{"id":741,"date":"2025-10-09T15:35:48","date_gmt":"2025-10-09T15:35:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=741"},"modified":"2025-11-20T17:19:14","modified_gmt":"2025-11-20T17:19:14","slug":"vector-calculus-and-vector-fields-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/vector-calculus-and-vector-fields-background-youll-need-3\/","title":{"raw":"Vector Calculus and Vector Fields: Background You'll Need 3","rendered":"Vector Calculus and Vector Fields: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Use a direction field to sketch solution curves<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Direction Fields<\/h2>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Direction fields (also called slope fields) provide a visual tool for understanding first-order differential equations without solving them algebraically. They're particularly useful when you need to predict solution behavior quickly or when an exact solution is difficult to find.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">We work with differential equations of the form:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]y^{\\prime} = f(x,y)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The key idea is straightforward: the derivative [latex]y^{\\prime}[\/latex] at any point represents the slope of the solution curve passing through that point. A direction field shows these slopes as short line segments across the coordinate plane.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>direction field<\/h3>\r\nA <strong>direction field (slope field)<\/strong> is a mathematical object used to graphically represent solutions to a first-order differential equation. At each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point.\r\n\r\n<\/section>\r\n<h3>Creating a Direction Field<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Let's build a direction field for [latex]y^{\\prime} = 3x + 2y - 4[\/latex].<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The process involves picking points [latex](x_0, y_0)[\/latex] in the coordinate plane and evaluating the right-hand side of the differential equation at each point. This gives you the slope of any solution curve that passes through that point.<\/p>\r\n&nbsp;\r\n<p class=\"whitespace-normal break-words\">Let's work through the process of creating a direction field using one of our examples: [latex]y^{\\prime} = 3x + 2y - 4[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">The key idea is to pick any point [latex](x_0, y_0)[\/latex] in the coordinate plane and substitute these coordinates into the right-hand side of the differential equation.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">At point [latex](1, 2)[\/latex], we substitute into the equation:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]y^{\\prime} = 3x + 2y - 4 = 3(1) + 2(2) - 4 = 3[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">This means any solution passing through [latex](1, 2)[\/latex] has slope [latex]3[\/latex] at that point. We draw a short line segment with slope [latex]3[\/latex] at [latex](1, 2)[\/latex].<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Repeat this process for multiple points across the plane to create the complete direction field. The domain consists of all ordered pairs [latex](x, y)[\/latex] where [latex]f(x, y)[\/latex] is defined.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Figure 2 shows the complete direction field for [latex]y^{\\prime} = 3x + 2y - 4[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233929\/CNX_Calc_Figure_08_02_001.jpg\" alt=\"A graph of the direction field for the differential equation y\u2019 = 3 x + 2 y \u2013 4 in all four quadrants. In quadrants two and three, the arrows point down and slightly to the right. On a diagonal line, roughly y = -x + 2, the arrows point further and further to the right, curve, and then point up above that line.\" width=\"487\" height=\"467\" data-media-type=\"image\/jpeg\" \/> Figure 2. Direction field for the differential equation [latex]y^{\\prime} =3x+2y - 4[\/latex].[\/caption]<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">You can create direction fields for any first-order differential equation of the form [latex]y^{\\prime} = f(x,y)[\/latex]. Common examples include:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]y^{\\prime} = 3x + 2y - 4[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y^{\\prime} = x^2 - y^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y^{\\prime} = \\frac{2x + 4}{y - 2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3 data-type=\"title\">Using Direction Fields<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Direction fields let you sketch approximate solutions without finding an exact formula. The line segments act as guides showing which direction the solution curve should follow.<\/p>\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>How to sketch a solution curve<\/strong>:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Start at any initial point [latex](x_0, y_0)[\/latex] in the field<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Follow the direction indicated by the line segment at that point<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Move a small step in that direction to reach a new point<\/li>\r\n \t<li class=\"whitespace-normal break-words\">At the new point, follow its line segment direction<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Continue this process to trace out the solution curve<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The curve you create is called a <strong>solution curve<\/strong> passing through the initial point.