@ Vasiliy Knyshev ,

What is your question ?

Kindly state and explain clearly the Newton's Second Law of the third order.

If x=sin(y), then y=arcsin(x) , so you see inverse functions are known and in use.

Newtons second law only involves a second derivative(with respect to time), so is of second order, not third order.

Only in radiation problems, ie. accelerating an electron , would the problem become (approximately) third order or higher.

The problems you mention are unrelated.

A non-degenerate function is a generalization of the inverse function. This allows equivalent use of direct and reverse function on each of at least a countable number of intervals. So get a mathematical proof the invasion of Newton's law of the second and third orders. --- See also .https: //sites.google.com/site/knysa1/home/ver2

Вывод закона Ньютона можно найти в pdf файле.

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The conclusion of Newton's law can be found in the pdf file.

Addressing inverse function only: If you were to draw a graph of a function y=f(x), the same graph draws the inverse function x=arc f(y). Rotate the graph 90 degrees and flip the graph over,

Невырожденная функция есть сумма интервалов, на каждом из которых она обратима(строго монотонна).Ничто не мешает рассматривать ее отдельно на каждом таком интервале.А не только на одном как sin.

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A non-degenerate function is the sum of intervals, on each

of which it is reversible (strictly monotonic). Nothing

prevents us from considering it separately on each such

interval. And not just on one, as sin.

what do you mean by degenerate function? one with more than one y for each x?

In that case you simply find the inverse function, for each branch of the direct function.

yes

Dear Juan,

You can not define any function which take one x as input parameter and give two y as output.. I mean this definition is the against the function definition itself....Come on Juan, you were professor...How can you support this claims?

Если arcsin определена на -90,+90, мы не можем ее рассматривать на других интервалах? Так и для любой функции.

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If arcsin is defined at -90, + 90, we can not consider it at

other intervals? So for any function.

Dear Vasiliy,

you wrote "arcsin is defined at -90, + 90, we can not consider it at other intervals?"

arcsin is an inverse function .. Inverse of any function only define if function is monotonic in certain domain.... sin function is monotonic in -90 to +90 .... Thats why arcsin is define between -90 to +90...

What is the point that you are clarifying?

I call them multi valued functions.

You can define whatever you like, as long as you do not confuse someone.

Dear Juan,

Please name any multivalued function...

На интервале -pi/2,+pi/2 sin имеет обратную функцию.По определению это arcsin.Почему ее не определили например на +pi/2,3pi/2? Остальные интервалы как-бы забыты?Поэтому остается допускать многозначность, но рассматривать функцию на данном интервале строгой монотонности.

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On the interval -pi / 2, + pi / 2 sin has an inverse function.

By definition, it is arcsin. Why wasn’t it defined, for

example, on + pi / 2,3pi / 2? The remaining intervals are

somehow forgotten? Therefore, it remains to admit polysemy,

but to consider a function on a given interval of strict

monotony.

Dear Vasiliy,

you wrote : "Why wasn’t it defined, for

example, on + pi / 2,3pi / 2?"

It is not defined between your mentioned interval because, you want minimum interval which can not be reduced further... If you subtract out Pi in your interval you will get exactly the same -pi/2 to pi/2.... Better you read how general solution writes in trigonometry... Link is here ... https://www.math-only-math.com/sin-theta-equals-sin-alpha.html

Я говорю о sin как о хорошем примере невырожденной функции.На интервале ее строгой монотонности можно дифференцировать прямую и обратную функцию не учитывая другие интервалы.Посмотрите мой файл pdf пункт 2.

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I speak of sin as a good example of a non-degenerate function. In the interval of its strict monotony, we can differentiate the direct and inverse functions without considering other intervals. Look at my pdf file, point 2.

Dear Vasiliy,

You can only differentiate two function if they have common domains .. Function and inverse function does not share a common domains ... This feature is known from ages... What new feature you are discovering here?

В каком смысле общий домен?Прямая функция дифференцируется по x, обратная по y, у них даже законы разные.

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In what sense is a common domain? A direct function is

differentiated in x, inverse in y, they even have different

laws.

Dear Vasiliy,

X and Y are overlapping? If not then you should not compare the two function ... Two function can only be compared if their domain intersects...

Меня в школе учили, что функция это не только закон, а также ее область определения и область значений.Какой домен у них общий?Возможно он там, где есть прямая и обратная функция?

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In school, I was taught that a function is not only a law,

but also its domain of definition and a domain of values.

What domain do they have in common? Perhaps it is where there

is a direct and inverse function?

Dear Vasiliy,

Your X unit is in theta/radian whereas Y unit is output of Sine function, which have no unit.... Both can not be compared... See some standard book on inverse trigonometric function ..

Dear Gokaran Shukla ,

Я вас абсолютно не понимаю.Возможно домен, о котором вы говорите-это как раз то куда можно свести определение arcsin для любой функции?

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Я надеюсь, что так как делаю я делать МОЖНО.

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Dear Gokaran Shukla, I absolutely do not understand you. Perhaps the domain you

are talking about is just where you can reduce the definition

of arcsin for any function? ---- I hope that as I do, you CAN.

Dear Gokaran Shukla ,

Большое вам спасибо за критику.

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Dear Gokaran Shukla, Thank you so much for the criticism.

Gokoran

y=sqrt(x) has two solutions for a positive value of x

I could give you many more examples

Dear Juan,

Question was one X but many Y as you have claimed.Thats why I told that this definition is against the basic definition of function .. .. Your example y =sqrt(X) will only give one y with one X... Your example is exactly the same as parabola with positive y, about x axis .

The square root of 1 is a number which squared gives you 1

1 time 1 is 1

-1times -1 is 1

What is your problem?

