What for

When reading a monograph, textbook or article, sometimes you come across a text that for some reason causes bewilderment, it is somehow incomprehensible, suspicious. After some more or less lengthy reflection, either understanding comes, or you start to suspect the text in some kind of blunder, misunderstanding, “squiggle”. I will cite several such texts.

In general, it seems to me that it would be nice to have an updated collection of “Squiggles and Bloopers”, which is freely available on the Internet. In my opinion, this would greatly help the pedagogy of textbooks.

And here are the definitions:

Blooper - a blatant or veiled mistake, not bearing, however, a fundamental nature.

Zagogulina is a phrase, a theme set forth in such a way that to understand it one needs to smash one’s head (ordinary, not genius and not talent).

And also, it would be nice to have a Curious study heading. Surely among the Habrovites most of the former or just students. And many have something to tell about their studies, both funny and sad. I will give an example from my studies.

And now let's get down to bloopers and squiggles.

Scattering cross section

I really like the book “Subatomic Physics” by Frauenfelder and Henley, Mir, Moscow, 1979. But, however, take pages 151-152. It is a definition of the concept of section. This concept generalizes the visual perception of the contour and area of the subject. A particle beam incident on a target is considered.

Let the particles scattered by the target be registered by a counter that includes all particles scattered by an angle θ within a certain solid angle dΩ. The number of particle registrations dR in such a counter per unit time is proportional to the flux F of incident particles, the solid angle dΩ and the number N of independent scattering centers present in the target and located on the path of the incident beam:

dR = FNσ (θ) dΩ

The proportionality coefficient is denoted by σ (θ); it is called the differential effective scattering cross section; we can write

σ (θ) dΩ = dσ (θ), i.e. σ (θ) = (dσ (θ)) / dΩ

Then I was perplexed. What is the trick? Consider the last formula. Following the same logic, we can write for the coordinate x (t) of the material point: x (t) dt = dx. But we know that dx = v (t) dt = x '(t) dt is the speed times dt.

I seem to need something like this:

dR = FNσ '(θ) dΩ

The proportionality coefficient is denoted by σ '(θ); it is called the differential effective scattering cross section; we can write

σ '(θ) dΩ = dσ (θ), i.e. σ '(θ) = dσ (θ) / dΩ

Where σ (θ) is the integral section. σ (2π) is the total cross section.

Here is such a blunder, in my opinion.

PS The smoking room is alive.

A lot of time has passed. It threw more material. I take the book “Introduction to the physics of the nucleus and particles. The author - Kapitonov. On the title page it is written “Approved by the Ministry of Education of the Russian Federation as a teaching aid for students of physics departments of classical universities, and also for students of other universities enrolled in the specialty“ Nuclear Physics ”and the direction“ Physics ”. I take p. 16 and what do I see? But that: σ (θ) = dσ (θ) / dΩ. Again the notorious formula.

Yes, they broke up uncritically.

Poor students!

Lyap Landau and Lifshits

The following mistake was noticed by my teacher at the seminar on the general theory of relativity during my studies at BSU. Take the “Field Theory” of Landau and Lifshitz for 1967. This is a covariant derivative. It is based on the concept of parallel transfer of a scalar, vector, tensor. In the usual derivative:

the difference between the values of the function in x and in x + dx is calculated
this difference is divided by dx
make the limit dx-> 0

In local field theory, the indicated subtraction cannot be done, because the difference in values, for example, of a scalar at different points is no longer a scalar field, but we would like to have a scalar one. To do this, transfer the scalar in parallel from x + dx to x and subtract the value of the transferred scalar from the value of the one sitting at x. It will already be a scalar. So, you need to know the parallel transfer rule. Usually it is not given by nature, but determined by the physicist. We assume that the parallel transfer rule is defined. Then we can determine the covariant derivative like the usual one, only taking into account the parallel transfer. Let DA be the above-mentioned difference in the values of the vector A, which is parallel transferred from x + dx to the point x and the vector A at the point x. And here is what Landau writes with Lifshitz:

Well, what's the blunder?

Back to

D A_{i} = g_{i},_{k} D A^{k}

$DA_i = g_i, _k DA ^ k$ . This can be understood in two ways:

or

(D A)_{i} = g_{i},_{k} (D A)^{k}

$(DA) _i = g_i, _k (DA) ^ k$ or

D (A_{i}) = g_{i},_{k} D (A^{k})

$D (A_i) = g_i, _k D (A ^ k)$

We defined

D A

$DA$ , but not

D (A_{i})

$D (A_i)$ , then the first understanding is correct. So we have

(D A)_{i} = g_{i},_{k} (D A)^{k}

$(DA) _i = g_i, _k (DA) ^ k$

but it doesn’t follow from here

g_{i},_{k} (D A)^{k} = D (g_{i},_{k} A^{k})

$g_i, _k (DA) ^ k = D (g_i, _k A ^ k)$

Then record

D g_{i},_{k} A^{k} = g_{i},_{k} D A^{k} + A^{k} D g_{i},_{k}

$Dg_i, _k A ^ k = g_i, _k DA ^ k + A ^ k Dg_i, _k$ completely out of nowhere. D is defined by an action on A, not an action on its components.

