jacobian determinant in a sentence

is the Jacobian determinant, which has the geometric interpretation of the deformation in volume from the infinitesimal cube d " x " d " y " d " z " to the infinitesimal curved volume in the orthogonal coordinates.
In order to correct for the volume changes due to the registration, the gray matter volume ( GMV ) in the original brain can be calculated by multiplying the GMD with the Jacobian determinants of the deformations used to register the brain to the template.
Restricting to changes of coordinates with positive Jacobian determinant is possible on orientable manifolds, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle of " n "-forms are distinct.
What I'm trying to prove is that if F is a continuous function of x, y, and z on R, x = a ( u, v, w ), y = b ( u, v, w ), z = c ( u, v, w ), which form a transformation from R to S, and the Jacobian determinant of x, y, and z with respect to u, v, and w exists and doesn't change sign on R, then:

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