{"id":705,"date":"2025-10-09T15:33:10","date_gmt":"2025-10-09T15:33:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=705"},"modified":"2025-11-20T17:05:03","modified_gmt":"2025-11-20T17:05:03","slug":"triple-integration-and-applications-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/triple-integration-and-applications-background-youll-need-3\/","title":{"raw":"Triple Integration and Applications: Background You'll Need 3","rendered":"Triple Integration and Applications: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate determinants of 2\u00d72 and 3\u00d73 matrice<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Evaluate the Determinant of a 2\u00d72 Matrix<\/h2>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">A <strong>determinant<\/strong> is a real number calculated from the entries of a square matrix. Determinants have practical applications in calculating area and volume, solving systems of equations, and even cryptography\u2014where encoded messages can only be decrypted using an invertible matrix and its determinant.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For us, the determinant serves a specific purpose: it tells us whether a matrix is invertible. If the determinant equals zero, the matrix is not invertible. If it's nonzero, we can find an inverse.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>the determinant of a 2 \u00d7 2 matrix<\/h3>\r\nThe <strong>determinant<\/strong> of a [latex]2\\text{ }\\times \\text{ }2[\/latex] matrix, given\r\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right][\/latex]<\/div>\r\nis defined as\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181431\/CNX_Precalc_Figure_09_08_0012.jpg\" alt=\"\" width=\"487\" height=\"59\" \/>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Notation for Determinants<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">You'll see determinants written in several ways:<\/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]\\text{det}(A)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]|A|[\/latex] (vertical bars replacing the matrix brackets)<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">These all mean the same thing\u2014the determinant of matrix [latex]A[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the determinant of the given matrix.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"149250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149250\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\mathrm{det}\\left(A\\right)&amp;=\\left\\rvert\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right\\rvert\\hfill \\\\ &amp;=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ &amp;=27\\hfill \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]6397[\/ohm_question]<\/section>\r\n<h2>Evaluate the Determinant of a 3 \u00d7 3 Matrix<\/h2>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Finding the determinant of a [latex]2 \\times 2[\/latex] matrix is straightforward, but the [latex]3 \\times 3[\/latex] case requires more steps. We'll use a method called <strong>diagonal expansion<\/strong>.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The process: augment the matrix by repeating its first two columns on the right, creating a [latex]3 \\times 5[\/latex] array. Then calculate products along diagonals\u2014adding products that go down-right and subtracting products that go up-right.<\/p>\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: Find the Determinant of a [latex]3 \\times 3[\/latex] Matrix<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Given matrix [latex]A=\\left[\\begin{array}{ccc}{a}_{1}&amp; {b}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {b}_{2}&amp; {c}_{2}\\\\ {a}_{3}&amp; {b}_{3}&amp; {c}_{3}\\end{array}\\right][\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Step 1:<\/strong> Augment [latex]A[\/latex] with its first two columns:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{det}(A) = \\left|\\begin{array}{ccc}a_1 &amp; b_1 &amp; c_1\\\\ a_2 &amp; b_2 &amp; c_2\\\\ a_3 &amp; b_3 &amp; c_3\\end{array}\\right| \\left.\\begin{array}{cc}a_1 &amp; b_1\\\\ a_2 &amp; b_2\\\\ a_3 &amp; b_3\\end{array}\\right|[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Step 2:<\/strong> Add products down the three diagonals (upper left to lower right):<\/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\">First diagonal: [latex]a_1 b_2 c_3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Second diagonal: [latex]b_1 c_2 a_3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Third diagonal: [latex]c_1 a_2 b_3[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Step 3:<\/strong> Subtract products up the three diagonals (lower left to upper right):<\/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\">First diagonal: [latex]a_3 b_2 c_1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Second diagonal: [latex]b_3 c_2 a_1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Third diagonal: [latex]c_3 a_2 