{"id":20177,"date":"2024-04-15T05:49:24","date_gmt":"2024-04-15T05:49:24","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20177"},"modified":"2024-04-15T05:49:24","modified_gmt":"2024-04-15T05:49:24","slug":"iint-left-nabla-times-textp-right-cdot-textds-where-p-is-a-vector-is-equal-to-a-oint-textp-cdot-textdl-b-oint-nabla-times-n","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/iint-left-nabla-times-textp-right-cdot-textds-where-p-is-a-vector-is-equal-to-a-oint-textp-cdot-textdl-b-oint-nabla-times-n\/","title":{"rendered":"\\[\\iint {\\left( {\\nabla \\times {\\text{P}}} \\right) \\cdot {\\text{ds,}}}\\] where P is a vector, is equal to A. \\[\\oint {{\\text{P}} \\cdot {\\text{d}}l} \\] B. \\[\\oint {\\nabla \\times \\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\] C. \\[\\oint {\\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\] D. \\[\\iiint {\\nabla \\cdot {\\text{Pdv}}}\\]"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a9b2e2d0ae54\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    &#8221;[oint                <\/div>\r\n                                                                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    &#8221; option2=&#8221;\\[\\oint {\\nabla \\times \\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\]&#8221; option3=&#8221;\\[\\oint {\\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\]&#8221; option4=&#8221;\\[\\iiint {\\nabla \\cdot {\\text{Pdv}}}\\]&#8221; correct=&#8221;option1&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\text{C}}$.<\/p>\n<p>The triple integral $\\iiint {\\nabla \\cdot {\\text{Pdv}}}$ is the divergence theorem, which states that the integral of the divergence of a vector field over a closed surface is equal to the integral of the vector field over the surface&#8217;s boundary.<\/p>\n<p>The line integral $\\oint {{\\text{P}} \\cdot {\\text{d}}l}$ is the circulation of a vector field around a closed loop.<\/p>\n<p>The line integral $\\oint {\\nabla \\times \\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l$ is the curl of a vector field around a closed loop.<\/p>\n<p>The curl of a vector field is a measure of how much the vector field rotates around a point. The curl of a vector field is zero if the vector field does not rotate around a point.<\/p>\n<p>The divergence of a vector field is a measure of how much the vector field spreads out from a point. The divergence of a vector field is zero <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> if the vector field does not spread out from a point.<\/p>\n<p>In this case, the vector field $P$ is not specified, so we cannot determine whether the integral is the divergence theorem, the circulation of a vector field, or the curl of <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> a vector field. However, we can determine that the integral is not the triple integral $\\iiint {\\nabla \\cdot {\\text{Pdv}}}$, because the triple integral is only valid for closed surfaces, and the surface in this case is not closed.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>&#8221; option2=&#8221;\\[\\oint {\\nabla \\times \\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\]&#8221; option3=&#8221;\\[\\oint {\\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\]&#8221; option4=&#8221;\\[\\iiint Subscribe on YouTube {\\nabla \\cdot {\\text{Pdv}}}\\]&#8221; correct=&#8221;option1&#8243;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\\[\\iint {\\left( {\\nabla \\times {\\text{P}}} \\right) \\cdot {\\text{ds,}}}\\] where P is a vector, is equal to A. \\[\\oint {{\\text{P}} \\cdot {\\text{d}}l} \\] B. \\[\\oint {\\nabla \\times \\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\] C. \\[\\oint {\\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\] D. \\[\\iiint {\\nabla \\cdot {\\text{Pdv}}}\\]<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/iint-left-nabla-times-textp-right-cdot-textds-where-p-is-a-vector-is-equal-to-a-oint-textp-cdot-textdl-b-oint-nabla-times-n\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\\[\\iint {\\left( {\\nabla \\times {\\text{P}}} \\right) \\cdot {\\text{ds,}}}\\] where P is a vector, is equal to A. \\[\\oint {{\\text{P}} \\cdot {\\text{d}}l} \\] B. \\[\\oint {\\nabla \\times \\nabla 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