| Author: Cao Zexian
( Institute of Physics, Chinese Academy of Sciences )
This article is from Physics, No. 10, 2020
Never mind when.1)
——Sir William Rowan Hamilton in 1859
Abstract Quaternions is Hamilton's promotion of binary numbers, that is, plurals, which successfully opened the door to modern algebra. Hamilton referred to the pure imaginary part of the quaternion as vector, a Vector in Chinese. The quaternion product of the three-dimensional world vector introduces the concepts of point multiplication and fork multiplication. Maxwell learned quaternions from Tate, invented the concepts of divergence and curl for differential vector operations, and the three-component ordinary quaternion world vector was used by Maxwell and Hewiside for the formulation of electromagnetism, so that with the form of Maxwell's equations that we are familiar with today, Gibbs and Hewiside developed vector analysis independently. Vector analysis is a pragmatic cut of rigorous quaternion algebra, and its usefulness is obvious and the harm is enormous. The messy ∇ point multiplication-fork multiplication makes electrodynamics a nightmare for most physics students. Tate fought hard to defend quaternions, but vector analysis became popular. Hamilton pursued the establishment of general multialgebras, gibbs tried to generalize three-dimensional vector analysis, coupled with the study of linear unfolding founded by Grassmann and the linear combined algebra created by Pierce, resulting in linear algebra. Matrix theories, Grassmann algebras, and Clifford algebras, which were born around the same time, are intimately related to them and are mathematical foundations for physical formulation. Figuring out the algebraic knowledge and relationship behind quaternions, vector analysis, and linear algebra may be less confusing in the mathematical formulations of general physics textbooks, and why vector fork multiplication and fork multiplication in electrodynamics are in quantum mechanics— it is said that wave functions are also vectors— and they are gone. By the way, a vector is a vector because it follows an algebraic structure that doesn't need to have a direction or even a length.
Key figures Hamilton, Tait, Maxwell, Heaviside, Cayley, Gibbs, Grassmann, Peirce, Clifford
01
Confused electric-dynamic symbols
The author is stupid, and the university physics results are unbearable, especially electrodynamics 2). There is a saying that "I haven't cried for a long night, it's not enough to talk about love." To paraphrase this saying, the author would like to say that "not for the mess of ∇ point multiplication-fork multiplication formula doubts life, not to learn electro-dynamics." "The author had to memorize those inexplicable formulas purely to survive, and as for what mathematical meaning they had, what physical images they corresponded to, and what the truth was, the author perfectly inherited the ignorance of the textbook author and the teacher. To add to the sense of nausea and direct gaze, John David Jackson is now working on Classical Electrodynamics, 3rdedition, John Wiley & Sons, Inc. (1999), I gave a highly bad rating! The appendix to the so-called vector formulas in the book is reproduced as follows:
For vector x in 3D, there is ∇ ∙x = 3; ∇ × x = 0.
A, b, c, and x in the table above are vectors. This table is usually listed in Electrodynamics textbooks, I guess it is to make it easier for the reader to find. So why is it easy to find? Of course, it's because you know you don't understand and can't remember! This is probably the presupposition of the authors of the average Electrodynamics textbook. Interestingly, these authors generally don't tell us that we're talking about a specific three-dimensional vector rather than an arbitrary vector in linear algebra, nor is it a wave function in quantum mechanics ψ the kind of vector that takes multiple values but has a total modulo of 1, and that the two-component monster of two wave functions is a spin but is always mistaken for a vector. As for the ∇ point multiplication-fork multiplication formula here comes from the algebraic properties of this particular three-dimensional vector itself, and it is even more rare in such books. It is worth being wary that the vector is defined as a quantity that has both length and direction in general (English) books, so there is a Chinese vector and an Arrow notation in English
, even more outrageously wrong. Vectors belong to such sets, where the linear superposition of any two elements a1, a2 λ1a1 + λ2a2, λ is a scalar belonging to some number domain, all in this set, i.e. the set is closed for linear superposition. As for whether the vector has a mold (length) and whether the angle (direction) can be defined between the two vectors, that is not necessarily true.
