Jacobian determinant


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Jacobian determinant

[jə′kō·bē·ən di′tər·mə·nənt]
(mathematics)
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One should notice that the possibility of solving ([bar.z], [bar.w]) is expressed by the nonvanishing of exactly and precisely the same Jacobian determinant as the one expressing Levi nondegeneracy.
The Jacobian determinant of the inverse transformation, from the rectangular region to the irregular region, defined as (7b), is a combination of partial derivatives in the rectangular region.
This problem not only precludes the derivatives and the transformation Jacobian determinants from being evaluated exactly on the boundaries, but also decreases the accuracy of the finite difference computation after the transformation.
In this paper, by applying the rectangular properties of the transformed region and the Cauchy-Riemann equations, the derivatives and the Jacobian determinants of the transformation can be evaluated on the boundaries.
To perform FDM on the grid in the rectangular region, the governing equation usually has to transform into a formulation with Jacobian determinants [18-20].
The Jacobian determinants for the forward transformation and inverse transformation are defined, respectively, as below:
The Jacobian determinant for the quasi-Newton method is only computed once in the first iteration with the linear convergence, while the Jacobian determinant Newton's method presents quadratic convergence.
Then, outcome of coordinates is inserted into the Jacobian determinant [J.sub.kk], in (7) to have the new [[lambda].sub.k](t) that are inserted into (1) and 2) again.
In order to calculate Jacobian determinant in (7), we need to build up relation equations between Cartesian and internal coordinates for constraint equations.
For [phi](x) = x + [epsilon]u(x), the equality J([phi])(x) [congruent to] 1 + div ([epsilon]w(x)) + O([[epsilon].sup.2]) holds for every e> 0 in Q where J([phi])(x) is the Jacobian determinant of [phi](x).
/(x) is related to the Jacobian determinant and is chosen as 1 initially.
The non-vanishing of the Jacobian determinant discussed in equation (7) satisfied that factor intensity condition.