Abstract
This chapter is devoted to vector analysis and to elements of field theory. It provides applications of the machinery of integral calculus, developed above, to physics, mathematics and mechanics.
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Notes
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On this subject the famous American physicist and mathematician R. Feynman (1918–1988) writes, with his characteristic acerbity, “From a long view of the history of mankind – seen from, say, ten thousand years from now – there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.” Richard R. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics: Mainly Electromagnetism and Matter, Addison-Wesley, Reading, MA, 1964.
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J.C. Maxwell (1831–1879) – outstanding Scottish physicist; he created the mathematical theory of the electromagnetic field and is also famous for his research in the kinetic theory of gases, optics and mechanics.
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W.R. Hamilton (1805–1865) – famous Irish mathematician and specialist in mechanics; he stated the variational principle (Hamilton’s principle) and constructed a phenomenological theory of optic phenomena; he was the creator of the theory of quaternions and the founder of vector analysis (in fact, the term “vector” is due to him).
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P.S. Laplace (1749–1827) – famous French astronomer, mathematician, and physicist; he made fundamental contributions to the development of celestial mechanics, the mathematical theory of probability, and experimental and mathematical physics.
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In the triorthogonal system (14.26) we have \(|{\boldsymbol{\xi}}_{i}|=\sqrt{E_{i}}=H_{i}\), \(i=1,2,3\). The quantities \(H_{1}\), \(H_{2}\), \(H_{3}\) are usually called the Lamé’ coefficient or Lamé’ parameters. G. Lamé (1795–1870) French engineer, mathematician, and physicist.
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Ch.O. Coulomb (1736–1806) – French physicist. He discovered experimentally the law (Coulomb’s law) of interaction of charges and magnetic fields using a torsion balance that he invented himself.
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Orienting arrows are shown along certain curves in Fig. 14.4. These arrows will be used a little later; for the time being the reader should not pay any attention to them.
- 9.
S.D. Poisson (1781–1849) – French scientist, specializing in mechanics and physics; his main work was on theoretical and celestial mechanics, mathematical physics, and probability theory. The Poisson equation arose in his research into gravitational potential and attraction by spheroids.
- 10.
H.L.F. Helmholtz (1821–1894) – German physicist and mathematician; one of the first to discover the general law of conservation of energy. Actually, he was the first to make a clear distinction between the concepts of force and energy.
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Zorich, V.A. (2016). Elements of Vector Analysis and Field Theory. In: Mathematical Analysis II. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48993-2_6
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DOI: https://doi.org/10.1007/978-3-662-48993-2_6
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