Derivation of Jacobian Formula with Dirac Delta Function

TL;DR

This paper introduces a systematic algebraic method using Dirac delta functions to derive Jacobian formulas for coordinate transformations, demonstrated through explicit examples without relying on standard formulas.

Contribution

It presents a novel, recursive approach employing Dirac delta functions for deriving Jacobians in coordinate transformations, offering explicit examples and insights.

Findings

Method successfully derives Jacobians algebraically

Examples demonstrate the approach's effectiveness

Provides alternative to traditional Jacobian calculation methods

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.