Scientific Papers of J. Willard Gibbs: Dynamics. Vector analysis and multiple algebra. Electromagnetic theory of light, etcLongmans, Green, 1906 - Electromagnetic waves |
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a₁ angles Ausdehnungslehre axes axis B₁ B₂ biscalar bivector called components consider constant coordinates corrections definite denote determined differential coefficients direction displacement dyadic elastic elastic theory electrical theory electromotive force ellipse ellipsoid equal equation evidently expressed factors flux force formula function of position geometrical geometrical algebra give given Grassmann Hamilton heliocentric hypothesis indeterminate product integral J. J. Thomson kinetic energy light linear function matrices medium meteoroids method molecules motion multiple algebra multiple quantities notations observed obtain optic orbit ordinary algebra paper parallel perpendicular plane position in space potential principle quadratic function quaternion R₁ R₂ reciprocals refraction regarded relations represented respect rotation scalar substitution suppose surface surface-integral unit vectors values vector analysis vector function velocity versor vibration volume wave-length wave-normal wave-plane waves zero
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Page 262 - Annalen alone, and about half as many others elsewhere. But such work as that of Clausius is not measured by counting titles or pages. His true monument lies not on the shelves of libraries, but in the thoughts of men, and in the history of more than one science.
Page 87 - Clebsch has said that it can never be admired enough ,t — is the use of equations in which the terms consist of letters representing points with numerical coefficients, to express barycentric relations between the points. Thus, that the point S is the centre of gravity of weights, a, b, c, d, placed at the points A, B, C, D, respectively, is expressed by the equation (a+b+c+d)S=aA + bB+cC+dD.
Page 90 - Grassmann's ideas upon the course of mathematical thought. The transcendent importance of these ideas was fully appreciated by the author, whose very able work seems to have had considerable influence in calling the attention of mathematicians to the subject. In 1870, Professor Benjamin Peirce published his Linear Associative Algebra, subsequently developed and enriched by his son, Professor CS Peirce. The fact that the edition was lithographed seems to indicate that even at this late date a work...
Page 151 - Also in Nature, Vol. XLIII, p. 512, " How much more deepl-y rooted in the nature of things are the functions Sufl and Vaft than any which depend on the definition of a quaternion, will appear in a strong light, if we try to extend our formulae to space of four or more dimensions. It will not be claimed that the notions of quaternions will apply to such a space, except indeed in such a limited and artificial manner as to rob them of their value in a system of geometrical algebra. But vectors exist...
Page 86 - ... of machines. That such has been the case, none will question. The improvement has been in every part. Even to enumerate the principal lines of advance would be a task for any one; for me an impossibility. But if we should ask, in what direction the advance has been made, which is to characterize the development of algebra in our day, we may, I think, point to that broadening of its field and methods, which gives us multiple algebra.
Page 101 - ... seven oranges into fifty apples -|- 100 oranges, or that of one vector into another. Now an operator has, of course, one characteristic relation, viz., its relation to the operand. This needs no especial definition, since it is contained in the definition of the operator. If the operation is distributive, it may not inappropriately be called multiplication, and the result is par excellence the product of the operator and operand. The sum of operators qua operators, is an operator which gives...
Page 256 - RUDOLF JULIUS EMANUEL CLAUSIUS was born at Coslin in Pomerania, January 2, 1822. His studies, after 1840, were pursued at Berlin, where he became Privat-docent in the University, and Instructor in Physics in the School of Artillery. He was Professor of Physics at Zurich in the Polytechnicum (1855-67) and in the University (1857-67), at Wiirzburg (1867-69), and finally at Bonn (1869-88), where he died on the 24th of August, 1888. His literary activity commenced in 1847, with the publication of a memoir...
Page 111 - It is often convenient to represent in the form of a single differential coefficient, as dr dp* a block or matrix of ordinary differential coefficients. In this expression, p may be a multiple quantity representing say n independent variables, and r another representing perhaps the same number of dependent variables. Then dp represents the n differentials of the former, and dr the n differentials of the latter. The whole expression represents an operator which turns dp into pr, so that we may write...
Page 110 - Grassmann's point analysis. The distinction of the problems is very marked, and corresponds precisely to the distinction familiar to all analysts between problems which are suitable for Cartesian coordinates, and those which are suitable for the use of tetrahedral, or, in plane geometry, triangular coordinates. Thus, in mechanics, kinematics, astronomy, physics, or crystallography, Grassmann's point analysis will rarely be wanted. One might teach these subjects for years by a vector analysis, and...
Page 151 - Sa/3 represent the sine and cosine of the angle included between a and /3, combined in each case with certain other simple notions. But the sine and cosine combined with these auxiliary notions are incomparably more amenable to analytical transformation than the simple sine and cosine of trigonometry, exactly as numerical quantities combined (as in algebra) with the notion of positive or negative quality are incomparably more amenable to analytical transformation than the simple numerical quantities...