{"id":149,"date":"2022-02-22T22:02:12","date_gmt":"2022-02-23T03:02:12","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/multivariatecalculus\/?post_type=chapter&#038;p=149"},"modified":"2022-03-03T22:52:34","modified_gmt":"2022-03-04T03:52:34","slug":"unit-10-vector-fields-in-2d-and-3d","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/chapter\/unit-10-vector-fields-in-2d-and-3d\/","title":{"raw":"Unit 11 : Vector Fields in 2D and 3D","rendered":"Unit 11 : Vector Fields in 2D and 3D"},"content":{"raw":"<h2>The Concept<\/h2>\r\nA <strong>vector field<\/strong> is an assignment of a vector to each point in a subset of space. In other words, if we are given a vector [latex] \\langle x,y\\rangle [\/latex], then the vector is simply the mapping in 2D of each point.\r\n\r\nVector fields can be written in two equivalent notations shown below for both 2D and 3D:\r\n<ul>\r\n \t<li><strong>2D Notation:<\/strong> [latex] \\vec{F}(x, y) = P(x, y)\\vec{i} + Q(x, y)\\vec{j} = \\langle P(x, y) , Q(x, y)\\rangle[\/latex]<\/li>\r\n \t<li><strong>3D Notation:<\/strong> [latex] \\vec{F}(x,y,z) =\u00a0 P(x, y,z)\\vec{i}+ Q(x,y,z)\\vec{j} + R(x,y,z)\\vec{k} = \\langle P(x, y,z) , Q(x, y,z) , R(x,y,z)\\rangle[\/latex]<\/li>\r\n<\/ul>\r\nWhere [latex] \\vec{i} = \\langle 1 , 0 \\rangle [\/latex] and [latex] \\vec{j} = \\langle 0,1\\rangle [\/latex] represent <strong>unit vectors<\/strong> in 2D, and [latex] \\vec{i} = \\langle1,0,0\\rangle ,\\, \\vec{j} = \\langle 0,1,0 \\rangle [\/latex] and [latex] \\vec{k} = \\langle 0,0,1\\rangle [\/latex] are unit vectors in 3D. A real life example that can be modeled as a vector field would be a fluid dynamics problem such as a river, where the velocity of the liquid is a vector at any given point. The magnitude (i.e., amplitude) of the vector represents the speed and the direction represents the direction of the flow at any given point.\r\n<h2>The Plot<\/h2>\r\nNow, you should engage with the 2D and 3D plots below to understand 2D and 3D vector fields[footnote]Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.[\/footnote]. Follow the steps below to apply changes to the plot and observe the effects:\r\n<ol>\r\n \t<li>The vector definition is done using [latex]P[\/latex] and [latex]Q[\/latex].<\/li>\r\n \t<li>Y Grid and X Grid control the number of arrows that will appear in the 2D plot.<\/li>\r\n \t<li>Xmin and Ymin set the minimum boundaries for the plot.<\/li>\r\n \t<li>YMax and Xmax set the maximum boundaries for the plot.<\/li>\r\n<\/ol>\r\n<h3>2D Vector Field Plot<\/h3>\r\n[h5p id=\"14\"]\r\n<h3>3D Vector Field Plot<\/h3>\r\n[h5p id=\"15\"]\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<h2 class=\"textbox__title\">Self-Checking Questions<\/h2>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCheck your understanding by solving the following questions<span>[footnote]Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax,\u00a0Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).[\/footnote]<\/span>:\r\n<ol>\r\n \t<li>Draw the following vector field [latex]\\vec{F}(x,y)=x\\vec{i} + y\\vec{j}[\/latex]<\/li>\r\n \t<li>Draw the following vector field [latex] \\vec{F}(x, y, z) = 2x\\vec{i} \u2212 2y\\vec{j} \u2212 2z\\vec{k} [\/latex]<\/li>\r\n<\/ol>\r\nUse the graphs to find the answers to these questions.\r\n\r\n<\/div>\r\n<\/div>","rendered":"<h2>The Concept<\/h2>\n<p>A <strong>vector field<\/strong> is an assignment of a vector to each point in a subset of space. In other words, if we are given a vector [latex]\\langle x,y\\rangle[\/latex], then the vector is simply the mapping in 2D of each point.