The formulas listed above are scientifically in the category of vector analysis. Since I have not studied the basics of algebra, let alone modern algebra, there is no other way to memorize the above formulas than to memorize them by rote. If there is a bit of an algebraic foundation, one will find that the formula above is problematic. First, notice that there are two kinds of multiplication, one is for easy, a∙b = b ∙ a, and one is against easy, a ×b = -b × a. Since it involves opposing easy relations, this must be a special kind of non-exchange algebra. Looking at the formula a ×(b×c) = b(c∙a)-c(a∙b), the parentheses in the formula are important, emphasizing that the expression is non-associative, a ×(b×c) ≠(a× b) × c. That is to say, this vector analysis involves non-commutative and unbound computation, which is not a trap. Note that since the average textbook author does not know the meaning of these formulas, it is not known how to place the four items on the right. For example, the ∇×(a× b) = a(∇∙b)b (∇∙a) + (b∙∇) a- (a∙ ∇) b in the aforementioned formula, if we know the origin and meaning of a particular item, write ∇×(a×b) = (∇∙b+b∙∇)a-(∇∙a+a∙∇) b is much easier to understand, the plus sign in each parenthesis on the right comes from the nature of the differential operator, and the subtraction in the middle comes from the anti-easy nature of the cross multiplication, which is clear at a glance.
Many years after I first came into contact with the above list of electrodynamic formulas, I noticed that the vector here is just a special case of the concept of vectors, ai+bj+ck, but an ordinary quaternion world vector, which is inherited from the special hypercomplex number of quaternions and is intrinsically related to linear algebra. Having figured out the relationship between quaternions, vector analysis, and linear algebra, with my limited knowledge of physics, I can assert that many of the contents of general physics in college will become clear and lovely. For example, in the differential form of Maxwell's equations in electromagnetism, the rotational representation and angular momentum of the rigid body, the spin in quantum mechanics, and the rotational representation in relativistic quantum mechanics, you can find an intrinsic, consistent, and simple understanding in the quaternion, which is the parent of vector analysis.
02
Hamiltonian, complex and quaternions
When people solve unary cubic equations, they will encounter the problem of negative numbers under the root number but cannot be thrown away (when solving unary quadratic equations, they can be thrown away), so they have to retain the square root of the negative number. induct
As a unit imaginary number (wrong, it should be ii = -1), the number of structures such as z=a+bi is called a complex number. Around 1830, the 25-year-old Irish mathematician and astronomer Sir William Rowan Hamilton (1805-1865, the Hamiltonian, see Figure 1) thought that writing plurals as a real number plus an imaginary number was misleading. At that time, I also noticed that the addition sign in the plural addition (a + ib) + (c+ id) = (a + c) + i(b + d) has different meanings, but it is over here, and I dare not question or have the ability to question [blame me!] 】。 Hamilton believed that the addition symbol in z=a+bi had only formal meaning, and what mattered about the complex number was the algorithm it followed, not what you represented it. For example, we can represent complex numbers in the form of matrices.
, which follows the same addition and multiplication as complex numbers and can represent the geometry of a two-dimensional plane. If the complex number is written as a matrix, then the complex number of the module is 1, and its matrix generally takes the form z=
This is the rotational transformation of the two-dimensional space, and the complex product has the function of representing the rotation in the two-dimensional plane. Of course
This form is also related to the angle-preserving transformation, the Hamiltonian equation and symplectic geometry, and so on, and so on, without mentioning [sigh: Don't underestimate the addition you learned in first grade, there's so much you've never heard of]. In summary, Hamilton realized that a complex number is a number with two elements that follows a specific algorithm, and can be written as (a, b), which he called algebraic couple, now also called binarion.
Figure 1 Sir Hamilton, a man whose mathematical and physical achievements I believe are higher than Newton's
Complex numbers, or binary numbers, algebraic even elements, are very powerful mathematical concepts that, in addition to solving algebraic equations, can also describe the rotation of two-dimensional space. But we live in three-dimensional space, Hamilton thought, we should construct triplets, triple numbers or ternary numbers to describe the phenomena that occur in physical space. Modeled on binary numbers, ternary numbers z=a+ ib+ jc, there are two imaginary parts i2 = j2 = -1, which seems to be not much difficulty. However, when the product of the ternary numbers is found, the ij and ji terms appear, which are new elements and bring new problems. Neither ij = ji = 0 nor ji = -ij eliminates the problems that arise in any ternary product and any ternary modulo square product. This made Hamilton very distressed, and the relevant research was repeatedly put down and picked up, and the 13 years of turning his face had not been completed.
