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2D Jacobian

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Let , be a transformation of the - plane into the - plane. Let be a small rectangle with sides parallel to the and axes with side lengths and . The image of this rectangle in the - plane is a curved rectangle . The coordinates of these points are:

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,

,

,

.

These points can be approximated by the points of the parallelogram

,

,

,

.

The area of this parallelogram is the absolute value of

.

The last determinant is called the Jacobian and is usually denoted by .

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Contributed by: Izidor Hafner (October 2014)
Open content licensed under CC BY-NC-SA

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Reference

[1] E. J. Borowski and J. M. Borwein, Collins Dictionary of Mathematics, New York: Collins, 1989 p. 314.

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