{"id":758,"date":"2025-10-09T15:37:01","date_gmt":"2025-10-09T15:37:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=758"},"modified":"2025-11-06T13:49:02","modified_gmt":"2025-11-06T13:49:02","slug":"greens-theorem-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/greens-theorem-learn-it-1\/","title":{"raw":"Green\u2019s Theorem: Learn It 1","rendered":"Green\u2019s Theorem: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Apply the circulation form of Green's theorem<\/li>\r\n \t<li>Apply the flux form of Green's theorem<\/li>\r\n \t<li>Calculate circulation and flux on general regions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Green's Theorem<\/h2>\r\n<p class=\"whitespace-normal break-words\">Green's theorem extends the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region [latex]D[\/latex] in the double integral to be simply connected. Later, we'll extend Green's theorem to regions that are not simply connected.<\/p>\r\n<p class=\"whitespace-normal break-words\">Green's theorem relates a line integral around a simply closed plane curve [latex]C[\/latex] and a double integral over the region enclosed by [latex]C[\/latex]. The theorem is useful because it allows us to translate difficult line integrals into simpler double integrals, or difficult double integrals into simpler line integrals.<\/p>\r\n<p class=\"whitespace-normal break-words\">Recall that the Fundamental Theorem of Calculus says that:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int_a^bF^\\prime(x)dx=F(b)-F(a)[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">Geometrically, this equation tells us that the integral over the region below the graph of [latex]F^\\prime(x)[\/latex] and above the line segment [latex][a,b][\/latex] depends only on the value of [latex]F[\/latex] at the endpoints [latex]a[\/latex] and [latex]b[\/latex]. Since the numbers [latex]a[\/latex] and [latex]b[\/latex] form the boundary of line segment [latex][a,b][\/latex], we can calculate the integral based on information about the boundary (Figure 1).<\/p>\r\n\r\n[caption id=\"attachment_1399\" align=\"aligncenter\" width=\"454\"]<img class=\"size-full wp-image-1399\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17144723\/1fd7145a16a0b05f17857a2d38eae6e520680739.webp\" alt=\"A graph in quadrant 1 of a generic function f(x). It is an increasing concave up function for the first quarter, an increasing concave down function for the second quarter, a decreasing concave down function for the third quarter, and an increasing concave down function for the last quarter. In the second quarter, a point a is marked on the x axis, and in the third quarter, a point b is marked on the x axis. The area under the curve and between a and b is shaded. This area is labeled the integral from a to b of f(x) dx.\" width=\"454\" height=\"275\" \/> Figure 1. The Fundamental Theorem of Calculus says that the integral over line segment [latex][a,b][\/latex] depends only on the values of the antiderivative at the endpoints of [latex][a,b][\/latex].[\/caption]<section id=\"fs-id1167794163661\" data-depth=\"1\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<p class=\"whitespace-normal break-words\">The Fundamental Theorem for Line Integrals works the same way:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int_C\\nabla f\\cdot d\\mathbf{r}=f(\\mathbf{r}(b))-f(\\mathbf{r}(a))[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">When we have a potential function (an \"antiderivative\"), we can calculate the line integral based solely on information about the boundary of curve [latex]C[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">Green's theorem extends this idea to double integrals. It says we can calculate a double integral over region [latex]D[\/latex] based solely on information about the boundary of [latex]D[\/latex]. It also says we can calculate a line integral over a simple closed curve [latex]C[\/latex] based solely on information about the region that [latex]C[\/latex] encloses. Green's theorem connects a double integral over region [latex]D[\/latex] to a line integral around the boundary of [latex]D[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Apply the circulation form of Green&#8217;s theorem<\/li>\n<li>Apply the flux form of Green&#8217;s theorem<\/li>\n<li>Calculate circulation and flux on general regions<\/li>\n<\/ul>\n<\/section>\n<h2>Green&#8217;s Theorem<\/h2>\n<p class=\"whitespace-normal break-words\">Green&#8217;s theorem extends the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region [latex]D[\/latex] in the double integral to be simply connected. Later, we&#8217;ll extend Green&#8217;s theorem to regions that are not simply connected.