{"id":108,"date":"2021-07-30T17:14:30","date_gmt":"2021-07-30T17:14:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=108"},"modified":"2022-11-01T05:27:52","modified_gmt":"2022-11-01T05:27:52","slug":"summary-of-divergence-and-curl","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-divergence-and-curl\/","title":{"raw":"Summary of Divergence and Curl","rendered":"Summary of Divergence and Curl"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The divergence of a vector field is a scalar function. Divergence measures the \u201coutflowing-ness\u201d of a vector field. If\u00a0[latex]{\\bf{v}}[\/latex]\u00a0is the velocity field of a fluid, then the divergence of\u00a0[latex]{\\bf{v}}[\/latex]\u00a0at a point is the outflow of the fluid less the inflow at the point.<\/li>\r\n \t<li>The curl of a vector field is a vector field. The curl of a vector field at point\u00a0[latex]P[\/latex]\u00a0measures the tendency of particles at\u00a0[latex]P[\/latex]\u00a0to rotate about the axis that points in the direction of the curl at\u00a0[latex]P[\/latex].<\/li>\r\n \t<li>A vector field with a simply connected domain is conservative if and only if its curl is zero.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Curl\r\n<\/strong>[latex]\\nabla\\times{\\bf{F}}=(R_{y}-Q_{z}){\\bf{i}}+(P_{z}-R_{x}){\\bf{j}}+(Q_{x}+P_{y}){\\bf{k}}[\/latex]<\/li>\r\n \t<li><strong>Divergence\r\n<\/strong>[latex]\\nabla\\cdot{\\bf{F}}=P_{x}+Q_{y}+R_{z}[\/latex]<\/li>\r\n \t<li><strong>Divergence fo curl is zero\r\n<\/strong>[latex]\\nabla\\cdot(\\nabla\\times{\\bf{F}})=0[\/latex]<\/li>\r\n \t<li><strong>Curl of a gradient is the zero vector\r\n<\/strong>[latex]\\nabla\\times(\\nabla{f})=0[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>curl<\/dt>\r\n \t<dd>the curl of vector field [latex]{\\bf{F}}=\\langle{P,Q,R}\\rangle[\/latex], denoted [latex]\\nabla\\times{\\bf{F}}[\/latex]\u00a0is the \u201cdeterminant\u201d of the matrix [latex]\\begin{vmatrix}{\\bf{i}} &amp; {\\bf{j}} &amp; {\\bf{k}}\\\\ \\frac{d}{dx} &amp; \\frac{d}{dy} &amp; \\frac{d}{dz}\\\\P &amp; Q &amp; R\\end{vmatrix}[\/latex] and is given by the expression\u00a0[latex](R_{y}-Q_{z}){\\bf{i}}+(P_{z}-R_{x}){\\bf{j}}+(Q_{x}+P_{y}){\\bf{k}}[\/latex];\u00a0it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>divergence<\/dt>\r\n \t<dd>the divergence of a vector field [latex]{\\bf{F}}=\\langle{P,Q,R}\\rangle[\/latex], denoted [latex]\\nabla\\times{\\bf{F}}[\/latex] is [latex]P_{x}+Q_{y}+R_{z}[\/latex]; it measures the \u201coutflowing-ness\u201d of a vector field<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The divergence of a vector field is a scalar function. Divergence measures the \u201coutflowing-ness\u201d of a vector field. If\u00a0[latex]{\\bf{v}}[\/latex]\u00a0is the velocity field of a fluid, then the divergence of\u00a0[latex]{\\bf{v}}[\/latex]\u00a0at a point is the outflow of the fluid less the inflow at the point.<\/li>\n<li>The curl of a vector field is a vector field. The curl of a vector field at point\u00a0[latex]P[\/latex]\u00a0measures the tendency of particles at\u00a0[latex]P[\/latex]\u00a0to rotate about the axis that points in the direction of the curl at\u00a0[latex]P[\/latex].<\/li>\n<li>A vector field with a simply connected domain is conservative if and only if its curl is zero.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Curl<br \/>\n<\/strong>[latex]\\nabla\\times{\\bf{F}}=(R_{y}-Q_{z}){\\bf{i}}+(P_{z}-R_{x}){\\bf{j}}+(Q_{x}+P_{y}){\\bf{k}}[\/latex]<\/li>\n<li><strong>Divergence<br \/>\n<\/strong>[latex]\\nabla\\cdot{\\bf{F}}=P_{x}+Q_{y}+R_{z}[\/latex]<\/li>\n<li><strong>Divergence fo curl is zero<br \/>\n<\/strong>[latex]\\nabla\\cdot(\\nabla\\times{\\bf{F}})=0[\/latex]<\/li>\n<li><strong>Curl of a gradient is the zero vector<br \/>\n<\/strong>[latex]\\nabla\\times(\\nabla{f})=0[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>curl<\/dt>\n<dd>the curl of vector field [latex]{\\bf{F}}=\\langle{P,Q,R}\\rangle[\/latex], denoted [latex]\\nabla\\times{\\bf{F}}[\/latex]\u00a0is the \u201cdeterminant\u201d of the matrix [latex]\\begin{vmatrix}{\\bf{i}} & {\\bf{j}} & {\\bf{k}}\\\\ \\frac{d}{dx} & \\frac{d}{dy} & \\frac{d}{dz}\\\\P & Q & R\\end{vmatrix}[\/latex] and is given by the expression\u00a0[latex](R_{y}-Q_{z}){\\bf{i}}+(P_{z}-R_{x}){\\bf{j}}+(Q_{x}+P_{y}){\\bf{k}}[\/latex];\u00a0it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>divergence<\/dt>\n<dd>the divergence of a vector field [latex]{\\bf{F}}=\\langle{P,Q,R}\\rangle[\/latex], denoted [latex]\\nabla\\times{\\bf{F}}[\/latex] is [latex]P_{x}+Q_{y}+R_{z}[\/latex]; it measures the \u201coutflowing-ness\u201d of a vector field<\/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-108\">\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":24,"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-108","chapter","type-chapter","status-publish","hentry"],"part":24,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/108","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\/108\/revisions"}],"predecessor-version":[{"id":3791,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/108\/revisions\/3791"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/24"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/108\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=108"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=108"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=108"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}