<\/p>\n<p>Vector fields can be written in two equivalent notations shown below for both 2D and 3D:<\/p>\n<ul>\n<li><strong>2D Notation:<\/strong> [latex]\\vec{F}(x, y) = P(x, y)\\vec{i} + Q(x, y)\\vec{j} = \\langle P(x, y) , Q(x, y)\\rangle[\/latex]<\/li>\n<li><strong>3D Notation:<\/strong> [latex]\\vec{F}(x,y,z) =\u00a0 P(x, y,z)\\vec{i}+ Q(x,y,z)\\vec{j} + R(x,y,z)\\vec{k} = \\langle P(x, y,z) , Q(x, y,z) , R(x,y,z)\\rangle[\/latex]<\/li>\n<\/ul>\n<p>Where [latex]\\vec{i} = \\langle 1 , 0 \\rangle[\/latex] and [latex]\\vec{j} = \\langle 0,1\\rangle[\/latex] represent <strong>unit vectors<\/strong> in 2D, and [latex]\\vec{i} = \\langle1,0,0\\rangle ,\\, \\vec{j} = \\langle 0,1,0 \\rangle[\/latex] and [latex]\\vec{k} = \\langle 0,0,1\\rangle[\/latex] are unit vectors in 3D. A real life example that can be modeled as a vector field would be a fluid dynamics problem such as a river, where the velocity of the liquid is a vector at any given point. The magnitude (i.e., amplitude) of the vector represents the speed and the direction represents the direction of the flow at any given point.<\/p>\n<h2>The Plot<\/h2>\n<p>Now, you should engage with the 2D and 3D plots below to understand 2D and 3D vector fields<a class=\"footnote\" title=\"Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.\" id=\"return-footnote-149-1\" href=\"#footnote-149-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>. Follow the steps below to apply changes to the plot and observe the effects:<\/p>\n<ol>\n<li>The vector definition is done using [latex]P[\/latex] and [latex]Q[\/latex].<\/li>\n<li>Y Grid and X Grid control the number of arrows that will appear in the 2D plot.<\/li>\n<li>Xmin and Ymin set the minimum boundaries for the plot.<\/li>\n<li>YMax and Xmax set the maximum boundaries for the plot.<\/li>\n<\/ol>\n<h3>2D Vector Field Plot<\/h3>\n<div id=\"h5p-14\">\n<div class=\"h5p-content\" data-content-id=\"14\"><\/div>\n<\/div>\n<h3>3D Vector Field Plot<\/h3>\n<div id=\"h5p-15\">\n<div class=\"h5p-content\" data-content-id=\"15\"><\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<h2 class=\"textbox__title\">Self-Checking Questions<\/h2>\n<\/header>\n<div class=\"textbox__content\">\n<p>Check your understanding by solving the following questions<span><a class=\"footnote\" title=\"Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax,\u00a0Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).\" id=\"return-footnote-149-2\" href=\"#footnote-149-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/span>:<\/p>\n<ol>\n<li>Draw the following vector field [latex]\\vec{F}(x,y)=x\\vec{i} + y\\vec{j}[\/latex]<\/li>\n<li>Draw the following vector field [latex]\\vec{F}(x, y, z) = 2x\\vec{i} \u2212 2y\\vec{j} \u2212 2z\\vec{k}[\/latex]<\/li>\n<\/ol>\n<p>Use the graphs to find the answers to these questions.<\/p>\n<\/div>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-149-1\">Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0. <a href=\"#return-footnote-149-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-149-2\">Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax,\u00a0Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA). <a href=\"#return-footnote-149-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":396,"menu_order":11,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-149","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/149","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/users\/396"}],"version-history":[{"count":13,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/149\/revisions"}],"predecessor-version":[{"id":553,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/149\/revisions\/553"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/149\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/media?parent=149"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=149"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/contributor?post=149"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/license?post=149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}