Curvilinear coordinates. General idea of ​​coordinates. Lengths and angles

On surface.

Local properties of curvilinear coordinates

When considering curvilinear coordinates in this section, we will assume that we are considering a three-dimensional space (n = 3), equipped with Cartesian coordinates x, y, z. The case of other dimensions differs only in the number of coordinates.

In the case of Euclidean space, the metric tensor, also called the square of the arc differential, will in these coordinates have the form corresponding to the identity matrix:

$dS^2\; =\; \backslash mathbf(dx)^2\; +\; \backslash mathbf(dy)^2\; +\; \backslash mathbf(dz)^2.$

General case

Let $q\_1$, $q\_2$, $q\_3$- certain curvilinear coordinates, which we will consider as given smooth functions of x, y, z. To have three functions $q\_1$, $q\_2$, $q\_3$ served as coordinates in a certain region of space, the existence of an inverse mapping is necessary:

$\backslash left\backslash (\backslash begin(matrix)\; x\; =\; \backslash varphi\_1\backslash left(q\_1,\backslash ;q\_2,\backslash ;q\_3\backslash right);\backslash \backslash \; y=\; \backslash varphi\_2\backslash left(q\_1,\backslash ;q\_2,\backslash ;q\_3\backslash right)\; ;\; \backslash \backslash \; z\; =\; \backslash varphi\_3\backslash left(q\_1,\backslash ;q\_2,\backslash ;q\_3\backslash right),\backslash end(matrix)\backslash right.$

Where $\backslash varphi\_1,\backslash ;\; \backslash varphi\_2,\backslash ;\; \backslash varphi\_3$- functions defined in some domain of sets $\backslash left(q\_1,\backslash ;q\_2,\backslash ;q\_3\backslash right)$ coordinates

Local basis and tensor analysis

In tensor calculus, we can introduce local basis vectors: $\backslash mathbf(R\_j)=\backslash frac(d\backslash mathbf\; r)(dy^j)=\; \backslash frac(dx^i)(dy^j)\; \backslash mathbf\; e\_i=Q^i\_j\; \backslash mathbf\; e\_i$, Where $\backslash mathbf\; e\_i$- unit vectors of the Cartesian coordinate system, $Q^i\_j$- Jacobi matrix, $x^i$ coordinates in the Cartesian system, $y^i$- entered curvilinear coordinates.
It is not difficult to see that curvilinear coordinates, generally speaking, change from point to point.
Let us indicate the formulas for the connection between curvilinear and Cartesian coordinates:
$\backslash mathbf\; R\_i=Q^j\_i\; \backslash mathbf\; e\_j$
$\backslash mathbf\; e\_i=P^j\_i\; \backslash mathbf\; R\_j$ Where $P^j\_i\; Q^i\_j=E$, where E is the identity matrix.
The product of two local basis vectors forms a metric matrix:
$\backslash mathbf\; R\_i\; \backslash mathbf\; R\_j\; =\; Q^n\_i\; Q^m\_j\; d\_(nm)\; =\; g\_(ij)$
$\backslash mathbf\; R^i\; \backslash mathbf\; R^j\; =\; P^i\_n\; P^j\_m\; d^(nm)=g^(ij)$
$g\_(ij)\; g^(jk)=g^(jk)\; g\_(ij)\; =d\_i^k$, Where $d\_(ij),\; d^(ij),\; d^i\_j$ contravariant, covariant and mixed Kronecker symbol
Thus, any tensor field $\backslash mathbf\; T$ rank n can be expanded into a local polyadic basis:
$\backslash mathbf\; T=\; T^(i\_1\; ...\; i\_n)\; \backslash mathbf\; e\_i\; \backslash otimes\; ...\; \backslash otimes\; \backslash mathbf\; e\_n\; =T^(i\_1\; ...i\_n)\; P^(j\_1)\_(i\_1)\; ...\; P^\; (j\_n)\_(i\_n)\; \backslash mathbf\; R\_(j\_1)\; \backslash otimes...\; \backslash otimes\; \backslash mathbf\; R\_(j\_n)$
For example, in the case of a first-rank tensor field (vector):
$\backslash mathbf\; v=v^i\; \backslash mathbf\; e\_i=v^i\; P^j\_i\; \backslash mathbf\; R\_j$

