Back to the roots of vector and tensor calculus: Heaviside versus Gibbs
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Back to the roots of vector and tensor calculus: Heaviside versus Gibbs Alessio Rocci1 Received: 4 June 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In June 1888, Oliver Heaviside received by mail an officially unpublished pamphlet, which was written and printed by the American author Willard J. Gibbs around 1881– 1884. This original document is preserved in the Dibner Library of the History of Science and Technology at the Smithsonian Institute in Washington DC. Heaviside studied Gibbs’s work very carefully and wrote some annotations in the margins of the booklet. He was a strong defender of Gibbs’s work on vector analysis against quaternionists, even if he criticised Gibbs’s notation system. The aim of our paper is to analyse Heaviside’s annotations and to investigate the role played by the American physicist in the development of Heaviside’s work.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 General remarks . . . . . . . . . . . . . . . . . . . . . . 2.1 Proof of authenticity . . . . . . . . . . . . . . . . . . 2.2 Dating the annotations . . . . . . . . . . . . . . . . 2.3 The structure of Gibbs’s work and a unifying notation 3 Heaviside’s annotations on vector algebra . . . . . . . . . 3.1 Comparing notations . . . . . . . . . . . . . . . . . 3.2 On the reciprocal of a vector . . . . . . . . . . . . . 4 Heaviside’s annotations on linear operators . . . . . . . . 4.1 On the ‘indeterminate’ product . . . . . . . . . . . . 4.2 Heaviside’s new triadic . . . . . . . . . . . . . . . . 4.3 On the curl of a tensor . . . . . . . . . . . . . . . . 5 Divergence and Stokes theorems generalised . . . . . . . 5.1 From surface to volume integrals . . . . . . . . . . . 5.2 From line to surface integrals . . . . . . . . . . . . .
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Communicated by Jesper Lützen.
B 1
Alessio Rocci [email protected] Department of Physics and Astronomy “G. Galilei”, University of Padova, via Marzolo 8, 35131 Padua, Italy
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A. Rocci 6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: On Heaviside’s generalisations of the Divergence theorem: some technical details. Appendix B: Back cover transcription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C: Other annotations by Heaviside . . . . . . . . . . .
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