{"id":755,"date":"2025-10-09T15:36:53","date_gmt":"2025-10-09T15:36:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=755"},"modified":"2025-11-05T22:52:47","modified_gmt":"2025-11-05T22:52:47","slug":"conservative-vector-fields-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/conservative-vector-fields-learn-it-1\/","title":{"raw":"Conservative Vector Fields: Learn It 1","rendered":"Conservative Vector Fields: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify simple and closed curves in various regions<\/li>\r\n \t<li>Find potential functions for conservative fields<\/li>\r\n \t<li>Apply the Fundamental Theorem for Line Integrals<\/li>\r\n \t<li>Test a vector field to determine whether it is conservative<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Curves and Regions<\/h2>\r\n<p class=\"whitespace-normal break-words\">Before we can work with the theorems in this section, we need to establish some geometric vocabulary about curves and regions. The definitions here will set us up for generalizing the Fundamental Theorem of Calculus to line integrals.<\/p>\r\n<p class=\"whitespace-normal break-words\">Let's start with two important types of curves. A <strong>closed curve<\/strong> begins and ends at the same point. A <strong>simple curve<\/strong> doesn't cross itself. When a curve is both closed and simple, we call it a <strong>simple closed curve<\/strong> (Figure 1).<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>closed and simple curves<\/h3>\r\n<p class=\"whitespace-normal break-words\">A curve [latex]C[\/latex] is <strong>closed<\/strong> if it has a parameterization [latex]\\mathbf{r}(t)[\/latex], [latex]a\\leq t\\leq b[\/latex], where [latex]\\mathbf{r}(a)=\\mathbf{r}(b)[\/latex] and the parameterization traverses the curve exactly once.<\/p>\r\n[latex]\\\\[\/latex]\r\n<p class=\"whitespace-normal break-words\">A curve [latex]C[\/latex] is <strong>simple<\/strong> if it doesn't cross itself. More precisely, [latex]C[\/latex] is simple if it has a parameterization [latex]\\mathbf{r}(t)[\/latex], [latex]a\\leq t\\leq b[\/latex], where [latex]\\mathbf{r}[\/latex] is one-to-one over [latex](a, b)[\/latex]. Note that we can have [latex]\\mathbf{r}(a)=\\mathbf{r}(b)[\/latex], which would make the simple curve also closed.<\/p>\r\n\r\n<\/section>\r\n\r\n[caption id=\"attachment_1375\" align=\"aligncenter\" width=\"929\"]<img class=\"size-full wp-image-1375\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143218\/6b7ab47eeaf8153b1a5df319d257a4e13e034e1d.webp\" alt=\"An image showing eight curves and their types. The first curve is neither simple nor closed; it has two endpoints and crosses itself twice. The second curve is simple but not closed; it does not cross itself and has two endpoints. The third curve is closed but is not simple; it crosses itself a few times. The fourth is a simple closed curve; it does not cross itself and has no endpoints. The fifth is a simple, not closed curve; it does not cross itself, but it has endpoints. The sixth is a simple, closed curve; it does not cross itself and has no endpoints. The seventh is closed but not a simple curve; it crosses itself but has no endpoints. The last is not simple and not closed; it crosses itself and has endpoints.\" width=\"929\" height=\"379\" \/> Figure 1. Types of curves that are simple or not simple and closed or not closed.[\/caption]\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Is the curve with parameterization [latex]{\\bf{r}}(t)=\\left\\langle\\cos{t},\\frac{\\sin{(2t)}}2\\right\\rangle, \\ 0\\leq{t}\\leq2\\pi[\/latex] a simple closed curve?[reveal-answer q=\"834725833\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834725833\"]\r\n\r\nNote that [latex]{\\bf{r}}(0)=\\langle0,1\\rangle[\/latex]; therefore, the curve is closed. The curve is not simple, however. To see this, note that [latex]{\\bf{r}}\\left(\\frac{\\pi}2\\right)=\\langle0,0\\rangle={\\bf{r}}\\left(\\frac{3\\pi}2\\right)[\/latex], and therefore the curve crosses itself at the origin (Figure 2).\r\n\r\n[caption id=\"attachment_1377\" align=\"aligncenter\" width=\"642\"]<img class=\"size-full wp-image-1377\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143308\/e3c0f528ccac9907f5cd1fba1bcfa807ed43ebe7.webp\" alt=\"A diagram in the (x,y) coordinate plane that shows a closed but not simple curve. It looks like a horizontal figure eight with the crossing point at the origin.\" width=\"642\" height=\"371\" \/> Figure 2. A curve that is closed but not simple.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">Now we need to define two types of regions. Many theorems in this module relate an integral over a region to an integral over its boundary. For these theorems to work, we need the region to be either <strong>connected<\/strong> or <strong>simply connected<\/strong>.