Orthogonal curvilinear coordinates

In Euclidean space, the use of orthogonal curvilinear coordinates is of particular importance, since formulas related to length and angles look simpler in orthogonal coordinates than in the general case. This is due to the fact that the metric matrix in systems with an orthonormal basis will be diagonal, which will significantly simplify calculations.
An example of such systems is a spherical system in $\backslash mathbb(R)^2$

Lamé coefficients

Let us write the differential of the arc in curvilinear coordinates in the form (we use Einstein’s summation rule):

$dS^2\; =\; \backslash left(\backslash frac(\backslash partial\; \backslash varphi\_1)(\backslash partial\; q\_i)\backslash mathbf(dq)\_i\; \backslash right)^2\; +$

\left(\frac(\partial \varphi_2)(\partial q_i)\mathbf(dq)_i \right)^2 + \left(\frac(\partial \varphi_3)(\partial q_i)\mathbf(dq)_i \right)^2 , ~ i=1,2,3

Taking into account the orthogonality of coordinate systems ( $\backslash mathbf(dq)\_i\; \backslash cdot\; \backslash mathbf(dq)\_j\; =\; 0$ at $i\; \backslash ne\; j$) this expression can be rewritten as

$dS^2\; =\; H\_1^2dq\_1^2\; +\; H\_2^2dq\_2^2\; +\; H\_3^2dq\_3^2,$

$H\_i\; =\; \backslash sqrt(\backslash left(\backslash frac(\backslash partial\; \backslash varphi\_1)(\backslash partial\; q\_i)\backslash right)^2\; +\; \backslash left(\backslash frac(\backslash partial\; \backslash varphi\_2)(\backslash partial\; q\_i)\backslash right)^2\; +\; \backslash \; left(\backslash frac(\backslash partial\; \backslash varphi\_3)(\backslash partial\; q\_i)\backslash right)^2);\backslash \; i=1,\backslash ;2,\backslash ;3$

Positive quantities $H\_i\backslash$, depending on a point in space, are called Lamé coefficients or scale factors. Lamé coefficients show how many units of length are contained in a unit of coordinates for a given point and are used to transform vectors when moving from one coordinate system to another.

Riemannian metric tensor written in coordinates $(q\_i)$, is a diagonal matrix, on the diagonal of which are the squares of the Lamé coefficients:

Examples

Polar coordinates ( n=2)

Polar coordinates on a plane include the distance r to the pole (origin) and the direction (angle) φ.

Relationship between polar coordinates and Cartesian coordinates:

$\backslash left\backslash (\backslash begin(matrix)\; x\; =\; r\backslash cos(\backslash varphi);\backslash \backslash \; y\; =\; r\backslash sin(\backslash varphi).\backslash end(matrix)\backslash right.$

Lamé coefficients:

$\backslash begin(matrix)H\_r\; =\; 1;\; \backslash \backslash \; H\_\backslash varphi\; =\; r.\; \backslash end(matrix)$

Arc differential:

$dS^2\backslash \; =\backslash \; dr^2\backslash \; +\backslash \; r^2d\backslash varphi^2.$

At the origin, the function φ is not defined. If the coordinate φ is considered not as a number, but as an angle (a point on the unit circle), then the polar coordinates form a coordinate system in the area obtained from the entire plane by removing the origin point. If we still consider φ to be a number, then in the designated area it will be multi-valued, and the construction of a strictly mathematical coordinate system is possible only in a simply connected area that does not include the origin of coordinates, for example, on a plane without a ray.

