Project Gutenberg’s Utility of Quaternions in Physics potx

Project Gutenberg’s Utility of Quaternions in Physics, by Alexander McAulay This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Utility of Quaternions in Physics Author: Alexander McAulay Release Date: August 11, 2008 [EBook #26262] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK UTILITY OF QUATERNIONS IN PHYSICS *** UTILITY  QUATERNIONS IN PHYSICS. [...]... account of their many applications in what follows it is expedient to place in this preliminary section If R is some function of ρ − ρ where ρ is the vector coordinate of some point under consideration and ρ the vector coordinate of any point in space, we have R = −ρ R ρ Now let Q(R) be any function of R, the coordinates of Q being functions of ρ only Consider the integral Q(R) ds the variable of integration... expression is in nite the added surface integral will be generally but not always in nite Similarly in the case of an added line integral if LtT (ρ−α)=0 T (ρ − α)QU(ρ − α) is zero or finite, the added line integral will be zero or finite respectively (of course including in the term finite a possibility of zero value) If this expression be in nite, the added line integral will generally also be in nite This... genius of the language (the laws of Quaternions) are placed on the relative positions in a product of operators and operands With this warning the reader ought to find no difficulty One of Prof Tait’s criticisms already alluded to appears in the third edition of his Quaternions. ’ The process held up in § 500 of this edition as an example of “how not to do it” is contained in § 6 below and was first given in. .. might an instrument of which nobody has attempted to master the principles be blamed for not being of much use Workers naturally find themselves while still inexperienced in the use of Quaternions incapable of clearly thinking through them and of making them do the work of Cartesian Geometry, and they conclude that Quaternions do not provide suitable treatment for what they have in hand The fact is... account of the extreme simplicity of Quaternion notation, large advances in the parts of Physics now indicated, are to be expected Expressions which are far too cumbrous to be of much use in the Cartesian shape become so simple when translated into Quaternions, that they admit of easy interpretation and, what is perhaps of more importance, of easy manipulation Compare for instance the particular case of. .. equation (9) may be put in the following form:—Q being any linear function (varying from point to point) of R1 and 1 , R being a function of the position of a point Q(R1 , 1 ) ds =− Q1 (R, 1 ) ds + Q(R, dΣ) (16) Potentials 9 We proceed at once to the application of these theorems in integration to Potentials Although the results about to be obtained are well-known ones in Cartesian Geometry or are easily... on the x-interface becomes P, U − N/2J, T + M/2J and similarly for the other interfaces To express the part of the stress (P &c.) which depends on the strain in terms of that strain, consider w the potential energy per unit volume of the unstrained solid as a function of E &c In the general thermodynamic case w may be defined by saying that w × (the element of volume) = (the intrinsic energy of the element)... slight development in order readily to apply to the practical consideration of most physical subjects The first steps of this, which consist chiefly in the invention of new symbols of operation and a slight examination of their chief properties, I have endeavoured to give in the following pages I may now state what I hold to be the mission of Quaternions to Physics I believe that Physics would advance... the bounding surface of the volume, and again surface integrals with integrals taken round the boundary of the surface For this purpose we shall use the following notations for linear, surface and volume integrals respectively Q dρ, Q dΣ, Q ds where Q is any function of the position of a point Here dρ is a vector element of the curve, dΣ a vector element of the surface, and ds an element of volume... it will occur twice, i.e once for the part of the space on each side In the case of the isolated points, if the surface integral or line integral round this added boundary vanish, we can of course cease to consider these points as singular Suppose Q becomes in nite at the point ρ = α Draw a small sphere of radius a and also a sphere of unit radius with the point α for centre, and consider 20 [ § 8  . English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK UTILITY OF QUATERNIONS IN PHYSICS *** UTILITY  QUATERNIONS IN PHYSICS.