{"id":13707,"date":"2021-04-06T20:25:04","date_gmt":"2021-04-06T14:55:04","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=13707"},"modified":"2021-07-25T20:04:12","modified_gmt":"2021-07-25T14:34:12","slug":"application-of-vector-differentiation","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=13707","title":{"rendered":"APPLICATION OF VECTOR DIFFERENTIATION"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\  \\overrightarrow{F}={F_1}\\overrightarrow{i} + {F_2}\\overrightarrow{j} + {F_3}\\overrightarrow{k}\\] <script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<p class=\"has-text-align-justify\">is a vector function, defined and differentiable at each point (x, y, z)in a certain region of space [i.e., A defines a vector field], then the divergence of &nbsp;(abbreviated as &#8216;Div &#8216;) is defined as,&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Div\\ \\overrightarrow{F} = \\nabla\\ . \\overrightarrow{F}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = (\\overrightarrow{i}\\frac{\\partial}{\\partial\\ x} + \\overrightarrow{j}\\frac{\\partial}{\\partial\\ y} + \\overrightarrow{k}\\frac{\\partial}{\\partial\\ z})\\ . \\ ({F_1}\\overrightarrow{i} + {F_2}\\overrightarrow{j} + {F_3}\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = (\\frac{\\partial {F_1}}{\\partial\\ x} + \\frac{\\partial {F_2}}{\\partial\\ y} + \\frac{\\partial {F_3}}{\\partial\\ z})\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- sidebar ad 1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:326px;height:280px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6703350399\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<h3 class=\"wp-block-heading\">Basic properties of Divergence:<\/h3>\n\n\n\n<p>If A, B are vector functions and \u2018f\u2019 is a scalar function, then<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1) \\nabla\\ . ( A + B) = \\nabla\\ .A + \\nabla\\ .B\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2) \\nabla\\ . ( fA) = (\\nabla\\ f) . A + (f.\\nabla\\ A)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3)\\overrightarrow{F}\\ is\\ solenoidal\\ if\\ \\nabla\\ . \\overrightarrow{F}=0\\] <\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"2812384453\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p> <strong>Example:<\/strong>   <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\  \\overrightarrow{F}=xyz\\overrightarrow{i} + 3x^2y\\overrightarrow{j} + (xy^2 &#8211; zy^3)\\overrightarrow{k}, then\\ find\\ div\\ \\overrightarrow{F}\\] <\/div>\n\n\n\n<p> Soln:    <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\  \\overrightarrow{F}=xyz\\overrightarrow{i} + 3x^2y\\overrightarrow{j} + (xy^2 &#8211; zy^3)\\overrightarrow{k}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = \\frac{\\partial {F_1}}{\\partial\\ x} + \\frac{\\partial {F_2}}{\\partial\\ y} + \\frac{\\partial {F_3}}{\\partial\\ z}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{\\partial }{\\partial\\ x} {(xyz)}+ \\frac{\\partial}{\\partial\\ y} {(3x^2y)}+ \\frac{\\partial}{\\partial\\ z} {(xy^2-zy^3)}\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp; yz&nbsp; + 3x<sup>2<\/sup> ( 1 )&nbsp; + ( 0 &#8211; y<sup>3<\/sup> )<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp; yz&nbsp; + 3x<sup>2<\/sup> &#8211; y<sup>3<\/sup><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"fluid\" data-ad-layout-key=\"-6t+ed+2i-1n-4w\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9770958327\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p> <strong>Example:<\/strong> <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\  \\overrightarrow{F}=x^2y\\overrightarrow{i} + xy^2z\\overrightarrow{j} + xyyz\\overrightarrow{k}, then\\ find\\ div\\ \\overrightarrow{F}\\ at\\ the\\ point\\ (1,-1,2)\\] <\/div>\n\n\n\n<p> Soln:  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\  \\overrightarrow{F}=x^2y\\overrightarrow{i} + xy^2z\\overrightarrow{j} + xyyz\\overrightarrow{k}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = \\frac{\\partial {F_1}}{\\partial\\ x} + \\frac{\\partial {F_2}}{\\partial\\ y} + \\frac{\\partial {F_3}}{\\partial\\ z}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{\\partial }{\\partial\\ x} {(x^2y)}+ \\frac{\\partial}{\\partial\\ y} {(xy^2z)}+ \\frac{\\partial}{\\partial\\ z} {(xyyz)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = 2xy  + 2xyz  + xy\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">At ( 1, -1, 2)<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = 2(1)(-1)  + 2(1)(-1)(2)  + (1)(-1)\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp;&nbsp; -2&nbsp; &#8211; 4&nbsp; &#8211; 1<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp; -7<\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3040040190\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p>  <strong>Example:<\/strong>    <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Show\\ that\\  \\overrightarrow{F}=3y^4z^2\\overrightarrow{i} + 4x^3z^2\\overrightarrow{j} + 6x^2y^3\\overrightarrow{k} is\\ solenoidal\\] <\/div>\n\n\n\n<p> Soln:   <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ \\overrightarrow{F}=3y^4z^2\\overrightarrow{i} + 4x^3z^2\\overrightarrow{j} + 6x^2y^3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = \\frac{\\partial {F_1}}{\\partial\\ x} + \\frac{\\partial {F_2}}{\\partial\\ y} + \\frac{\\partial {F_3}}{\\partial\\ z}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{\\partial }{\\partial\\ x} {(3y^4z^2)}+ \\frac{\\partial}{\\partial\\ y} {(4x^3z^2)}+ \\frac{\\partial}{\\partial\\ z} {(6x^2y^3)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = 0+ 0  + 0 = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}\\ is\\ solenoidal\\] <\/div>\n\n\n\n<p> <strong>Example:<\/strong> <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Show\\ that\\  \\overrightarrow{F}=(x + 3y)\\overrightarrow{i} + (x &#8211; 3z)\\overrightarrow{j} + (x -z)\\overrightarrow{k} is\\ solenoidal\\] <\/div>\n\n\n\n<p> Soln:   <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ \\overrightarrow{F}=(x + 3y)\\overrightarrow{i} + (x &#8211; 3z)\\overrightarrow{j} + (x -z)\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = \\frac{\\partial {F_1}}{\\partial\\ x} + \\frac{\\partial {F_2}}{\\partial\\ y} + \\frac{\\partial {F_3}}{\\partial\\ z}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{\\partial }{\\partial\\ x} {(x + 3y)}+ \\frac{\\partial}{\\partial\\ y} {(x &#8211; 3z)}+ \\frac{\\partial}{\\partial\\ z} {(x -z)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = (1+ 0) + (0 -0) + (0 -1) = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}\\ is\\ solenoidal\\] <\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p>   <strong>Example:<\/strong> <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\  \\overrightarrow{F}=2xy\\overrightarrow{i} + 3x^2y\\overrightarrow{j} &#8211; 3pyz\\overrightarrow{k} is\\ solenoidal\\ at\\ (1,1,1)\\, find\\ &#8216;p&#8217;\\] <\/div>\n\n\n\n<p> Soln:  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ \\overrightarrow{F}=2xy\\overrightarrow{i} + 3x^2y\\overrightarrow{j} &#8211; 3pyz\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = \\frac{\\partial {F_1}}{\\partial\\ x} + \\frac{\\partial {F_2}}{\\partial\\ y} + \\frac{\\partial {F_3}}{\\partial\\ z}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{\\partial }{\\partial\\ x} {(2xy)}+ \\frac{\\partial}{\\partial\\ y} {(3x^2y)}- \\frac{\\partial}{\\partial\\ z} {(3pyz)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = 2y  + 3x^2- 3py\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> At ( 1, 1, 1) <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = 2(1)  + 3(1)^2- 3p(1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ . \\overrightarrow{F} = 2+ 3 &#8211; 3p\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ \\overrightarrow{F}\\ is\\ solenoidal\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i.e\\ \\nabla\\ . \\overrightarrow{F} =0\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">2&nbsp; + 3 \u2013 3p =&nbsp;&nbsp; 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">5 \u2013 3p = 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">3p&nbsp; = 5<\/p>\n\n\n\n<p class=\"has-text-align-center\">p&nbsp; =&nbsp; 5\/3<\/p>\n\n\n\n<!