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=582"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}