[amp_mcq option1=”\\[\\oint {{\\text{P}} \\cdot {\\text{d}}l} \\]” option2=”\\[\\oint {\\nabla \\times \\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\]” option3=”\\[\\oint {\\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\]” option4=”\\[\\iiint {\\nabla \\cdot {\\text{Pdv}}}\\]” correct=”option1″]<\/p>\n
The correct answer is $\\boxed{\\text{C}}$.<\/p>\n
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’s boundary.<\/p>\n
The line integral $\\oint {{\\text{P}} \\cdot {\\text{d}}l}$ is the circulation of a vector field around a closed loop.<\/p>\n
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
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
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 if the vector field does not spread out from a point.<\/p>\n
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 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":"
[amp_mcq option1=”\\[\\oint {{\\text{P}} \\cdot {\\text{d}}l} \\]” option2=”\\[\\oint {\\nabla \\times \\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\]” option3=”\\[\\oint {\\nabla \\times {\\text{P}}} \\cdot {\\text{d}}l\\]” option4=”\\[\\iiint {\\nabla \\cdot {\\text{Pdv}}}\\]” correct=”option1″]<\/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":[],"class_list":["post-20177","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"yoast_head":"\n