Cylindrical coordinates ( n=3)

Cylindrical coordinates are a trivial generalization of polar ones to the case three-dimensional space by adding a third z coordinate. Relationship between cylindrical coordinates and Cartesian ones:

$\backslash left\backslash (\backslash begin(matrix)\; x\; =\; r\backslash cos(\backslash varphi);\backslash \backslash \; y\; =\; r\backslash sin(\backslash varphi).\; \backslash \backslash \; z\; =\; z.\; \backslash end(matrix)\backslash right.$

Lamé coefficients:

$\backslash begin(matrix)H\_r\; =\; 1;\; \backslash \backslash \; H\_\backslash varphi\; =\; r;\; \backslash \backslash \; H\_z\; =\; 1.\; \backslash end(matrix)$

Arc differential:

$dS^2\backslash \; =\backslash \; dr^2\backslash \; +\backslash \; r^2d\backslash varphi^2\; +\; dz^2.$

Spherical coordinates ( n=3)

Spherical coordinates are related to the latitude and longitude coordinates on the unit sphere. Relationship between spherical coordinates and Cartesian coordinates:

$\backslash left\backslash (\backslash begin(matrix)\; x\; =\; r\backslash sin(\backslash theta)\backslash cos(\backslash varphi);\backslash \backslash \; y\; =\; r\backslash sin(\backslash theta)\backslash sin(\backslash varphi);\; \backslash \backslash \; z\; =\; r\backslash cos\; (\backslash theta).\backslash end(matrix)\backslash right.$

Lamé coefficients:

$\backslash begin(matrix)H\_r\; =\; 1;\; \backslash \backslash H\_\backslash theta\; =\; r;\; \backslash \backslash H\_\backslash varphi\; =\; r\backslash sin(\backslash theta).\; \backslash end(matrix)$

Arc differential:

$dS^2\backslash \; =\backslash \; dr^2\backslash \; +\backslash \; r^2d\backslash theta^2\; +\; r^2\backslash sin^2(\backslash theta)d\backslash varphi^2.$

Spherical coordinates, like cylindrical ones, do not work on the z axis ( x =0, y =0), since the φ coordinate is not defined there.

Various exotic coordinates on the plane ( n=2) and their generalizations

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Literature

• Korn G., Korn T. Handbook of mathematics (for scientists and engineers). - M.: Nauka, 1974. - 832 p.

An excerpt characterizing the Curvilinear coordinate system

The military council, at which Prince Andrei was not able to express his opinion, as he had hoped, left a vague and alarming impression on him. He did not know who was right: Dolgorukov and Weyrother or Kutuzov and Langeron and others who did not approve of the attack plan. “But was it really impossible for Kutuzov to directly express his thoughts to the sovereign? Can't this really be done differently? Is it really necessary to risk tens of thousands and my, my life for the sake of court and personal considerations?” he thought.
“Yes, it’s very possible they’ll kill you tomorrow,” he thought. And suddenly, at this thought of death, a whole series of memories, the most distant and most intimate, arose in his imagination; he remembered the last farewell to his father and wife; he remembered the first times of his love for her! He remembered her pregnancy, and he felt sorry for both her and himself, and in a nervously softened and excited state he left the hut in which he had stood with Nesvitsky and began to walk in front of the house.
The night was foggy, and moonlight mysteriously broke through the fog. “Yes, tomorrow, tomorrow! - he thought. “Tomorrow, perhaps, everything will be over for me, all these memories will no longer exist, all these memories will no longer have any meaning for me.” Tomorrow, maybe, even probably, tomorrow, I foresee it, for the first time I will finally have to show everything that I can do.” And he imagined the battle, its loss, the concentration of the battle on one point and the confusion of all the commanders. And now that happy moment, that Toulon, which he had been waiting for so long, finally appears to him. He firmly and clearly speaks his opinion to Kutuzov, Weyrother, and the emperors. Everyone is amazed at the correctness of his idea, but no one undertakes to carry it out, and so he takes a regiment, a division, pronounces a condition so that no one will interfere with his orders, and leads his division to the decisive point and alone wins. What about death and suffering? says another voice. But Prince Andrei does not answer this voice and continues his successes. The disposition of the next battle is made by him alone. He holds the rank of army duty officer under Kutuzov, but he does everything alone. The next battle was won by him alone. Kutuzov is replaced, he is appointed... Well, and then? another voice speaks again, and then, if you are not wounded, killed or deceived ten times before; Well, then what? “Well, and then,” Prince Andrei answers himself, “I don’t know what will happen next, I don’t want and can’t know: but if I want this, I want fame, I want to be famous people, I want to be loved by them, then it’s not my fault that I want this, that I want this alone, for this alone I live. Yes, for this alone! I'll never tell anyone this, but oh my God! What should I do if I love nothing but glory, human love? Death, wounds, loss of family, nothing scares me. And no matter how dear and dear many people are to me - my father, sister, wife - the most dear people to me - but, no matter how scary and unnatural it seems, I will give them all now for a moment of glory, triumph over people, for love for to myself people whom I do not know and will not know, for the love of these people,” he thought, listening to the conversation in Kutuzov’s yard. In Kutuzov's yard the voices of the orderlies were heard; one voice, probably the coachman, teasing the old Kutuzovsky cook, whom Prince Andrei knew, and whose name was Titus, said: “Titus, what about Titus?”
“Well,” answered the old man.
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“Ugh, to hell with it,” a voice rang out, covered by the laughter of the orderlies and servants.
“And yet I love and treasure only the triumph over all of them, I treasure this mysterious power and glory that floats above me here in this fog!”