On October 16, 1843, Hamilton had an epiphany in his mind: if it were a quaternion, it would probably be that the product of the number and the product of the square of the number modulus would also have the form of a number and a square of the number, respectively. That is to say, he needs to study the number of the form z=a + ib + jc + kd, for which he needs to introduce a third imaginary number k2 = -1, which satisfies the relationship ij = k, jk = i, ki = j; ij = -ji, jk = -kj, ki = -ik [remember these two relationships, this is the relationship to be inherited by vector analysis focus], i2 = j2 = k2 = ijk = -1. Constructing ternary and quaternions, abandoning the entrenched law of multiplicative commutation is a crucial step that does require courage and boldness. Later we learned that Hamilton wanted a ternary number with division, which did not exist at all, and that the algebra of the quaternion was the division algebra — the division algebra had only four cases: real, binary, quaternary, and octalotype, which is Hurwitz's theorem. The product of two quaternions is still a quaternion, and with this property it is easy to prove that the product of the sum of squares of any two groups of four integers or the sum of squares of four integers is just a calculus, and neither can be proved.
The pure imaginary part of the quaternion, r=xi +yj +zk, can describe three-dimensional space, which Hamilton called vector (Chinese translation vector. Please note again that the vector here is the number! Ordinary quaternion world vector, ordinary quaternion world vector 3). We need to remember that vectors such as r=xi +yj +zk are quaternions with zero realities, not directional quantities. Computing (xi +yj +zk) (ai +bj +ck) = -(xa+yb+zc) + (yc-zb)i + (za-xc)j + (xb-ya)k, the quaternion product of the visible vector consists of the real part (Hamilton gave the name scalar, Chinese translation scalar) and the vector part, which Hamilton called the vector's dot multiplication (scalar product) and fork multiplication, respectively— the result of Hamilton's construction of the quaternion. The vector multiplication of the above equation uses the properties of the imaginary numbers of quaternions ij = k, jk = i, ki = j, which reflects the right-hand rule of the so-called three-dimensional space vector cross multiplication. The A× B= -B× A properties of vector fork multiplication actually come from the rules of quaternion multiplication ij = -ji, jk = -kj, ki = -ik, which is related to the positive and negative sign alternation of permutation, and the positive and negative sign alternation in the expression when evaluating the matrix value. In 1846, Hamilton even introduced it
symbol. We see that Hamilton actually had what he could manage the full panoply of quaternion operations with ease, but Hamilton was a philosophical, rigorous mathematician who could not afford to sacrifice the elegance and rigor of mathematics to cater to the practicality of physics, and he was reluctant to make a simplified version of his quaternions. If you are replaced, you are reluctant to do it - if you use everyone as a fire burner, it must be a rough person! The development of science, especially applied science, may require such a crude person, of course, not to the extent that mathematics is not pity.
Hamilton's ideas on quaternions are found in his two books Lectures on the quaternions and Elements of quaternions (Figure 2), the latter of which was intended to be a simplified version of the previous one, but the deeper and thicker the writing became, Hamilton's attitude toward mathematics is evident. For more information, see the chapters on complex numbers, quaternions, and Hamilton in the humble works Majestic for One and At the Foot of the Clouds 4).
