THIS well-written book gives the usual definitions, of scalar and vector products, introduces the now familiar differential operators “div” and “rot” (or “curl”), and uses them skilfully in the simpler applications of the line, surface, and volume integrals, associated with the names of Green, Gauss, and Stokes. The necessity for vector analysis in electromagnetic work is becoming more generally recognised, and Dr. Gans deserves the thanks of all for his able presentation of the outlines of the method, which, nevertheless, is at its best a “Quaternionen-stenographie,” as C. Neumann felicitously nicknames it. One has only to compare the demonstrations here given, which are primarily Cartesian and are then transformed into the concise vector notation, with corresponding quaternion demonstrations, such as may be found in Joly's “Manual,” to see plainly the analytical gulf which separates Hamilton's calculus from other vector analyses, which are essentially shorthand notations. The mathematical historian of the future will find much food for thought in the mental shortsightedness of many vector analysts who delight in the use of contraction symbols like grad, rot, div, but despise the Hamiltonian selective symbols V and S, which with the real ∇ give the-whole theory in exquisite compactness and flexibility. On a folding sheet at the end Dr. Gans gives a table of eighteen transformation formulæ, which presumably must all be learned off by rote. There does not seem to be any resemblance among the formulæ (h), (o), (q), which give respectively the equivalents of [A[BC]], rot rot A, rot [AB]. In the quaternion notation VAVBC, V∇V∇A, V∇VAB, they are seen to be of the same “form,” and are, indeed, analytically amenable to the same treatment. This is but one illustration of the inferiority of the “Quaternionenstenographie” to the real quaternion analysis. Dr. Gans gives interesting applications in hydrodynamics and in Maxwell's electromagnetic theory, but is limited somewhat by the fact that in this introduction there is no account taken of the linear vector function or matrix.
Einführung in die Vektoranalysis mit Anwendungen auf die mathematische Physik.
By Dr. Richard Gans. Pp. ix + 226. (Leipzig: Teubner, 1905.) Price 8 marks.
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Einführung in die Vektoranalysis mit Anwendungen auf die mathematische Physik . Nature 72, 483 (1905). https://doi.org/10.1038/072483c0