{"id":5025,"date":"2014-03-26T06:05:54","date_gmt":"2014-03-25T21:05:54","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5025"},"modified":"2014-08-01T18:42:36","modified_gmt":"2014-08-01T09:42:36","slug":"gradient-divergence-and-curl","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5025","title":{"rendered":"Gradient, divergence and curl"},"content":{"rendered":"<div class=\"theContentWrap-ccc\"><blockquote>\n<p>Consider the vector operator <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cnabla%5C+%28del%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\nabla\\ (del)' title='\\nabla\\ (del)' class='latex' \/> defined by<\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cnabla+%5Cequiv+%5Cbold%7Bi%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%7D+%2B+%5Cbold%7Bj%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+y%7D+%2B+%5Cbold%7Bk%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+z%7D%5Ccdots%2813%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\nabla \\equiv \\bold{i}\\frac{\\partial}{\\partial x} + \\bold{j}\\frac{\\partial}{\\partial y} + \\bold{k}\\frac{\\partial}{\\partial z}\\cdots(13)' title='\\displaystyle \\nabla \\equiv \\bold{i}\\frac{\\partial}{\\partial x} + \\bold{j}\\frac{\\partial}{\\partial y} + \\bold{k}\\frac{\\partial}{\\partial z}\\cdots(13)' class='latex' \/><\/p>\n<p>Then if <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cphi%28x%2C+y%2C+z%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\phi(x, y, z)' title='\\phi(x, y, z)' class='latex' \/> and <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7BA%7D%28x%2C+y%2C+z%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{A}(x, y, z)' title='\\bold{A}(x, y, z)' class='latex' \/> have continuous first partial derivatives in a region (a condition which is in many cases stronger than necessary), we can define the following. <\/p>\n<h4>1. Gradient<\/h4>\n<p>The <em>gradient<\/em> of &phi; is defined by<\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+grad%5Cphi+%3D+%5Cnabla%5Cphi+%3D+%5Cleft%28%5Cbold%7Bi%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%7D+%2B+%5Cbold%7Bj%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+y%7D+%2B+%5Cbold%7Bk%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+z%7D%5Cright%29%5Cphi%5C%5C++++%3D+%5Cbold%7Bi%7D%5Cfrac%7B%5Cpartial%5Cphi%7D%7B%5Cpartial+x%7D+%2B+%5Cbold%7Bj%7D%5Cfrac%7B%5Cpartial%5Cphi%7D%7B%5Cpartial+y%7D+%2B+%5Cbold%7Bk%7D%5Cfrac%7B%5Cpartial%5Cphi%7D%7B%5Cpartial+z%7D%5C%5C++++%3D+%5Cfrac%7B%5Cpartial%5Cphi%7D%7B%5Cpartial+x%7D%5Cbold%7Bi%7D+%2B+%5Cfrac%7B%5Cpartial%5Cphi%7D%7B%5Cpartial+y%7D%5Cbold%7Bj%7D+%2B+%5Cfrac%7B%5Cpartial%5Cphi%7D%7B%5Cpartial+z%7D%5Cbold%7Bk%7D%5Ccdots%2814%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle grad\\phi = \\nabla\\phi = \\left(\\bold{i}\\frac{\\partial}{\\partial x} + \\bold{j}\\frac{\\partial}{\\partial y} + \\bold{k}\\frac{\\partial}{\\partial z}\\right)\\phi\\\\    = \\bold{i}\\frac{\\partial\\phi}{\\partial x} + \\bold{j}\\frac{\\partial\\phi}{\\partial y} + \\bold{k}\\frac{\\partial\\phi}{\\partial z}\\\\    = \\frac{\\partial\\phi}{\\partial x}\\bold{i} + \\frac{\\partial\\phi}{\\partial y}\\bold{j} + \\frac{\\partial\\phi}{\\partial z}\\bold{k}\\cdots(14)' title='\\displaystyle grad\\phi = \\nabla\\phi = \\left(\\bold{i}\\frac{\\partial}{\\partial x} + \\bold{j}\\frac{\\partial}{\\partial y} + \\bold{k}\\frac{\\partial}{\\partial z}\\right)\\phi\\\\    = \\bold{i}\\frac{\\partial\\phi}{\\partial x} + \\bold{j}\\frac{\\partial\\phi}{\\partial y} + \\bold{k}\\frac{\\partial\\phi}{\\partial z}\\\\    = \\frac{\\partial\\phi}{\\partial x}\\bold{i} + \\frac{\\partial\\phi}{\\partial y}\\bold{j} + \\frac{\\partial\\phi}{\\partial z}\\bold{k}\\cdots(14)' class='latex' \/><\/p>\n<p>An interesting interpretation is that if <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cphi%28x%2C+y%2C+z%29+%3D+c&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\phi(x, y, z) = c' title='\\phi(x, y, z) = c' class='latex' \/> is the equation of a surface, then <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cnabla%5Cphi&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\nabla\\phi' title='\\nabla\\phi' class='latex' \/> is a normal to this surface. <\/p>\n<h4>2. Divergence<\/h4>\n<p>The <em>divergence<\/em> of <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7BA%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{A}' title='\\bold{A}' class='latex' \/> is defined by<\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+div%5Cbold%7BA%7D+%3D+%5Cnabla%5Ccdot%5Cbold%7BA%7D+%3D+%5Cleft%28%5Cbold%7Bi%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%7D+%2B+%5Cbold%7Bj%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+y%7D+%2B+%5Cbold%7Bk%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+z%7D%5Cright%29%5Ccdot%28A_1%5Cbold%7Bi%7D+%2B+A_2%5Cbold%7Bj%7D+%2B+A_3%5Cbold%7Bk%7D%29%5C%5C+%3D+%5Cfrac%7B%5Cpartial+A_1%7D%7B%5Cpartial+x%7D+%2B+%5Cfrac%7B%5Cpartial+A_2%7D%7B%5Cpartial+y%7D+%2B+%5Cfrac%7B%5Cpartial+A_3%7D%7B%5Cpartial+z%7D%5Ccdots%2815%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle div\\bold{A} = \\nabla\\cdot\\bold{A} = \\left(\\bold{i}\\frac{\\partial}{\\partial x} + \\bold{j}\\frac{\\partial}{\\partial y} + \\bold{k}\\frac{\\partial}{\\partial z}\\right)\\cdot(A_1\\bold{i} + A_2\\bold{j} + A_3\\bold{k})\\\\ = \\frac{\\partial A_1}{\\partial x} + \\frac{\\partial A_2}{\\partial y} + \\frac{\\partial A_3}{\\partial z}\\cdots(15)' title='\\displaystyle div\\bold{A} = \\nabla\\cdot\\bold{A} = \\left(\\bold{i}\\frac{\\partial}{\\partial x} + \\bold{j}\\frac{\\partial}{\\partial y} + \\bold{k}\\frac{\\partial}{\\partial z}\\right)\\cdot(A_1\\bold{i} + A_2\\bold{j} + A_3\\bold{k})\\\\ = \\frac{\\partial A_1}{\\partial x} + \\frac{\\partial A_2}{\\partial y} + \\frac{\\partial A_3}{\\partial z}\\cdots(15)' class='latex' \/><\/p>\n<h4>3. Curl<\/h4>\n<p>The <em>curl<\/em> of <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7BA%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{A}' title='\\bold{A}' class='latex' \/> is defined by <\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+curl%5Cbold%7BA%7D+%3D+%5Cnabla%5Ctimes%5Cbold%7BA%7D+%3D+%5Cleft%28%5Cbold%7Bi%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%7D+%2B+%5Cbold%7Bj%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+y%7D+%2B+%5Cbold%7Bk%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+z%7D%5Cright%29%5Ctimes%28A_1%5Cbold%7Bi%7D+%2B+A_2%5Cbold%7Bj%7D+%2B+A_3%5Cbold%7Bk%7D%29%5C%5C++++%3D+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D+%5Cbold%7Bi%7D+%26+%5Cbold%7Bj%7D+%26+%5Cbold%7Bk%7D+%5C%5C+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%7D+%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+y%7D+%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+z%7D+%5C%5C+A_1+%26+A_2+%26+A_3+%5Cend%7Barray%7D%5Cright%7C+%5C%5C+++%3D+%5Cbold%7Bi%7D%5Cleft%7C%5Cbegin%7Barray%7D%7Bcc%7D+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+y%7D+%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+z%7D+%5C%5C+A_2+%26+A_3+%5Cend%7Barray%7D%5Cright%7C+-+%5Cbold%7Bj%7D%5Cleft%7C%5Cbegin%7Barray%7D%7Bcc%7D+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+z%7D+%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+z%7D+%5C%5C+A_1+%26+A_3+%5Cend%7Barray%7D%5Cright%7C+%2B+%5Cbold%7Bk%7D%5Cleft%7C%5Cbegin%7Barray%7D%7Bcc%7D+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%7D+%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+y%7D+%5C%5C+A_1+%26+A_2+%5Cend%7Barray%7D%5Cright%7C%5C%5C+++%3D+%5Cleft%28%5Cfrac%7B%5Cpartial+A_3%7D%7B%5Cpartial+y%7D+-+%5Cfrac%7B%5Cpartial+A_2%7D%7B%5Cpartial+z%7D%5Cright%29%5Cbold%7Bi%7D+%2B+%5Cleft%28%5Cfrac%7B%5Cpartial+A_1%7D%7B%5Cpartial+z%7D+-+%5Cfrac%7B%5Cpartial+A_3%7D%7B%5Cpartial+x%7D%5Cright%29%5Cbold%7Bj%7D+%2B+%5Cleft%28%5Cfrac%7B%5Cpartial+A_2%7D%7B%5Cpartial+x%7D+-+%5Cfrac%7B%5Cpartial+A_1%7D%7B%5Cpartial+y%7D%5Cright%29%5Cbold%7Bk%7D%5Ccdots%2816%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle curl\\bold{A} = \\nabla\\times\\bold{A} = \\left(\\bold{i}\\frac{\\partial}{\\partial x} + \\bold{j}\\frac{\\partial}{\\partial y} + \\bold{k}\\frac{\\partial}{\\partial z}\\right)\\times(A_1\\bold{i} + A_2\\bold{j} + A_3\\bold{k})\\\\    = \\left|\\begin{array}{ccc} \\bold{i} &amp; \\bold{j} &amp; \\bold{k} \\\\ \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ A_1 &amp; A_2 &amp; A_3 \\end{array}\\right| \\\\   = \\bold{i}\\left|\\begin{array}{cc} \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ A_2 &amp; A_3 \\end{array}\\right| - \\bold{j}\\left|\\begin{array}{cc} \\frac{\\partial}{\\partial z} &amp; \\frac{\\partial}{\\partial z} \\\\ A_1 &amp; A_3 \\end{array}\\right| + \\bold{k}\\left|\\begin{array}{cc} \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} \\\\ A_1 &amp; A_2 \\end{array}\\right|\\\\   = \\left(\\frac{\\partial A_3}{\\partial y} - \\frac{\\partial A_2}{\\partial z}\\right)\\bold{i} + \\left(\\frac{\\partial A_1}{\\partial z} - \\frac{\\partial A_3}{\\partial x}\\right)\\bold{j} + \\left(\\frac{\\partial A_2}{\\partial x} - \\frac{\\partial A_1}{\\partial y}\\right)\\bold{k}\\cdots(16)' title='\\displaystyle curl\\bold{A} = \\nabla\\times\\bold{A} = \\left(\\bold{i}\\frac{\\partial}{\\partial x} + \\bold{j}\\frac{\\partial}{\\partial y} + \\bold{k}\\frac{\\partial}{\\partial z}\\right)\\times(A_1\\bold{i} + A_2\\bold{j} + A_3\\bold{k})\\\\    = \\left|\\begin{array}{ccc} \\bold{i} &amp; \\bold{j} &amp; \\bold{k} \\\\ \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ A_1 &amp; A_2 &amp; A_3 \\end{array}\\right| \\\\   = \\bold{i}\\left|\\begin{array}{cc} \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ A_2 &amp; A_3 \\end{array}\\right| - \\bold{j}\\left|\\begin{array}{cc} \\frac{\\partial}{\\partial z} &amp; \\frac{\\partial}{\\partial z} \\\\ A_1 &amp; A_3 \\end{array}\\right| + \\bold{k}\\left|\\begin{array}{cc} \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} \\\\ A_1 &amp; A_2 \\end{array}\\right|\\\\   = \\left(\\frac{\\partial A_3}{\\partial y} - \\frac{\\partial A_2}{\\partial z}\\right)\\bold{i} + \\left(\\frac{\\partial A_1}{\\partial z} - \\frac{\\partial A_3}{\\partial x}\\right)\\bold{j} + \\left(\\frac{\\partial A_2}{\\partial x} - \\frac{\\partial A_1}{\\partial y}\\right)\\bold{k}\\cdots(16)' class='latex' \/><\/p>\n<p>Note that in the expansion of the determinant, the operators <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cpartial%2F%5Cpartial+x&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\partial\/\\partial x' title='\\partial\/\\partial x' class='latex' \/>, <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cpartial%2F%5Cpartial+y&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\partial\/\\partial y' title='\\partial\/\\partial y' class='latex' \/>, <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cpartial%2F%5Cpartial+z&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\partial\/\\partial z' title='\\partial\/\\partial z' class='latex' \/> must precede <img src='https:\/\/s0.wp.com\/latex.php?latex=A_1&#038;bg=T&#038;fg=000000&#038;s=0' alt='A_1' title='A_1' class='latex' \/>, <img src='https:\/\/s0.wp.com\/latex.php?latex=A_2&#038;bg=T&#038;fg=000000&#038;s=0' alt='A_2' title='A_2' class='latex' \/>, <img src='https:\/\/s0.wp.com\/latex.php?latex=A_3&#038;bg=T&#038;fg=000000&#038;s=0' alt='A_3' title='A_3' class='latex' \/>. <\/p>\n<\/blockquote>\n<p><iframe src=\"\/\/rcm-fe.amazon-adsystem.com\/e\/cm?lt1=_blank&#038;bc1=000000&#038;IS2=1&#038;bg1=FFFFFF&#038;fc1=000000&#038;lc1=0000FF&#038;t=fujiitoshiki-22&#038;o=9&#038;p=8&#038;l=as4&#038;m=amazon&#038;f=ifr&#038;ref=ss_til&#038;asins=0071635408\" style=\"width:120px;height:240px;\" scrolling=\"no\" marginwidth=\"0\" marginheight=\"0\" frameborder=\"0\"><\/iframe><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Consider the vector operator defined by Then if and have continuous first partial derivatives in a region (a c &hellip; <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5025\" class=\"more-link\"><span class=\"screen-reader-text\">&#8220;Gradient, divergence and curl&#8221; \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":6040,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-5025","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.9 - aioseo.com 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