{"id":795,"date":"2025-10-09T15:39:41","date_gmt":"2025-10-09T15:39:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=795"},"modified":"2025-12-15T14:53:55","modified_gmt":"2025-12-15T14:53:55","slug":"the-divergence-theorem-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/the-divergence-theorem-apply-it\/","title":{"raw":"The Divergence Theorem: Apply It","rendered":"The Divergence Theorem: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Understand the divergence theorem<\/li>\r\n \t<li>Calculate flux using the divergence theorem<\/li>\r\n \t<li>Apply the divergence theorem to physical problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid gap-4 [&amp;_&gt;_*]:min-w-0\">\r\n<h2 class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Analyzing Heat Distribution in Your Microwave Burrito<\/h2>\r\n<\/div>\r\n<\/div>\r\nYou just got back to your apartment after a long day of classes and grab a frozen burrito from the freezer. As you heat it in the microwave, you notice a familiar problem: some parts come out scorching hot while others remain frozen. Your roommate, who is taking an electromagnetic fields course, mentions that microwaves generate an energy field inside the appliance that can be modeled mathematically. Curious, you decide to analyze the energy flux through the burrito's surface to understand why it heats so unevenly. The burrito can be approximated as a cylinder, and the microwave's energy field in the region around your food can be modeled using vector fields. By applying the divergence theorem, you can relate the energy absorption inside the burrito to the flux across its surface, helping explain why that first bite is always a gamble between lava and ice.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317073 [\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317074 [\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317075 [\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1] 317076[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Understand the divergence theorem<\/li>\n<li>Calculate flux using the divergence theorem<\/li>\n<li>Apply the divergence theorem to physical problems<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid gap-4 [&amp;_&gt;_*]:min-w-0\">\n<h2 class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Analyzing Heat Distribution in Your Microwave Burrito<\/h2>\n<\/div>\n<\/div>\n<p>You just got back to your apartment after a long day of classes and grab a frozen burrito from the freezer. As you heat it in the microwave, you notice a familiar problem: some parts come out scorching hot while others remain frozen. Your roommate, who is taking an electromagnetic fields course, mentions that microwaves generate an energy field inside the appliance that can be modeled mathematically. Curious, you decide to analyze the energy flux through the burrito&#8217;s surface to understand why it heats so unevenly. The burrito can be approximated as a cylinder, and the microwave&#8217;s energy field in the region around your food can be modeled using vector fields. By applying the divergence theorem, you can relate the energy absorption inside the burrito to the flux across its surface, helping explain why that first bite is always a gamble between lava and ice.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317073\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317073&theme=lumen&iframe_resize_id=ohm317073&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317074\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317074&theme=lumen&iframe_resize_id=ohm317074&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317075\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317075&theme=lumen&iframe_resize_id=ohm317075&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317076\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317076&theme=lumen&iframe_resize_id=ohm317076&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","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-795","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\/795","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":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/795\/revisions"}],"predecessor-version":[{"id":2809,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/795\/revisions\/2809"}],"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\/795\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=795"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=795"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=795"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=795"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}