How Maxwell's Equations (and Quaternions) Led to Vector Analysis

Transcript

So in my last video which was a history of quitterians and how they work and a biography of its inventor William Rowan Hamilton I stated I hope pretty convincingly that all the elements of vector analysis came from William Rowan Hamilton. the scalar, the vector of the dotproduct, or at least the negative of the dotproduct, the crossroduct, the dell function, also called the knob function, the divergence, or at least the negative of the divergence, and the curl. All were created, if not named, by Hamilton between 1846 and 1847, specifically to help with describing physics systems. Although quaternians are quite popular among pure mathematicians and computer programmers, the majority, and I do mean the vast majority of physicists, only know William Rowan Hamilton for the Hamiltonian and have no idea that he invented the preponderance of mathematics that we use in advanced physics. But why? Well, I feel like the answer to that is intertwined with the history of Maxwell's equations and the life and personality of a man named Peter Tate.

So, in order to explain the history of vector analysis, I'd like to start with when a teenage Peter Tate met a teenage James Clerk Maxwell in 1846. Ready? Let's go. Part one. Peter Tate, James Clerk Maxwell, 1846 to 1856. When Peter Tate met James Clerk Maxwell at a prestigious high school called the Edinburg Academy in 1846, 15-year-old Tate agreed with his classmates that Maxwell was an odd duck.

Tate even joined his classmates in calling Maxwell dafted or dafty, meaning crazy. Tate said that the big problem was that he and his classmates were quite innocent of mathematics and Maxwell's absorption and such pursuits caused the other students to avoid Maxwell so that Maxwell made no friends. However, Maxwell was championed by a teacher named Dr. James Forbes. And after Maxwell surprised his companions by winning awards, Tate decided to start talking to Maxwell.

Soon they became best buds and Tate recalled fondly how from this time forward I became very intimate with him and we discussed together with school boy enthusiasm numerous curious problems. Although Tate already had a vague interest in math and physics before he met Maxwell. It was arguably this friendship and the influence of Dr. Forbes that transformed Tate into a mathematical physicist. Inspired before he even turned 18, Tate left the academy in 1848 and moved to Cambridge to focus on becoming a mathematician and a mathematical scientist.

Maxwell followed him two years later in 1850. Tate was very successful at Cambridge, winning first wrangler or first in the tripost test in 1852. But he found the work involved to be painful. scrawling his frustration about studying for this test that it was purgatory, torment, and at one time in all capitals or hell. Maxwell also didn't have a great time at Cambridge, writing his friend Lewis Cample, "Facts are very scarce here and that his teachers grind up these subjects intellectually." Maxwell complained, "The Cambridge student starves while being crammed.

He wants man's meat, not college pudding. is truth nowhere but in mathematics. Still, Maxwell was too engrossed in his studies to give up, even though he felt that the fellow students, aside from Tate and a few others, were just as disdainful of him as Cambridge as they were at the academy. Maxwell then came in second in the mathematical tripost in January of 1854 and was finally free to study what he wanted to which Maxwell referred to as entering the unholy state of bachelorhood. It was for that reason that in February 1854 22-year-old Maxwell wrote his friend and mentor William Thompson asking for advice on how to attack electricity.

Specifically, Maxwell wanted to know if he should begin with Faraday, who use no mathematics, or with people that Faraday seemed to be in conflict with, like Aier, who were math-based. Thompson then told Maxwell that Faraday was not in any way in conflict with the other scientists and that he should start with Faraday. When Maxwell read Faraday's experimental researches, he not only agreed with Thompson but also felt like Faraday's method of conceiving the phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols. So, Maxwell just decided to try to convert Faraday's ideas from Faraday's math into the ordinary mathematical forms. By December 1855 and February 1856, Maxwell published a series of papers titled on Faraday's lines of force where he modeled Faraday's electric and magnetic lines of force as theoretical tubes of an fluid.

So he could use fluid mechanics with charges either being sources or syncs of these fluids. Among the many ideas that Maxwell mathematically modeled in these papers was the idea of areas of equal pressure P around the sources or sinks that was equivalent to both the voltage of a battery they called the electric tension and the potential in static electricity. In addition, inspired by how Thompson modeled pressures and temperatures, Maxwell decided that it is easy to see that these surfaces of equal pressure must be perpendicular to the lines of fluid motion. Mathematically, Maxwell realized that these lines of fluid motion were a vector with values in the x, y, and z direction, which he designated as alpha, beta, and gamma. where alpha is dpdx, beta is dpdy and gamma is dpdz.

