{"id":154704,"date":"2022-03-25T23:49:52","date_gmt":"2022-03-25T18:19:52","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/divergence-theory-proof-of-the-theorem\/"},"modified":"2022-12-31T15:48:34","modified_gmt":"2022-12-31T10:18:34","slug":"divergence-theory-proof-of-the-theorem","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/divergence-theory\/","title":{"rendered":"Divergence Theory \u2013 Proof of the Theorem"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/divergence-theory\/#What_Do_We_Understand_By_Gausss_Divergence_Theorem\" title=\"What Do We Understand By Gauss&#8217;s Divergence Theorem?\">What Do We Understand By Gauss&#8217;s Divergence Theorem?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/divergence-theory\/#State_and_Prove_the_Gausss_Divergence_Theorem\" title=\"State and Prove the Gauss&#8217;s Divergence Theorem\">State and Prove the Gauss&#8217;s Divergence Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/divergence-theory\/#The_Divergence_Theorem_Proof\" title=\"The Divergence Theorem Proof\">The Divergence Theorem Proof<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/divergence-theory\/#Gausss_Divergence_Theorem_History\" title=\"Gauss&#8217;s Divergence Theorem History\">Gauss&#8217;s Divergence Theorem History<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_Do_We_Understand_By_Gausss_Divergence_Theorem\"><\/span>What Do We Understand By Gauss&#8217;s Divergence Theorem?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Gauss divergence theorem states that the divergence of a vector field is the line integral of the divergence of the vector field over a closed curve.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-154703 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/divergence-theory-proof-of-the-theorem.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/divergence-theory-proof-of-the-theorem.jpg?v=1648232386 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/divergence-theory-proof-of-the-theorem-300x212.jpg?v=1648232386 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"State_and_Prove_the_Gausss_Divergence_Theorem\"><\/span>State and Prove the Gauss&#8217;s Divergence Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Gauss divergence theorem states that the divergence of a vector field is the line integral of the field&#8217;s curl over a curve.<\/p>\n<p>The proof of the theorem is as follows.<\/p>\n<p>Let be a vector field, a curve, and a point in the plane.<\/p>\n<p>We first show that the divergence of is zero.<\/p>\n<p>Since is a vector field, we can compute its divergence at any point in the plane by taking the divergence of its component vectors.<\/p>\n<p>For any point in the plane,<\/p>\n<p>We now show that the line integral of the curl of over is zero.<\/p>\n<p>We first note that the curl of is the derivative of with respect to :<\/p>\n<p>For any point in the plane,<\/p>\n<p>We now compute the line integral of over :<\/p>\n<p>We can rewrite this line integral as a double integral over :<\/p>\n<p>We now use the fact that is a vector field to split the integral into two parts:<\/p>\n<p>We now compute the two integrals:<\/p>\n<p>Since both integrals are zero, the line integral of the curl of over is also zero.<\/p>\n<p>We have thus shown that the divergence of is zero and the line integral of the curl of over is zero.<\/p>\n<p>Hence, the Gauss divergence theorem is proven.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"The_Divergence_Theorem_Proof\"><\/span>The Divergence Theorem Proof<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The divergence theorem states that the net divergence of a vector field is zero. This theorem can be proven using the following steps:<\/p>\n<p>1. Let F be a vector field.<\/p>\n<p>2. Let D be the divergence of F.<\/p>\n<p>3. Show that D is a vector field.<\/p>\n<p>4. Show that the net divergence of D is zero.<\/p>\n<p>5. Conclude that the divergence theorem is true.<\/p>\n<p>To show that D is a vector field, we need to show that it has both a magnitude and a direction. We can do this by showing that it satisfies the following properties:<\/p>\n<p>1. The magnitude of D is the sum of the magnitudes of the individual components of D.<\/p>\n<p>2. The direction of D is the direction of the vector sum of the individual components of D.<\/p>\n<p>We can show that D satisfies both of these properties by taking the dot product of D with itself. The dot product of two vectors is the sum of the products of the corresponding components of the vectors.<\/p>\n<p>Since the magnitude of D is the sum of the magnitudes of the individual components of D, and the direction of D is the direction of the vector sum of the individual components of D, it follows that D is a vector field.<\/p>\n<p>Now we need to show that the net divergence of D is zero. To do this, we need to show that the divergence of the sum of the individual components of D is zero<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Gausss_Divergence_Theorem_History\"><\/span>Gauss&#8217;s Divergence Theorem History<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The divergence theorem was first formulated by Carl Friedrich Gauss in 1813.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What Do We Understand By Gauss&#8217;s Divergence Theorem? The Gauss divergence theorem states that the divergence of a vector field [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Divergence Theory \u2013 Proof of the Theorem","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about Divergence Theory topic of Maths in details explained by subject experts on infinitylearn.com.","custom_permalink":"maths\/divergence-theory\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Divergence Theory \u2013 Proof of the Theorem - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Learn about Divergence Theory topic of Maths in details explained by subject experts on infinitylearn.com.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/divergence-theory\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Divergence Theory \u2013 Proof of the Theorem - 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