Derivation of Jacobian formula with Dirac delta function

Dohyun Kim, June Haak Ee, Chaehyun Yu, Jungil Lee

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number035006
JournalEuropean Journal of Physics
Volume42
Issue number3
DOIs
Publication statusPublished - 2021 May

Keywords

  • chain rule of partial derivatives
  • coordinate transformation
  • Dirac delta function
  • Jacobian

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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