{"id":234444,"date":"2025-06-14T06:23:53","date_gmt":"2025-06-14T06:23:53","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=234444"},"modified":"2025-06-14T06:23:55","modified_gmt":"2025-06-14T06:23:55","slug":"find-the-number-of-independent-components-of-an-antisymmetric-tensor-of-rank-2-indimensions","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/14\/find-the-number-of-independent-components-of-an-antisymmetric-tensor-of-rank-2-indimensions\/","title":{"rendered":"Find the number of independent components of an antisymmetric tensor of rank 2 indimensions"},"content":{"rendered":"\n<p>Find the number of independent components of an antisymmetric tensor of rank 2 in<br>dimensions<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the <strong>number of independent components<\/strong> of an <strong>antisymmetric tensor of rank 2<\/strong> in <strong>n dimensions<\/strong>, we use the following principle:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Correct Answer:<\/strong><\/h3>\n\n\n\n<p>The number of independent components of an antisymmetric tensor of rank 2 in nnn dimensions isn(n\u22121)2\\frac{n(n-1)}{2}2n(n\u22121)\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation:<\/strong><\/h3>\n\n\n\n<p>An <strong>antisymmetric tensor of rank 2<\/strong>, often written as A\u03bc\u03bdA^{\\mu\\nu}A\u03bc\u03bd, is a tensor for which the components satisfy the condition:A\u03bc\u03bd=\u2212A\u03bd\u03bcA^{\\mu\\nu} = -A^{\\nu\\mu}A\u03bc\u03bd=\u2212A\u03bd\u03bc<\/p>\n\n\n\n<p>This means that the component flips sign when its two indices are exchanged.<\/p>\n\n\n\n<p>Additionally, the diagonal components must satisfy:A\u03bc\u03bc=\u2212A\u03bc\u03bc\u21d22A\u03bc\u03bc=0\u21d2A\u03bc\u03bc=0A^{\\mu\\mu} = -A^{\\mu\\mu} \\Rightarrow 2A^{\\mu\\mu} = 0 \\Rightarrow A^{\\mu\\mu} = 0A\u03bc\u03bc=\u2212A\u03bc\u03bc\u21d22A\u03bc\u03bc=0\u21d2A\u03bc\u03bc=0<\/p>\n\n\n\n<p>Hence, <strong>all diagonal elements are zero<\/strong>.<\/p>\n\n\n\n<p>This antisymmetry significantly reduces the number of independent components compared to a general (non-symmetric) rank 2 tensor, which would have n2n^2n2 components in nnn dimensions.<\/p>\n\n\n\n<p>To count how many independent components remain:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>We only consider <strong>pairs of indices (\u03bc,\u03bd)(\\mu, \\nu)(\u03bc,\u03bd)<\/strong> such that \u03bc&lt;\u03bd\\mu &lt; \\nu\u03bc&lt;\u03bd, because the components with \u03bc>\u03bd\\mu > \\nu\u03bc>\u03bd are determined by antisymmetry: A\u03bc\u03bd=\u2212A\u03bd\u03bcA^{\\mu\\nu} = -A^{\\nu\\mu}A\u03bc\u03bd=\u2212A\u03bd\u03bc.<\/li>\n\n\n\n<li>The number of such unordered pairs (\u03bc,\u03bd)(\\mu, \\nu)(\u03bc,\u03bd) with \u03bc&lt;\u03bd\\mu &lt; \\nu\u03bc&lt;\u03bd from nnn items is given by the <strong>binomial coefficient<\/strong>: (n2)=n(n\u22121)2\\binom{n}{2} = \\frac{n(n-1)}{2}(2n\u200b)=2n(n\u22121)\u200b<\/li>\n<\/ul>\n\n\n\n<p>This formula gives the exact count of <strong>independent elements<\/strong> because each independent component corresponds to one unique pair of indices (\u03bc,\u03bd)(\\mu, \\nu)(\u03bc,\u03bd) with \u03bc&lt;\u03bd\\mu &lt; \\nu\u03bc&lt;\u03bd.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Examples:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>In <strong>3 dimensions<\/strong>: 3(3\u22121)2=62=3independent\u00a0components\\frac{3(3-1)}{2} = \\frac{6}{2} = 3 \\quad \\text{independent components}23(3\u22121)\u200b=26\u200b=3independent\u00a0components<\/li>\n\n\n\n<li>In <strong>4 dimensions<\/strong>: 4(4\u22121)2=122=6\\frac{4(4-1)}{2} = \\frac{12}{2} = 624(4\u22121)\u200b=212\u200b=6<\/li>\n\n\n\n<li>In <strong>5 dimensions<\/strong>: 5(5\u22121)2=202=10\\frac{5(5-1)}{2} = \\frac{20}{2} = 1025(5\u22121)\u200b=220\u200b=10<\/li>\n<\/ul>\n\n\n\n<p>Thus, for any nnn-dimensional space, the antisymmetric rank-2 tensor has n(n\u22121)2\\frac{n(n-1)}{2}2n(n\u22121)\u200b independent components.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-410.jpeg\" alt=\"\" class=\"wp-image-234445\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the number of independent components of an antisymmetric tensor of rank 2 indimensions The Correct Answer and Explanation is: To find the number of independent components of an antisymmetric tensor of rank 2 in n dimensions, we use the following principle: Correct Answer: The number of independent components of an antisymmetric tensor of rank [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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