Now those of you who are familiar with vector physics can clearly see that this is equivalent to the modern equation for the relationship between the electric field and the electric potential in a static system i.e electric field is negative gradient of phi if phi is used instead of p and the electric field e is defined as negative alpha beta and gamma. Now Maxwell was an excellent mathematician but he didn't know that the special triangle operator that is needed for the gradient had already been invented by William Rowan Hamilton as part of his tricky quitterians. Instead, it was Tate, not Maxwell, who first got into Quirnians. Which brings me to part two. Tate Hamilton and the Quatians, 1854 to 1867.

In September 1854, just after Maxwell asked Thompson for his advice on how to attack electricity, Tate was offered a job as a mathematician and a mathematical physicist at Queens College London. It was at this new job with the freedom to study how he wanted to that Tate started to discuss with a coworker about his struggles with quturnians which led to a friendship with the inventor of the quitterians William Rowan Hamilton. Hamilton initially created quitterians as an extension of the complex number. So instead of having a complex number x + i y where i is the of -1 and i^2 = -1, he had what you might call a very complex number w + ix + j y + kz with a real number and three imaginary numbers i, j and k. Which is why in quitterians i^2= j^2= k^2=1.

In addition, in 1846, Hamilton defined the real term as the scalar and the terms with an i, j or k as the vector. In this way, Hamilton could use his quitterian multiplication rules to multiply vectors, which would give two results. A scalar that was equivalent to the negative of the modern dotproduct and a vector that was equivalent to the modern crossroduct. Even more intriguing to Tate, in 1847, Hamilton created a new operator that I'm going to call the dell operator for reasons I'll explain later that he symbolized with a triangle first as an upside down delta and then he turned it to the side. Anyway, this unnamed triangle was defined as ddxi plus ddy j plus ddz k.

The reason he created this operator is because in physics the term ddx^2 plus ddy^2 + ddz^ 2 came up regularly and in Hamilton quitterian mathematics dell squar has no vector term and the scalar term is the negative of this function. Interestingly, Hamilton also noted that if you took the dell function times a vector Q, then you get two results. A scalar which is equivalent to the modern definition of negative dell. Q and a vector result which is equivalent to the modern definition of dell cross q. However, Hamilton did not, as far as I can tell, take the simple step of seeing what happened if you took the dell of a scalar P, which would according to Hamilton's rules, result in a simple vector, which is equivalent to the gradient of a function, i.e.

delp equals dp dx i plus dp dy j plus dp dz k which clearly would be interesting to anyone dealing with electric or temperature potentials. Hamilton then combined many of his articles about quitterians into a book titled unimaginatively lectures in quitterians which he published in 1853 which is how Tate initially learned of quitter as Tate told Hamilton 5 years later I attacked your volume on quitterians immediately on its appearance. However, when it came to this triangle or dell function, Tate had trouble. In August of 1858, Tate wrote to Hamilton that since I left Cambridge, I've been busy at the theories of heat, electricity, etc., "Your remarkable formula for ddx² + ddy^2 plus ddz² as the square of a vector form appear to me to offer the very instrument I seek for some general investigations and potentials." Soon the letters were flying back and forth between Oxford, England, where Tate was working, and outside Dublin, Ireland, where Hamilton was the royal astronomer. By April 1859, Hamilton wrote, Tate, do you not feel as well as think that we are on the right track, and shall be thanked hereafter? When I created Quitterians, a friend wrote to me nearly as follows.

I suspect, Hamilton, that you have caught the right s by the ear between us, dear Mr. Tate, I think we shall begin the shearing of it. However, Hamilton was very busy writing his second book, Elements of Quitterons, which he was sure he was going to be finished with very soon, telling Tate that the only thing I asked was that you would not publish a separate work before the appearance of the elements. I shall be charmed for both our sakes to set you free as soon as possible. Meanwhile, Tate was diverted from Caternians by Maxwell's mentor, William Thompson.

See, in 1860, Tate and Maxwell's old teacher, Dr. Forbes, who had moved on to Edinburg University, had retired, and Tate beat out Maxwell for the position, mostly because Maxwell was known as being a terrible teacher. Tate's new job pleased Thompson, who'd been worried that a mere nobody had a good chance for the chair. Thompson then decided to collaborate with Tate on a series of physics books, writing his brother in January of 1862. I've been projecting a book on natural philosophy, elementary and non-mathematical, along with Professor Tate, and he has a very good executive energy and facility in writing.

