{"id":20258,"date":"2024-04-15T05:50:30","date_gmt":"2024-04-15T05:50:30","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20258"},"modified":"2024-04-15T05:50:30","modified_gmt":"2024-04-15T05:50:30","slug":"let-nabla-cdot-left-textfoverrightarrow-textv-right-textx2texty-texty2textz-textz2textx-where-f-and-v-are-scalar-and","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/let-nabla-cdot-left-textfoverrightarrow-textv-right-textx2texty-texty2textz-textz2textx-where-f-and-v-are-scalar-and\/","title":{"rendered":"Let $$\\nabla \\cdot \\left( {{\\text{f}}\\overrightarrow {\\text{v}} } \\right) = {{\\text{x}}^2}{\\text{y}} + {{\\text{y}}^2}{\\text{z}} + {{\\text{z}}^2}{\\text{x}},$$ where f and v are scalar and vector fields respectively. If $$\\overrightarrow {\\text{v}} = {\\text{y}}\\overrightarrow {\\text{i}} + {\\text{z}}\\overrightarrow {\\text{j}} + {\\text{x}}\\overrightarrow {\\text{k}} ,$$ then $$\\overrightarrow {\\text{v}} \\cdot \\nabla {\\text{f}}$$ is A. x2y + y2z + z2x B. 2xy + 2yz + 2zx C. x + y + z D. 0"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;x2y + y2z + z2x&#8221; option2=&#8221;2xy + 2yz + 2zx&#8221; option3=&#8221;x + y + z&#8221; option4=&#8221;0&#8243; correct=&#8221;option3&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{0}$.<\/p>\n<p>The divergence of a product of a scalar field and a vector field is the sum of the product of the scalar field and the divergence of the vector field, plus the vector field times the curl of the scalar field. In this case, the scalar field is $f$ and the vector field is $\\mathbf{v}$. The divergence of $\\mathbf{v}$ is $0$, since $\\mathbf{v}$ is a constant vector field. The curl of $f$ is also $0$, since $f$ is a scalar field. Therefore, the divergence of the product of $f$ and $\\mathbf{v}$ is $0$.<\/p>\n<p>Here is a more detailed explanation of each option:<\/p>\n<ul>\n<li>Option A: $x^2y + y^2z + z^2x$. This is the product of the scalar field $f$ and the vector field $\\mathbf{v}$. However, the divergence of a product of a scalar field and a vector field is the sum of the product of the scalar field and the divergence of the vector field, plus the vector field times the curl of the scalar field. In this case, the scalar field is $f$ and the vector field is $\\mathbf{v}$. The divergence of $\\mathbf{v}$ is $0$, since $\\mathbf{v}$ is a constant vector field. The curl of $f$ is also $0$, since $f$ is a scalar field. Therefore, the divergence of the product of $f$ and $\\mathbf{v}$ is $0$, which is not equal to $x^2y + y^2z + z^2x$.<\/li>\n<li>Option B: $2xy + 2yz + 2zx$. This is the curl of the vector field $\\mathbf{v}$. However, the divergence of a product of a scalar field and a vector field is the sum of the product of the scalar field and the divergence of the vector field, plus the vector field times the curl of the scalar field. In this case, the scalar field is $f$ and the vector field is $\\mathbf{v}$. The divergence of $\\mathbf{v}$ is $0$, since $\\mathbf{v}$ is a constant vector field. The curl of $f$ is also $0$, since $f$ is a scalar field. Therefore, the divergence of the product of $f$ and $\\mathbf{v}$ is $0$, which is not equal to $2xy + 2yz + 2zx$.<\/li>\n<li>Option C: $x + y + z$. This is the sum of the components of the vector field $\\mathbf{v}$. However, the divergence of a product of a scalar field and a vector field is the sum of the product of the scalar field and the divergence of the vector field, plus the vector field times the curl of the scalar field. In this case, the scalar field is $f$ and the vector field is $\\mathbf{v}$. The divergence of $\\mathbf{v}$ is $0$, since $\\mathbf{v}$ is a constant vector field. The curl of $f$ is also $0$, since $f$ is a scalar field. Therefore, the divergence of the product of $f$ and $\\mathbf{v}$ is $0$, which is not equal to $x + y + z$.<\/li>\n<li>Option D: $0$. This is the correct answer. The divergence of a product of a scalar field and a vector field is the sum of the product of the scalar field and the divergence of the vector field, plus the vector field times the curl of the scalar field. In this case, the scalar field is $f$ and the vector field is $\\mathbf{v}$. The divergence of $\\mathbf{v}$ is $0$, since $\\mathbf{v}$ is a constant vector field. The curl of $f$ is also $0$, since $f$ is a scalar field. Therefore, the divergence of the product of $f$ and $\\mathbf{v}$ is $0$.