Conditions for A Transformation of Coordinates to Be Well Defined

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

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