## Conditions for A Transformation of Coordinates to Be Well Defined

Suppose we have coordinate systems
$S(x,y,z)$
and
$S'(u,v,w)$
.
What are the conditions for a the transformation from
$S$
to
$S'$
to be well defined?
Each of the coordinates in
$S'$
is a function of the coordinates in

so
$(u,v,w)=(u(x,y,z), y(u,v,w), z(u,v,w))$
.
The Jacobian matrix is
$\left| \begin{array}{ccc} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{array} \right| = \frac{\partial (u,v,w)}{\partial (x,y,z)}$

The determinant of this matrix cannot be zero for a well defined transformation. This is equivalent to saying the transformation is one to one and onto,,, or that the gradient of any of the functions
$u, v, w$
is never zero. The matrix is then invertible and the inverse transformation, from
$S'$
to
$S$
is well defined.