The absolute value of the Jacobian determinant at gives us the factor by which the function expands or shrinks volumes near; this is why it occurs in the general substitution rule.
22.
From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates.
23.
In particular, the function has locally in the neighborhood of a point an inverse function that is differentiable if and only if the Jacobian determinant is nonzero at ( see Jacobian conjecture ).
24.
The degree of f at a regular value p \ in B ( 0 ) is defined as the sum of the signs of the Jacobian determinant of f over the preimages of p under f:
25.
It asserts that, if the Jacobian determinant is a non-zero constant ( or, equivalently, that it does not have any complex zero ), then the function is invertible and its inverse is a polynomial function.
26.
While locally the more general transformation law can indeed be used to recognise these tensors, there is a global question that arises, reflecting that in the transformation law one may write either the Jacobian determinant, or its absolute value.
27.
Densities can be generalized into " "'s "-densities "', whose coordinate representations become multiplied by the " s "-th power of the absolute value of the jacobian determinant.
28.
Conversely, if the Jacobian determinant is not zero at a point, then the function is " locally invertible " near this point, that is, there is a neighbourhood of this point in which the function is invertible.
29.
Similarly, under a change of coordinates a differential-form changes by the Jacobian determinant, while a measure changes by the " absolute value " of the Jacobian determinant, } }, which further reflects the issue of orientation.
30.
Similarly, under a change of coordinates a differential-form changes by the Jacobian determinant, while a measure changes by the " absolute value " of the Jacobian determinant, } }, which further reflects the issue of orientation.