b_1[\/latex]<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181433\/CNX_Precalc_Figure_09_08_0022.jpg\" alt=\"\" width=\"487\" height=\"89\" \/><\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Result:<\/strong><\/p>\r\n<p style=\"text-align: center;\">[latex]|A| = a_1 b_2 c_3 + b_1 c_2 a_3 + c_1 a_2 b_3 - a_3 b_2 c_1 - b_3 c_2 a_1 - c_3 a_2 b_1[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Think of it as:<\/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\"><strong>Add<\/strong> the three products going down-right (\u2198)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Subtract<\/strong> the three products going up-right (\u2197)<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">This pattern makes the calculation systematic and reduces errors.<\/p>\r\n\r\n<\/section>\r\n<div><section class=\"textbox example\" aria-label=\"Example\">Find the determinant of the given 3 \u00d7 3 matrix:\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}0&amp; 2&amp; 1\\\\ 3&amp; -1&amp; 1\\\\ 4&amp; 0&amp; 1\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"915069\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"915069\"]\r\n\r\nAugment the matrix with the first two columns and then follow the formula. Thus,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}|A|&amp;=\\left\\rvert\\begin{array}{ccc}0&amp; 2&amp; 1\\\\ 3&amp; -1&amp; 1\\\\ 4&amp; 0&amp; 1\\end{array}\\right\\rvert\\left.\\begin{array}{c}0 &amp; 2\\\\ 3 &amp; -1\\\\ 4 &amp; 0\\end{array}\\right\\rvert\\hfill \\\\ &amp;=0\\left(-1\\right)\\left(1\\right)+2\\left(1\\right)\\left(4\\right)+1\\left(3\\right)\\left(0\\right)-4\\left(-1\\right)\\left(1\\right)-0\\left(1\\right)\\left(0\\right)-1\\left(3\\right)\\left(2\\right)\\hfill \\\\ &amp;=0+8+0+4 - 0-6\\hfill \\\\ &amp;=6\\hfill \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the determinant of the 3 \u00d7 3 matrix.\r\n<p style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}1&amp; -3&amp; 7\\\\ 1&amp; 1&amp; 1\\\\ 1&amp; -2&amp; 3\\end{array}\\right\\rvert[\/latex]<\/p>\r\n[reveal-answer q=\"612489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"612489\"]\r\n\r\n[latex]-10[\/latex]\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]19398[\/ohm_question]<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate determinants of 2\u00d72 and 3\u00d73 matrice<\/li>\n<\/ul>\n<\/section>\n<h2>Evaluate the Determinant of a 2\u00d72 Matrix<\/h2>\n<p class=\"font-claude-response-body whitespace-normal break-words\">A <strong>determinant<\/strong> is a real number calculated from the entries of a square matrix. Determinants have practical applications in calculating area and volume, solving systems of equations, and even cryptography\u2014where encoded messages can only be decrypted using an invertible matrix and its determinant.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For us, the determinant serves a specific purpose: it tells us whether a matrix is invertible. If the determinant equals zero, the matrix is not invertible. If it&#8217;s nonzero, we can find an inverse.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>the determinant of a 2 \u00d7 2 matrix<\/h3>\n<p>The <strong>determinant<\/strong> of a [latex]2\\text{ }\\times \\text{ }2[\/latex] matrix, given<\/p>\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right][\/latex]<\/div>\n<p>is defined as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181431\/CNX_Precalc_Figure_09_08_0012.jpg\" alt=\"\" width=\"487\" height=\"59\" \/><\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Notation for Determinants<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">You&#8217;ll see determinants written in several ways:<\/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]\\text{det}(A)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]|A|[\/latex] (vertical bars replacing the matrix brackets)<\/li>\n<\/ul>\n<p class=\"font-claude-response-body whitespace-normal break-words\">These all mean the same thing\u2014the determinant of matrix [latex]A[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the determinant of the given matrix.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q149250\">Show Solution<\/button><\/p>\n<div id=\"q149250\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\mathrm{det}\\left(A\\right)&=\\left\\rvert\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right\\rvert\\hfill \\\\ &=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ &=27\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm6397\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6397&theme=lumen&iframe_resize_id=ohm6397&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Evaluate the Determinant of a 3 \u00d7 3 Matrix<\/h2>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Finding the determinant of a [latex]2 \\times 2[\/latex] matrix is straightforward, but the [latex]3 \\times 3[\/latex] case requires more steps. We&#8217;ll use a method called <strong>diagonal expansion<\/strong>.