A coordinate transformation is a change from one coordinate system to another. If 
is an open set in 
and 
is a 
diffeomorphism onto its
image, its local linear behavior is described by the Jacobian
matrix
 |
The absolute value of the Jacobian determinant gives the local factor by which volumes change,
so the change of variables theorem for
an integrable function 
has the form
 |
Linear coordinate transformations are linear transformations and include changes of basis, rotations, reflections, and scalings.
