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The Nabla of Sir William Rowan Hamilton

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The Nabla of Sir William Rowan Hamilton Tutorial, Electric Geometry - Nabla Del Hamilton Page 1 of 27 e-mail: math@ H Y P E R C O M P L E X .COM H << | >> T.Nabla 1846 THE NABLA OF SIR WILLIAM ROWAN HAMILTON ( revision .02, 2001-07-19, new section 2002-05-07 ) Sir William Rowan Hamilton (1805-1865), Irish Mathematician and Physicist, discovered Quaternions, q = w + ix + jy + kz, in 18431, and in 18462 he introduced the differential operator, = i@/@x + j@/@y + k@/@z, to facilitate the special vector differentiation of quaternions. His original symbol, the "horizontal wedge," , was later turned over after his death by mathematicians3 and written as a "vertical wedge," , instead. The came into widspread use, while the names4 "nabla" and "del" were assigned to this operator, and beginning in 1881 the definition of the vector basis (i,j,k) was also changed, this time by physicists5 who objected to Hamilton's mathematical definition of vectors and sought what they thought was a more physically meaningful definition, so that today, the we know, and are so familiar with, bears little resemblence to Hamilton's original differential operator. While Hamilton's Operator may be somewhat unfamiliar to us, we can easily grasp the essential difference between this original operator and its modern incarnation as found in our modern vector calculus. If we agree to use the simple trick6 of matching up only the components of the old and new vectors, then for a vector A = iA1 + jA2 + kA3, we can write; A = -div(A) + curl(A) A = -div(A) - curl(A) where "div()" and "curl()" are the usual differential operations in modern vector analysis. Care must be taken when using these expressions on either side of the equal signs as equivalents, but with the appropriate warnings6 heeded they can be very useful in facilitating easy recognition of quaternion results in terms of things we already know from modern vectors. This allows us to quickly grasp the results of quaternion analysis, something that could otherwize be a rather tedious affair. A glance is all we need to understand the significance of the result. Hamilton's operator produces the sum of a scalar and vector. The vector part changes sign when the direction of the operation is reversed, because quaternions do not commute, but the scalar part is unchanged in sign. This means one cannot indicate the difference of differentiation from left and right with a simple overall minus sign like in cross products7. Mysteriously, although Hamilton dies as much as nineteen years after he first introduces this operator, during which time he is mostly devoted to the development of quaternions, and spends all of his time in the last years working on the Elements8, almost nothing can be found in Hamilton's writings on the topic, apart from two brief references to this operator. Charles Jasper Joly writes9 - "Hamilton's writings on the operator consist, so far as I am aware, and I have searched through his manuscripts in the library of Trinity College, of a communication to the Royal Irish Academy (July 20, 1846), which is published in the proceedings, Vol.iii, p.291, and practically reprinted in the Phil.Mag. of the following year, and of Art.620 of the Lectures on Quaternions."10 Apparently, Hamilton dies before he gets the opportunity to write down what his thoughts are on this differential operator. As Joly says in his preface to Hamilton's Elements11 - "Sir William Rowan Hamilton died on the 2nd of September, 1865, leaving his great work on Quaternions unfinished. He intended to have added some account of the operator , an Index, and an Appendix containing notes on Anharmonic Coordinates, on the Barycentric Calculus, and on proofs of his geometrical theorems stated in Nichol's Cyclopaedia. At the time of his death, with the exception of a fragment of the preface, and a small portion of the table of contents, all the manuscript he had prepared was in type. As he rarely commenced writing before his thoughts were fully matured, he has left no outline http://web.archive.org/web/20070928163722/http://www.hypercomplex.com/educatio... 