That night Rostov was with a platoon in the flanker chain, ahead of Bagration’s detachment. His hussars were scattered in chains in pairs; he himself rode on horseback along this line of chain, trying to overcome the sleep that was irresistibly pushing him over. Behind him he could see a huge expanse of our army’s fires burning dimly in the fog; ahead of him was foggy darkness. No matter how much Rostov peered into this foggy distance, he saw nothing: sometimes it turned gray, sometimes something seemed black; then lights seemed to flash where the enemy should be; then he thought that it was only shining in his eyes. His eyes closed, and in his imagination he imagined first the sovereign, then Denisov, then Moscow memories, and again he hastily opened his eyes and close in front of him he saw the head and ears of the horse on which he was sitting, sometimes the black figures of the hussars when he was six steps away I ran into them, and in the distance there was still the same foggy darkness. "From what? It’s very possible,” Rostov thought, “that the sovereign, having met me, will give an order, like any officer: he will say: “Go, find out what’s there.” Many people told how, quite by accident, he recognized some officer and brought him closer to him. What if he brought me closer to him! Oh, how I would protect him, how I would tell him the whole truth, how I would expose his deceivers,” and Rostov, in order to vividly imagine his love and devotion to the sovereign, imagined an enemy or deceiver of the German whom he enjoyed not only killed, but hit him on the cheeks in the eyes of the sovereign. Suddenly a distant scream woke up Rostov. He shuddered and opened his eyes.
"Where I am? Yes, in a chain: slogan and password – drawbar, Olmütz. What a shame that our squadron will be in reserves tomorrow... - he thought. - I’ll ask you to get involved. This may be the only opportunity to see the sovereign. Yes, it won't be long until the shift. I’ll go around again and when I return, I’ll go to the general and ask him.” He adjusted himself in the saddle and moved his horse to once again ride around his hussars. It seemed to him that it was brighter. On the left side one could see a gentle illuminated slope and the opposite, black hillock, which seemed steep, like a wall. On this hillock there was a white spot that Rostov could not understand: was it a clearing in the forest, illuminated by the moon, or the remaining snow, or white houses? It even seemed to him that something was moving along this white spot. “The snow must be a spot; spot – une tache,” thought Rostov. “Here you go…”

Until now, wanting to know the position of a point on a plane, or in space, we used the Cartesian coordinate system. So, for example, we determined the position of a point in space using three coordinates. These coordinates were the abscissa, ordinate and applicate of a variable point in space. However, it is clear that specifying the abscissa, ordinate and applicate of a point is not the only way to determine the position of a point in space. This can be done in another way, for example, using curvilinear coordinates.