<\/p>\r\n<p class=\"whitespace-normal break-words\">A connected region is one where you can draw a path between any two points without leaving the region. A simply connected region goes further\u2014it's a connected region with no holes in it.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>connected and simply connected regions<\/h3>\r\n<p class=\"whitespace-normal break-words\">A region [latex]D[\/latex] is <strong>connected<\/strong> if, for any two points [latex]P_1[\/latex] and [latex]P_2[\/latex] in [latex]D[\/latex], there exists a path from [latex]P_1[\/latex] to [latex]P_2[\/latex] that stays entirely inside [latex]D[\/latex].<\/p>\r\n[latex]\\\\[\/latex]\r\n<p class=\"whitespace-normal break-words\">A region [latex]D[\/latex] is <strong>simply connected<\/strong> if [latex]D[\/latex] is connected and any simple closed curve [latex]C[\/latex] inside [latex]D[\/latex] can be continuously shrunk to a point while staying entirely inside [latex]D[\/latex]. In two dimensions, this means the region is connected and has no holes.<\/p>\r\n\r\n<\/section>All simply connected regions are connected, but not all connected regions are simply connected (Figure 3).\r\n\r\n[caption id=\"attachment_1378\" align=\"aligncenter\" width=\"548\"]<img class=\"size-full wp-image-1378\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143348\/8d0eb3f345f768a2c295aa0f1991aaf19d3bcca2.webp\" alt=\"A diagram showing simply connected, connected, and not connected regions. The simply connected regions have no holes. The connected regions may have holes, but a path can still be found between any two points in the region. The not connected region has some points that cannot be connected by a path in the region. Here, this is illustrated by showing two circular shapes that are defined as part of region D1 but are separated by white space.\" width=\"548\" height=\"657\" \/> Figure 3. Not all connected regions are simply connected. (a) Simply connected regions have no holes. (b) Connected regions that are not simply connected may have holes but you can still find a path in the region between any two points. (c) A region that is not connected has some points that cannot be connected by a path in the region.[\/caption]\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Is the region in the below image connected? Is the region simply connected?<img class=\"aligncenter size-full wp-image-5249\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/31223004\/6.25a.jpg\" alt=\"&lt;img src=&quot;\/apps\/archive\/20220422.171947\/resources\/4fa7f7d2717cf20dc660c0bca880c7c7de267432&quot; data-media-type=&quot;image\/jpeg&quot; alt=&quot;A shaded circle with an open space in the shape of a circle inside it but very close to the boundary.&quot; id=&quot;9&quot;&gt;\" width=\"254\" height=\"254\" \/>\r\n\r\n[reveal-answer q=\"977374235\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"977374235\"]\r\n\r\nThe region in the figure is connected. The region in the figure is not simply connected.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]253265[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify simple and closed curves in various regions<\/li>\n<li>Find potential functions for conservative fields<\/li>\n<li>Apply the Fundamental Theorem for Line Integrals<\/li>\n<li>Test a vector field to determine whether it is conservative<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Curves and Regions<\/h2>\n<p class=\"whitespace-normal break-words\">Before we can work with the theorems in this section, we need to establish some geometric vocabulary about curves and regions. The definitions here will set us up for generalizing the Fundamental Theorem of Calculus to line integrals.<\/p>\n<p class=\"whitespace-normal break-words\">Let&#8217;s start with two important types of curves. A <strong>closed curve<\/strong> begins and ends at the same point. A <strong>simple curve<\/strong> doesn&#8217;t cross itself. When a curve is both closed and simple, we call it a <strong>simple closed curve<\/strong> (Figure 1).<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>closed and simple curves<\/h3>\n<p class=\"whitespace-normal break-words\">A curve [latex]C[\/latex] is <strong>closed<\/strong> if it has a parameterization [latex]\\mathbf{r}(t)[\/latex], [latex]a\\leq t\\leq b[\/latex], where [latex]\\mathbf{r}(a)=\\mathbf{r}(b)[\/latex] and the parameterization traverses the curve exactly once.