Jacobians

    With a function of one variable we often use a change of variable (a substitution) to simplify an integral. By reversing the roles of and an integral can be rewritten,



where and This factor is crucial for determining whether a change of variable will be successful, or not. In order to have a change of variables for functions of two or more variables, a factor which is defined in terms of partial derivatives will be needed.

Definition (Jacobian) If and then the Jacobian of and with respect to and denoted by is  



More generally, for



the Jacobian of with respect to is  

Example (Jacobian: Rectangular to Polar Coordinates) Compute the Jacobian for the conversion from the rectangular plane to the polar plane.

    Solution. The conversion formulas are and So the Jacobian is,
    

Example (Jacobian: Rectangular to Cylindrical Coordinates) Compute the Jacobian for the conversion from rectangular coordinates to cylindrical coordinates.

    Solution.  The conversion formulas are and So the Jacobian is,
    

Example (Jacobian: Rectangular to Spherical Coordinates) Compute the Jacobian for the conversion from rectangular coordinates to spherical coordinates.

    Solution. The conversion formulas are and So the Jacobian is,
    

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Jacobians
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/jacobians.html