Jacobian determinant


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Jacobian determinant

[jə′kō·bē·ən di′tər·mə·nənt]
(mathematics)
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By applying the geometric property of the transformed rectangular region and the Cauchy-Riemann equations, the derivatives and Jacobian determinant of the numerical conformal mapping method developed by Tsay and Hsu [16] can be evaluated on the boundaries, and more accurate results are obtained.
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.
The Jacobian determinants for the forward transformation and inverse transformation are defined, respectively, as below:
With the help of Chain rule and these Jacobian determinants, the relationships between the partial derivatives of x-y and [xi]-[eta] can be built as follows:
The Derivatives and Jacobian Determinants on the Boundaries
The numerical boundary derivatives and Jacobian determinants were examined with their analytical values.
Finally, the analytical Jacobian determinants can be derived using above equations and shown as below:
x) is related to the Jacobian determinant and is chosen as 1 initially.
In 1953, Samuelson refocused his attention on the Jacobian determinant of the [a.