{"id":100,"date":"2021-07-30T17:13:36","date_gmt":"2021-07-30T17:13:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=100"},"modified":"2022-11-01T05:01:57","modified_gmt":"2022-11-01T05:01:57","slug":"summary-of-vector-fields","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-vector-fields\/","title":{"raw":"Summary of Vector Fields","rendered":"Summary of Vector Fields"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>A vector field assigns a vector [latex]{\\bf{F}}(x,y)[\/latex] to each point\u00a0[latex](x,y)[\/latex] in a subset [latex]D[\/latex] of\u00a0[latex]\\mathbb{R}^{2}[\/latex] or\u00a0[latex]\\mathbb{R}^{3}[\/latex].\u00a0[latex]{\\bf{F}}(x,y,z)[\/latex] to each point\u00a0[latex](x,y,z)[\/latex] in a subset [latex]D[\/latex] of\u00a0[latex]\\mathbb{R}^{3}[\/latex].<\/li>\r\n \t<li>Vector fields can describe the distribution of vector quantities such as forces or velocities over a region of the plane or of space. They are in common use in such areas as physics, engineering, meteorology, oceanography.<\/li>\r\n \t<li>We can sketch a vector field by examining its defining equation to determine relative magnitudes in various locations and then drawing enough vectors to determine a pattern.<\/li>\r\n \t<li>A vector field [latex]{\\bf{F}}[\/latex] is called conservative if there exists a scalar function [latex]f[\/latex] such that [latex]\\nabla{f}={\\bf{F}}[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Vector field in [latex]\\mathbb{R}^{2}[\/latex]\r\n<\/strong>[latex]{\\bf{F}}(x,y)=\\langle{P(x,y),Q(x,y)}\\rangle[\/latex]\u00a0 or\u00a0\u00a0[latex]{\\bf{F}}(x,y)=P(x,y){\\bf{i}},Q(x,y){\\bf{j}}[\/latex]<\/li>\r\n \t<li><strong>Vector field in [latex]\\mathbb{R}^{3}[\/latex]\r\n<\/strong>[latex]{\\bf{F}}(x,y,z)=\\langle{P(x,y,z),Q(x,y,z),R(x,y,z)}\\rangle[\/latex]\u00a0 or\u00a0\u00a0[latex]{\\bf{F}}(x,y,z)=P(x,y,z){\\bf{i}},Q(x,y,z){\\bf{j}},R(x,y,z){\\bf{k}}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>conservative field<\/dt>\r\n \t<dd>a vector field for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>gradient field<\/dt>\r\n \t<dd>a vector field [latex]{\\bf{F}}[\/latex]\u00a0for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]\u00a0in other words, a vector field that is the gradient of a function; such vector fields are also called <em data-effect=\"italics\">conservative<\/em><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>potential function<\/dt>\r\n \t<dd>a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>radial field<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a vector field in which all vectors either point directly toward or directly away from the origin; the magnitude of any vector depends only on its distance from the origin<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>rotational field<\/dt>\r\n \t<dd>a vector field in which the vector at point\u00a0<span style=\"font-size: 1em;\">[latex](x,y)[\/latex]<\/span>\u00a0is tangent to a circle with radius\u00a0<span style=\"font-size: 1em;\">[latex]r=\\sqrt{x^{2}+y^{2}}[\/latex]<\/span>\u00a0in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>unit vector field<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a vector field in which the magnitude of every vector is [latex]1[\/latex]<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector field<\/dt>\r\n \t<dd>measured in [latex]\\mathbb{R}^{2}[\/latex],\u00a0an assignment of a vector <span style=\"font-size: 1em;\">[latex]{\\bf{F}}(x,y)[\/latex]<\/span>\u00a0to each point\u00a0<span style=\"font-size: 1em;\">[latex](x,y)[\/latex]<\/span>\u00a0of a subset\u00a0<span style=\"font-size: 1em;\">[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{2}[\/latex]; in [latex]\\mathbb{R}^{3}[\/latex],\u00a0an assignment of a vector <span style=\"font-size: 1em;\">[latex]{\\bf{F}}(x,y,z)[\/latex]<\/span> to each point\u00a0<span style=\"font-size: 1em;\">[latex](x,y,z)[\/latex]<\/span>\u00a0of a subset\u00a0<span style=\"font-size: 1em;\">[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{3}[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>A vector field assigns a vector [latex]{\\bf{F}}(x,y)[\/latex] to each point\u00a0[latex](x,y)[\/latex] in a subset [latex]D[\/latex] of\u00a0[latex]\\mathbb{R}^{2}[\/latex] or\u00a0[latex]\\mathbb{R}^{3}[\/latex].