Let, according to some well-defined rule, each point M space uniquely corresponds to a certain triple of numbers ( q 1 , q 2 , q 3), and different triplets of numbers correspond to different points. Then they say that a coordinate system is given in space; numbers q 1 , q 2 , q 3 that correspond to the point M, are called coordinates (or curvilinear coordinates) this point.

Depending on the rule according to which the triple of numbers ( q 1 , q 2 , q 3) is put in correspondence with a point in space, they talk about one or another coordinate system.

If they want to note that in a given coordinate system the position of point M is determined by the numbers q 1 , q 2 , q 3, then it is written as follows M(q 1 , q 2 , q 3).

Example 1. Let some fixed point be marked in space ABOUT(origin of coordinates), and three mutually perpendicular axes with the scale selected on them are drawn through it. (Axes Oh, Oh, Oz). Three of numbers x, y, z let's match the point M, such that the projections of its radius vector OM on the axis Oh, Oh, Oz will be equal respectively x, y, z. This method of establishing a relationship between triplets of numbers ( x, y, z) and dots M leads us to the well-known Cartesian coordinate system.

It is easy to see that in the case of a Cartesian coordinate system, not only does each triple of numbers correspond to a certain point in space, but also vice versa, each point in space corresponds to a certain triple of coordinates.

Example 2. Let the coordinate axes be drawn again in space Oh, Oh, Oz passing through a fixed point ABOUT(origin).

Consider a trio of numbers r, j, z, Where r³0; £0 j£2 p, –¥<z<¥, и поставим в соответствие этой тройке чисел точку M, such that its applicate is equal to z, and its projection onto the plane Oxy has polar coordinates r And j(see Fig. 4.1). It is clear that here each three numbers r, j, z corresponds to a certain point M and back, to each point M corresponds to a certain triple of numbers r, j, z. The exception is points lying on the axis Oz: in this case r And z are uniquely defined, and the angle j any meaning can be assigned. Numbers r, j, z are called the cylindrical coordinates of the point M.

It is easy to establish a relationship between cylindrical and Cartesian coordinates:

x = r×cos j; y = r×sin j; z = z.

And back ; ; z = z.

Example 3. Let's introduce a spherical coordinate system. Let's set three numbers r, q, j, characterizing the position of the point M in space as follows: r– distance from the origin to the point M(length of the radius vector), q Oz and radius vector OM(latitude of point M) j– angle between the positive direction of the axis Oh and the projection of the radius vector onto the plane Oxy(longitude of point M). (See Figure 4.2).

It is clear that in this case not only each point M corresponds to a certain triple of numbers r, q, j, Where r³0.0£ q £ p, 0£ j£2 p, but also vice versa, each such triple of numbers corresponds to a certain point in space (again with the exception of the points of the axis Oz, where this uniqueness is violated).

It is easy to find the connection between spherical and Cartesian coordinates:

x = r sin q cos j; y = r sin q sin j; z = r cos q.

Let's return to an arbitrary coordinate system ( Oq 1 , Oq 2 , Oq 3). We will assume that not only each point in space corresponds to a certain triple of numbers ( q 1 , q 2 , q 3), but also vice versa, each triple of numbers corresponds to a certain point in space. Let us introduce the concept of coordinate surfaces and coordinate lines.

Definition. The set of those points for which the coordinate q 1 is constant, called the coordinate surface q 1 . Coordinate surfaces are defined similarly q 2 , and q 3 (see Fig. 4.3).

Obviously, if point M has coordinates WITH 1 , WITH 2 , WITH 3 then at this point the coordinate surfaces intersect q 1 =C 1 ; q 2 =C 2 ; q 3 =C 3 .

