{"id":170,"date":"2021-07-30T17:24:16","date_gmt":"2021-07-30T17:24:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=170"},"modified":"2022-10-29T02:13:29","modified_gmt":"2022-10-29T02:13:29","slug":"summary-of-directional-derivatives-and-the-gradient","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-directional-derivatives-and-the-gradient\/","title":{"raw":"Summary of Directional Derivatives and the Gradient","rendered":"Summary of Directional Derivatives and the Gradient"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167793299690\" data-bullet-style=\"bullet\">\r\n \t<li>A directional derivative represents a rate of change of a function in any given direction.<\/li>\r\n \t<li>The gradient can be used in a formula to calculate the directional derivative.<\/li>\r\n \t<li>The gradient indicates the direction of greatest change of a function of more than one variable.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Directional derivative (two dimensions)\r\n<\/strong>[latex]D_{\\bf{u}} f(a,b)=\\underset{h \\to {0}}{\\lim} \\frac{f(a+h\\cos{\\theta}, b+h\\sin{\\theta})-f(a,b)}{h}[\/latex]\u00a0 or\u00a0 [latex]D_{\\bf{u}} f(x,y)=f_{x}(x,y)\\cos{\\theta}+f_{y}(x,y)\\sin{\\theta}[\/latex]<\/li>\r\n \t<li><strong>Gradient (two dimensions)\r\n<\/strong>[latex]\\nabla f(x,y)=f_{x}(x,y){\\bf{i}}+f_{y}(x,y){\\bf{j}}[\/latex]<\/li>\r\n \t<li><strong>Gradient (three dimensions)\r\n<\/strong>[latex]\\nabla f(x,y,z)=f_{x}(x,y,z){\\bf{i}}+f_{y}(x,y,z){\\bf{j}}+f_{z}(x,y,z){\\bf{k}}[\/latex]<\/li>\r\n \t<li><strong>Directional derivative (three dimensions)\r\n<\/strong>[latex]\\begin{array}{c} D_{\\bf{u}} f(x,y,z) &amp;=\\nabla f(x,y,z)\\cdot{\\bf{u}} \\hfill \\\\ \\hfill &amp;=f_{x}(x,y,z)\\cos{\\alpha}+f_{y}(x,y,z)\\cos{\\beta}+f_{z}(x,y,z)\\cos{\\gamma}\\end{array}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>directional derivative<\/dt>\r\n \t<dd>the derivative of a function in the direction of a given unit vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>gradient<\/dt>\r\n \t<dd>the gradient of the function [latex]f(x,y)[\/latex]\u00a0is defined to be [latex]\\nabla f(x,y)=(\\partial{f}{\/}\\partial{x}){\\bf{i}}+(\\partial{f}{\/}\\partial{y}){\\bf{j}}[\/latex]\u00a0which can be generalized to a function of any number of independent variables<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167793299690\" data-bullet-style=\"bullet\">\n<li>A directional derivative represents a rate of change of a function in any given direction.<\/li>\n<li>The gradient can be used in a formula to calculate the directional derivative.<\/li>\n<li>The gradient indicates the direction of greatest change of a function of more than one variable.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Directional derivative (two dimensions)<br \/>\n<\/strong>[latex]D_{\\bf{u}} f(a,b)=\\underset{h \\to {0}}{\\lim} \\frac{f(a+h\\cos{\\theta}, b+h\\sin{\\theta})-f(a,b)}{h}[\/latex]\u00a0 or\u00a0 [latex]D_{\\bf{u}} f(x,y)=f_{x}(x,y)\\cos{\\theta}+f_{y}(x,y)\\sin{\\theta}[\/latex]<\/li>\n<li><strong>Gradient (two dimensions)<br \/>\n<\/strong>[latex]\\nabla f(x,y)=f_{x}(x,y){\\bf{i}}+f_{y}(x,y){\\bf{j}}[\/latex]<\/li>\n<li><strong>Gradient (three dimensions)<br \/>\n<\/strong>[latex]\\nabla f(x,y,z)=f_{x}(x,y,z){\\bf{i}}+f_{y}(x,y,z){\\bf{j}}+f_{z}(x,y,z){\\bf{k}}[\/latex]<\/li>\n<li><strong>Directional derivative (three dimensions)<br \/>\n<\/strong>[latex]\\begin{array}{c} D_{\\bf{u}} f(x,y,z) &=\\nabla f(x,y,z)\\cdot{\\bf{u}} \\hfill \\\\ \\hfill &=f_{x}(x,y,z)\\cos{\\alpha}+f_{y}(x,y,z)\\cos{\\beta}+f_{z}(x,y,z)\\cos{\\gamma}\\end{array}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>directional derivative<\/dt>\n<dd>the derivative of a function in the direction of a given unit vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>gradient<\/dt>\n<dd>the gradient of the function [latex]f(x,y)[\/latex]\u00a0is defined to be [latex]\\nabla f(x,y)=(\\partial{f}{\/}\\partial{x}){\\bf{i}}+(\\partial{f}{\/}\\partial{y}){\\bf{j}}[\/latex]\u00a0which can be generalized to a function of any number of independent variables<\/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-170\">\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 <\/section>","protected":false},"author":17533,"menu_order":28,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at 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-170","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/170","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":15,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/170\/revisions"}],"predecessor-version":[{"id":3768,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/170\/revisions\/3768"}],"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\/170\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=170"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=170"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=170"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}