<\/p>\n<p>[latex]\\\\[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">A curve [latex]C[\/latex] is <strong>simple<\/strong> if it doesn&#8217;t cross itself. More precisely, [latex]C[\/latex] is simple if it has a parameterization [latex]\\mathbf{r}(t)[\/latex], [latex]a\\leq t\\leq b[\/latex], where [latex]\\mathbf{r}[\/latex] is one-to-one over [latex](a, b)[\/latex]. Note that we can have [latex]\\mathbf{r}(a)=\\mathbf{r}(b)[\/latex], which would make the simple curve also closed.<\/p>\n<\/section>\n<figure id=\"attachment_1375\" aria-describedby=\"caption-attachment-1375\" style=\"width: 929px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1375\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143218\/6b7ab47eeaf8153b1a5df319d257a4e13e034e1d.webp\" alt=\"An image showing eight curves and their types. The first curve is neither simple nor closed; it has two endpoints and crosses itself twice. The second curve is simple but not closed; it does not cross itself and has two endpoints. The third curve is closed but is not simple; it crosses itself a few times. The fourth is a simple closed curve; it does not cross itself and has no endpoints. The fifth is a simple, not closed curve; it does not cross itself, but it has endpoints. The sixth is a simple, closed curve; it does not cross itself and has no endpoints. The seventh is closed but not a simple curve; it crosses itself but has no endpoints. The last is not simple and not closed; it crosses itself and has endpoints.\" width=\"929\" height=\"379\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143218\/6b7ab47eeaf8153b1a5df319d257a4e13e034e1d.webp 929w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143218\/6b7ab47eeaf8153b1a5df319d257a4e13e034e1d-300x122.webp 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143218\/6b7ab47eeaf8153b1a5df319d257a4e13e034e1d-768x313.webp 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143218\/6b7ab47eeaf8153b1a5df319d257a4e13e034e1d-65x27.webp 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143218\/6b7ab47eeaf8153b1a5df319d257a4e13e034e1d-225x92.webp 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143218\/6b7ab47eeaf8153b1a5df319d257a4e13e034e1d-350x143.webp 350w\" sizes=\"auto, (max-width: 929px) 100vw, 929px\" \/><figcaption id=\"caption-attachment-1375\" class=\"wp-caption-text\">Figure 1. Types of curves that are simple or not simple and closed or not closed.<\/figcaption><\/figure>\n<section class=\"textbox example\" aria-label=\"Example\">Is the curve with parameterization [latex]{\\bf{r}}(t)=\\left\\langle\\cos{t},\\frac{\\sin{(2t)}}2\\right\\rangle, \\ 0\\leq{t}\\leq2\\pi[\/latex] a simple closed curve?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q834725833\">Show Solution<\/button><\/p>\n<div id=\"q834725833\" class=\"hidden-answer\" style=\"display: none\">\n<p>Note that [latex]{\\bf{r}}(0)=\\langle0,1\\rangle[\/latex]; therefore, the curve is closed. The curve is not simple, however. To see this, note that [latex]{\\bf{r}}\\left(\\frac{\\pi}2\\right)=\\langle0,0\\rangle={\\bf{r}}\\left(\\frac{3\\pi}2\\right)[\/latex], and therefore the curve crosses itself at the origin (Figure 2).<\/p>\n<figure id=\"attachment_1377\" aria-describedby=\"caption-attachment-1377\" style=\"width: 642px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1377\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143308\/e3c0f528ccac9907f5cd1fba1bcfa807ed43ebe7.webp\" alt=\"A diagram in the (x,y) coordinate plane that shows a closed but not simple curve. It looks like a horizontal figure eight with the crossing point at the origin.\" width=\"642\" height=\"371\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143308\/e3c0f528ccac9907f5cd1fba1bcfa807ed43ebe7.webp 642w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143308\/e3c0f528ccac9907f5cd1fba1bcfa807ed43ebe7-300x173.webp 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143308\/e3c0f528ccac9907f5cd1fba1bcfa807ed43ebe7-65x38.webp 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143308\/e3c0f528ccac9907f5cd1fba1bcfa807ed43ebe7-225x130.webp 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143308\/e3c0f528ccac9907f5cd1fba1bcfa807ed43ebe7-350x202.webp 350w\" sizes=\"auto, (max-width: 642px) 100vw, 642px\" \/><figcaption id=\"caption-attachment-1377\" class=\"wp-caption-text\">Figure 2. A curve that is closed but not simple.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<p class=\"whitespace-normal break-words\">Now we need to define two types of regions. Many theorems in this module relate an integral over a region to an integral over its boundary. For these theorems to work, we need the region to be either <strong>connected<\/strong> or <strong>simply connected<\/strong>.