Maybe it will be even more intelligible like this:

Yes, there is a linearity property

D (\sum_{i} V^{i}) = \sum_{i} D (V^{i})

$D (∑_iV ^ i) = ∑_iD (V ^ i)$

Where

V^{i}

$V ^ i$ Is the i-th vector, but not the i-component

A-priory

A_{i} = g_{i},_{k} A^{k} = \sum_{k} g_{i},_{k} A^{k}

$A_i = g_i, _k A ^ k = ∑_kg_i, _k A ^ k$

Where

A^{k}

$A ^ k$ Is the kth component of the vector A

But that doesn’t mean that

D A_{i} = g_{i},_{k} D A^{k} = \sum_{k} g_{i},_{k} D A^{k} = D (\sum_{k} g_{i},_{k} A^{k}) = D (g_{i},_{k} A^{k})

$DA_i = g_i, _k DA ^ k = ∑_kg_i, _k DA ^ k = D (∑_kg_i, _k A ^ k) = D (g_i, _k A ^ k)$

The first equality follows from the definition, the second and fourth equality is Einstein’s rule, but the third equality is incorrect : here you cannot use the additivity property, since we are dealing with components, not vectors. This is a defect in the notation when we do not parse where the kth vector, and where the kth component of one vector. In a componentless record, such a blunder would not have passed.

And what is the correct proof? But this is not. Mathematical, at least. This is a matter of definition.

Take the textbook Mishchenko and Fomenko “A Short Course in Differential Geometry and Topology”. Paragraph “Connectivity and covariant differentiation”. The property Dg = 0 is postulated there.
In the excellent book "Geometric Methods in Physics" Schutz Dg = 0 is derived as a consequence of additional conditions - the consistency of the metric and volume.
In the course “Gravity” (Mizner, Thorne, Wheeler), the equality Dg = 0 is motivated by physical considerations. This is done something like this. Locally, space time is as close as possible to the flat Minkowski space. In it, the derivatives of the metric tensor are 0. And in flat space, the components of the covariant derivative are ordinary derivatives of the metric tensor. Therefore, the covariant derivative in the local flat space is equal to zero. But this equality is tensor in nature. So this equality is true in any other coordinate system.

I’m watching the latest edition of Field Theory, and there everything is the same.

Big squiggle mass

Newtonian mechanics

At Newton, mass was defined as an additive measure of the quantity of a substance: a system of three hydrogen atoms has a mass three times that of one hydrogen atom. Therefore, taking the mass of the hydrogen atom as a unit of mass, we get the opportunity to measure the masses of all bodies. Further, Newton’s second law says that mass is a measure of the inertia of the body.

And, further, the law of universal gravitation and the equality of gravitational and inert mass indicate that mass serves as a gravitational charge.

So, in Newtonian mechanics, mass serves as:

Substance quantity measure
Inertia measure
Measure of gravitational interaction, gravitational charge

Mass is a scalar. In whatever system it is measured, we get the same value. Mass and energy are measured differently. But there is a connection between them:

kinetic energy of a particle is expressed in terms of mass: $T = mv ^ 2/2$
potential energy of a particle in a gravitational field $φ$ expressed in terms of mass: $U = -mφ$

So, the properties of mass in classical mechanics:

Mass scalar
Mass is additive
Mass is a measure of inertia
Mass is a measure of the amount of a substance.
Mass - gravitational charge, source of gravitational field

Relativistic mechanics

The theory of relativity has made its complications. The concept of mass defect appeared - a manifestation of mass nonadditivity , when energy is generated during the formation of the system. This will manifest itself as a decrease in the mass of the system compared to the sum of the masses of the subsystems. So a positron and an electron in a collision can turn into photons and, therefore, matter will disappear altogether. This means that mass is not suitable as a measure of the quantity of a substance .

Now about the measure of inertia. Consider a moving material point. If we try to measure mass as a measure of inertia (applying force to a point, we look at the acceleration of the point), we find that depending on the direction of the force, mass as a measure of inertia will be different. We can talk about the longitudinal mass (the force is parallel to the velocity), about the transverse mass (the force is perpendicular to the velocity). So, mass as a measure of inertia is also not good .

