Dot - abstract object in space that does not have any measurable characteristics (a zero-dimensional object). The dot is one of fundamental concepts in mathematics.
Point in Euclidean geometry
Euclid defined a point as "an object without parts". In the modern axiomatics of Euclidean geometry, a point is a primary concept, given only by a list of its properties - axioms.
In the chosen coordinate system, any point of the two-dimensional Euclidean space can be represented as an ordered pair ( x; y) real numbers. Likewise, point n-dimensional Euclidean space (as well as vector or affine space) can be represented as a tuple ( a 1 , a 2 , … , a n) from n numbers.
Links
- point(English) on the PlanetMath website.
- Weisstein, Eric W. Point on the Wolfram MathWorld website.
point is:
dot dot noun, well., use Often Morphology: (no) what? dots, what? dot, (see) what? dot, how? dot, about what? about the point; pl. what? dots, (no) what? points, what? points, (see) what? dots, how? dots, about what? about points 1. Dot- this is a small round speck, a trace from a touch with something sharp or writing.Dot pattern. | Puncture point. | The city on the map is indicated by a small dot and the availability bypass road one can only guess.
2. Dot- this is something very small, poorly visible due to remoteness or for other reasons.
Point on the horizon. | As the ball approached the horizon in the western part of the sky, it began to slowly decrease in size until it turned into a dot.
3. Dot- a punctuation mark that is placed at the end of a sentence or when abbreviating words.
Put a point. | Don't forget to put a dot at the end of the sentence
4. In mathematics, geometry and physics dot is a unit having a position in space, the boundary of a line segment.
5. dot called certain place in space, on the ground or on the surface of something.
placement point. | Pain point.
6. dot name the place where something is located or carried out, a certain node in the system or network of any points.
Each outlet must have its own sign.
7. dot they call the limit of development of something, a certain level or moment in development.
Nai highest point. | point in development. | The state of affairs has reached a critical point. | This is the highest point of manifestation of the spiritual power of man.
8. dot called the temperature limit at which the transformation of a substance from one state of aggregation into another.
Boiling point. | Freezing point. | Melting point. | How more height the lower the boiling point of water.
9. Semicolon (;) called a punctuation mark used to separate common, more independent parts compound sentence.
AT English language practically the same punctuation marks are used as in Russian: dot, comma, semicolon, dash, apostrophe, brackets, ellipsis, interrogative and exclamation marks, hyphen.
10. When they talk about point of view, mean someone's opinion about a certain problem, a look at things.
Less popular now is another point of view, previously almost universally recognized. | Nobody shares this point of view today.
11. If people are said to have points of contact so they have common interests.
We may be able to find common ground.
12. If something is said dot to dot, meaning an absolutely exact match.
Dot to dot in the place where it was indicated, there was a coffee-colored car.
13. If a person is said to be reached the point, which means that he has reached the extreme limit in the manifestation of some negative qualities.
We've reached the point! You can't live like this anymore! | You can't tell him that the secret services have reached the point under his wise leadership.
14. If someone puts an end in some business, it means that he stops it.
Then he returned from emigration to his homeland, to Russia, to Soviet Union, and this put an end to all his searches and thoughts.
15. If someone dot the "and"(or over i), which means that he brings the matter to its logical conclusion, leaves nothing unsaid.
Let's dot the i's. I didn't know anything about your initiative.
16. If someone hits one point, which means that he concentrated all his forces on achieving one goal.
That is why his images are so distinct; he always hits one point, never getting carried away by secondary details. | He understands very well what the task of his business is, and purposefully hits one point.
17. If someone hit the spot, which means that he said or did exactly what was needed, guessed it.
The very first letter that came to the next round of the competition pleasantly surprised the editors - in one of the listed options, our reader immediately hit the mark!
point adj.Acupressure.
Explanatory dictionary of the Russian language Dmitriev. D.V. Dmitriev. 2003.
