\[S(x,y,z)\]

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.