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If $\\Phi$ is a scalar function and $\\bold{A} = A_1\\bold{e_1} + A_2\\bold{e_2} + A_3\\bold{e_3}$ a vector function of orthogonal curvilinear coordinates $u_1$ , $u_2$ , $u_3$ , we have the following results. <\/p>\n
1. $\\displaystyle \\nabla\\Phi = grad\\Phi = \\frac{1}{h_1}\\frac{\\partial\\Phi}{\\partial u_1}\\bold{e_1} + \\frac{1}{h_2}\\frac{\\partial\\Phi}{\\partial u_2}\\bold{e_2} + \\frac{1}{h_3}\\frac{\\partial\\Phi}{\\partial u_3}\\bold{e_3}$ <\/h4>\n
2. $\\displaystyle \\nabla\\cdot\\bold{A} = div\\bold{A} = \\frac{1}{h_1h_2h_3}\\left[ \\frac{\\partial}{\\partial u_1}(h_2h_3A_1) + \\frac{\\partial}{\\partial u_2}(h_3h_1A_2) + \\frac{\\partial}{\\partial u_3}(h_1h_2A_3) \\right]$ <\/h4>\n
3. $\\displaystyle \\nabla\\times\\bold{A} = curl\\bold{A} = \\frac{1}{h_1h_2h_3}\\left| \\begin{array}{ccc} h_1\\bold{e_1} & h_2\\bold{e_2} & h_3\\bold{e_3} \\\\ \\frac{\\partial}{\\partial u_1} & \\frac{\\partial}{\\partial u_2} & \\frac{\\partial}{\\partial u_3} \\\\ h_1A_1 & h_2A_2 & h_3A_3 \\end{array} \\right|$ <\/h4>\n
4. $\\displaystyle \\nabla^2\\Phi = Laplacian\\ of\\ \\Phi\\\\ = \\frac{1}{h_1h_2h_3}\\left[ \\frac{\\partial}{\\partial u_1}\\left( \\frac{h_2h_3}{h_1}\\frac{\\partial\\Phi}{\\partial u_1} \\right) + \\frac{\\partial}{\\partial u_2}\\left( \\frac{h_3h_1}{h_2}\\frac{\\partial\\Phi}{\\partial u_2} \\right) + \\frac{\\partial}{\\partial u_3}\\left( \\frac{h_1h_2}{h_3}\\frac{\\partial\\Phi}{\\partial u_3} \\right) \\right]$ <\/h4>\n
These reduce to the usual expressions in rectangular coordinates if we replace $(u_1, u_2, u_3)$ by $(x, y, z)$ , in which case $\\bold{e_1}$ , $\\bold{e_2}$ and $\\bold{e_3}$ are replaced by $\\bold{i}$ , $\\bold{j}$ and $\\bold{k}$ and $h_1 = h_2 = h_3 = 1$ . <\/p>\n<\/blockquote>\n
<\/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 … <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5112\" class=\"more-link\"><span class=\"screen-reader-text\">“Gradient, divergence, curl and Laplacian in orthogonal curvilinear coordinates” \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":{"_crdt_document":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-5112","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5112","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5112"}],"version-history":[{"count":10,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5112\/revisions"}],"predecessor-version":[{"id":5357,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5112\/revisions\/5357"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/media\/6040"}],"wp:attachment":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5112"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5112"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5112"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

1. <\/h4>\n

2. <\/h4>\n

3. <\/h4>\n

1. $\\displaystyle \\nabla\\Phi = grad\\Phi = \\frac{1}{h_1}\\frac{\\partial\\Phi}{\\partial u_1}\\bold{e_1} + \\frac{1}{h_2}\\frac{\\partial\\Phi}{\\partial u_2}\\bold{e_2} + \\frac{1}{h_3}\\frac{\\partial\\Phi}{\\partial u_3}\\bold{e_3}$ <\/h4>\n

2. $\\displaystyle \\nabla\\cdot\\bold{A} = div\\bold{A} = \\frac{1}{h_1h_2h_3}\\left[ \\frac{\\partial}{\\partial u_1}(h_2h_3A_1) + \\frac{\\partial}{\\partial u_2}(h_3h_1A_2) + \\frac{\\partial}{\\partial u_3}(h_1h_2A_3) \\right]$ <\/h4>\n