{"id":5093,"date":"2014-04-14T06:05:41","date_gmt":"2014-04-13T21:05:41","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5093"},"modified":"2014-08-01T18:38:52","modified_gmt":"2014-08-01T09:38:52","slug":"orthogonal-curvilinear-coordinates-jacobians","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5093","title":{"rendered":"Orthogonal curvilinear coordinates. Jacobians"},"content":{"rendered":"
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The transformation equations<\/em><\/p>\n

x = f(u_1, u_2, u_3)\\ y = g(u_1, u_2, u_3)\\ z = h(u_1, u_2, u_3)\\cdots(17)<\/p>\n

where we assume that f, g, h are continuous, have continuous partial derivatives and have a single-valued inverse establish a one to one correspondence between points in an xyz u_1u_2u_3 rectangular coordinate system. In vector notation the transformation (17) can be written<\/p>\n

\\bold{r} = x\\bold{i} + y\\bold{j} + z\\bold{k} = f(u_1, u_2, u_3)\\bold{i} + g(u_1, u_2, u_3)\\bold{j} + h(u_1, u_2, u_3)\\bold{k}\\cdots (18)<\/p>\n

A point P can be defined not only by rectangular coordinates<\/em> (x, y, z) but by coordinates (u_1, u_2, u_3) as well. We call (u_1, u_2, u_3) the curvilinear coordinates<\/em> of the point. <\/p>\n

If u_2 and u_3 are constant, then as u_1 varies, \\bold{r} describes a curve which we call the u_1 coordinate curve<\/em>. Similarly we define the u_2 and u_3 coordinate curves through P. <\/p>\n

From (18), we have <\/p>\n

\\displaystyle d\\bold{r} = \\frac{\\partial\\bold{r}}{\\partial u_1}du_1 + \\frac{\\partial\\bold{r}}{\\partial u_2}du_2 + \\frac{\\partial\\bold{r}}{\\partial u_3}du_3 \\cdots (19)<\/p>\n

The vector \\partial\\bold{r}\/\\partial u_1 is tangent to the u_1 coordinate curve at P. If \\bold{e_1} is a unit vector at P in this direction, we can write \\partial \\bold{r} \/ \\partial u_1 = h_1\\bold{e_1} where h_1 = |\\partial\\bold{r}\/\\partial u_1|. Similarly we can write \\partial\\bold{r} \/ \\partial u_2 = h_2\\bold{e_2} and \\partial\\bold{r}\/\\partial u_3 = h_3 \\bold{e_3}, where h_2 = |\\partial\\bold{r}\/\\partial u_2| and h_3 = |\\partial\\bold{r}\/\\partial u_3| respectively. Then (19) can be written <\/p>\n

d\\bold{r} = h_1du\\bold{e_1} + h_2du\\bold{e_2} + h_3du\\bold{e_3}\\cdots(20)<\/p>\n

The quantities h_1, h_2, h_3 are sometimes called scale factors<\/em>. <\/p>\n

If \\bold{e_1}, \\bold{e_2}, \\bold{e_3} are mutually perpendicular at any point P, the curvilinear coordinates are called orthogonal<\/em>. In such case the element of arc length ds is given by <\/p>\n

ds^2 = d\\bold{r} \\cdot d\\bold{r} = h_1^2du_1^2 + h_2^2du_2^2 + h_3^2du_3^2 \\cdots(21)<\/p>\n

and corresponds to the square of the length of the diagonal in the above parallelepiped. <\/p>\n

Also, in the case of orthogonal coordinates the volume of the parallelepiped is given by <\/p>\n

dV = |(h_1du_1\\bold{e_1}) \\cdot (h_2du_2\\bold{e_2}) \\times (h_3du_3\\bold{e_3})| = h_1h_2h_3du_1du_2du_3 \\cdots (22) <\/p>\n

which can be written by <\/p>\n

\\displaystyle dV = \\left| \\frac{\\partial\\bold{r}}{\\partial u_1} \\cdot \\frac{\\partial\\bold{r}}{\\partial u_2} \\times \\frac{\\partial\\bold{r}}{\\partial u_3} \\right| du_1du_2du_3 = \\left| \\frac{\\partial(x, y, z)}{\\partial(u_1, u_2, u_3)} \\right|du_1du_2du_3 \\cdots (23)<\/p>\n

where <\/p>\n

\\displaystyle \\frac{\\partial(x, y, z)}{\\partial(u_1, u_2, u_3)} = \\left| \\begin{array}{ccc} \\frac{\\partial x}{\\partial u_1} & \\frac{\\partial x}{\\partial u_2} & \\frac{\\partial x}{\\partial u_3} \\\\ \\frac{\\partial y}{\\partial u_1} & \\frac{\\partial y}{\\partial u_2} & \\frac{\\partial y}{\\partial u_3} \\\\ \\frac{\\partial z}{\\partial u_1} & \\frac{\\partial z}{\\partial u_2} & \\frac{\\partial z}{\\partial u_3} \\end{array} \\right|\\cdots (24) <\/p>\n

is called the Jacobian<\/em> of the transformation. <\/p>\n

It is clear that when the Jacobian is identically zero there is no parallelepiped. In such case there is a functional relationship between x, y and z, i.e. there is a function \\phi such that \\phi(x, y, z) = 0 identically. <\/p>\n<\/blockquote>\n