Proof of Cramer's rule with Dirac delta function. (30th September 2020)

• Record Type: Journal Article
• Authors: Ee, June-Haak; Lee, Jungil; Yu, Chaehyun
• Is Part Of: European journal of physics. Volume 41:Number 6(2020:Nov.)
• Publication Date: 2020-09-30
• Subjects: Cramer's rule -- Dirac delta function -- generalized coordinates; Physics -- Periodicals; Physics -- Study and teaching (Higher) -- Europe -- Periodicals; 530
• DOI: 10.1088/1361-6404/aba455
• Languages: English
• ISSNs: 0143-0807

Abstract

Abstract: We present a new proof of Cramer's rule by interpreting a system of linear equations as a transformation of n -dimensional Cartesian-coordinate vectors. To find the solution, we carry out the inverse transformation by convolving the original coordinate vector with Dirac delta functions and changing integration variables from the original coordinates to new coordinates. As a byproduct, we derive a generalized version of Cramer's rule that applies to a partial set of variables, which is new to the best of our knowledge. Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized coordinates of a mechanical system.