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The process: augment the matrix by repeating its first two columns on the right, creating a [latex]3 \\times 5[\/latex] array. Then calculate products along diagonals\u2014adding products that go down-right and subtracting products that go up-right.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: Find the Determinant of a [latex]3 \\times 3[\/latex] Matrix<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Given matrix [latex]A=\\left[\\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\\\ {a}_{2}& {b}_{2}& {c}_{2}\\\\ {a}_{3}& {b}_{3}& {c}_{3}\\end{array}\\right][\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Step 1:<\/strong> Augment [latex]A[\/latex] with its first two columns:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{det}(A) = \\left|\\begin{array}{ccc}a_1 & b_1 & c_1\\\\ a_2 & b_2 & c_2\\\\ a_3 & b_3 & c_3\\end{array}\\right| \\left.\\begin{array}{cc}a_1 & b_1\\\\ a_2 & b_2\\\\ a_3 & b_3\\end{array}\\right|[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Step 2:<\/strong> Add products down the three diagonals (upper left to lower right):<\/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\">First diagonal: [latex]a_1 b_2 c_3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Second diagonal: [latex]b_1 c_2 a_3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Third diagonal: [latex]c_1 a_2 b_3[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Step 3:<\/strong> Subtract products up the three diagonals (lower left to upper right):<\/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\">First diagonal: [latex]a_3 b_2 c_1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Second diagonal: [latex]b_3 c_2 a_1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Third diagonal: [latex]c_3 a_2 b_1[\/latex]<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181433\/CNX_Precalc_Figure_09_08_0022.jpg\" alt=\"\" width=\"487\" height=\"89\" \/><\/li>\n<\/ul>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Result:<\/strong><\/p>\n<p style=\"text-align: center;\">[latex]|A| = a_1 b_2 c_3 + b_1 c_2 a_3 + c_1 a_2 b_3 - a_3 b_2 c_1 - b_3 c_2 a_1 - c_3 a_2 b_1[\/latex]<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"font-claude-response-body whitespace-normal break-words\">Think of it as:<\/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\"><strong>Add<\/strong> the three products going down-right (\u2198)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Subtract<\/strong> the three products going up-right (\u2197)<\/li>\n<\/ul>\n<p class=\"font-claude-response-body whitespace-normal break-words\">This pattern makes the calculation systematic and reduces errors.<\/p>\n<\/section>\n<div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the determinant of the given 3 \u00d7 3 matrix:<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}0& 2& 1\\\\ 3& -1& 1\\\\ 4& 0& 1\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q915069\">Show Solution<\/button><\/p>\n<div id=\"q915069\" class=\"hidden-answer\" style=\"display: none\">\n<p>Augment the matrix with the first two columns and then follow the formula. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}|A|&=\\left\\rvert\\begin{array}{ccc}0& 2& 1\\\\ 3& -1& 1\\\\ 4& 0& 1\\end{array}\\right\\rvert\\left.\\begin{array}{c}0 & 2\\\\ 3 & -1\\\\ 4 & 0\\end{array}\\right\\rvert\\hfill \\\\ &=0\\left(-1\\right)\\left(1\\right)+2\\left(1\\right)\\left(4\\right)+1\\left(3\\right)\\left(0\\right)-4\\left(-1\\right)\\left(1\\right)-0\\left(1\\right)\\left(0\\right)-1\\left(3\\right)\\left(2\\right)\\hfill \\\\ &=0+8+0+4 - 0-6\\hfill \\\\ &=6\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the determinant of the 3 \u00d7 3 matrix.<\/p>\n<p style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}1& -3& 7\\\\ 1& 1& 1\\\\ 1& -2& 3\\end{array}\\right\\rvert[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q612489\">Show Solution<\/button><\/p>\n<div id=\"q612489\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm19398\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=19398&theme=lumen&iframe_resize_id=ohm19398&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\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-705","chapter","type-chapter","status-publish","hentry"],"part":434,"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\/705","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":2,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/705\/revisions"}],"predecessor-version":[{"id":2669,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/705\/revisions\/2669"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/434"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/705\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=705"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=705"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=705"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=705"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}