Abstract
We demonstrate how to make the coordinate transformation or change of variables from Cartesian coordinates to curvilinear coordinates by making use of a convolution of a function with Dirac delta functions whose arguments are determined by the transformation functions between the two coordinate systems. By integrating out an original coordinate with a Dirac delta function, we replace the original coordinate with a new coordinate in a systematic way. A recursive use of Dirac delta functions allows the coordinate transformation successively. After replacing every original coordinate into a new curvilinear coordinate, we find that the resultant Jacobian of the corresponding coordinate transformation is automatically obtained in a completely algebraic way. In order to provide insights on this method, we present a few examples of evaluating the Jacobian explicitly without resort to the known general formula.
Highlights
A coordinate transformation or change of variables from a coordinate system to another in multi-dimensional integrals has widely been applied to a variety of fields in mathematics and physics
We present our strategy to derive the Jacobian for a coordinate transformation or change of variables from the Cartesian coordinates xi to the curvilinear coordinates qi with transformation functions qi = qi(x1, · · ·, xn) for i = 1 through n, where n is a positive integer
We have derived the general formula for the Jacobian of the transformation from the n-dimensional Cartesian coordinates to arbitrary curvilinear coordinates by making use of Dirac delta functions, whose arguments correspond to the transformation functions between the two coordinate systems
Summary
A coordinate transformation or change of variables from a coordinate system to another in multi-dimensional integrals has widely been applied to a variety of fields in mathematics and physics. It turns out that the method with a convolution of a coordinate vector with Dirac delta functions provides a systematic way to change integration variables from original coordinates to new coordinates. We exploit a simple concept of integration of the one-dimensional Dirac delta function repeatedly in combination with a purely algebraic manipulation in reorganizing the extra factor by applying chain rules. This intuitive and systematic approach is expected to be pedagogically useful in upper-level mathematics or physics courses in practice of the recursive use of both the Dirac delta function and the chain rule of partial derivatives.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