Dot
Dot Can mean:
Wiktionary has an article "dot"- A point is an abstract object in space that does not have any measurable characteristics other than coordinates.
- Dot - diacritic, which can be placed above, below or in the middle of the letter.
- Point - a unit of distance measurement in Russian and English systems measures.
- The dot is one of the representations of the decimal separator.
- Dot (network technologies) - designation of the root domain in the hierarchy of global network domains.
- Tochka - chain of electronics and entertainment stores
- Tochka - album of the group "Leningrad"
- Point - Russian film of 2006 based on the story of the same name by Grigory Ryazhsky
- Dot is the second studio album by rapper Sten.
- Tochka is a divisional missile system.
- Tochka - Krasnoyarsk Youth and Subcultural Journal.
- Tochka is a club and concert venue in Moscow.
- The dot is one of the characters in Morse code.
- The point is the place of combat duty.
- Point (processing) - the process of machining, turning, sharpening.
- POINT - Information and analytical program on NTV.
- Tochka is a rock band from the city of Norilsk, founded in 2012.
Toponym
Kazakhstan
- Dot- until 1992, the name of the village Bayash Utepov in the Ulan district of the East Kazakhstan region.
Russia
- Tochka is a village in the Sheksninsky district of the Vologda region.
- Tochka is a village in the Volotovsky district of the Novgorod region.
- Tochka is a village in the Lopatinsky district of the Penza region.
Can you give a definition of such concepts as a point and a line?
Our schools and universities did not have these definitions, although they are key in my opinion (I don’t know how this is in other countries). We can define these concepts as "successful and unsuccessful" and consider whether this is useful for the development of thinking.
Wrestler
Strange, but we were given the definition of a point. This is an abstract object (convention) located in space, which has no dimensions. This is the first thing that was hammered into our heads at school - a point has no dimensions, it is a "zero-dimensional" object. A conditional concept, like everything else in geometry.
Straight lines are even more difficult. First of all, it's a line. Secondly, it is a set of points located in space in a certain way. In the very simple definition it is a line defined by the two points through which it passes.
Medivh
A point is some kind of abstract object. A point has coordinates but no mass or dimensions. In geometry, everything begins precisely from a point, this is the beginning of all other figures. (In writing, by the way, too, without a point there will be no beginning of a word). A straight line is the distance between two points.
Leonid Kutny
You can define anything and anything. But there is a question: will this definition "work" in a particular science? Based on what we have, it makes no sense to define a point, a line and a plane. I really liked Arthur's remarks. I would like to add that a point has many properties: it has no length, width, height, no mass and weight, etc. But the main property of a point is that it clearly indicates the location of an object, an object on plane, in space. That's why we need a point! But, a smart reader will say that then a book, a chair, a watch and other things can be taken as a point. Absolutely right! Therefore, it makes no sense to define a point. Sincerely, L.A. Kutniy
A straight line is one of the basic concepts of geometry.
The period is a punctuation mark in writing in many languages.
Also, the dot is one of the symbols of Morse code
So many definitions :D
The definitions of a point, a line, a plane were given by me back in the late 80s and early 90s of the 20th century. I give a link:
https://yadi.sk/d/bn5Cr4iirZwDP
In a 328-page volume, the cognitive essence of these concepts is described in a completely new aspect, which are explained on the basis of a real physical worldview and a sense of I exist, which means "I" exist, just as the Universe itself to which I belong exists.
Everything written in this work is confirmed by the knowledge of mankind about nature and its properties long ago discovered and still being studied on this moment time. Mathematics has become so complex to understand and comprehend in order to apply its abstract images to the practice of technological breakthroughs. Having revealed the Foundations, which are the fundamental principles, it is possible to explain even to a student elementary school reasons underlying the existence of the universe. Read and come closer to the Truth. Dare, the world in which we exist opens before you in a new light.
Is there a definition of the concept of "point" in mathematics, geometry.
Mikhail Levin
"indefinable concept" is a definition?
