{"id":5121,"date":"2014-04-16T06:05:29","date_gmt":"2014-04-15T21:05:29","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5121"},"modified":"2014-08-01T18:37:23","modified_gmt":"2014-08-01T09:37:23","slug":"special-curvilinear-coordinates","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5121","title":{"rendered":"Special curvilinear coordinates"},"content":{"rendered":"
\n

1. Cylindrical coordinates (\\rho, \\phi, z)<\/h4>\n

Transformation equations<\/em>: x = \\rho\\cos\\phi ,\\ y = \\rho\\sin\\phi ,\\ z = z <\/p>\n

where \\rho \\ge 0 , 0 \\le \\phi \\le 2\\pi, -\\infty < z < \\infty [\/latex]. <em>Scale factors<\/em>: <img src='https:\/\/s0.wp.com\/latex.php?latex&bg=T&fg=000000&s=0' alt='' title='' class='latex' \/>h_1 = 1,\\ h_2 = 1,\\ h_3 = 1<\/p>\n

Element of arc length<\/em>: ds^2 = d\\rho^2 + \\rho^2 d\\phi^2 + dz^2 <\/p>\n

Jacobian<\/em>: \\displaystyle \\frac{\\partial(x, y, z)}{\\partial(\\rho, \\phi, z)} = \\rho<\/p>\n

Element of volume<\/em>: dV = \\rho d\\rho d\\phi dz <\/p>\n

Laplacian<\/em>: \\displaystyle \\nabla^2U = \\frac{1}{\\rho}\\frac{\\partial}{\\partial\\rho}\\left( \\rho\\frac{\\partial U}{\\partial\\rho} \\right) + \\frac{1}{\\rho^2}\\frac{\\partial^2U}{\\partial\\phi^2} + \\frac{\\partial^2U}{\\partial z^2} = \\frac{\\partial^2U}{\\partial\\rho^2} + \\frac{1}{\\rho}\\frac{\\partial U}{\\partial\\rho} + \\frac{1}{\\rho^2}\\frac{\\partial^2U}{\\partial\\phi^2} + \\frac{\\partial^2U}{\\partial z^2} <\/p>\n

Note that corresponding results can be obtained for polar coordinates in the plane by omitting z dependence. In such case for example, ds^2 = d\\rho^2 + \\rho^2d\\phi^2, while the element of volume is replaced by the element of area, dA = \\rho d\\rho d\\phi. <\/p>\n

2. Spherical coordinates (r, \\theta, \\phi)<\/h4>\n

Transformation equations<\/em>: x = r\\sin\\theta,\\ y = r\\sin\\theta\\sin\\phi,\\ z = r\\cos\\theta<\/p>\n

where r \\ge 0,\\ 0 \\le \\theta \\le \\pi,\\ 0 \\le \\phi \\le 2\\pi . <\/p>\n

Scale factors<\/em>: h_1 = 1,\\ h_2 = r,\\ h_3 = r\\sin\\theta <\/p>\n

Element of arc length<\/em>: ds^2 = dr^2 + r^2d\\theta^2 + r^2\\sin^2\\theta d\\phi^2 <\/p>\n

Jacobian<\/em>: \\displaystyle \\frac{\\partial(x, y, z)}{\\partial(r, \\theta, \\phi)} = r^2\\sin\\theta <\/p>\n

Element of volume<\/em>: dV = r^2\\sin\\theta drd\\theta d\\phi <\/p>\n

Laplacian<\/em>: \\displaystyle \\nabla^2U = \\frac{1}{r^2}\\frac{\\partial}{\\partial r}\\left(r^2\\frac{\\partial U}{\\partial r}\\right) + \\frac{1}{r^2\\sin\\theta}\\frac{\\partial}{\\partial\\theta}\\left(\\sin\\theta\\frac{\\partial U}{\\partial\\theta}\\right) + \\frac{1}{r^2\\sin^2\\theta}\\frac{\\partial^2U}{\\partial\\phi^2} <\/p>\n

Other types of coordinate systems are possible. <\/p>\n<\/blockquote>\n