-- admitad.banner: 9a1p2gsrb0109003a68856637026d8 Redmagic WW -->\n<a target=\"_blank\" rel=\"nofollow noopener\" href=\"https:\/\/ad.admitad.com\/g\/9a1p2gsrb0109003a68856637026d8\/?i=4\"><img fetchpriority=\"high\" decoding=\"async\" width=\"300\" height=\"250\" border=\"0\" src=\"https:\/\/ad.admitad.com\/b\/9a1p2gsrb0109003a68856637026d8\/\" alt=\"Redmagic WW\"><\/a>\n<!-- \/admitad.banner -->\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Curl of a vector function:<\/strong><\/h2>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\  \\overrightarrow{F}={F_1}\\overrightarrow{i} + {F_2}\\overrightarrow{j} + {F_3}\\overrightarrow{k}\\] <\/div>\n\n\n\n<p class=\"has-text-align-center\">is a vector function,<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[then\\ curl\\ \\overrightarrow{F}=\\nabla\\ \u00d7 \\overrightarrow{F}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[curl\\ \\overrightarrow{F} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n\\frac{\\partial }{\\partial\\ x} &amp; \\frac{\\partial }{\\partial\\ y} &amp; \\frac{\\partial }{\\partial\\ z}\\\\\n{F_1} &amp; {F_2} &amp; {F_3}\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Irrotational vector&nbsp; :&nbsp;<\/strong><\/h4>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ vector\\ \\overrightarrow{F}\\ is\\ said\\ to\\ be\\ irrotational\\ if\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[curl\\ \\overrightarrow{F} =0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i.e\\ \\nabla\\ \u00d7 \\overrightarrow{F} =0\\]<\/div>\n\n\n\n<a target=\"_blank\" href=\"https:\/\/shareasale.com\/r.cfm?b=1483247&amp;u=2863840&amp;m=16058&amp;urllink=&amp;afftrack=\" rel=\"noopener\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/static.shareasale.com\/image\/16058\/300X250_06.gif?ssl=1\" border=\"0\" alt=\"Puzzle Box 06\"><\/a>\n\n\n\n<p> <strong>Example:<\/strong>  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\  \\overrightarrow{F}=xyz\\overrightarrow{i} + 3x^2y\\overrightarrow{j} +(x y ^2- zy^3)\\overrightarrow{k}\\ then\\ find\\ curl\\ \\overrightarrow{F}\\]<\/div>\n\n\n\n<p> Soln: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ \\overrightarrow{F}=xyz\\overrightarrow{i} + 3x^2y\\overrightarrow{j} + (x y ^2- zy^3)\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \u00d7 \\overrightarrow{F} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n\\frac{\\partial }{\\partial\\ x} &amp; \\frac{\\partial }{\\partial\\ y} &amp; \\frac{\\partial }{\\partial\\ z}\\\\\n{xyz} &amp; {3x^2y} &amp; {x y ^2- zy^3}\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( \\frac{\\partial}{\\partial\\ y} {(x y ^2- zy^3)} &#8211; \\frac{\\partial}{\\partial\\ z} {(3x^2y)})- \\overrightarrow{j}( \\frac{\\partial}{\\partial\\ x} {(x y ^2- zy^3)} &#8211; \\frac{\\partial}{\\partial\\ z} {(xyz)}) + \\overrightarrow{k}( \\frac{\\partial}{\\partial\\ x} {(3x^2y)} &#8211; \\frac{\\partial}{\\partial\\ y} {(xyz)})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 2xy &#8211; 3zy^2  ) -\\overrightarrow{j}((y^2 &#8211; 0) &#8211; xy)+\\overrightarrow{k}(6xy &#8211; xz))\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \u00d7 \\overrightarrow{F} = [2xy &#8211; 3zy^2] \\overrightarrow{i} &#8211; [y^2 &#8211; xy] \\overrightarrow{j} + [ 6xy &#8211; xz] \\overrightarrow{k}\\]<\/div>\n\n\n\n<!-- admitad.banner: pl7xom3as7109003a688a663530cb9 Ajio [CPS] IN -->\n<a target=\"_blank\" rel=\"nofollow noopener\" href=\"https:\/\/ad.admitad.com\/g\/pl7xom3as7109003a688a663530cb9\/?i=4&amp;subid=https%3A%2F%2Fad.admitad.com%2Fg%2Fgobb106sd9109003a688a663530cb9%2F%3Fsubid%3Dhttps%253A%252F%252Fad.admitad.com%252Fg%252Fgobb106sd9109003a688a663530cb9%252F%253Fsubid%253Dhttps%25253A%25252F%25252Fad.admitad.com%25252Fg%25252Fgobb106sd9109003a688a663530cb9%25252F\"><img decoding=\"async\" width=\"120\" height=\"600\" border=\"0\" src=\"https:\/\/ad.admitad.com\/b\/pl7xom3as7109003a688a663530cb9\/\" alt=\"Ajio [CPS] IN\"><\/a>\n<!