<\/p>\r\n\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Connection to Linear Approximation<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Direction fields build on linear approximation. The linear approximation of a function [latex]f(x)[\/latex] at point [latex](x_0, y_0)[\/latex] is:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]L(x) = y_0 + f'(x_0)(x - x_0)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For small changes in [latex]x[\/latex], we can approximate the change in [latex]y[\/latex]:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\Delta y \\approx f'(x_0) \\Delta x[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Each line segment in the direction field represents this local linear approximation.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1170571119933\" data-type=\"problem\">\r\n<p id=\"fs-id1170573295181\">Create a direction field for the differential equation [latex]y^{\\prime} ={x}^{2}-{y}^{2}[\/latex] and sketch a solution curve passing through the point [latex]\\left(-1,2\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558898\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558898\"]\r\n<div id=\"fs-id1170573593135\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1170573262665\">Use [latex]x[\/latex] and [latex]y[\/latex] values ranging from [latex]-5[\/latex] to [latex]5[\/latex]. For each coordinate pair, calculate [latex]y^{\\prime} [\/latex] using the right-hand side of the differential equation.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558899\"]\r\n<div id=\"fs-id1170573299269\" data-type=\"solution\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"459\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233936\/CNX_Calc_Figure_08_02_003.jpg\" alt=\"A graph of the direction field for the differential equation y\u2019 = x ^ 2 \u2013 y ^ 2. Along y = x and y = -x, the lines are horizontal. On either side of y = x and y = -x, the lines slant and direct solutions along those two functions. The rest of the lines are vertical. The solution going through (-1, 2) is shown. It curves down from about (-2.75, 10), through (-1, 2) and about (0, 1.5), and then up along the diagonal to (10, 10).\" width=\"459\" height=\"464\" data-media-type=\"image\/jpeg\" \/> Figure 4.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">[caption]Watch the following videos to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6722736&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=NmwwRLHZtkw&amp;video_target=tpm-plugin-zrpsnm0j-NmwwRLHZtkw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.2.1_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for \"4.2.1\" here (opens in new window)<\/a>.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6722737&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=8HNOR0-GDe8&amp;video_target=tpm-plugin-0tskvtbs-8HNOR0-GDe8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.2.2_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for \"4.2.2\" here (opens in new window)<\/a>.<\/section>&nbsp;","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Use a direction field to sketch solution curves<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Direction Fields<\/h2>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Direction fields (also called slope fields) provide a visual tool for understanding first-order differential equations without solving them algebraically. They&#8217;re particularly useful when you need to predict solution behavior quickly or when an exact solution is difficult to find.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">We work with differential equations of the form:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]y^{\\prime} = f(x,y)[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The key idea is straightforward: the derivative [latex]y^{\\prime}[\/latex] at any point represents the slope of the solution curve passing through that point. A direction field shows these slopes as short line segments across the coordinate plane.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>direction field<\/h3>\n<p>A <strong>direction field (slope field)<\/strong> is a mathematical object used to graphically represent solutions to a first-order differential equation. At each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point.<\/p>\n<\/section>\n<h3>Creating a Direction Field<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Let&#8217;s build a direction field for [latex]y^{\\prime} = 3x + 2y - 4[\/latex].<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The process involves picking points [latex](x_0, y_0)[\/latex] in the coordinate plane and evaluating the right-hand side of the differential equation at each point. This gives you the slope of any solution curve that passes through that point.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"whitespace-normal break-words\">Let&#8217;s work through the process of creating a direction field using one of our examples: [latex]y^{\\prime} = 3x + 2y - 4[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">The key idea is to pick any point [latex](x_0, y_0)[\/latex] in the coordinate plane and substitute these coordinates into the right-hand side of the differential equation.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body whitespace-normal break-words\">At point [latex](1, 2)[\/latex], we substitute into the equation:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]y^{\\prime} = 3x + 2y - 4 = 3(1) + 2(2) - 4 = 3[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">This means any solution passing through [latex](1, 2)[\/latex] has slope [latex]3[\/latex] at that point. We draw a short line segment with slope [latex]3[\/latex] at [latex](1, 2)[\/latex].<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Repeat this process for multiple points across the plane to create the complete direction field. The domain consists of all ordered pairs [latex](x, y)[\/latex] where [latex]f(x, y)[\/latex] is defined.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Figure 2 shows the complete direction field for [latex]y^{\\prime} = 3x + 2y - 4[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233929\/CNX_Calc_Figure_08_02_001.jpg\" alt=\"A graph of the direction field for the differential equation y\u2019 = 3 x + 2 y \u2013 4 in all four quadrants. In quadrants two and three, the arrows point down and slightly to the right. On a diagonal line, roughly y = -x + 2, the arrows point further and further to the right, curve, and then point up above that line.