Definition. The set of those points along which only the coordinate changes q 1 (and the remaining two coordinates q 2 and q 3 remain constant) is called a coordinate line q 1 .

Obviously, every coordinate line q 1 is the line of intersection of the coordinate planes q 2 and q 3 .

The coordinate lines are determined similarly q 2 and q 3 .

Example 1. Coordinate surfaces (along the coordinate x) in the Cartesian coordinate system are all planes x= const. (They are parallel to the plane Оyz). Coordinate surfaces are determined similarly by coordinates y And z.

Coordinate x-line is a straight line parallel to the axis Oh. Coordinate y-line ( z-line) – straight, parallel to the axis OU(axes Oz).

Example 2. Coordinate surfaces in a cylindrical system are: any plane parallel to the plane Oxy(coordinate surface z= const), the surface of a circular cylinder whose axis is directed along the axis Oz(coordinate surface r= const) and a half-plane limited by the axis Oz(coordinate surface j= const) (see Fig. 4.4).

The name cylindrical coordinate system is explained by the fact that among its coordinate surfaces there are cylindrical surfaces.

The coordinate lines in this system are z-line – straight, parallel to the axis Oz; j-line – a circle lying in a horizontal plane with its center on the axis Oz; And r-line – a ray emanating from an arbitrary point on the axis Oz, parallel to the plane Oxy.

Rice. 4.5

Since there are spheres among the coordinate surfaces, this coordinate system is called spherical.

The coordinate lines here are: r-line – a ray emerging from the origin, q-line – a semicircle with a center at the origin, connecting two points on an axis Oz; j-line – a circle lying in a horizontal plane, centered on an axis Oz.

In all the examples discussed above, coordinate lines passing through any point M, are orthogonal to each other. This does not happen in every coordinate system. However, we will limit ourselves to studying only those coordinate systems for which this occurs; such coordinate systems are called orthogonal.

Definition. Coordinate system ( Oq 1 , Oq 2 , Oq 3) is called orthogonal if at each point M coordinate lines passing through this point intersect at right angles.

Let us now consider some point M and draw unit vectors touching the corresponding coordinate lines at this point and directed towards increasing the corresponding coordinate. If these vectors form a right-handed triple at each point, then we are given a right-handed coordinate system. So, for example, the Cartesian coordinate system x, y, z(with the usual arrangement of axes) is right. Also are right-handed cylindrical coordinate systems r, j, z(but precisely with this order of coordinates; if you change the order of coordinates, taking, for example, r, z, j, we will no longer get the right system).

The spherical coordinate system is also right-handed (if you set this order r, q, j).

Note that in the Cartesian coordinate system the direction of the unit vector does not depend on the point at which M we carry out this vector; the same is true for vectors. We observe something different in curvilinear coordinate systems: for example, in a cylindrical coordinate system, vectors at a point M and at some other point M 1 no longer have to be parallel to each other. The same applies to a vector (at different points it has, generally speaking, different directions).

Thus, the triple of unit orthogonal vectors in a curvilinear coordinate system depends on the position of the point M, in which these vectors are considered. A triple of unit orthogonal vectors is called a moving frame, and the vectors themselves are called unit vectors (or simply vectors).

Corresponding to such a vector space. In this article, the first definition will be taken as the starting point.

N (\displaystyle n)-dimensional Euclidean space is denoted by E n , (\displaystyle \mathbb (E) ^(n),) the notation is also often used (if it is clear from the context that the space has a Euclidean structure).

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✪ 04 - Linear algebra. Euclidean space

✪ Non-Euclidean geometry. Part one.