Figure 2 Hamilton's classic book The Origin of Quaternions
03
Maxwell, Gibbs, Hayside and Vector Analysis
Hamilton had a pro-Scottish Thai (1831-1901, Fig. 3), a famous mathematical physicist and one of the founders of thermodynamics5). Tate was of course a strong advocate of quaternions, a propagator of quaternions, and wrote about Treatise on quaternions. Tate had a classmate named James Clerk Maxwell (1831-1979), and maxwell learned quaternions from Tate, and of course understood the physical significance of quaternions, so he supported quaternions. Maxwell wrote an essay in 1871 on the mathematical classification of physical quantities, which I thought was an important document in the history of science—how many electromagnetic and electrodynamic literature did not understand Maxwell at all, and mistakenly regarded E and B as the same mathematical objects! Maxwell pointed out that It is of great physical significance that Hamilton divided the product result of the imaginary part of the quaternion into a scalar part and a vector part! Maxwell even thought that quaternions were a big step toward gaining knowledge of the quantities of space, comparable to Descartes' introduction of coordinate systems. Many physical phenomena have similar mathematical expressions, and if you pay attention to these mathematical forms, you can have a better understanding of physical phenomena6) - It has to be said that Maxwell was a really insightful physicist. Maxwell then discussed Hamilton's differential vector operator ∇, creating convergence and curl, i.e. curl
Acting on another vector of dot multiplication and fork multiplication, these two concepts. Maxwell saw the advantages of Hamilton's world vector representation of physical phenomena in space, rather than a simple calculation method "... but it is a method of thinking... It calls upon us at every step to form a mental image of geometrical features represented by symbols！ ”
Figure 3 Tate
In Maxwell's view, quaternions represent electromagnetic phenomena more directly than with coordinates, so that mathematics can explain the properties of electromagnetism more. Maxwell's scheme was to use both coordinates and quaternions, the bilingual scheme. One of the inconvenient things about this is that, according to the convention of quaternions, the vector is negative with its own point multiplication (modulo square), but when expressed in coordinates, it must be a positive number, which is a bit of a twist. In Maxwell's own words, the whole thing is ploughing with an ox and an ass together. Well, this sentence the author understands as a cropper second.
The inventors of vector analysis were Josiah Willard Gibbs (1839-1903) and Oliver Heaviside (1850-1925) of the English, see Figure 4. Gibbs, the Gibbs of Gibbs's free energy, one of the founders of statistical physics, coined the term statistical physics, made his field a well-nigh finished theoretical structure. Hewiside introduced complex numbers into circuit analysis, so there is no psychological obstacle to the use of quaternions in physics, and the current form of Maxwell's equations is what he wrote, which is a major achievement of vector analysis! Both men, while reading Maxwell's 1873 Treatise on Electricity and Magnetism, felt that there was a simplification or cutting of quaternions to express the practical needs of electromagnetism, and finally developed vector analysis independently. Gibbs, who had read Maxwell's work, noted that there was no need to preserve the entire set of quaternion algebras for electromagnetism, and in an 1888 letter he indicated his determination to invent vector analysis. Gibbs says he sees that as far as electromagnetism is concerned, it's not a good idea to keep the dot multiplication and fork multiplication of the vector in one equation (Gibbs was wrong at this point). Strict mathematics leads to correct physics), and he treats cross multiplication and point multiplication as two separate vector operations. Gibbs then constructed vector analysis with two kinds of multiplication, and differential operators ∇ different roles of scalars and vectors. Gibbs was also inspired by Hamilton's own writings and Tate's tretteise on quaternions,7 but was fundamentally Maxwell's thought. Gibbs's Elements of Vector Analysis was published in 1881, but it was not officially published until 1901, edited by his student Edwin Bidwell Wilson. During the same period, Hewieseld also developed vector analysis in England under the influence of Maxwell's writings, and in 1881 and 1883 published proposals on vector analysis under the title Of Relations between magnetic force and electric current, which he published in his 1893 issue of Electromagnetic theory. A book of volumes one mile. Hewiside heard in 1888 that the American Gibbs had also developed vector analysis.
Vector analysis (the world of three-dimensional physical space) is a pragmatic tailoring of quaternions for the expression of electromagnetism and electrodynamics, which has brought some convenience and development. However, because vector analysis is a pragmatic cut of rigorous quaternion algebra, its harm is also enormous. The numerous plastery of electro-dynamics textbooks and the many people like this author who have learned electro-dynamics but have learned a brain paste are proof!
