Suppose the equations map a plane region in the -plane to a region in the -plane. Suppose further that the mapping is invertible with equations that map back to .
Let the Jacobian of the map from to be given by
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Then a double integral over is related to an equivalent double over by the "formula"
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where or , and or .
The Jacobian of the map from to , that is , is the reciprocal of the Jacobian . This relationship can be exploited in cases where inverting the mapping equations is algebraically tedious. In some such cases, obtaining the reciprocal of and expressing it in terms of and might involve simpler manipulations than a direct calculation of .