In fact, it is the uncertainty of concepts that makes it possible to apply mathematics to different objects.
A mathematician can even say "by a point I will mean a Euclidean plane, by a plane - a Euclidean point" - check all the axioms and get new geometry or new theorems.
The point is that to define term A, you need to use term B. To define B, you need term C. And so on ad infinitum. And in order to be saved from this infinity, one has to accept some terms without definitions and build definitions of others on them. ©
Grigory Piven
In mathematics, Piven Grigory A point is a part of space that is abstractly (mirrored) taken as the minimum length segment equal to 1, which is used to measure other parts of space. Therefore, a person chooses the scale of a point for convenience, for a productive measurement process: 1mm, 1cm, 1m, 1km, 1a. e., 1 St. year. etc.
See also: http://akotlin.com/index.php?sec=1&lnk=2_07
Abstraction has been used in mathematics for two and a half millennia. dimensionless point, which contradicts not only common sense, but also knowledge about the surrounding world, obtained by such sciences as physics, chemistry, quantum mechanics and informatics.
Unlike other abstractions, the abstraction of a dimensionless mathematical point does not idealize reality, simplifying its cognition, but deliberately distorts it, giving it the opposite meaning, which, in particular, makes it fundamentally impossible to understand and study spaces of higher dimensions!
The use of the abstraction of a dimensionless point in mathematics can be compared with the use of the basic monetary unit with zero cost. Fortunately, the economy did not think of this.
Let us prove the absurdity of the abstraction of a dimensionless point.
Theorem. The mathematical point is voluminous.
Proof.
Since in mathematics
Point_size = 0,
For a segment of finite (nonzero) length, we have
Segment_size = 0 + 0 + ... + 0 = 0.
The obtained zero size of the segment, as a sequence of its constituent points, contradicts the condition of finite length of the segment. In addition, the zero point size is absurd in that the sum of zeros does not depend on the number of terms, that is, the number of "zero" points in the segment does not affect the size of the segment.
Therefore, the original assumption about the zero size of a mathematical point is WRONG.
Thus, it can be argued that a mathematical point has a non-zero (finite) size. Since the point belongs not only to the segment, but also to the space in which the segment is located, it has the dimension of space, that is, the mathematical point is volumetric. Q.E.D.
Consequence.
The above proof, performed using the mathematical apparatus junior group kindergarten instills pride in the boundless wisdom of the priests and adepts of the “queen of all sciences”, who managed to carry through the millennia and preserve for posterity in its original form the archancient delusion of mankind.
Reviews
Dear Alexander! I'm not strong in mathematics, but maybe YOU can tell me where and by whom it is stated that the point is equal to zero? Another thing, she has infinite small amount, up to the convention, but not zero at all. Thus, any segment can be considered zero, since there is another segment that contains infinite set initial segments, roughly speaking. Maybe we should not confuse mathematics and physics. Mathematics is the science of being, physics is about the existing. Sincerely.
I mentioned Achilles twice in detail and many times in passing:
"Why won't Achilles catch up with the tortoise"
"Achilles and the tortoise - a paradox in a cube"
Maybe one solution to Zeno's paradox is that space is discrete and time is continuous. He considered, as it is possible for you, that both are discrete. The body can remain at some point in space for some time. But it cannot be in different places at the same time at the same time. This is all, of course, amateurishness, like our entire dialogue. Sincerely.
By the way, if a point is 3D, what are its dimensions?
The discreteness of time follows, for example, from the aporia "Arrow". “Simultaneously stay in different places” can only be an electron for physicists who, in principle, do not understand and do not accept either the structure of the ether or the structure of 4-dimensional space. I don't know of any other examples of this phenomenon. I see no "amateurism" in our conversation. On the contrary, everything is extremely simple: a point is either dimensionless or has a size; continuity and infinity either exist or they do not. The third is not given - either TRUE or FALSE! Fundamentals mathematicians, unfortunately, are built on false dogmas, accepted out of ignorance 2500 years ago.
