Orthogonal curvilinear coordinates. Jacobians

The transformation equations

$x = f(u_1, u_2, u_3)\ y = g(u_1, u_2, u_3)\ z = h(u_1, u_2, u_3)\cdots(17)$

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

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

A point $P$ can be defined not only by rectangular coordinates $(x, y, z)$ but by coordinates $(u_1, u_2, u_3)$ as well. We call $(u_1, u_2, u_3)$ the curvilinear coordinates of the point.

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. Similarly we define the $u_2$ and $u_3$ coordinate curves through $P$ .

From (18), we have

$\displaystyle d\bold{r} = \frac{\partial\bold{r}}{\partial u_1}du_1 + \frac{\partial\bold{r}}{\partial u_2}du_2 + \frac{\partial\bold{r}}{\partial u_3}du_3 \cdots (19)$

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

$d\bold{r} = h_1du\bold{e_1} + h_2du\bold{e_2} + h_3du\bold{e_3}\cdots(20)$

The quantities $h_1$ , $h_2$ , $h_3$ are sometimes called scale factors.

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

$ds^2 = d\bold{r} \cdot d\bold{r} = h_1^2du_1^2 + h_2^2du_2^2 + h_3^2du_3^2 \cdots(21)$

and corresponds to the square of the length of the diagonal in the above parallelepiped.

Also, in the case of orthogonal coordinates the volume of the parallelepiped is given by

$dV = |(h_1du_1\bold{e_1}) \cdot (h_2du_2\bold{e_2}) \times (h_3du_3\bold{e_3})| = h_1h_2h_3du_1du_2du_3 \cdots (22)$

which can be written by

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

where

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

is called the Jacobian of the transformation.

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.