{"id":759,"date":"2025-10-09T15:37:03","date_gmt":"2025-10-09T15:37:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=759"},"modified":"2025-12-12T16:51:14","modified_gmt":"2025-12-12T16:51:14","slug":"greens-theorem-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/greens-theorem-apply-it\/","title":{"raw":"Green\u2019s Theorem: Apply It","rendered":"Green\u2019s Theorem: Apply It"},"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>Analyzing Airflow Patterns in Your Apartment<\/h2>\r\nYou recently moved into a small apartment and noticed that some areas feel stuffy while others have a nice breeze when your window fan is running. Curious about the airflow patterns, you borrow a handheld [pb_glossary id=\"2791\"]anemometer[\/pb_glossary] from a friend in the engineering program to measure air velocity at different points in your rectangular living space. The room measures [latex]4[\/latex] meters by [latex]3[\/latex] meters. Based on your measurements, you model the air velocity as a vector field [latex]\\mathbf{F}(x,y) = \\langle P(x,y), Q(x,y) \\rangle[\/latex] where [latex]x[\/latex] and [latex]y[\/latex] are positions in meters and the components are in meters per second. You want to understand how air circulates around the room and how much air flows across certain boundaries. This information will help you decide where to place a second fan or reposition furniture for better airflow.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]316987[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]316988[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]316990[\/ohm_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]316991[\/ohm_question]<\/section><\/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>Analyzing Airflow Patterns in Your Apartment<\/h2>\n<p>You recently moved into a small apartment and noticed that some areas feel stuffy while others have a nice breeze when your window fan is running. Curious about the airflow patterns, you borrow a handheld <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_759_2791\">anemometer<\/a> from a friend in the engineering program to measure air velocity at different points in your rectangular living space. The room measures [latex]4[\/latex] meters by [latex]3[\/latex] meters. Based on your measurements, you model the air velocity as a vector field [latex]\\mathbf{F}(x,y) = \\langle P(x,y), Q(x,y) \\rangle[\/latex] where [latex]x[\/latex] and [latex]y[\/latex] are positions in meters and the components are in meters per second. You want to understand how air circulates around the room and how much air flows across certain boundaries. This information will help you decide where to place a second fan or reposition furniture for better airflow.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm316987\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316987&theme=lumen&iframe_resize_id=ohm316987&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm316988\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316988&theme=lumen&iframe_resize_id=ohm316988&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm316990\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316990&theme=lumen&iframe_resize_id=ohm316990&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm316991\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316991&theme=lumen&iframe_resize_id=ohm316991&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_759_2791\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_759_2791\"><div tabindex=\"-1\"><p>an instrument for measuring the speed of the wind, or of any current of gas.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":15,"menu_order":26,"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-759","chapter","type-chapter","status-publish","hentry"],"part":436,"module-header":"apply_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\/759","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\/759\/revisions"}],"predecessor-version":[{"id":2794,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/759\/revisions\/2794"}],"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\/759\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=759"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=759"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=759"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=759"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}