The point size depends on the condition of the problem being solved and on the required accuracy. For example, if a gear is designed for wrist watch, then the accuracy can be limited by the size of the atom, that is, eight decimal places. The atom itself here will be the physical analogue of the mathematical point. You may need 16-character precision somewhere; then the role of a point will be played by a particle of ether. Note that talk about allegedly "infinite" accuracy in practice turns into wild nonsense, or, to put it mildly, absurdity.
I still don't understand: does the point exist? If it exists objectively, therefore it has a certain physical value, if it exists subjectively, in the form of an abstraction of our mind, then it has a mathematical value. Zero has NOTHING, it does not exist, this is the abstract definition of Non-existence in mathematics or emptiness in physics. The point does not exist by itself outside of the relationship. As soon as the second point appears, a segment appears - Something, etc. This topic can be developed endlessly. With uv.
It seemed to me that I brought good example, but probably not detailed enough. Objectively, there is a World that science cognizes, and at the present time cognizes mainly mathematical methods. Mathematics cognizes the world by constructing mathematical models. To build these models, the basic mathematical abstractions, in particular, such as: point, line, continuity, infinity. These abstractions are basic because it is no longer possible to further subdivide and simplify them. Each of the basic abstractions can be either adequate objective reality(true) or not (false). All of the above abstractions are initially false, because they contradict the latest knowledge about the real world. So these abstractions prevent correct understanding real world. One could somehow put up with this while science was studying the 3-dimensional world. However, the abstractions of a dimensionless point and continuity make all worlds of higher dimension unknowable in principle!
The brick of the universe - a point - cannot be a void. Everyone knows that nothing comes from emptiness. Physicists, declaring the ether non-existent, filled the world with emptiness. I believe that mathematics with its empty point pushed them to this stupidity. I'm not talking about atoms-points of worlds of higher dimension than 4D. So, for each dimension the role of an indivisible (conditionally) mathematical point is played by the (conditionally) indivisible atom of this world (space, matter). For 3D - a physical atom, for 4D - an ether particle, for 5D - an astral atom, for 6D - a mental atom, and so on. Sincerely,
So, does the brick of the universe have any absolute value? And what does it represent, in your opinion, in the ethereal or mental world. I'm afraid to ask about the worlds themselves. With interest...
Ether particles (these are not atoms!) are electron-positron pairs, in which the particles themselves rotate relative to each other at the speed of light. This fully explains the structure of all nucleons, the propagation electromagnetic oscillations and all the effects of the so-called physical vacuum. The structure of the atom of thought is unknown to anyone. There is only evidence that ALL the most higher worlds material, that is, they have their own atoms. Up to the matter of the Absolute. You're being ironic, though. Really wormholes and big bangs Do you find it more believable?
What is the irony here, just a little taken aback after such an avalanche of information. I, unlike you, am not a professional and I find it difficult to say anything about the five- or six-dimensionality of spaces. I'm all about our long-suffering point ... As far as I understand, you are against material continuity, and the point is that you have a really existing "democratic" atom. "Brick of the Universe". Maybe I was inattentive, but still, do not hesitate to repeat what its structure, physical parameters, dimensions, etc. are.
And also answer, does the unit exist in itself, as such, outside of any relations? Thank you.
Having dealt with what units of measurement and dimension are, we can now move on to the actual measurements. AT school mathematics two measuring instrument- (1) a ruler for measuring distances and (2) a protractor for measuring angles.
Dot
Distance is always measured between any two points. From a practical point of view, a dot is a small speck that remains on paper when you poke it with a pencil or pen. Another, more preferred way to specify a point is to draw a cross with two thin lines, which sets dot their intersections. On drawings in books, the dot is often depicted as a small black circle. But these are all just approximations. visual images, but in the strict mathematical sense, dot - it is an imaginary object whose size in all directions is zero. For mathematicians, the whole world is made up of dots. The dots are everywhere. When we poke a pen on paper or draw a cross, we are not creating new point, but only put a mark on an existing one in order to draw someone's attention to it. Unless otherwise stated, it is understood that the points are fixed and do not change their relative position. But it is not difficult to imagine a moving point that moves from place to place, as if merging with one fixed point, then on the other.
