\[S(u,v,w)\]
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.