I hope we may soon get volume one out. However, both Tates and Thompson's elementary physics book and Hamilton's Quitterian books were taking a lot longer than anyone expected. Now, Tate and Thompson were in their late 20s to early 30s and in pretty good health, but Hamilton was in ill health in his late 50s and had trouble with gout and possibly overwork, but not drunkenness as I described in my last video. In addition, despite Tates and Hamilton's enthusiasm for quitterians, didn't seem like anyone else was very interested. And in 1865, Maxwell quipped to Tate, "Does anyone write quitterians but Sir William Hamilton and you?" Then tragically in August of 1865, Hamilton got a terrible bronchial infection and realized he wouldn't live very longer and reached out to Tate.

According to Tate, Sir William Hamilton, a few days before his death, urged me to prepare my work as soon as possible. His being almost ready for publication, he then expressed his profound conviction of the importance of quitterians to the progress of physical science and his desire that a really elementary treatis on the subject should soon be published. Peter Tate was torn. He was being pressured by Thompson to finish this physics book, but he felt a need to honor Hamilton's deathbed request. Perhaps because of this, Thompson started to loathe quitterians, a feeling he kept for the remainder of his life.

Thompson even recalled that he basically taunted Tate multiple times that they could include quitterians in their book if he could only show that in any case our work would be helped by their use. But as this was supposed to be an elementary and non-mathematical textbook, that didn't happen. Although, of course, it turned out to be far less elementary than Thompson had initially planned. Years later, in Tate's obituary, Thompson recalled that this war was part of their friendship, and that even though he and Tate often had keen differences on every conceivable subject, including Quitians, we never agreed to differ, always fought it out. But it was almost as great a pleasure to fight with Tate as it was to agree with him.

Anyway, back in 1867, Tate and Thompson finally finished their first and it turns out only book on natural philosophy. As Thompson would sign his letters T, Tate started to sign his letters from T prime and Maxwell started to refer to their book as the TNT Prime book and the name stuck. The same year as the TNT Prime book, 1867, Tate finished his first book on quitterianons. In this book, Tate expanded the uses of quitterians and of Hamilton's triangle operator Dell, which he for some reason rotated back to be in its current orientation. However, Tate felt that due to his work with Thompson on the TNT Prime book, his Quatronian book was incomplete, writing, "I regret I have so imperfectly fulfilled this last request of my reverend friend." Despite this, or perhaps because of his simpler style and his plea for help, right after the tragic death of Hamilton, people started to slowly get interested in quitterians, which brings me back to James Clerk Maxwell and his equations.

Part three, Maxwell, his equations, and quitterians, 1856 to 1879. Back in February 1856 when Maxwell was just publishing his papers on Faraday's lines of force he got a job as a professor at Mauricial College in Aberdine Scotland and grew a beard to make himself look more professorial. He then wrote to his friend Cample No one here seems to think me odd or dafted. Some did at Cambridge but here I have escaped. Mauricio College is also where Maxwell met the love of his life, Katherine Der, and in June 1858 they married, a relationship that his friend Cample called one of unexampled devotion.

Maxwell's wife had no background in math and science, but she was very interested and soon started contributing especially to experiments in heat transfer and statistical mechanics. Then in 1860, Maxwell's college was combined with another to form the University of Aberdine and Maxwell found himself without a job. However, Maxwell quickly found a new job at King's College in London. It was at King's College at the end of 1861 and the beginning of 1862 when Maxwell first wrote Maxwell's equations. What happened is that Maxwell heard about a model of the atom as a charged nucleus with an electric atmosphere from a fellow Scotsman named William Ranken.

With this more realistic molecular vortices model, Maxwell wrote the four modern Maxwell's equations, albeit with different units and without vectors in a sea of 165 equations. Additionally, with his laws, Maxwell realized that in a vacuum, you could have a vibrating electric field that makes a perpendicular vibrating magnetic field and a vibrating magnetic field that would make a perpendicular vibrating electric field. And that electromagnetic wave would move at a speed that depends on the square root of an electric constant K divided by a magnetic constant mu. This was particularly inspiring to Maxwell as in 1857 a German team named Dr. Veber and Kouch experimentally measured this ratio and found that the square root of K over mu is 3.1 * 10 8 m/s.

In February 1862 Maxwell stated outright quote the velocity of transverse undilations in our hypothetical medium calculated from the electromagnetic experiments of Mr. Kus and Veber agree so exactly with the velocity of light that we can scarcely avoid the inference that light consists of transverse bundillations of the same medium which is the cause of electric and magnetic phenomena. These ideas were so exciting that in 1864 Maxwell wrote another series of paper expounding on what these ideas meant about light and about the electromagnetic field. a term Maxwell created for this paper. Note that Faraday had initially come up with the idea that maybe light was a vibration of electric and/or magnetic lines of force 18 years earlier in 1846.