08.01.2012 Tutorial, Electric Geometry - Nabla Del Hamilton Page 2 of 27 of the additions contemplated." Not only could Joly not find any further reference to , but even in refering to this operator he is already using the modern vertical wedge symbol, , a symbol Hamilton chose not to use, as if they were totally interchangeable. And although he does remark later in footnotes of the 2nd edition of the Elements that Hamilton's symbol was different, and renders the symbol correctly then as , Joly nevertheless adopts the modern mathematicians vertical wedge in all his own writings, consistent with everyone else of the times, and never seems to be aware of the subtle difference in meaning introduced by the new symbol, even though he is forced to invent an alternate mechanism to distinguish left and right differentiations later in Art.57 of his own 1905 work A Manual of Quaternions12. This modern vertical wedge was in fact considered by Hamilton himself, and actually used for the differential operator in Hamilton's very first communication on the subject. The paper, prepared for a meeting of the British Association at Southampton, never made it to its intended destination, and so was never published. But, we know of it because Hamilton refers to this first paper when discussing the operator in the October 1847 edition of the Philosophical Magazine. On page 291, Hamilton writes13, "In the paper* designed for Southampton it was remarked, as an illustration, that this result enables us to put the known thermological equation, d2v/dx2 + d2v/dy2 + d2v/dz2 + a dv/dt = 0, under the new and more symbolic form, ( 2. - a d/dt)v = 0; . (d.) when v denotes, in quantity and in direction, the flux of heat, at the time t and at the point xyz." And on this same page, in the corresponding footnote to the referenced Southampton 'paper*', he explains, "* In that paper itself, the characteristic was written ; but this more common sign has been so often used with other meanings, that it seems desirable to abstain from appropriating it to the new signification here proposed." Hamilton feels the need to explain here why he has just changed the symbol from , which he had only very recently used in that Southampton Paper, and is now writing instead, giving a plausible argument. But, his explanation is a bit puzzling, since no one else is using the symbol for anything. Where are all these other meanings that cause confusion? Only Hamilton has used this symbol before, and he used it in two different contexts, once as a permutation symbol, and once as an arbitrary function, but certainly "so often used with other meanings" is definitely an exaggeration on his part here. Why did he feel the need to exaggerate? He was constructing a plausable argument for his readers, because he was not prepared to reveal his ideas and the real reason for making the change, as we shall argue below.9 The Southampton paper was never published. So in fact, all Hamilton's public writings use , and were it not for that one footnote, we would not have known that Hamilton actually considered and rejected it in favor of the rotated . We get an explanation of what happened to the Southampton Paper from Hamilton's very extensive biography written by Graves; it was misdirected to Sir John Herschel's home at Collingwood and never made it to Southampton. In discussing the year 1846, Graves writes14, "From the letters which passed at this time between Hamilton and Herschel, I gather that Hamilton was prevented by the illness, which proved the mortal illness, of his sister Eliza from attending the Meeting of the Association at Southampton, and that he sent to Herschel, as President of Section A, a communication 'respecting the application of my new algebraic geometry to the mathematics of Heat, and to some parts of the general theory of curves and surfaces, especially as respects Ellipsoids and Cones of the Second Degree.' The communication, through mistake, was not forwarded from Collingwood; and Herschel, finding it there on his return home, writes to express his regret, and adds, 'it looks very beautiful. Go on and prosper.'" Peter Guthrie Tait (1831-1901), the Scottish Mathematician and Physicist, close friend of James Clerk Maxwell, and understudy of Hamilton, was the first person to take up Hamilton's Operator and attempt to develop the theory. But, Tait seems not to have understood why Hamilton used the http://web.archive.org/web/20070928163722/http://www.hypercomplex.com/educatio... 08.01.2012 Tutorial, Electric Geometry - Nabla Del Hamilton Page 3 of 27 symbol, for he was one of the first to change this symbol to the modern , if not the very first person to do so, and it seems he never considered the profound implications of the switch. In his 1898 Scientific Papers15 Tait writes - "I was originally attracted to the study of Quaternions by Sir W. R. Hamilton's = i d/dx + j d/dy + k d/dz to which he called special attention (Lectures on Quaternions, $ 620) on account of its promise of usefulness in physical applications. But I soon found that in order that its full power may be applied, in general investigations, it is necessary that we should have processes of definite integration, of the kinds required in physics, applicable to quaternion symbols and not merely to scalar variables. I often consulted Hamilton about this want, and he promised to endeavour to supply it at some future time.
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