Quaternions satisfy all algebraic operation rules except the commutation law of ordinary and binary numbers, and most importantly it has division. However, taking out the vector parts as a separate mathematical system, the problem can be troublesome: (1) the vector is a bit multiplicative and cross multiplication (even mistaken by some people as independent), and the resulting nature is not the same, in principle, they are not vectors (easy to understand the physics, the product of two identical objects, its dimension and itself are not the same, it must be a physical quantity of different properties!). The result of the fork multiplication only happens to be in three-dimensional space because the duality can be thought of as a vector 8) ;(2) the vector multiplication does not satisfy the law of union, which is a big thing, a big problem that quaternions do not have; (3) the vector does not have division; (4) the vector modulo square does not satisfy the identity of the modulo square product; (5) the vector is not zero but the cross multiplication may be zero (too terrible!), which is the most important thing algebra to avoid. These problems are all because the vector (in the world of ordinary quaternions) is only a part of the quaternion, so there is a possibility of fork multiplication, while vectors in the general sense have no cross multiplication operation. For example, a wave function in quantum mechanics can be thought of as a vector, but without the cross multiplication of a wave function. Later authors of electrodynamics textbooks did not understand the sources and properties of the vectors there (the world of ordinary quaternions) and their algorithms, and the more they copied, the more chaotic they became.
By 1893, a contradiction had emerged between those who embraced quaternions and those who embraced the new vector analyst. Tate, a mathematician who was the founder of quaternions and a researcher of quaternions, called vector analysis hermaphrodite monsters– monsters that are neither male nor female! That's true. Tate fights on two fronts, supporting the battlefield of quaternion pair coordinates and the battlefield of quaternion pair vector analysis. In his 1890 essay On the importance of quaternionsin physics, Tate praised "the natural qualities of quaternions." It directly explains the physical properties of space in the most obvious way by removed the arbitrariness of coordinatesand reveal the physical properties of space in the most obvious way." Quaternions are modes of representation. This is made clearer when quaternions later become operators to indicate rotation. Tate compares coordinates to sledgehammers, where there is work to be done with a hammer, while quaternions can be likened to elephant trunks, flexible, able to do anything, and active. Quaternions are natural! In the preface to the third edition of his Treatise on quaternions, Tate argues that Gibbs was "one of the retarders of quaternion progress", which now seems unfortunate to be his word. A large number of physics students do not understand quaternions and therefore do not understand the algebraic ideas in vector analysis, of course, they can only memorize the formula of the electro-dynamics textbook, which greatly hinders the teaching of electro-dynamics!
On the other hand, Hewiside, because he knew quaternions, did not have any negative evaluation of quaternions, he simply thought that quaternions were too difficult to understand and could only be understood by consummately profound metaphysicomathematicians. But the geometer Arthur Cayley (1821-1895) was less polite, declaring that quaternions were merely a special method or theory using coordinates, for which quaternions could only be considered special computational or analytical geometries. Gloria is an algebraic master, how can he not know the great significance of quaternions (this is the pioneering work of recent generational numbers), and his attitude towards quaternion algebra makes the author elusive. In fact, the dispute between quaternions and vector analysis is also a strange thing. Vector analysis is born out of the multiplication of the imaginary parts of the quaternion, and the special thing is that they are only differential operators, and it is said that they should not become two methods that are hostile to each other. It is said that vector analysis is supported by physicists because it is more directly related to physical problems, and this may not seem to make sense today. First of all, vector analysis has internal injuries and is also very troublesome to learn; as for being closer to the physical problem, it depends on what the physical problem is. Describe the rotation or use quaternions!
To add, because divisional algebras have only binary, quaternions, and octavians, vector cross multiplication holds only for three-dimensional vectors; if the result uniqueness requirement is abandoned, there is also a fork multiplication for seven-dimensional vectors. The operation corresponding to vector cross multiplication in other dimensional spaces is the outer product.