Now about the mass as a gravitational charge. The general theory of relativity is based on the fact that the source of the gravitational field is not a scalar (rest mass, for example), but a tensor - an energy-momentum tensor. This means that mass loses the role of a measure of gravitational interaction . So a photon with energy

h ν

$hν$ gravitational field is by no means measured

h ν / c^{2}

$hν / c ^ 2$ . Otherwise, a general theory of relativity would not be needed to explain the effect of a deflection of a light beam in a gravitational field.

And only when the body is at rest does its mass of rest serve as a measure of inertia.

Mass is not a scalar, and, in general, is not an independent physical quantity. Only mass of rest makes sense. The so-called relativistic mass

m = E / c^{2}

$m = E / c ^ 2$ it does not have a specific dimension, but is expressed through energy, which means that the concept of relativistic mass is redundant.

In this regard, modern physics avoids the concept of relativistic mass, defined as

m = E / c^{2}

$m = E / c ^ 2$ . This value is not a measure of inertia and is not a measure of gravity. Then the measure of what? A measure of energy? So for this there is energy itself and it is not necessary to introduce another equivalent measure. Occam's razor cuts off unnecessary concepts.

The rest mass is included in the relativistic relation for a particle:

E^{2} - (p ⃗)^{2} c^{2} = (m_{0}^{2} c^{2})^{2}

$E ^ 2- (p ⃗) ^ 2 c ^ 2 = (m_0 ^ 2 c ^ 2) ^ 2$ Where

m_{0}

$m_0$ - mass of rest.

(E, p c) is a relativistic 4-vector. This shows that

m = E / c^{2}

$m = E / c ^ 2$ - only another measure E. It can be interpreted in terms of the rest mass as follows: if the system at rest had an energy reserve E, then its rest mass would equal

m = E / c^{2}

$m = E / c ^ 2$ . But she is not alone. So its rest mass will be less, down to zero (photon, for example).

We summarize. In relation to mass, there is only the concept of rest mass. The concept of relativistic mass

m = E / c^{2}

$m = E / c ^ 2$ not necessary, it is conceptually redundant and only confusing. And in the modern presentation, m is always understood as the mass of rest (which was previously designated as

m_{0}

$m_0$ ), but they do not use the concept of relativistic mass and write the relativistic invariant for the energy-momentum in the form

E^{2} - (p ⃗)^{2} c^{2} = (m^{2} c^{2})^{2}

$E ^ 2- (p ⃗) ^ 2 c ^ 2 = (m ^ 2 c ^ 2) ^ 2$

The famous physicist Perch wrote a number of convincing articles on this subject.

but

And now, attention, questions. Apply the logic of Perch to these 4 vectors:

4th event vector: (ct, x ). We have an invariant $(ct) ^ 2- (x ⃗) ^ 2 = (cτ) ^ 2$ . Everything is similar to the 4th energy-momentum vector. Have your own time $τ$ , analogous to the rest mass, is t - relativistic time, analogous to relativistic mass. So you need to abandon the concept of relativistic time t between events, preserving only your own $τ$ : after all, this relativistic time is similar to the relativistic mass. But nobody seems to refuse the concept of time dilation. But the relativistic increase in mass must be abandoned. What is the difference?
Wave 4th vector (ω / s, k ) of a plane relativistic wave. We have an invariant $(ω / s) ^ 2- (k ⃗) ^ 2 = (ω_0 / s) ^ 2$ . Everything is similar to the 4th energy-momentum vector. So you can’t talk about the relativistic frequency of the ω wave, and, therefore, the Doppler effect for the movement of the observer relative to the wave source?

So is Perch right? And what does rightness mean here?

Little scalp squiggle

Take the mathematical encyclopedia. Here is the definition of a scalar.

A scalar is a quantity, each value of which can be expressed by one (real) number. In the general case, the scalar is an element of some field

Following this definition, the x-coordinate of a point is a scalar. However, in physics this is not so. The numbers there are different from the scalars . Physicists call a quantity whose value is expressed by a single number and this value does not depend on the choice of a reference frame .

The scalars in this sense are in Newtonian mechanics:

• number of items

• temperature

• distance

• electric charge

• weight

• volume

• duration

• scalar product of two vectors

• scalar convolution of any tensor

In the special theory of relativity, mass, volume, duration, distance already depend on the reference frame and are not scalars. Examples of scalars in service stations:

• number of items

• temperature

• interval between events

• electric charge

• scalar product of two 4 vectors

• scalar convolution of any tensor

The coordinate depends on the reference frame and is not a scalar. So grains are separated from the chaff - essential from non-essential, due to the choice of a reference frame.

