{"id":5112,"date":"2014-04-15T06:05:51","date_gmt":"2014-04-14T21:05:51","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5112"},"modified":"2014-08-01T18:38:07","modified_gmt":"2014-08-01T09:38:07","slug":"gradient-divergence-curl-and-laplacian-in-orthogonal-curvilinear-coordinates","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5112","title":{"rendered":"Gradient, divergence, curl and Laplacian in orthogonal curvilinear coordinates"},"content":{"rendered":"<div class=\"theContentWrap-ccc\"><blockquote>\n<p>If <img src='https:\/\/s0.wp.com\/latex.php?latex=%5CPhi&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\Phi' title='\\Phi' class='latex' \/> is a scalar function and <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7BA%7D+%3D+A_1%5Cbold%7Be_1%7D+%2B+A_2%5Cbold%7Be_2%7D+%2B+A_3%5Cbold%7Be_3%7D+&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{A} = A_1\\bold{e_1} + A_2\\bold{e_2} + A_3\\bold{e_3} ' title='\\bold{A} = A_1\\bold{e_1} + A_2\\bold{e_2} + A_3\\bold{e_3} ' class='latex' \/> a vector function of orthogonal curvilinear coordinates <img src='https:\/\/s0.wp.com\/latex.php?latex=u_1&#038;bg=T&#038;fg=000000&#038;s=0' alt='u_1' title='u_1' class='latex' \/>, <img src='https:\/\/s0.wp.com\/latex.php?latex=u_2&#038;bg=T&#038;fg=000000&#038;s=0' alt='u_2' title='u_2' class='latex' \/>, <img src='https:\/\/s0.wp.com\/latex.php?latex=u_3&#038;bg=T&#038;fg=000000&#038;s=0' alt='u_3' title='u_3' class='latex' \/>, we have the following results. <\/p>\n<h4>1. <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cnabla%5CPhi+%3D+grad%5CPhi+%3D+%5Cfrac%7B1%7D%7Bh_1%7D%5Cfrac%7B%5Cpartial%5CPhi%7D%7B%5Cpartial+u_1%7D%5Cbold%7Be_1%7D+%2B+%5Cfrac%7B1%7D%7Bh_2%7D%5Cfrac%7B%5Cpartial%5CPhi%7D%7B%5Cpartial+u_2%7D%5Cbold%7Be_2%7D+%2B+%5Cfrac%7B1%7D%7Bh_3%7D%5Cfrac%7B%5Cpartial%5CPhi%7D%7B%5Cpartial+u_3%7D%5Cbold%7Be_3%7D+&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\nabla\\Phi = grad\\Phi = \\frac{1}{h_1}\\frac{\\partial\\Phi}{\\partial u_1}\\bold{e_1} + \\frac{1}{h_2}\\frac{\\partial\\Phi}{\\partial u_2}\\bold{e_2} + \\frac{1}{h_3}\\frac{\\partial\\Phi}{\\partial u_3}\\bold{e_3} ' title='\\displaystyle \\nabla\\Phi = grad\\Phi = \\frac{1}{h_1}\\frac{\\partial\\Phi}{\\partial u_1}\\bold{e_1} + \\frac{1}{h_2}\\frac{\\partial\\Phi}{\\partial u_2}\\bold{e_2} + \\frac{1}{h_3}\\frac{\\partial\\Phi}{\\partial u_3}\\bold{e_3} ' class='latex' \/><\/h4>\n<h4>2. <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cnabla%5Ccdot%5Cbold%7BA%7D+%3D+div%5Cbold%7BA%7D+%3D+%5Cfrac%7B1%7D%7Bh_1h_2h_3%7D%5Cleft%5B+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+u_1%7D%28h_2h_3A_1%29+%2B+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+u_2%7D%28h_3h_1A_2%29+%2B+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+u_3%7D%28h_1h_2A_3%29+%5Cright%5D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\nabla\\cdot\\bold{A} = div\\bold{A} = \\frac{1}{h_1h_2h_3}\\left[ \\frac{\\partial}{\\partial u_1}(h_2h_3A_1) + \\frac{\\partial}{\\partial u_2}(h_3h_1A_2) + \\frac{\\partial}{\\partial u_3}(h_1h_2A_3) \\right]' title='\\displaystyle \\nabla\\cdot\\bold{A} = div\\bold{A} = \\frac{1}{h_1h_2h_3}\\left[ \\frac{\\partial}{\\partial u_1}(h_2h_3A_1) + \\frac{\\partial}{\\partial u_2}(h_3h_1A_2) + \\frac{\\partial}{\\partial u_3}(h_1h_2A_3) \\right]' class='latex' \/><\/h4>\n<h4>3. <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cnabla%5Ctimes%5Cbold%7BA%7D+%3D+curl%5Cbold%7BA%7D++++%3D+%5Cfrac%7B1%7D%7Bh_1h_2h_3%7D%5Cleft%7C+%5Cbegin%7Barray%7D%7Bccc%7D+h_1%5Cbold%7Be_1%7D+%26+h_2%5Cbold%7Be_2%7D+%26+h_3%5Cbold%7Be_3%7D+%5C%5C+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+u_1%7D+%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+u_2%7D+%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+u_3%7D+%5C%5C+h_1A_1+%26+h_2A_2+%26+h_3A_3+%5Cend%7Barray%7D+%5Cright%7C&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\nabla\\times\\bold{A} = curl\\bold{A}    = \\frac{1}{h_1h_2h_3}\\left| \\begin{array}{ccc} h_1\\bold{e_1} &amp; h_2\\bold{e_2} &amp; h_3\\bold{e_3} \\\\ \\frac{\\partial}{\\partial u_1} &amp; \\frac{\\partial}{\\partial u_2} &amp; \\frac{\\partial}{\\partial u_3} \\\\ h_1A_1 &amp; h_2A_2 &amp; h_3A_3 \\end{array} \\right|' title='\\displaystyle \\nabla\\times\\bold{A} = curl\\bold{A}    = \\frac{1}{h_1h_2h_3}\\left| \\begin{array}{ccc} h_1\\bold{e_1} &amp; h_2\\bold{e_2} &amp; h_3\\bold{e_3} \\\\ \\frac{\\partial}{\\partial u_1} &amp; \\frac{\\partial}{\\partial u_2} &amp; \\frac{\\partial}{\\partial u_3} \\\\ h_1A_1 &amp; h_2A_2 &amp; h_3A_3 \\end{array} \\right|' class='latex' \/><\/h4>\n<h4>4. <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cnabla%5E2%5CPhi+%3D+Laplacian%5C+of%5C+%5CPhi%5C%5C++++%3D+%5Cfrac%7B1%7D%7Bh_1h_2h_3%7D%5Cleft%5B+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+u_1%7D%5Cleft%28+%5Cfrac%7Bh_2h_3%7D%7Bh_1%7D%5Cfrac%7B%5Cpartial%5CPhi%7D%7B%5Cpartial+u_1%7D+%5Cright%29+%2B+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+u_2%7D%5Cleft%28+%5Cfrac%7Bh_3h_1%7D%7Bh_2%7D%5Cfrac%7B%5Cpartial%5CPhi%7D%7B%5Cpartial+u_2%7D+%5Cright%29+%2B+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+u_3%7D%5Cleft%28+%5Cfrac%7Bh_1h_2%7D%7Bh_3%7D%5Cfrac%7B%5Cpartial%5CPhi%7D%7B%5Cpartial+u_3%7D+%5Cright%29+%5Cright%5D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\nabla^2\\Phi = Laplacian\\ of\\ \\Phi\\\\    = \\frac{1}{h_1h_2h_3}\\left[ \\frac{\\partial}{\\partial u_1}\\left( \\frac{h_2h_3}{h_1}\\frac{\\partial\\Phi}{\\partial u_1} \\right) + \\frac{\\partial}{\\partial u_2}\\left( \\frac{h_3h_1}{h_2}\\frac{\\partial\\Phi}{\\partial u_2} \\right) + \\frac{\\partial}{\\partial u_3}\\left( \\frac{h_1h_2}{h_3}\\frac{\\partial\\Phi}{\\partial u_3} \\right) \\right]' title='\\displaystyle \\nabla^2\\Phi = Laplacian\\ of\\ \\Phi\\\\    = \\frac{1}{h_1h_2h_3}\\left[ \\frac{\\partial}{\\partial u_1}\\left( \\frac{h_2h_3}{h_1}\\frac{\\partial\\Phi}{\\partial u_1} \\right) + \\frac{\\partial}{\\partial u_2}\\left( \\frac{h_3h_1}{h_2}\\frac{\\partial\\Phi}{\\partial u_2} \\right) + \\frac{\\partial}{\\partial u_3}\\left( \\frac{h_1h_2}{h_3}\\frac{\\partial\\Phi}{\\partial u_3} \\right) \\right]' class='latex' \/><\/h4>\n<p>These reduce to the usual expressions in rectangular coordinates if we replace <img src='https:\/\/s0.wp.com\/latex.php?latex=%28u_1%2C+u_2%2C+u_3%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='(u_1, u_2, u_3)' title='(u_1, u_2, u_3)' class='latex' \/> by <img src='https:\/\/s0.wp.com\/latex.php?latex=%28x%2C+y%2C+z%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='(x, y, z)' title='(x, y, z)' class='latex' \/>, in which case <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7Be_1%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{e_1}' title='\\bold{e_1}' class='latex' \/>, <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7Be_2%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{e_2}' title='\\bold{e_2}' class='latex' \/> and <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7Be_3%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{e_3}' title='\\bold{e_3}' class='latex' \/> are replaced by <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7Bi%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{i}' title='\\bold{i}' class='latex' \/>, <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7Bj%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{j}' title='\\bold{j}' class='latex' \/> and <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7Bk%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{k}' title='\\bold{k}' class='latex' \/> and <img src='https:\/\/s0.wp.com\/latex.php?latex=h_1+%3D+h_2+%3D+h_3+%3D+1&#038;bg=T&#038;fg=000000&#038;s=0' alt='h_1 = h_2 = h_3 = 1' title='h_1 = h_2 = h_3 = 1' 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>If is a scalar function and a vector function of orthogonal curvilinear coordinates , , , we have the followin &hellip; <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5112\" class=\"more-link\"><span class=\"screen-reader-text\">&#8220;Gradient, divergence, curl and Laplacian in orthogonal curvilinear coordinates&#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,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-5112","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.0.1 - aioseo.com -->\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"admin\"\/>\n\t<meta name=\"keywords\" content=\"gradient,divergence,curl,laplacian,orthogonal curvilinear coordinates\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5112\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.0.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"ja_JP\" \/>\n\t\t<meta 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