Dear Juan,

Thats why Y^2 =X does not qualify the function definition ... So, parabolic equation or any such equation where you get two value of y on same value of x does not qualify the function definition ... This has been taught when function topic introduce..

See how the function is defined in definition section https://www.wikiwand.com/en/Function_(mathematics)

See the relational approaches section where (2,B) and (2,C) has been written that it does not qualify for function definition ...

Non productive discussion...

У нас был писатель Достоевский.Буквальный перевод его фамилии-достал, в смысле заебал.Вот так Достоевский вас сделал.

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We had a writer Dostoevsky. A literal translation of his

last name-got it, in the sense of arrive at. That's what

Dostoevsky did to you.

Dear Juan,

Function has very strict and precise definition in mathematics ... You should not take it lightly..

I admire Dostoyevski; feodor

What does his last name mean?

Gokoran

Yes I know, but there is a difference if you say FUNCTION and MANY VALUED

FUNCTION. If your in science you should have an open mind.

Берем интервал на котором функция монотонна. И забываем про оставшуюся функцию. Работаем функцией на интервале. Так делать не запрещено! И так по всем интервалам невырожденной функции. Есть ли тут многозначность? Не знаю. Во всяком случае работая на каждом интервале монотонности можно ее избежать.

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We take the interval at which the function is monotone.

And forget about the remaining function. We work as a

function on the interval. So do not prohibited! And so

on for all intervals of nondegenerate function. Is there

ambiguity? I do not know. In any case, working on each

interval of monotony can be avoided.

Dear Juan,

Seems you did not read my shared wikipedia link... please read... Function definition is clearly given there....Multi valued function means that you can have same output with many input...Example... Y =x^2....Not the other way round that one input and multi output....

Dear Vasiliy,

Yes, if you know interval in which function is monotone, then you can define inverse function in that restricted range...

Наконец-то. Я очень рад. Возможно я объяснял плохо, либо с первого раза непонятно. Теперь далее. Все предыдущее - это уровень слов и идей. Этому есть математическое обоснование в виде теоремы. Интервал есть сумма конечного или счетного числа попарно не пересекающихся интервалов. Интервал это интервал определения функции. Если существует сумма интервалов на каждом из которых функция обратима, то ее называем невырожденной. Теперь отрицание невырожденной. Для всякой суммы интервалов существует интервал на котором функция необратима.

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Finally. I am very happy. Maybe I explained poorly, or from

the first time it is not clear. Now further. All previous is

the level of words and ideas. This is a mathematical

justification in the form of a theorem. The interval is the

sum of a finite or countable number of pairwise disjoint

intervals. An interval is a function definition interval.

If there is a sum of intervals on each of which the function

is reversible, then we call it nondegenerate. Now the negation

is nondegenerate. For any sum of intervals, there is an

interval at which the function is irreversible.

Gokoran

For the last time, if you have the same output with many different inputs, you

have a constant function and there is no problem with this. It is a proper function. f(x) =C

x is input, y is output

Please do not repeat again that there should be just one output y for FUNCTION,

for a given input x

I know this

MULTI VALUED FUNCTION means more than one output y for one input x

at least that is my understanding and definition.

Dear Juan,

Can you name any multivalued function according to your definition?...Your function should qualify the present understanding of mathematics..

I already gave you the square root. It does give two y for each x.

If x is negative you have two complex values for output.

Ahhhhh... I thought you are talking about real function ...Are you aware of the fact that comparison and inverse are not allowed for complex value function?

It is a real multivalued function in the positive x domain.

You can invert any complex number which is different from zero, and I do not understand what you mean by comparison.

So you can invert the complex function at any point except when it is zero.

Both the real and complex numbers are fields. (look up the technical definition)

Dear Juan,

Seems you are getting confused between inverse function as it defined in trigonometry and the inverse of any complex number ... Inverse of any number can not be inverse function.... Please share any link if you have..

Прекратите заниматься онанизмом, около математическим.

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Stop doing masturbation, about math.

Yes Gokoran

since you simply said inverse and not inverse function, I thought you just meant division

Inverse functions exist also for functions over complex variables.

Невырожденная функция дифференцируема почти всюду, по Лебегу.

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A non-degenerate function is differentiable almost everywhere,

according to Lebesgue.

Я такого не ожидал, от Dhritikesh Chakrabarty копай глубже, дорогой товарищ.

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I did not expect this, from Dhritikesh Chakrabarty dig deeper,

dear comrade.

The definition of function, one y for each x, allows all sorts of sharp corners to happen, or jumps. Therefore a function may not be differential at some points, or not be differential anywhere.

Привет Жан.

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Вполне согласен.. Жан не переживай таких точек, перегибов, экстремум умом слабый есть не более чем счетное число. Этаж над, какой переводчик .ИХ НЕ БОЛЕЕ ЧЕМ СЧЕТНОЕ ЧИСЛО, извращений нормального числа.

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Hi Jean. ----- I completely agree .. Jean do not worry about such points,

kinks, the extremum of the weak mind is no more than a

countable number. The floor is over, what a translator.

THERE ARE NO MORE THAN AN ACCOUNT NUMBER, perversions of a

normal number.

По теореме Лебега строго монотонная функция имеет почти всюду конечную производную. Если она равна нулю почти всюду, обратная функция будет бесконечной почти всюду, а это противоречит теореме Лебега. Поэтому невырожденная функция почти всюду имеет ненулевую конечную производную. А точек с нулевой производной будет не более счетного числа.

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By the Lebesgue theorem, a strictly monotone function has a

finite derivative almost everywhere. If it is zero almost

everywhere, the inverse function will be infinite almost

everywhere, and this contradicts Lebesgue's theorem.

Therefore, a non-degenerate function has a nonzero finite

derivative almost everywhere. And points with zero derivative

will be no more than a countable number.