Straight
By attaching a ruler to two points, we can draw a straight line through them, and, moreover, the only way. imaginary mathematical straight, drawn along an imaginary ideal ruler, has zero thickness and extends in both directions to infinity. In a real drawing, this imaginary design takes the form:
As a matter of fact, everything in this picture is wrong. The thickness of the line here is clearly greater than zero, and there is no way to say that the line extends to infinity. Nevertheless, such incorrect drawings are very useful as a support for the imagination, and we will use them constantly. In order to make it more convenient to distinguish one point from another, they are usually marked capital letters Latin alphabet. In this figure, for example, the points are marked with letters A and B. Line passing through points A and B, automatically receives the name "direct AB". For brevity, the notation ( AB), where the word "straight" is omitted and round brackets. Lines can also be labeled lower case. In the figure above, the straight line AB marked with a letter n.
Beyond the dots A and B on a straight line n there are a huge number of other points, each of which can be represented as an intersection with some other line. Many lines can be drawn through the same point.
If we know that there are non-coincident points on a line A, B, C and D, then it can rightly be denoted not only as ( AB), but also how ( AC), (BD), (CD) etc.
Line segment. Cut length. Distance between points
The part of a line bounded by two points is called segment. These limiting points also belong to the segment and are called it. ends. A segment whose endpoints are at points A and B, denoted as "segment AB' or, somewhat shorter, [ AB].
Each segment is characterized long- the number (possibly fractional) of "steps" that must be taken along the segment in order to get from one end to the other. In this case, the length of the "step" itself is a strictly fixed value, which is taken as a unit of measurement. The lengths of line segments drawn on a sheet of paper are most conveniently measured in centimeters. If the endpoints of the segment fall on the points A and B, then its length is denoted as | AB|.
Under distance between two points is the length of the segment connecting them. In fact, however, it is not required to draw a segment to measure the distance - it is enough to attach a ruler to both points (on which traces of “steps” are pre-marked). Since a point is a fictional object in mathematics, nothing prevents us from using in our imagination an ideal ruler that measures distance with absolute accuracy. However, one should not forget that a real ruler applied to spots or centers of crosses on paper allows you to set the distance only approximately - with an accuracy of one millimeter. Distance is always non-negative.
Position of a point on a line
Let us be given some straight line. We mark an arbitrary point on it and denote it by the letter O. Let's put the number 0 next to it. One of the two possible directions along the straight line we will call "positive", and the opposite to it - "negative". Usually, the positive direction is taken from left to right or from bottom to top, but this is not necessary. Mark the positive direction with an arrow, as shown in the figure:
Now for any point located on the line, we can determine it position. Point position A is given by a value that can be negative, zero or positive. Her absolute value equal to the distance between points O and A(that is, the length of the segment OA), and the sign is determined by the direction from the point O you have to move to get to the point A. If you need to move in a positive direction, then the sign is positive. If it is negative, then the sign is negative. Instead of the word "position", the word " coordinate».
Irrational and real (real) numbers
When we are dealing with a real drawing and determine the position of a real point on a real hole using a school ruler, we get a value rounded to the nearest millimeter. In other words, the result is a value taken from the following series:
0 mm, 1 mm, −1 mm, 2 mm, −2 mm, 3 mm, −3 mm etc.
The result cannot be equal to, for example, 1/3 cm, because, as we know, one third of a centimeter can be represented as an infinite periodic fraction
0,333333333... cm,
which after rounding should be equal to 0.3 cm.
It is a different matter when we manipulate ideal mathematical objects in our imagination.
