Project Gutenberg’s Utility of Quaternions in Physics, by Alexander McAulay pptx

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... 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)... 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. .. 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... 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 that the subject requires a slight development in order readily... in value by changing them into φ ζ and ζ respectively.] 16 [ § 3a   which gives m in terms of φ That S ζ1 ζ2 ζ3 S ζ1 ζ2 ζ3 = 6 is seen by getting rid of each pair of ζ’s in succession thus:— 2 S ζ1 (S ζ1 Vζ2 ζ3 ) ζ2 ζ3 = −S Vζ2 ζ3 ζ2 ζ3 = S (ζ2 ζ3 − ζ2 S ζ2 ζ3 ) ζ3 = −2ζ3 ζ3 = 6 Next observe that S φω φζ1 φζ2 = mS ω ζ1 ζ2 Multiplying by Vζ1 ζ2 and again on the right getting rid of. .. result of this part of the essay is to lead to a presumption against Sir William Thomson’s Vortex-Atom Theory and in favour of Hicks’s As one of the objects of this introduction is to give a bird’s-eye view of the merits of Quaternions as opposed to Cartesian Geometry, it will not be out of place to give side by side the Quaternion and the Cartesian forms of most of the new results I have been speaking... for our notation for linear surface and volume integrals we will now prove that if Q be any linear function of a vector∗ ∫ Q dρ = Q dΣ = Q (V dΣ ∆), (8) (9) Q∆ ds To prove the first divide the surface up into a series of elementary parallelograms by two families of lines—one or more members of one family coinciding with the given boundary,—apply the line integral to the boundary of each parallelogram... Vαβ Adding for the whole surface we get equation (8) Equation (9) is proved in an exactly similar way by splitting the volume up into elementary parallelepipeda by three families of surfaces one or more members of one of the families coinciding with the given boundary If α, β, γ be the vector edges of one such parallelepiped we get a term corresponding to Qβ − Qα viz Q(vector sum of surface of parallelepiped) . set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK UTILITY OF QUATERNIONS IN PHYSICS *** UTILITY  QUATERNIONS IN PHYSICS.