Which is why Maxwell wrote that the electromagnetic theory of light as proposed by Faraday is the same in substance as that which I have begun to develop in this paper except that in 1846 there was no data to calculate the velocity of propagation. In April 1865, Maxwell decided that he was so busy with all his theories and experiments which he could not undertake as long as I had public duties. And so he retired from teaching and he and his wife Katherine moved back to Scotland. As Thompson and Tate were completing their TNT Prime book on natural philosophy in 1867, Maxwell started contemplating writing his own elementary treatise of electricity or magnetism or what Tate called a Senate House Treatise. Tate was thrilled, writing Maxwell, "I'm delighted to hear you are going to do a Senate House Treatise on electricity.

The sooner the better." Soon Maxwell was communicating endlessly with both Tate and Thompson, arguing about the science, the mathematics, the concepts, everything. Then after multiple distractions in 1870, Maxwell started consider using quitterons. Max will then rotate with a series of proposed names for quitterian operators as I am unlearned in quaternium idioms and I want phrases of this kind to make statements in electromagnetisms and I do not wish to expose either myself to the contempt of the initiated or quitterians to the scorn of the profane. For example, Max will name the triangle operator the alted function for delta backwards and the scalar part of the alted times the vector or negative dell do a to be the convergence. Maxwell then went through several names for the vector part of alted times the vector or dell cross a.

What did Tate think of twist, turn, or version? Maybe twirl is sufficiently racy. or perhaps a pure mathematician might prefer curl after the fashion of scroll. By October 1872, Maxwell wrote his friend Cample that I'm getting converted to quturnons and have put some in my book. He added that the Hamilton's function dell occurs continually. Amusingly, in these letters, Maxwell started to refer to the Dell function as the knob for a type of harp, as a type of joke.

so they could quip that he was a knoblotty for using it. He also liked to ask Tate if he was still harping on the naba. Maxwell finished his book in 1873 where he called it the scalar of an times a function the convergence, the vector of an times a function the curl and the dell squared as lelass's operator. All names we still use to this day. Then tragically in spring of 1877, Maxwell started to have health issues and by October 1879, 48-year-old Maxwell was told that like his late mother, he had terminal stomach cancer and only had a month to live.

Maxwell took the news stoically and was mostly concerned with his wife Catherine, who had chronically delicate health. By November 5th, he passed away. Tate and Thompson were among the many who were devastated. Tate cried, "I cannot adequately express in words the extent of the loss which his early death has inflicted not merely on his personal friends, on the University of Cambridge, on the whole scientific world, but also and most especially on the cause of common sense and true science." However, Tate consoled himself. Men of his stamp never live in vain.

And in one sense, at least they cannot die. The spirit of Clerk Maxwell still lives with us in his imperishable writings and will speak to the next generation by the lips of those who caught inspiration from his teachings and example. Now Tate felt like he had two great works to promote after their inventor's early deaths, Hamilton's and Maxwells, and he became very invested. one might even say territorial in Maxwell's equations being written in quitterians and in this he was about to be bitterly disappointed which brings us to an American scientist and mathematician named Josea Willard Gibbs part four Willard Gibbs 1873 to 1884 Gibbs who was 7 years younger than Maxwell and Tate was a longtime collaborator with both of them and on the subject of statistical mechanics, a term he coined. So it's no surprise that Gibbs read Maxwell's 1873 book on electricity magnetism.

That is where Gibbs learned about quitterians and Gibbs decided that it was necessary to commence to mastering these methods. However, he found that that in respect to the operator dell as applied to vector, I saw that the vector part and the scalar part of the result represented important operations, but their union generally to be separated afterwards did not seem a valuable idea. It was for that reason that Gibbs made sort of a mathematical shortorthhand from caterians where he separated the scalar multiplication from the vector multiplication. Then between 1881 and 1884, Gibbs put together a little pamphlet about his new mathematical methods to help his students understand Maxwell's equations. In this pamphlet, Gibbs called the scalar multiplication the direct product and noted it with a dot and called the vector multiplication the skew product and noted it with an X or a cross.

I think it's startling to compare Gibbs' scalar product and vector product to Hamilton's scalar and vector results of multiplying two vectors from his 1846 paper. You can see clearly how Gibbs's direct product is identical to the scalar product without the negative sign and that the skew product was identical to the vector result from a quitter. Shoot, Gibbs even called his directions I, J, and K after Hamilton's notation. But now I, J and K are unit vectors and not equal to the square root of negative1. With this new mathematics, Gibbs needed a new name for the negative convergence or dell dot of function.