Firstly, in this case, one can easily discard units of measurement and operate exclusively with dimensionless quantities. Then we come to the geometric construction that we met when we went through rational numbers, and which we named number line:
Since the word "line" in geometry is already heavily "loaded", the same construction is often called numerical axis or simply axis.
Secondly, we can well imagine that the coordinate of a point is given by some periodic decimal, like
Moreover, we can imagine an infinite non-periodic fraction, such as
1 ,01 001 0001 00001 000001 0000001 ...
1 ,23 45 67 89 1011 1213 1415 1617 1819 2021 ...
Such imaginary numbers, represented as infinite non-repeating decimal fractions, are called irrational. Irrational numbers, together with the rational numbers already familiar to us, form the so-called valid numbers. Instead of the word "valid" we also use the word " real". Any conceivable position of a point on a line can be expressed as a real number. And vice versa, if we are given some real number x, we can always imagine a point X, whose position is given by the number x.
Bias
Let be a- point coordinate A, a b- point coordinate B. Then the value
v = b − a
is an displacement, which translates the point A exactly B. This becomes especially obvious if the previous equality is rewritten as
b = a + v.
Sometimes instead of the word "displacement" they use the word " vector". It is easy to see that the position x arbitrary point X is nothing more than an offset that translates the dot O(with coordinate equal to zero) to a point X:
x= 0 + x.
Displacements can be added to each other, as well as subtracted from each other. So, if the offset ( b − a) translates point A exactly B, and the offset ( c − b) point B exactly C, then the offset
(b − a) + (c − b) = c − a
translates the point A exactly C.
Note. According to the logic of things, it should be clarified here how to add and subtract irrational numbers, since the bias may well be irrational. Of course, mathematicians took care to develop the appropriate formal procedures, but in practice we will not deal with this, since for the solution practical tasks approximate calculations with rounded values are always sufficient. For now, we will simply take it on faith that the concepts of "addition" and "subtraction" - as well as "multiplication" and "division" - are correctly defined for any two real numbers (with the caveat that you cannot divide by zero).
Here, perhaps, it would be appropriate to note the subtle difference between the concepts of "displacement" and "distance". Distance is always non-negative. It is, in fact, an offset taken from absolute value. So if the offset
v = b − a
translates the point A exactly B, then the distance s between points A and B equals
s = |v| = |b − a|.
This equality remains true regardless of which of the two numbers is greater - a or b.
Plane
In a practical sense, a plane is a sheet of paper on which we draw our geometric drawings. imaginary mathematical plane differs from a sheet of paper in that it has zero thickness and an unbounded surface that extends into different sides to infinity. In addition, unlike a sheet of paper, the mathematical plane is absolutely rigid: it never bends or wrinkles - even if it is torn off the desk and placed in space in any way.
The location of the plane in space is uniquely given by three points (unless they lie on any one straight line). To better visualize this, let's draw three arbitrary points, O, A and B, and draw two straight lines through them OA and OB, as it shown on the picture:
It is already somewhat easier to “stretch” a plane in the imagination on two intersecting lines than to “lean” it on three points. But for even greater clarity, we will do some more additional constructions. Let's take a couple of points at random: one anywhere on the line OA, and the other - anywhere on the line OB. Draw a new line through this pair of points. Next, in a similar way, we select another pair of points and draw another line through them. By repeating this procedure many times, we get something like a web:
Imposing a plane on such a structure is already quite simple - especially since this imaginary web can be made so thick that it will cover the entire plane without gaps.
Note that if we take a pair of non-coincident points on a plane and draw a line through them, then this line will necessarily lie in the same plane.
Abstract
Dot (A, B, etc.): an imaginary object whose size in all directions is zero.
Straight (n, m or ( AB)): infinitely thin line; passed through two points ( A and B) along the ruler in an unambiguous way; extends in both directions to infinity.
Line segment ([AB]): part of a line bounded by two points ( A and B) - the ends of the segment, which are also considered to belong to the segment.