-- \/admitad.banner -->\n\n\n\n<p>         <strong>Example:<\/strong>   <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Show\\ that\\  \\overrightarrow{F}= x\\overrightarrow{i} +y^2\\overrightarrow{j} +z^3\\overrightarrow{k}\\ is\\ irrotational\\]<\/div>\n\n\n\n<p>   Soln:  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ \\overrightarrow{F}=x\\overrightarrow{i} +y^2\\overrightarrow{j} +z^3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \u00d7 \\overrightarrow{F} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n\\frac{\\partial }{\\partial\\ x} &amp;  \\frac{\\partial }{\\partial\\ y} &amp; \\frac{\\partial }{\\partial\\ z}\\\\\n{x} &amp; y^2 &amp; z^3\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( \\frac{\\partial}{\\partial\\ y} {(z^3)} &#8211; \\frac{\\partial}{\\partial\\ z} {(y^2)})- \\overrightarrow{j}( \\frac{\\partial}{\\partial\\ x} {(z^3)} &#8211; \\frac{\\partial}{\\partial\\ z} {(x)}) + \\overrightarrow{k}( \\frac{\\partial}{\\partial\\ x} {(y^2)} &#8211; \\frac{\\partial}{\\partial\\ y} {(x)})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}[ 0- 0 ] -\\overrightarrow{j}[0 &#8211; 0]+\\overrightarrow{k}[0- 0]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \u00d7 \\overrightarrow{F} = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{F}\\ is\\ irrotational\\]<\/div>\n\n\n\n<iframe width=\"300\" height=\"250\" frameborder=\"0\" scrolling=\"no\" src=\"https:\/\/widget.cuelinks.com\/widgets\/63246?cid=138248\"><\/iframe>\n\n\n\n<p> <strong>Example:<\/strong> <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\  \\overrightarrow{F}=( x^2 &#8211; y^2 + 2xz)\\overrightarrow{i} + (xz &#8211; xy + yz)\\overrightarrow{j} + (z^2 + x^2)\\overrightarrow{k}\\  find\\  \\nabla\\ \u00d7 ( \\nabla\\ \u00d7 \\overrightarrow{F})\\]<\/div>\n\n\n\n<p> Soln: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ \\overrightarrow{F}=( x^2 &#8211; y^2 + 2xz)\\overrightarrow{i} + (xz &#8211; xy + yz)\\overrightarrow{j} + (z^2 + x^2)\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \u00d7 \\overrightarrow{F} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n\\frac{\\partial }{\\partial\\ x} &amp;  \\frac{\\partial }{\\partial\\ y} &amp; \\frac{\\partial }{\\partial\\ z}\\\\\nx^2 &#8211; y^2 + 2xz &amp; xz &#8211; xy + yz  &amp;  z^2 + x^2\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}[ \\frac{\\partial}{\\partial\\ y} {( z^2 + y^2 )} &#8211; \\frac{\\partial}{\\partial\\ z} {(xz &#8211; xy + yz)}]- \\overrightarrow{j}[ \\frac{\\partial}{\\partial\\ x} {(z^2 + y^2)} &#8211; \\frac{\\partial}{\\partial\\ z} {(x^2 &#8211; y^2 + 2xz)}] + \\overrightarrow{k}[ \\frac{\\partial}{\\partial\\ x} {(xz &#8211; xy + yz )} &#8211; \\frac{\\partial}{\\partial\\ y} {(x^2 &#8211; y^2 + 2xz)}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}[ (0 + 0) &#8211; (x &#8211; 0 +y)  ] -\\overrightarrow{j}[(0 + 0) \u2013 ( 0 &#8211; 0 + 2x)]+\\overrightarrow{k}[ (z &#8211; y + 0) \u2013 ( 0 &#8211; 2y + 0)]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \u00d7 \\overrightarrow{F} = \\overrightarrow{i} (-x &#8211; y ) &#8211; \\overrightarrow{j}(-2x) + \\overrightarrow{k}(z +2y)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \u00d7 \\overrightarrow{F} =- (x + y)\\overrightarrow{i} + 2x\\overrightarrow{j} + ( z + 2y)\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \u00d7 ( \\nabla\\ \u00d7 \\overrightarrow{F}) =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n\\frac{\\partial }{\\partial\\ x} &amp; \\frac{\\partial }{\\partial\\ y} &amp; \\frac{\\partial }{\\partial\\ z}\\\\\n-(x + y) &amp;  2x  &amp;  z + 2y \\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}[\\frac{\\partial}{\\partial\\ y} {(z + 2y)} &#8211; \\frac{\\partial}{\\partial\\ z} {(2x)}]- \\overrightarrow{j}[\\frac{\\partial}{\\partial\\ x} {(z + 2y )} &#8211; \\frac{\\partial}{\\partial\\ z} {-(x + y)}] + \\overrightarrow{k}[\\frac{\\partial}{\\partial\\ x} {(2x)} &#8211; \\frac{\\partial}{\\partial\\ y} {-(x + y)}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}[ (0 + 2) &#8211; (0)] -\\overrightarrow{j}[( 0 + 0) + ( 0 + 0)]+\\overrightarrow{k}[(2) + ( 0  +1 )]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}[ 2] -\\overrightarrow{j}[0]+\\overrightarrow{k}[3]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \u00d7 ( \\nabla\\ \u00d7 \\overrightarrow{F}) = 2\\overrightarrow{i} + 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<!-- admitad.banner: 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