Luckily, he had read a book from a mathematical physicist named William Clifford who wrote in 1878 about this very problem where he noted that Professor Clerk Maxwell calls the quantity negative E the convergent of sigma. We might perhaps therefore call E itself the divergence of sigma. This is why in Gibbs's pamphlet, Gibbs related Clifford's divergence and Maxwell's curl to Hamilton's dell function when he wrote that dell.W is called the divergence of W and Dell crossw is its curl. In this case, you can see how Gibbs related the divergence from the negative of Maxwell's convergence and the curl directly from Maxwell's definition in his 1873 book. This is clearly aside from a few name differences and the dot product being in the wrong position modern vector analysis.

Now after he came up with this idea, Gibbs read about another mathematician named Grassman who in 1844 had created his own separate vector multiplication and scalar multiplication. The Gibbs felt was superior to those of Hamilton. Although Gibbs admitted that I am not conscious that Grassman's writing exerted any particular influence on my vector analysis, Gibbs decided to add a little introductory paragraph to his pamphlet in 1888 where he specifically mentioned Clifford and Grassman, even though Clifford seemed to have a minor influence and Grassman not at all. Gibbs wrote to a friend that he was glad enough for the introductory paragraph to shelter myself behind one or two distinguished names in making changes of notation which I felt would be distasteful to quit on this. Gibbs had vector algebra right there with some pretty big names to back him up and he printed it but he delayed and delayed publishing it due to difficulty in making up my mind in respect to the details of notation.

Meanwhile, in England, a former telegraph operator came to the same idea, which brings us finally to part five. Oliver Heide, 1873 to 1887. Oliver Height's life was transformed by Maxwell's 1873 book. Years later, he recalled, "When I saw on the table in the library the work that just been published, I browsed through it and I was astonished. I saw that it was great, greater, and greatest with prodigious possibility in its power.

However, Hebite only had a high school education. He had to leave school at 16, and he only learned school algebra and trigonometry, which had largely forgotten. The following year, Hebite quit his job and moved back in with his parents and started to focus on studying mathematics. Then he published paper after paper after paper on the mathematical theories of telegraphs derived from his blossoming understanding of Maxwell's equations. Supposedly he worked mostly in the middle of the night with all the doors and windows closed and fires burning hotter than hell.

Starting in 1882 at the request of the editor height started to publish in a weekly trade journal called the electrician. And in November of 1882, Hebidites published his first paper on the mathematics of magnetic field from a current caring wire where he used Maxwell's term of curl in isolation without using any other forms of the quitterian. By December 2nd, 1882, Heide decided that although he felt quitterons are a very remarkable system of mathematics, which he said were superficially attractive as one equation takes the place of three, height decided that the operations met are much more difficult than the corresponding ones in the ordinary system. So that the saving of labor is in a great measure imaginary. Head therefore suggested that they should make a compromise.

Look behind the easily managed but complex scalar equations and see the single vector one behind them expressing the real thing. By April of the following year 1883, Heide seems to have independently conceived of the idea of divergence just as Clifford had done 5 years earlier writing as the expression with the negative sign prefixed Maxwell called the convergence. we may as well use the term divergence for the same quantity with a positive sign prefixed. Sorry that I attributed this to Heside in my last video and not to Clifford. I didn't realize Clifford came up with it first.

Anyway, in June 1885, Heide realized that his trick of just using the divergence and the curl and no other parts of quitterians was not good enough. As he put it, owing to the extraordinary complexity of the investigation which when written out in cartisian form which I began doing but gave up a ghast some abbreviated method of expression becomes desirable. Then like Gibbs before him height came up with two kinds of multiplication a scalar product and a vector product. However, unlike Gibbs, who used the modern notations of a dot for the scalar multiplication and an x for the vector multiplication, height used no term to be the scalar result and a v to represent the vector result. A big problem with this method is the confusion of what dell n represents.

If n is a scalar p, then like with modern vector algebra, delp p is a gradient of p, which is a vector. If on the other hand n is a vector a then del a is the divergence of a which results in a scalar. It was for this reason that he realized that his notation was confusing and the following year promoted the idea of using a different font the claridan font that looks bold to represent vectors. Even with this change it was still pretty confusing. Anyway, years later, Heidi said that Tate contacted him after this paper, and he appeased Tate considerably during a little correspondence we had by disclaiming any idea of discovering a new system.

Heside's word apparently mullified the invested Tate, especially as Heside Riley recalled, "I was too small to be seen at first." Heside wasn't kidding about being too small at first because at this time basically no one was interested in what he had to say. In 1887, Heide was told that the new owner of the magazine, The Electrician, wanted him to discontinue his work on Maxwell's equations altogether. Heid sadly noted that the owner had told him that he had made particular inquiries among students who would be likely to read my papers to find if anyone did so and he had been unable to discover a single one. Heid must have been desperate but then an event happened that converted almost everyone into a Maxwellian. that event at the discovery of an invisible wave by a German scientist named Hinrich Herz.