Cut length(|AB|): (fractional) number of centimeters (or other unit of measurement) that fit between the ends ( A and B).
Distance between two points: the length of the line segment ending at these points.
Position of a point on a line (coordinate): distance from a point to some pre-selected center (also lying on a straight line) with a plus or minus sign assigned, depending on which side of the center the point is located.
The position of a point on a straight line is given valid(real)number, namely, a decimal fraction, which can be either (1) finite or infinite periodic ( rational numbers), or (2) infinite non-periodic ( irrational numbers).
Bias, which translates the point A(with coordinate a) exactly B(with coordinate b): v = b − a.
The distance is equal to the displacement, taken in absolute value: | AB| = |b − a|.
Plane: an infinitely thin sheet of paper that extends in different directions to infinity; is uniquely defined by three points that do not lie on the same straight line.
The concept of a critical point can be generalized to the case of differentiable mappings , and to the case of differentiable mappings of arbitrary manifolds f: N n → M m (\displaystyle f:N^(n)\to M^(m)). In this case, the definition of a critical point is that the rank of the Jacobian matrix of the mapping f (\displaystyle f) it has less than the maximum possible value, equal to .
Critical points functions and mappings play important role in areas of mathematics such as differential equations, calculus of variations, stability theory, and in mechanics and physics. The study of critical points of smooth mappings is one of the main questions in catastrophe theory. The notion of a critical point is also generalized to the case of functionals defined on infinite-dimensional function spaces. The search for critical points of such functionals is important part calculus of variations. Critical points of functionals (which, in turn, are functions) are called extremals.
Formal definition
critical(or special or stationary) a point of a continuously differentiable mapping f: R n → R m (\displaystyle f:\mathbb (R) ^(n)\to \mathbb (R) ^(m)) a point is called at which the differential of this mapping f ∗ = ∂ f ∂ x (\displaystyle f_(*)=(\frac (\partial f)(\partial x))) is an degenerate linear transformation corresponding tangent spaces T x 0 R n (\displaystyle T_(x_(0))\mathbb (R) ^(n)) and T f (x 0) R m (\displaystyle T_(f(x_(0)))\mathbb (R) ^(m)), that is, the dimension of the transformation image f ∗ (x 0) (\displaystyle f_(*)(x_(0))) smaller min ( n , m ) (\displaystyle \min\(n,m\)). In the coordinate notation for n = m (\displaystyle n=m) this means that the jacobian is the determinant of the jacobi matrix of the mapping f (\displaystyle f), composed of all partial derivatives ∂ f j ∂ x i (\displaystyle (\frac (\partial f_(j))(\partial x_(i))))- vanishes at a point. Spaces and R m (\displaystyle \mathbb (R) ^(m)) in this definition can be replaced by varieties N n (\displaystyle N^(n)) and M m (\displaystyle M^(m)) the same dimensions.
Sard's theorem
The display value at the critical point is called its critical. According to Sard's theorem, the set of critical values of any sufficiently smooth mapping f: R n → R m (\displaystyle f:\mathbb (R) ^(n)\to \mathbb (R) ^(m)) has zero Lebesgue measure (although there can be any number of critical points, for example, for the identical mapping, any point is critical).
Constant rank mappings
If in the vicinity of the point x 0 ∈ R n (\displaystyle x_(0)\in \mathbb (R) ^(n)) rank of a continuously differentiable mapping f: R n → R m (\displaystyle f:\mathbb (R) ^(n)\to \mathbb (R) ^(m)) is equal to the same number r (\displaystyle r), then in the vicinity of this point x 0 (\displaystyle x_(0)) there are local coordinates centered at x 0 (\displaystyle x_(0)), and in the neighborhood of its image - points y 0 = f (x 0) (\displaystyle y_(0)=f(x_(0)))- there are local coordinates (y 1 , … , y m) (\displaystyle (y_(1),\ldots ,y_(m))) centered on f (\displaystyle f) is given by the relations:
Y 1 = x 1 , … , y r = x r , y r + 1 = 0 , … , y m = 0. (\displaystyle y_(1)=x_(1),\ \ldots ,\ y_(r)=x_(r ),\ y_(r+1)=0,\ \ldots ,\ y_(m)=0.)