<\/p>\n<p class=\"whitespace-normal break-words\">Green&#8217;s theorem relates a line integral around a simply closed plane curve [latex]C[\/latex] and a double integral over the region enclosed by [latex]C[\/latex]. The theorem is useful because it allows us to translate difficult line integrals into simpler double integrals, or difficult double integrals into simpler line integrals.<\/p>\n<p class=\"whitespace-normal break-words\">Recall that the Fundamental Theorem of Calculus says that:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int_a^bF^\\prime(x)dx=F(b)-F(a)[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Geometrically, this equation tells us that the integral over the region below the graph of [latex]F^\\prime(x)[\/latex] and above the line segment [latex][a,b][\/latex] depends only on the value of [latex]F[\/latex] at the endpoints [latex]a[\/latex] and [latex]b[\/latex]. Since the numbers [latex]a[\/latex] and [latex]b[\/latex] form the boundary of line segment [latex][a,b][\/latex], we can calculate the integral based on information about the boundary (Figure 1).<\/p>\n<figure id=\"attachment_1399\" aria-describedby=\"caption-attachment-1399\" style=\"width: 454px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1399\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17144723\/1fd7145a16a0b05f17857a2d38eae6e520680739.webp\" alt=\"A graph in quadrant 1 of a generic function f(x). It is an increasing concave up function for the first quarter, an increasing concave down function for the second quarter, a decreasing concave down function for the third quarter, and an increasing concave down function for the last quarter. In the second quarter, a point a is marked on the x axis, and in the third quarter, a point b is marked on the x axis. The area under the curve and between a and b is shaded. This area is labeled the integral from a to b of f(x) dx.\" width=\"454\" height=\"275\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17144723\/1fd7145a16a0b05f17857a2d38eae6e520680739.webp 454w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17144723\/1fd7145a16a0b05f17857a2d38eae6e520680739-300x182.webp 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17144723\/1fd7145a16a0b05f17857a2d38eae6e520680739-65x39.webp 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17144723\/1fd7145a16a0b05f17857a2d38eae6e520680739-225x136.webp 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17144723\/1fd7145a16a0b05f17857a2d38eae6e520680739-350x212.webp 350w\" sizes=\"auto, (max-width: 454px) 100vw, 454px\" \/><figcaption id=\"caption-attachment-1399\" class=\"wp-caption-text\">Figure 1. The Fundamental Theorem of Calculus says that the integral over line segment [latex][a,b][\/latex] depends only on the values of the antiderivative at the endpoints of [latex][a,b][\/latex].<\/figcaption><\/figure>\n<section id=\"fs-id1167794163661\" data-depth=\"1\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<p class=\"whitespace-normal break-words\">The Fundamental Theorem for Line Integrals works the same way:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int_C\\nabla f\\cdot d\\mathbf{r}=f(\\mathbf{r}(b))-f(\\mathbf{r}(a))[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">When we have a potential function (an &#8220;antiderivative&#8221;), we can calculate the line integral based solely on information about the boundary of curve [latex]C[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Green&#8217;s theorem extends this idea to double integrals. It says we can calculate a double integral over region [latex]D[\/latex] based solely on information about the boundary of [latex]D[\/latex]. It also says we can calculate a line integral over a simple closed curve [latex]C[\/latex] based solely on information about the region that [latex]C[\/latex] encloses. Green&#8217;s theorem connects a double integral over region [latex]D[\/latex] to a line integral around the boundary of [latex]D[\/latex].<\/p>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":22,"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-758","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\/758","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":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/758\/revisions"}],"predecessor-version":[{"id":2047,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/758\/revisions\/2047"}],"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\/758\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=758"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=758"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=758"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=758"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}