## Transformed Ordinate Functions of a Single Ordinate

Suppose we gave a coordinate system
$S(u,v,w)$
with
$u=u(x),\: v=v(y),\: w=w(z)($

We can take as basis vectors
$\mathbf{e_1} = \frac{\mathbf{\nabla} u}{\left| \mathbf{\nabla} u \right|} \mathbf{i} = \frac{\partial u / \partial x}{\left| \partial u / \partial x \right| } \mathbf{i}$

$\mathbf{e_2} = \frac{\mathbf{\nabla} v}{\left| \mathbf{\nabla} v \right|} \mathbf{j}= \frac{\partial v / \partial y}{\left| \partial v / \partial t \right|} \mathbf{j}$

$\mathbf{e_3} = \frac{\mathbf{\nabla} w}{\left| \mathbf{\nabla} w \right|} \mathbf{k}= \frac{\partial w / \partial z}{\left| \partial w / \partial z \right|} \mathbf{k}$

Then
$\mathbf{e_1} , \: \mathbf{e_2}, \{ \mathbf{e_3}$
form an orthonormal right handed coordinate system. In fact, as long as each of
$u, \: v, \: w$
is a function of only one of
$x, \: y, \:, z$
and each appears only once, the resulting system
$S$
as constructed above is always orthonormal - but not necessarily right handed.
Each of these transformations is in fact only a reflection or rotation, and left handed or right handed depending on whether the permutation of
$xyz$
is even or odd.