In particular, if r = n = m (\displaystyle r=n=m), then there are local coordinates (x 1 , … , x n) (\displaystyle (x_(1),\ldots ,x_(n))) centered on x 0 (\displaystyle x_(0)) and local coordinates (y 1 , … , y n) (\displaystyle (y_(1),\ldots ,y_(n))) centered on y 0 (\displaystyle y_(0)), such that they display f (\displaystyle f) is identical.
Happening m = 1
When this definition means that the gradient ∇ f = (f x 1 ′ , … , f x n ′) (\displaystyle \nabla f=(f"_(x_(1)),\ldots ,f"_(x_(n)))) vanishes at this point.
Let's assume that the function f: R n → R (\displaystyle f:\mathbb (R) ^(n)\to \mathbb (R) ) has a smoothness class of at least C 3 (\displaystyle C^(3)). Critical point of a function f called non-degenerate, if it contains a Hessian | ∂ 2 f ∂ x 2 | (\displaystyle (\Bigl |)(\frac (\partial ^(2)f)(\partial x^(2)))(\Bigr |)) different from zero. In a neighborhood of a nondegenerate critical point, there are coordinates in which the function f has a quadratic normal form (Morse's lemma).
A natural generalization of the Morse lemma for degenerate critical points is Toujron's theorem: in a neighborhood of a degenerate critical point of the function f, differentiable infinite number times() finite multiplicity µ (\displaystyle \mu ) there is a coordinate system in which smooth function has the form of a polynomial of degree μ + 1 (\displaystyle \mu +1)(as P μ + 1 (x) (\displaystyle P_(\mu +1)(x)) one can take the Taylor polynomial of the function f (x) (\displaystyle f(x)) at a point in the original coordinates) .
At m = 1 (\displaystyle m=1) it makes sense to ask about the maximum and minimum of a function. According to the famous statement mathematical analysis, a continuously differentiable function f (\displaystyle f), defined in the whole space R n (\displaystyle \mathbb (R) ^(n)) or in its open subset, can reach local maximum(minimum) only at critical points, and if the point is nondegenerate, then the matrix (∂ 2 f ∂ x 2) = (∂ 2 f ∂ x i ∂ x j) , (\displaystyle (\Bigl ()(\frac (\partial ^(2)f)(\partial x^(2)))( \Bigr))=(\Bigl ()(\frac (\partial ^(2)f)(\partial x_(i)\partial x_(j)))(\Bigr)),) i , j = 1 , … , n , (\displaystyle i,j=1,\ldots ,n,) must be negatively (positively) definite in it. The latter is also sufficient condition local maximum (respectively, minimum).
Happening n = m = 2
When n=m=2 we have a mapping f plane onto a plane (or two-dimensional manifold onto another two-dimensional manifold). Let's assume that the display f differentiable an infinite number of times ( C ∞ (\displaystyle C^(\infty ))). In this case, the typical critical points of the mapping f are those in which the determinant of the Jacobian matrix is equal to zero, but its rank is equal to 1, and hence the differential of the mapping f has a one-dimensional kernel at such points. The second condition of typicality is that in a neighborhood of the considered point on the inverse-image plane, the set of critical points forms a regular curve S, and at almost all points of the curve S core ker f ∗ (\displaystyle \ker \,f_(*)) does not concern S, while the points where this is not the case are isolated and the tangency at them is of the first order. Critical points of the first type are called crease points, and the second type assemblage points. Folds and folds are the only types of singularities of plane-to-plane mappings that are stable with respect to small perturbations: under a small perturbation, the fold and fold points move only slightly along with the deformation of the curve S, but do not disappear, do not degenerate, and do not fall apart into other singularities.