\" width=\"487\" height=\"467\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 2. Direction field for the differential equation [latex]y^{\\prime} =3x+2y - 4[\/latex].<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"font-claude-response-body whitespace-normal break-words\">You can create direction fields for any first-order differential equation of the form [latex]y^{\\prime} = f(x,y)[\/latex]. Common examples include:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]y^{\\prime} = 3x + 2y - 4[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y^{\\prime} = x^2 - y^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y^{\\prime} = \\frac{2x + 4}{y - 2}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<h3 data-type=\"title\">Using Direction Fields<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Direction fields let you sketch approximate solutions without finding an exact formula. The line segments act as guides showing which direction the solution curve should follow.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>How to sketch a solution curve<\/strong>:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Start at any initial point [latex](x_0, y_0)[\/latex] in the field<\/li>\n<li class=\"whitespace-normal break-words\">Follow the direction indicated by the line segment at that point<\/li>\n<li class=\"whitespace-normal break-words\">Move a small step in that direction to reach a new point<\/li>\n<li class=\"whitespace-normal break-words\">At the new point, follow its line segment direction<\/li>\n<li class=\"whitespace-normal break-words\">Continue this process to trace out the solution curve<\/li>\n<\/ol>\n<\/section>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The curve you create is called a <strong>solution curve<\/strong> passing through the initial point.<\/p>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Connection to Linear Approximation<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Direction fields build on linear approximation. The linear approximation of a function [latex]f(x)[\/latex] at point [latex](x_0, y_0)[\/latex] is:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]L(x) = y_0 + f'(x_0)(x - x_0)[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For small changes in [latex]x[\/latex], we can approximate the change in [latex]y[\/latex]:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\Delta y \\approx f'(x_0) \\Delta x[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Each line segment in the direction field represents this local linear approximation.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1170571119933\" data-type=\"problem\">\n<p id=\"fs-id1170573295181\">Create a direction field for the differential equation [latex]y^{\\prime} ={x}^{2}-{y}^{2}[\/latex] and sketch a solution curve passing through the point [latex]\\left(-1,2\\right)[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558898\">Hint<\/button><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573593135\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1170573262665\">Use [latex]x[\/latex] and [latex]y[\/latex] values ranging from [latex]-5[\/latex] to [latex]5[\/latex]. For each coordinate pair, calculate [latex]y^{\\prime}[\/latex] using the right-hand side of the differential equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558899\">Show Solution<\/button><\/p>\n<div id=\"q44558899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573299269\" data-type=\"solution\">\n<figure style=\"width: 459px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233936\/CNX_Calc_Figure_08_02_003.jpg\" alt=\"A graph of the direction field for the differential equation y\u2019 = x ^ 2 \u2013 y ^ 2. Along y = x and y = -x, the lines are horizontal. On either side of y = x and y = -x, the lines slant and direct solutions along those two functions. The rest of the lines are vertical. The solution going through (-1, 2) is shown. It curves down from about (-2.75, 10), through (-1, 2) and about (0, 1.5), and then up along the diagonal to (10, 10).\" width=\"459\" height=\"464\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 4.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following videos to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6722736&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=NmwwRLHZtkw&amp;video_target=tpm-plugin-zrpsnm0j-NmwwRLHZtkw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.2.1_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;4.2.1&#8221; here (opens in new window)<\/a>.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6722737&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=8HNOR0-GDe8&amp;video_target=tpm-plugin-0tskvtbs-8HNOR0-GDe8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.2.2_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;4.2.2&#8221; here (opens in new window)<\/a>.<\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":15,"menu_order":4,"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-741","chapter","type-chapter","status-publish","hentry"],"part":436,"module-header":"background_you_need","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\/741","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":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/741\/revisions"}],"predecessor-version":[{"id":2676,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/741\/revisions\/2676"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/436"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/741\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=741"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=741"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=741"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=741"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}