<\/p>\n<p class=\"whitespace-normal break-words\">A connected region is one where you can draw a path between any two points without leaving the region. A simply connected region goes further\u2014it&#8217;s a connected region with no holes in it.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>connected and simply connected regions<\/h3>\n<p class=\"whitespace-normal break-words\">A region [latex]D[\/latex] is <strong>connected<\/strong> if, for any two points [latex]P_1[\/latex] and [latex]P_2[\/latex] in [latex]D[\/latex], there exists a path from [latex]P_1[\/latex] to [latex]P_2[\/latex] that stays entirely inside [latex]D[\/latex].<\/p>\n<p>[latex]\\\\[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">A region [latex]D[\/latex] is <strong>simply connected<\/strong> if [latex]D[\/latex] is connected and any simple closed curve [latex]C[\/latex] inside [latex]D[\/latex] can be continuously shrunk to a point while staying entirely inside [latex]D[\/latex]. In two dimensions, this means the region is connected and has no holes.<\/p>\n<\/section>\n<p>All simply connected regions are connected, but not all connected regions are simply connected (Figure 3).<\/p>\n<figure id=\"attachment_1378\" aria-describedby=\"caption-attachment-1378\" style=\"width: 548px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1378\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143348\/8d0eb3f345f768a2c295aa0f1991aaf19d3bcca2.webp\" alt=\"A diagram showing simply connected, connected, and not connected regions. The simply connected regions have no holes. The connected regions may have holes, but a path can still be found between any two points in the region. The not connected region has some points that cannot be connected by a path in the region. Here, this is illustrated by showing two circular shapes that are defined as part of region D1 but are separated by white space.\" width=\"548\" height=\"657\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143348\/8d0eb3f345f768a2c295aa0f1991aaf19d3bcca2.webp 548w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143348\/8d0eb3f345f768a2c295aa0f1991aaf19d3bcca2-250x300.webp 250w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143348\/8d0eb3f345f768a2c295aa0f1991aaf19d3bcca2-65x78.webp 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143348\/8d0eb3f345f768a2c295aa0f1991aaf19d3bcca2-225x270.webp 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17143348\/8d0eb3f345f768a2c295aa0f1991aaf19d3bcca2-350x420.webp 350w\" sizes=\"auto, (max-width: 548px) 100vw, 548px\" \/><figcaption id=\"caption-attachment-1378\" class=\"wp-caption-text\">Figure 3. Not all connected regions are simply connected. (a) Simply connected regions have no holes. (b) Connected regions that are not simply connected may have holes but you can still find a path in the region between any two points. (c) A region that is not connected has some points that cannot be connected by a path in the region.<\/figcaption><\/figure>\n<section class=\"textbox example\" aria-label=\"Example\">Is the region in the below image connected? Is the region simply connected?<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-5249\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/31223004\/6.25a.jpg\" alt=\"&lt;img src=&quot;\/apps\/archive\/20220422.171947\/resources\/4fa7f7d2717cf20dc660c0bca880c7c7de267432&quot; data-media-type=&quot;image\/jpeg&quot; alt=&quot;A shaded circle with an open space in the shape of a circle inside it but very close to the boundary.&quot; id=&quot;9&quot;&gt;\" width=\"254\" height=\"254\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q977374235\">Show Solution<\/button><\/p>\n<div id=\"q977374235\" class=\"hidden-answer\" style=\"display: none\">\n<p>The region in the figure is connected. The region in the figure is not simply connected.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm253265\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=253265&theme=lumen&iframe_resize_id=ohm253265&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":16,"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-755","chapter","type-chapter","status-publish","hentry"],"part":436,"module-header":"learn_it","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\/755","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\/755\/revisions"}],"predecessor-version":[{"id":1374,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/755\/revisions\/1374"}],"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\/755\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=755"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=755"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=755"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}