Of course, definition is a matter of taste. But here the disagreement is annoying, since mathematicians often refer to physical laws, and the concept of a scalar there differs from the mathematical one. And references to physics are now more and more common. So even in a very bourgeois book by Kostrikin and Manin, “Linear Algebra and Geometry,” there are several references to the values of the space of quantum mechanics. Modern theoretical physics attracts a lot of mathematicians and a lot of mathematics and the disagreement of physical and mathematical interpretations is annoying. Moreover, for a number there is a name - "number". Why does he need another second name - a scalar?

I attribute this situation to the category of zagogulin.

And now

Curiosities of study.

Nuclear Physics Exam and Probability Theory

4th year physics department of BSU. I surrender nuclear physics. As I remember now, I got a problem on the Breit-Wigner formula. There is a formula that describes (if I have not forgotten) the energy curve of an unsteady state. Or something like that. The resonance curve actually. It looks like a bell. Everything seems to be done and waiting for me to be called. Professor P. took the exam. It must be said that he was an avid tennis player and before that he won some kind of republican level prize. On this occasion, as he put it more diplomatically, he "took to his chest." It smells like cognac and not Pliska (a Bulgarian Pliska could afford a student once a month: 7 rubles a bottle, and a scholarship 35 - regular or 42 - increased.) And then a Vietnamese student Huang is holding an exam in front of me. There were many Vietnamese on the course. The fact is that at that time the Americans fought in Vietnam on the south side, and the Chinese and our guys fought on the north side. And we helped Vietnam with weapons and education. So, Huang answers. And the professor listens and smokes. And suddenly I see: the professor takes a cigarette out of his mouth and instead of an ashtray, for some reason, he pokes in the offset of Huang. Then he says: “Cheeeepukhaaa! .. Huang! If I were you, I would not teach nuclear physics, but would sit in the jungle and shoot Americans. But you don’t know nuclear physics. Two points. Come to retake tomorrow. Who is next?". Huang was dumbfounded, and with bewilderment and tears in his eyes left the audience. The next one was me. I start talking about the Breit-Wigner formula. Professor: "Show the task." Looked, looked and "Cheeeepuuuhaaa! Two points!". I trembling voice: "When to retake?". He: "Tomorrow."

I went to the hostel. I'm getting ready for tomorrow. I’m wondering what is wrong with my decision? Thought, thought, but invented nothing else. And then he turned to probability theory: this ticket tomorrow I can’t get in any way. So there’s nothing to puzzle. And I relied on probability and calmed down.

I come tomorrow. The professor sits slightly bruised. But surprisingly vigorous, clean-shaven and smells of cologne ... However, it seems he has become sober. Offers to pull a ticket. I pull. Fathers !!! Yesterday’s ticket comes across. Here you have the theory of probability. I almost fainted. What to do? I haven’t come up with a new solution, and I start with yesterday’s trembling voice as before. The professor listened, listened and said: "Well, my friend is so fine. Why did you blunder so much yesterday?" I was taken aback a little and answer - they say, a little confused. - "That is OK. I can’t put five already, but I’ll put four. ” This is how the four in nuclear physics appeared in my record.

I must say, then I did not appreciate the humor of the situation. It was not funny. And now, after many years, I recall this with pleasure. And already funny.

And now about women

I remember how the same professor P. had a very pretty graduate student at the department. More precisely, a beautiful, slender blonde. She led a nuclear physics practice with us. We make labs, and do not take our eyes off the graduate student. And her eyes are sad. I still remember how she sat at the table, and in front of her is the thick, large-format volume of Schweber's “Introduction to Relativistic Quantum Field Theory”. And she looked with such longing at this mischief that she involuntarily aroused sympathy. Only later, as an intern at the research institute, did I study this volume and then, it seems, understood the reason for her longing. No, quantum field theory is contraindicated for girls. Especially beautiful.

In our third year, one student decided to study all the Landau quantum mechanics from A to Z. Very nice girl. The case ended in the fact that she could not stand the load and with a nervous breakdown was hospitalized. A year later, she recovered, but she was already sad all the time. Moral: Quantums and girls are incompatible. And the inner voice of programming experience whispers to me and such: "Programming and girls are incompatible." I remembered that once I presumptuously thought that I would understand any program of a project under development. But the programmer went on maternity leave, and then her program failed. I take the program and try to understand the logic. And it is crammed with GO TO statements with a dynamically changeable transition label. And these labels change in different parts in different ways. No matter how I tried to establish control over the transitions, but to my shame I could not cope with it. But I rewrote the program according to all the rules of structural programming without one GO TO and was very pleased with myself. Further, I was always wary of female logic in programming. In my opinion, this logic is not in the first place.

Blunders of textbooks and curiosities of study