04
Grassmann, Pierce and Linear Algebra
Hamilton set out from the binary number, the complex number, to construct the triplet, and the result fell on the quaternion. As an abstract scholar armed to the teeth by metaphysics, Hamilton was also always on the lookout for polylets of ever higher dimensions. While Hamilton developed quaternions, the possibility of extending vectors to high-dimensional linear spaces was advanced by Hermann Gunther Grassmann (1809–1877, linguist, mathematician, physicist), as evidenced by his 1844 book Die lineale Ausdehnungslehre (Wiegand, 1844), published in 1844. This is the originator of linear algebra! In 1862, Grassmann also published Die Ausdehnungslehre. vollständig und in strenger Form begründet (Enslin, 1862). However, due to the pursuit of abstraction and rigor, Grassman's books were only read by fearless mathematicians, so his books were almost unpopular in the 30 years after their publication. Glazmann angrily quit the mathematical community and concentrated on linguistics. Grassmann recognized mathematics as a theory of forms, and he studied directed lines in space in the most general and abstract ways. The purpose of Grassmann's writings was to abstract geometry. Note that the image of a line with direction was later given to the word vector, and even became the standard interpretation of vector, which is also the origin of the Chinese vector word. I would like to emphasize again that vector, as a word, means carrier, and as a mathematical concept it is not a quantity of both size and direction: the nature of vector is defined by the linear algorithm it follows, and it can have no direction, or even length! In addition, Grassmann also provides us with the Grassmann algebra, which satisfies 1 ∙ ξ = ξ; ξ ∙ ξ = 0, you can see, which can be used to describe fermions.
Fig. 5 Grassmann and his Unfolding as a New Branch of Mathematics (some versions have the word "linear", linear unfolding)
Gibbs had read Grassmann's book when he published Elements of Vector Analysis in 1881. Some of the vector properties in Gibbs vector analysis are not limited to three-dimensional vectors. Hamilton, Grassman, and Gibbs all pointed to multiple algebra. However, Gloria believes that multi-algebra began with the American Benjamin Peirce (1809-1880, Figure 6), which is also an enthusiastic quaternion supporter. Pierce wrote in 1870 and formally published linear associate algebra (Johns Hopkins University Press, 1881) in 1881, and published analytic mechanics in 1855, which shows that he was deeply influenced by Hamilton. Pers argues that "the greatest value of the square root of minus one was its' magical power of doubling the actual universe, and placing by its side an ideal universe, its exact counterpart, with which it can be comparedand contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry!’ seelinear associate algebra, p.216—217”. In the book Linear Bound Algebra, Pierce summarized all the supercomputers and linear bound algebras of less than 7 units at that time. Gibbs published an article called Multiple Algebra in 1886, and Gloria also published an article of the same name in 1887. Erdehnungslehre, vector analysis, multiple algebra, and multiple associate algebra, all came together to bring together the linear algebra we all learned in college.
Figure 6 Perse
Speaking of linear algebra, the word matrix must be mentioned. The key word for linear algebra is linear transformation, which is the root of multiple algebra. Transformations of linear vectors are often represented by a matrix. The general history of science believes that the term matrix was proposed by Gloria in 1858, and Gibbs believes that the basis of Gloria's 1858 essay Memoir on the theory of matrices was actually found in Grassmann's Unfolded Learning in 1844! In fact, in 1844, Gotthold Eisenstein (1823-1852), the Allgemeine Untersuchungen über die Formen dritten Grades (general study of the cubic) also contained the idea of matrices, which Eisenstein published after his visit to Hamilton in the summer of 1843, and Hamilton was frightened by the conversation with him. It took all his strength to pick up the triplet study he had put down many times and invent the quaternion on October 16 of that year. In fact, things like Block ofn2 quantities, discriminants of algebraic systems of equations, and Jacobian discriminants of function transformations are easily reminiscent of matrices, or matrices.
Interestingly, since 1890 physicists have accepted vector analysis, but the quaternions, the parent of vector analysis, have been discarded (there is an idiom that buys beads and is specially prepared for this purpose). I didn't touch quaternions in ten years of college. Sir Edmund Whittaker (1873-1956) called for the resurrection of Hamilton's quaternions in 1940, but there was little response at the time. However, how can the profound mathematics and physics of quaternions be buried? Hamilton was convinced of the value of quaternions, which were a reflection of nature, and which would surely bring more mathematics and physics. We're just waiting to see it. Never mind when, Hamilton said to Tate in 1859. In fact, no matter how long it takes, the emergence of willian Kingdon Clifford (1845-1879) algebra, as the object of the action of quaternions, the proposal of spins, immediately let quaternions in electromagnetism and classical mechanics find more magical applications, so that physicists have to seriously study quaternion 9). As for the influence of the invention of algebraic objects and algebraic rules opened by quaternions on mathematics, the author does not understand, regardless of.