Which brings us to part six, Herz changes the game. 1887 to 1890. That same year that he was told that no one was interested in his papers on Maxel's equations. 1887, a 30-year-old German scientist named Hinrich Herz published a paper titled on very rapid electric oscillations. This is significant because this was the paper where Hertz demonstrated that he could create an invisible electromagnetic wave that we now call a radio wave that moves at the speed of light from vibrating electronics.

Herz wasn't motivated by wireless telegraphy. Instead, according to Hertz, the object of these experiments was to test the fundamental hypothesis of the Faraday Maxwell theory. In other words, Herz discovered radio waves because he wanted to demonstrate just as Faraday and Maxwell postulated that light was an electromagnetic wave. Anyway, after Herz's discovery, everyone wanted to know about Maxwell's equations. But as almost no one knew quitterons, Maxwell's elementary book was a really hard read.

Heside wasted no time with the renewed interest and by February 1888 published the first of many papers on electromagnetic waves for the prestigious philosophical magazine where he mentioned his previous papers from a few years earlier in the electrician. Soon several important people started to reference heside. For example, in March of 1889, the Maxwellian Oliver Lodge gushed in a footnote, "One whose name is not yet on everybody's lips, but whose profound researches into electromagnetic waves have penetrated further than anybody yet understands into the depths of the subject, and whose papers have likely contributed largely to the theoretical inspiration of Herz." I mean that powerful mathematician Mr. Oliver Heide. Now it turns out that Lodge was incorrect on the idea that Herz was inspired by heide as Herz only heard of heside with this 1888 paper which happened after his experiments.

However, once Herz read he signed, he was inspired by him and soon they started a correspondence where they bonded over the thought that they didn't like that Maxwell focused on magnetic and electric potentials and instead believed as headside dramatically put it, it's best to murder the whole lot. In March of 1890, Herz published his own theoretical paper on the fundamental equations of electromagnetics where he admitted that Mr. Oliver Height has been working in the same direction as me ever since 1885. A fact he learned from the philosophical magazine from February 1888. Note that some people think that Herz was part of the development of vector analysis from this statement, but really what he was talking about here was their shared dislike of potentials.

And this paper does not include vectors in any form, although Herz did attempt to add a bit of vectors in later papers due to heaves influence. Meanwhile, Gibbs, who still hadn't published that pamphlet, was inspired by Herz's discovery to start sharing it around. It is unclear to me whether he directly gave it to Peter Tate or Tate got it secondhand, but either way, he didn't like it. Which leads me to part seven. The War of the Vectors begins 1890 to 1894.

Possibly inspired by Herz's result, Peter Tate began working on a third edition much enlarged of his elementary treatise of quitterians. However, the slow acceptance of quitterians was starting to grind tape down and he was particularly upset with Gibbs because he had known him for several years with their work on statistical mechanics and he knew as well as anyone that Gibbs could handle quitters. In the introduction to Tate's book, Tate basically snapped and wrote, "It is disappointing to find how little progress has recently been made with the development of Quernians, and that even Professor Willard Gibbs must be ranked as one of the retarders of Quitterian progress in virtue of his pamphlet on vector analysis, a sort of hermaphrodite monster composed of the notations of Hamilton and of Grassmen. Note that people who born interex are not monsters and I will ban anyone in the comments who is cruel to people for how they are born. Sorry, but this is my channel and there is enough cruelty in the world.

I will not have my channel be part of it. Tate's use of that inflammatory word, which he probably meant as a joke, grabbed the attention of many people, including Gibbs, who responded in Nature magazine on April 2nd, 1891. In this retort, Gibbs noted that if the issue was just his notation, then the objection should not be that he was monstrous, but instead more that his dress was uncouthed. Gibbs then tried to explain that though he agreed with Tate and Hamilton that the quitterian affords a convenient notation for rotation that using alpha. and alpha cross beta for what is expressed in quitterians by negative s alpha beta and v alpha beta and in a like manner dell do and del crossw for negative s delw and v dellw in quitterians have some substantial advantages over the quaternionic in point of convenience by the way how precient was gibbs that is exactly and I mean exactly what most physicists believe today.

