{"id":779,"date":"2025-10-09T15:38:12","date_gmt":"2025-10-09T15:38:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=779"},"modified":"2025-12-15T13:34:37","modified_gmt":"2025-12-15T13:34:37","slug":"divergence-and-curl-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/divergence-and-curl-apply-it\/","title":{"raw":"Divergence and Curl: Apply It","rendered":"Divergence and Curl: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Determine divergence from the formula for a given vector field<\/li>\r\n \t<li>Determine curl from the formula for a given vector field<\/li>\r\n \t<li>Use divergence and curl to identify conservative fields<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Analyzing Airflow in Your Dorm Room<\/h2>\r\nIt's the first hot week of the semester, and your small dorm room feels stuffy despite having two fans running. Your roommate, who is studying mechanical engineering, offers to help optimize your airflow setup. They place sensors at various points in the room and model the combined air velocity as vector fields based on different fan positions. The velocity vectors describe how fast and in what direction air moves at each point in the room. By analyzing these vector fields, you can determine where air is stagnating, where it's flowing effectively, and whether the fans create useful circulation patterns. Understanding these airflow characteristics will help you reposition the fans for maximum cooling comfort during the hot days ahead.\r\n\r\nYour roommate's first sensor reading models the airflow velocity in a horizontal cross-section of the room as the vector field [latex]\\mathbf{v}(x, y) = \\langle 3x + y,\u00a0 2x - 3y \\rangle[\/latex], where [latex]x[\/latex] and [latex]y[\/latex] are measured in meters from the center of the room, and velocity is in meters per second.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317056[\/ohm_question]<\/section>Your roommate repositions the oscillating fan and the new airflow in a vertical cross-section is modeled by the three-dimensional vector field [latex]\\mathbf{F}(x, y, z) = \\langle yz, xz,\u00a0 xy \\rangle[\/latex], where [latex]z[\/latex] represents height above the floor.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1] 317057[\/ohm_question]<\/section>Your roommate tries a different fan configuration. The resulting airflow is modeled by [latex]\\mathbf{G}(x, y, z) = \\langle 2xz,\u00a0 y^2, x^2 + z \\rangle[\/latex].\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317058 [\/ohm_question]<\/section>You want to know if certain airflow patterns could represent a conservative vector field, which would mean the air circulation is \"path-independent\" and easier to predict. Your roommate tests two different fan setups:\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317059 [\/ohm_question]<\/section>Your roommate proposes one final airflow model: [latex]\\mathbf{H}(x, y, z) = \\langle e^x \\sin y,\u00a0 e^x \\cos y,\u00a0 z^2 \\rangle[\/latex].\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1] 317060[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine divergence from the formula for a given vector field<\/li>\n<li>Determine curl from the formula for a given vector field<\/li>\n<li>Use divergence and curl to identify conservative fields<\/li>\n<\/ul>\n<\/section>\n<h2>Analyzing Airflow in Your Dorm Room<\/h2>\n<p>It&#8217;s the first hot week of the semester, and your small dorm room feels stuffy despite having two fans running. Your roommate, who is studying mechanical engineering, offers to help optimize your airflow setup. They place sensors at various points in the room and model the combined air velocity as vector fields based on different fan positions. The velocity vectors describe how fast and in what direction air moves at each point in the room. By analyzing these vector fields, you can determine where air is stagnating, where it&#8217;s flowing effectively, and whether the fans create useful circulation patterns. Understanding these airflow characteristics will help you reposition the fans for maximum cooling comfort during the hot days ahead.<\/p>\n<p>Your roommate&#8217;s first sensor reading models the airflow velocity in a horizontal cross-section of the room as the vector field [latex]\\mathbf{v}(x, y) = \\langle 3x + y,\u00a0 2x - 3y \\rangle[\/latex], where [latex]x[\/latex] and [latex]y[\/latex] are measured in meters from the center of the room, and velocity is in meters per second.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317056\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317056&theme=lumen&iframe_resize_id=ohm317056&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Your roommate repositions the oscillating fan and the new airflow in a vertical cross-section is modeled by the three-dimensional vector field [latex]\\mathbf{F}(x, y, z) = \\langle yz, xz,\u00a0 xy \\rangle[\/latex], where [latex]z[\/latex] represents height above the floor.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317057\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317057&theme=lumen&iframe_resize_id=ohm317057&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Your roommate tries a different fan configuration. The resulting airflow is modeled by [latex]\\mathbf{G}(x, y, z) = \\langle 2xz,\u00a0 y^2, x^2 + z \\rangle[\/latex].<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317058\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317058&theme=lumen&iframe_resize_id=ohm317058&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>You want to know if certain airflow patterns could represent a conservative vector field, which would mean the air circulation is &#8220;path-independent&#8221; and easier to predict. Your roommate tests two different fan setups:<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317059\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317059&theme=lumen&iframe_resize_id=ohm317059&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Your roommate proposes one final airflow model: [latex]\\mathbf{H}(x, y, z) = \\langle e^x \\sin y,\u00a0 e^x \\cos y,\u00a0 z^2 \\rangle[\/latex].<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317060\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317060&theme=lumen&iframe_resize_id=ohm317060&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":8,"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-779","chapter","type-chapter","status-publish","hentry"],"part":438,"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\/779","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\/779\/revisions"}],"predecessor-version":[{"id":2802,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/779\/revisions\/2802"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/438"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/779\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=779"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=779"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=779"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=779"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}