05
Redundant words
Hamilton was the first to remove the algebraic limits imposed on natural numbers, opening the door to near-generation numbers. The development of algebraics that removes these restrictions (commutative law, distribution law, union law) gives us a deeper understanding of the nature of these limits. This is a very philosophical thing, and it also inspires me to realize that too many things are only realized after they are lost. Algebra reflects the form, result, and logic of operations of numbers, which are natural and necessarily found in the physical world. Or conversely, to understand the mathematics that describes physics, it may be necessary for most of us to seriously learn algebra, the most basic discipline that we began to learn in the first grade of elementary school. Gibbs wrote a very sensational quote in Multiple algebra in 1886: "We begin by studying multiple algebra. We end, I think, by studying MULITPLE ALGEBRA!” Well, we started with multiple algebra, and when we ended up learning mulitple ALGEbra! If I had read this sentence in my freshman year, I am afraid that when I studied electricity-dynamics, I would not have let myself be confused.
Faced with the list of vector analysis formulas in the aforementioned electrodynamics textbook, my intuition at the time was that this thing must be wrong! What makes me sad and relieved is that my intuition is right. Where did this thing come from, how did it become popular and what are the reasons for its popularity, what are its inherent flaws and hazards, and how can we save ourselves? After many years of great waste, the author has a little rough understanding of these problems. The author shares it here, hoping to help young people learn physics, mathematics and, of course, other fields in the future. Regarding the study of knowledge, it is better to learn shallow than nothing. However, at different initial stages, people are following the teacher, and the teacher's level is crucial. Therefore, for teachers, I would like to say that "it is better to teach people to be shallow than nothing." The world is big, there are more occupations that can make a living, and the things that mislead people's children are not indispensable. If one chooses to be a teacher, one should not expect the miracle of "making one manifest oneself with his faintness." First-graders may only need to teach [4 × (3 + 2)- 6] ÷ 7 = 2, but if the teacher has practiced non-exchange algebra, non-combined algebra, or even non-cross-alternate numbers, the future of the child taught in the aforementioned formula may be different. Some people may say, how can you teach so much? I don't quite agree with that. As an old primary school student with 48 years of experience in school, I personally believe that learning should adopt a strategy of "cows eat grass". Cows eat grass, first eat a full stomach and then say, when free time, slowly digest through rumination. Knowledge is not a linear structure, and only under the premise of knowing more, especially knowing more about the framework of advanced learning, can we compare and confirm each other and finally achieve the effect of digestion. It may indeed not be possible to blindly pursue more teaching, but this is not an excuse to cover up the serious shortage of teachers' knowledge reserves. The main contradiction in education is always the contradiction between the learner's infinitely strong desire for knowledge and the educator's pitiful reserve of knowledge.
Teacher, in terms of ability, I may only be able to climb the small dirt slope behind the village, but I still hope that you will take me to see Everest.
2020.05.01Laber Day in Beijing
concentrate:
1) Why care what time it is! This is a phrase Hamilton said about when he invented the quaternion. Yes, good mathematics will always be useful, it just has to wait quietly for the advent of its time.
2) Electrodynamics, not electrodynamics, is electrodynamics. It is not theory of force, Kraftslehre. The so-called "four mechanics" in the context of Chinese is a distortion of physics.
3) Later, concepts such as world lines in relativity came from this, right?
4) It is expected to be published by the end of 2020 and the beginning of 2021.
5) No one pays attention to him in the thermodynamic world of Chinese.
6) This is presumably Maxwell's praise for Thomas Young.
7) Gibbs traveled to Europe for three years, but he was by no means a Fang Hongzhi figure.
8) Interested readers should refer to geometric algebra.
9) After learning the quaternion points, you will find J. J. Sakurai's quantum mechanics is a bit easier to understand. Besides, quaternions are at least useful for teaching electrodynamics.
Source: Journal of the Chinese Physical Society
Editor: Katsumi