Then in June, Oliver Heide added fuel to the fire in an article titled on the forces, stresses, and fluxes of energy. In this paper, Heide stood up and added that he rejected the quatronianic basis of vector analysis and that Hamilton's system was metaphysics. Although Hevside still rejected Gibbs's notation despite the fact that Gibbs showed him his obviously superior notation three years earlier in 1888. In this paper, Heide was still moderately polite about quitterians, writing, "The modifications I made are simply the elements of quitterians without the quitteron with the notation simplified to the uttermost and with a very inconvenient minus sign before the scalar products done away with. However, by November, Heide was invited back to write for the electrician, and by then he was done with nicities writing.

I came later to see that so far as the vector analysis I required was concerned, the quitterian was not only not required, but was a positive evil of no inconsiderable magnitude. Soon everyone was printing nasty things about each other and no one was taking a step back and realizing that not only are vector analysis and quitterians not in conflict, they actually complement each other. Vector analysis was derived from quitterians. But then in return, it's nearly impossible to learn queons without starting with vector analysis first. Well, at least if your name isn't Gibbs or Heside or a scant handful of other people.

Not to mention Maxwell's equations are a lot easier to learn with vector analysis than with quitterians. And frankly, we need all the help we can get. Then on January 1st, 1894, the scientific community was horrified to learn that Hinrich Herz, who was only 36 years old, had died of some mysterious ailment. In June in England, Oliver Lodge put together a talk on the life and work of his newly found friend. a talk where he changed Herz's receiver so that it could be demonstrated for a crowd which initiated the wireless revolution which of course vastly increased the interest in Maxwell's equations in Germany well it seemed like the whole country was in mourning and many scientists turned to heside as the representative of Maxwell's equations because after all the great Herz had recommended him.

Part 8, Tate loses the war. 1894 to 1901. The same year that Herz passed, a German scientist and mathematician named August Fel wrote a book on the theories and math of electromagnetic waves. In this book, he wrote that the works of Hevside influence my presentation more than those of any other physicist with the obvious exemption of Maxwell himself. I consider Hevside to be the most eminent successor to Maxwell in regards to theoretical developments just as it was Herz who alas was so quickly snatched from us who was Maxwell's most eminent successor in regards to experimental developments.

Fol also admitted that another motivation for him to promote heside is because he was deeply disdainful of quitterians and quitterians were invented by an Irishman William Rowan Hamilton and he preferred to promote a German mathematician Hermon Grassman. I'm not kidding. He literally said it like that. He said the country which produced the grassmen should no longer stand behind the country of Hamilton with the introduction of this important improvement in the mathematical means of theoretical physics. Meanwhile, in England, in December of 1893, Heide published a book titled Electromagnetic Theory along with Fel's book convinced many people of the merits of vector analysis and soon grassman's influence was exaggerated and quitterians were considered evil or at least distasteful.

You can see heides influence from this book in many of the names and conventions we use in modern Maxwell's equations. For example, Heide created the name permitivity in 1887 and restated it in his book in 1893 and is still in use today. Head also promoted the use of the word voltage which was used occasionally but rarely to replace the much more commonly used term of pressure. Amusingly, in 1893, he wrote how irritated he was by the utterly vicious misuse of pressure to indicate EMF for voltage by men who are old enough to know better. As a reviewer of Hebid's 1893 book named George Mansion noted, as everyone knows, Mr.

Hebide has advocated for many changes in scientific nomomenclature and has already succeeded in getting some of them adopted. Also, it was in the same review of Hebid's book where Mention noted how difficult it would be to handwrite the Claritin type and that Heidi's suggestion of a suffix would sometimes result in the double suffix and suggested perhaps a horizontal bar over the letters is best. Though this is not good, quickly mentions tenative suggestion of a bar was replaced with an arrow as it has remained ever since. After Heides and Fal's books, Heside was at the height of his fame and influence, but he instead became fascinated with geoysics and meteorology and didn't do much to influence the development of Maxwell's laws of vector analysis after this point. However, even after Heidi's influence was largely forgotten for a time, his insistence on the evil of Quernians remained.

Meanwhile, Gibbs had moved back to focusing on statistical mechanics and also didn't do much more on Maxwell's equations or vector analysis. This was particularly relevant in early 1901 or late 1900 when Gibbs was approached by the administrators of his university to turn his vector pamphlet into a book in connection with Yale's bicesentennial. Gibbs was too busy with his statistical mechanics, but was delighted when a former student named Edwin Wilson agreed to take a stab at it. In the introduction, Wilson said that although the greater part of the material used in the following pages has been taken from the course of lectures on vector analysis by professor Gibbs, he also used Oliver Height's 1893 book and August Fel's 1894 book and that his previous study of quitterians has also been a great assistance. You can see how popular Heide was at the time because Wilson justified the reason for his book by quoting Heside.