\u00a0[latex]{\\bf{F}}(x,y,z)[\/latex] to each point\u00a0[latex](x,y,z)[\/latex] in a subset [latex]D[\/latex] of\u00a0[latex]\\mathbb{R}^{3}[\/latex].<\/li>\n<li>Vector fields can describe the distribution of vector quantities such as forces or velocities over a region of the plane or of space. They are in common use in such areas as physics, engineering, meteorology, oceanography.<\/li>\n<li>We can sketch a vector field by examining its defining equation to determine relative magnitudes in various locations and then drawing enough vectors to determine a pattern.<\/li>\n<li>A vector field [latex]{\\bf{F}}[\/latex] is called conservative if there exists a scalar function [latex]f[\/latex] such that [latex]\\nabla{f}={\\bf{F}}[\/latex].<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Vector field in [latex]\\mathbb{R}^{2}[\/latex]<br \/>\n<\/strong>[latex]{\\bf{F}}(x,y)=\\langle{P(x,y),Q(x,y)}\\rangle[\/latex]\u00a0 or\u00a0\u00a0[latex]{\\bf{F}}(x,y)=P(x,y){\\bf{i}},Q(x,y){\\bf{j}}[\/latex]<\/li>\n<li><strong>Vector field in [latex]\\mathbb{R}^{3}[\/latex]<br \/>\n<\/strong>[latex]{\\bf{F}}(x,y,z)=\\langle{P(x,y,z),Q(x,y,z),R(x,y,z)}\\rangle[\/latex]\u00a0 or\u00a0\u00a0[latex]{\\bf{F}}(x,y,z)=P(x,y,z){\\bf{i}},Q(x,y,z){\\bf{j}},R(x,y,z){\\bf{k}}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>conservative field<\/dt>\n<dd>a vector field for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>gradient field<\/dt>\n<dd>a vector field [latex]{\\bf{F}}[\/latex]\u00a0for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]\u00a0in other words, a vector field that is the gradient of a function; such vector fields are also called <em data-effect=\"italics\">conservative<\/em><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>potential function<\/dt>\n<dd>a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>radial field<\/dt>\n<dd><span style=\"font-size: 1em;\">a vector field in which all vectors either point directly toward or directly away from the origin; the magnitude of any vector depends only on its distance from the origin<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>rotational field<\/dt>\n<dd>a vector field in which the vector at point\u00a0<span style=\"font-size: 1em;\">[latex](x,y)[\/latex]<\/span>\u00a0is tangent to a circle with radius\u00a0<span style=\"font-size: 1em;\">[latex]r=\\sqrt{x^{2}+y^{2}}[\/latex]<\/span>\u00a0in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>unit vector field<\/dt>\n<dd><span style=\"font-size: 1em;\">a vector field in which the magnitude of every vector is [latex]1[\/latex]<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector field<\/dt>\n<dd>measured in [latex]\\mathbb{R}^{2}[\/latex],\u00a0an assignment of a vector <span style=\"font-size: 1em;\">[latex]{\\bf{F}}(x,y)[\/latex]<\/span>\u00a0to each point\u00a0<span style=\"font-size: 1em;\">[latex](x,y)[\/latex]<\/span>\u00a0of a subset\u00a0<span style=\"font-size: 1em;\">[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{2}[\/latex]; in [latex]\\mathbb{R}^{3}[\/latex],\u00a0an assignment of a vector <span style=\"font-size: 1em;\">[latex]{\\bf{F}}(x,y,z)[\/latex]<\/span> to each point\u00a0<span style=\"font-size: 1em;\">[latex](x,y,z)[\/latex]<\/span>\u00a0of a subset\u00a0<span style=\"font-size: 1em;\">[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{3}[\/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-100\">\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":5,"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-100","chapter","type-chapter","status-publish","hentry"],"part":24,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/100","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\/100\/revisions"}],"predecessor-version":[{"id":3782,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/100\/revisions\/3782"}],"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\/100\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=100"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=100"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=100"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}