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;x2y + y2z + z2x&#8221; option2=&#8221;2xy + 2yz + 2zx&#8221; option3=&#8221;x + y + z&#8221; option4=&#8221;0&#8243; correct=&#8221;option3&#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":[],"class_list":["post-20258","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"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>Let $$\\nabla \\cdot \\left( {{\\text{f}}\\overrightarrow {\\text{v}} } \\right) = {{\\text{x}}^2}{\\text{y}} + {{\\text{y}}^2}{\\text{z}} + {{\\text{z}}^2}{\\text{x}},$$ where f and v are scalar and vector fields respectively. If $$\\overrightarrow {\\text{v}} = {\\text{y}}\\overrightarrow {\\text{i}} + {\\text{z}}\\overrightarrow {\\text{j}} + {\\text{x}}\\overrightarrow {\\text{k}} ,$$ then $$\\overrightarrow {\\text{v}} \\cdot \\nabla {\\text{f}}$$ is A. x2y + y2z + z2x B. 2xy + 2yz + 2zx C. x + y + z D. 0<\/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\/let-nabla-cdot-left-textfoverrightarrow-textv-right-textx2texty-texty2textz-textz2textx-where-f-and-v-are-scalar-and\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Let $$\\nabla \\cdot \\left( {{\\text{f}}\\overrightarrow {\\text{v}} } \\right) = {{\\text{x}}^2}{\\text{y}} + {{\\text{y}}^2}{\\text{z}} + {{\\text{z}}^2}{\\text{x}},$$ where f and v are scalar and vector fields respectively. If $$\\overrightarrow {\\text{v}} = {\\text{y}}\\overrightarrow {\\text{i}} + {\\text{z}}\\overrightarrow {\\text{j}} + {\\text{x}}\\overrightarrow {\\text{k}} ,$$ then $$\\overrightarrow {\\text{v}} \\cdot \\nabla {\\text{f}}$$ is A. x2y + y2z + z2x B. 2xy + 2yz + 2zx C. x + y + z D. 0\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=&#8221;x2y + y2z + z2x&#8221; option2=&#8221;2xy + 2yz + 2zx&#8221; option3=&#8221;x + y + z&#8221; option4=&#8221;0&#8243; correct=&#8221;option3&#8243;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/let-nabla-cdot-left-textfoverrightarrow-textv-right-textx2texty-texty2textz-textz2textx-where-f-and-v-are-scalar-and\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:50:30+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Let $$\\nabla \\cdot \\left( {{\\text{f}}\\overrightarrow {\\text{v}} } \\right) = {{\\text{x}}^2}{\\text{y}} + {{\\text{y}}^2}{\\text{z}} + {{\\text{z}}^2}{\\text{x}},$$ where f and v are scalar and vector fields respectively. If $$\\overrightarrow {\\text{v}} = {\\text{y}}\\overrightarrow {\\text{i}} + {\\text{z}}\\overrightarrow {\\text{j}} + {\\text{x}}\\overrightarrow {\\text{k}} ,$$ then $$\\overrightarrow {\\text{v}} \\cdot \\nabla {\\text{f}}$$ is A. x2y + y2z + z2x B. 2xy + 2yz + 2zx C. x + y + z D. 0","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/let-nabla-cdot-left-textfoverrightarrow-textv-right-textx2texty-texty2textz-textz2textx-where-f-and-v-are-scalar-and\/","og_locale":"en_US","og_type":"article","og_title":"Let $$\\nabla \\cdot \\left( {{\\text{f}}\\overrightarrow {\\text{v}} } \\right) = {{\\text{x}}^2}{\\text{y}} + {{\\text{y}}^2}{\\text{z}} + {{\\text{z}}^2}{\\text{x}},$$ where f and v are scalar and vector fields respectively. If $$\\overrightarrow {\\text{v}} = {\\text{y}}\\overrightarrow {\\text{i}} + {\\text{z}}\\overrightarrow {\\text{j}} + {\\text{x}}\\overrightarrow {\\text{k}} ,$$ then $$\\overrightarrow {\\text{v}} \\cdot \\nabla {\\text{f}}$$ is A. x2y + y2z + z2x B. 2xy + 2yz + 2zx C. x + y + z D. 0","og_description":"[amp_mcq option1=&#8221;x2y + y2z + z2x&#8221; option2=&#8221;2xy + 2yz + 2zx&#8221; option3=&#8221;x + y + z&#8221; option4=&#8221;0&#8243; correct=&#8221;option3&#8243;]","og_url":"https:\/\/exam.pscnotes.com\/mcq\/let-nabla-cdot-left-textfoverrightarrow-textv-right-textx2texty-texty2textz-textz2textx-where-f-and-v-are-scalar-and\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:50:30+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/let-nabla-cdot-left-textfoverrightarrow-textv-right-textx2texty-texty2textz-textz2textx-where-f-and-v-are-scalar-and\/","url":"https:\/\/exam.pscnotes.com\/mcq\/let-nabla-cdot-left-textfoverrightarrow-textv-right-textx2texty-texty2textz-textz2textx-where-f-and-v-are-scalar-and\/","name":"Let $$\\nabla \\cdot \\left( {{\\text{f}}\\overrightarrow {\\text{v}} } \\right) = {{\\text{x}}^2}{\\text{y}} + {{\\text{y}}^2}{\\text{z}} + {{\\text{z}}^2}{\\text{x}},$$ where f and v are scalar and vector fields respectively. 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