When Heide said that the work he had written would serve as a stop gap till regular vetorial treatises come to be written suitable for physicists. Wilson then made several minor changes to vector analysis that were to have a lasting impact on modern mathematics due to the clarity of the thought on how to improve instruction and learning. First, Wilson repeated height's idea of a different font, the Claritin font for vectors. So it's possible to pass from directed quantities to their scalar magnitudes by mere change in appearance of a letter without any conf confusing change in the letter itself like bold V for the velocity of a moving mass and unbold V for the magnitude of that velocity. In this way, Wilson noted that even when the method of multiplication will tell you whether you have a scalar or a vector, it was still very useful to have a visual signifier of whether the letter was a scalar or a vector, either bold in print or an arrow in handwriting.

Second, Wilson moved up the dot in the scalar product so it wouldn't be confused with a period and noted that as you read the scalar multiplication a dobb scalar multiplication may often be called the dotproduct instead of the direct product. Third, and you probably guessed this one, Wilson noted that the vector multiplication is read a crossb and for this reason it is often called the crossroduct. Finally, Wilson was the one who came up with the name Dell for Hamilton's triangle operator. His logic was that the symbolic operator was introduced by Sir William Rowan Hamilton is now in universal employment. There seems however to be no universally recognized name for it.

adding in a footnote that some use knob, some use alted and foil calls it die operation triangle causing Wilson to note dryly how this is to be read is not divulged. However, Wilson felt very strongly that for lecturing and purposes of instruction, something is required something too that does not confuse the speaker or hearer even when often repeated. It was for that reason that Wilson decided that the mono syllable dell is so short and easy to pronounce that even in complicated formula in which it occurs a number of times no inconvenience to the speaker or hearer arises from the repetition. And that ladies and gentlemen is why I call it the dell operator. One more comment about the dell operator or the knob operator or the alted operator.

There are a million debates about Hamilton versus Grassman versus Clifford in the history of vector algebra. But no one can debate that Hamilton was the one who created the dell function. This single function can be used to represent the leloian, the gradient, the divergence and the curl. This is the operator that attracted Tate in the first place. And it was Tate's work with this operator that attracted Maxwell to use it in his 1873 book.

And it was the quitterians in his book that caused both Gibbs and Heside to make vector analysis. Now could you make vector analysis from Grassman's theories? Yes, you definitely could. Would you do it without the Dell function? It doesn't seem likely to me and that's not what happened. Meanwhile, just before Wilson published his book on vector analysis on July 4th, 1901, Tate passed away at the age of 70. William Thompson, who was then known as Lord Kelvin, wrote his obituary and ended it with a personal note that Tate's death is a loss to me which cannot as long as I live be replaced.

Without Tate's push, quitterians were eventually abandoned and only semi recently have been revived, just as Gibbs foretold, mostly as a method for computing rotations. Now, you might notice that I didn't mention how Heside combined or emphasized the four modern Maxwell's equations. That's because he didn't. Heside like Maxwell before him mentioned the four equations that we now consider Maxwell's equations in his papers. between 1883 and 1885.

But that didn't cause anyone to think that Maxwell only had four equations. Heck, in Heatherside's influential 1893 book, Heide was considered to have eight equations, including terms for a magnetic density and a magnetic current, even though they were, according to Heside himself, fictitious. So if Maxwell didn't cause people to think he had four equations and Heside didn't cause people to think Maxwell had four equations, then who did? Well, it wasn't Maxwell and it wasn't Heide and it wasn't Gibbs and it wasn't Herz. It was actually Hendrickk Laurren in 1895. In fact, I contend that it was only due to the popularity of relativity that was derived from Lorenz's ideas that people started to think of Maxwell only having four equations.

My theory on how we started to think that Maxwell only got four equations, what it has to do with relativity, and why we falsely attributed to heside is next time on the evolution of wireless. Thanks for watching my video. If you go to my website, www.cathylovesphysics.com, you can get the script for this video with citations with links so you can read the information for yourself. Also on my website, you can learn more about my book, The Lightning Tamers. And also, you can learn about my first live appearance in the Bay Area on April 2nd, 2023 at 400 PM.

I have a link down below. My talk is going to be about basically chapter two and three, the wild electricity wizards of the 1700s and how they jumpstarted our electrical world. If you can't make it to my talk either because you are out of town or unavailable or, you know, from the future, don't worry. I'm going to live stream it and I will replace the link to tickets to a link to the talk. Big thanks to my patreons.

If you want to join their ranks, there is a link down below. Also, if you've never read Michael Crow's History of Vector Analysis, I highly advise it. Has a lot more details and I only read it recently and I was blown away. Okay, stay safe and curious, my friends. Also, on my website, you can learn more about my book, The Lightning Tamers.

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