Derivation of Jacobian formula with Dirac delta function. (10th March 2021)

• Record Type: Journal Article
• Authors: Kim, Dohyun; Ee, June-Haak; Yu, Chaehyun; Lee, Jungil
• Is Part Of: European journal of physics. Volume 42:Number 3(2021)
• Publication Date: 2021-03-10
• Subjects: Jacobian -- Dirac delta function -- coordinate transformation -- chain rule of partial derivatives; Physics -- Periodicals; Physics -- Study and teaching (Higher) -- Europe -- Periodicals; 530
• DOI: 10.1088/1361-6404/abdca9
• Languages: English
• ISSNs: 0143-0807

Abstract

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.