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.
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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.
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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.
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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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