✪ Non-Euclidean geometry. Part two

✪ 01 - Linear algebra. Linear (vector) space

✪ 8. Euclidean spaces

Subtitles

Formal definition

To define Euclidean space, the easiest way is to take as the main concept the scalar product. Euclidean vector space is defined as a finite-dimensional vector space over the field of real numbers, on whose vectors a real-valued function is specified (⋅ , ⋅) , (\displaystyle (\cdot ,\cdot),) having the following three properties:

Example of Euclidean space - coordinate space R n , (\displaystyle \mathbb (R) ^(n),) consisting of all possible tuples of real numbers (x 1 , x 2 , … , x n) , (\displaystyle (x_(1),x_(2),\ldots ,x_(n)),) scalar product in which is determined by the formula (x , y) = ∑ i = 1 n x i y i = x 1 y 1 + x 2 y 2 + ⋯ + x n y n .

(\displaystyle (x,y)=\sum _(i=1)^(n)x_(i)y_(i)=x_(1)y_(1)+x_(2)y_(2)+\cdots +x_(n)y_(n).)

Lengths and angles The scalar product defined on Euclidean space is sufficient to introduce the geometric concepts of length and angle. Vector length u (\displaystyle u) is defined as(u , u) (\displaystyle (\sqrt ((u,u)))) and is designated The positive definiteness of the scalar product guarantees that the length of the nonzero vector is nonzero, and from bilinearity it follows that | a u |

= | The scalar product defined on Euclidean space is sufficient to introduce the geometric concepts of length and angle. Vector length And a || u |, (\displaystyle |au|=|a||u|,) that is, the lengths of proportional vectors are proportional. Angle between vectors v (\displaystyle v)

determined by the formula

φ = arccos ⁡ ((x , y) | x | | y |) . (\displaystyle \varphi =\arccos \left((\frac ((x,y))(|x||y|))\right).) From the cosine theorem it follows that for a two-dimensional Euclidean space ( Euclidean plane) this definition of angle coincides with the usual one. Orthogonal vectors, as in three-dimensional space, can be defined as vectors the angle between which is equal to π 2.(\displaystyle (\frac (\pi )(2)).) The Cauchy-Bunyakovsky-Schwartz inequality and the triangle inequality There is one gap left in the definition of angle given above: in order to arccos ⁡ ((x , y) | x | | y |) (\displaystyle \arccos \left((\frac ((x,y))(|x||y|))\right)) And has been defined, it is necessary that the inequality| (x, y) | x | |

y |

|

⩽ 1. (\displaystyle \left|(\frac ((x,y))(|x||y|))\right|\leqslant 1.)

This inequality actually holds in an arbitrary Euclidean space; it is called the Cauchy-Bunyakovsky-Schwartz inequality. From this inequality, in turn, follows the triangle inequality: arccos ⁡ ((x , y) | x | | y |) (\displaystyle \arccos \left((\frac ((x,y))(|x||y|))\right))| u + v |⩽ | u | This mapping is an isomorphism between Euclidean space and

(a 11 − λ1 )(a 22 − λ1 ) − a 12 a 21 = 0 ;

λ 12 - (a 11 + a 22)λ 1 + (a 11a 22 - a 12 a 21) = 0.

The discriminant of these quadratic equations is ³ 0, i.e.

D = (a 11 + a 22) 2 - 4 (a 11a 22 - a 12 a 21) = (a 11 - a 22) 2 + 4a 122 ³ 0.

Equations (2.56), (2.57) are called characteristic equations

matrices, and the roots of these equations are eigenvalues matrix A. We substitute the eigenvalues ​​found from (2.57) into (2.39), we obtain

canonical equation.

Find the canonical form of this equation.

Since here a 11 = 5; a 21 = 2; a 22 = 2, then the characteristic equation (2.56) for this quadratic form will have the form

5 - λ 2

2 2 - λ 1

Equating the determinant of this matrix equation to zero

(5 – λ)(2 – λ) – 4 = λ2 – 7λ + 6 = 0

and solving this quadratic equation, we obtain λ1 = 6; λ2 = 1.

And then the canonical form of this quadratic form will have the form

F (x 1 , x 2 ) = 6 x 1 2 + x 2 2 .

2.3. Curvilinear coordinates

2.3.1. General information about curvilinear coordinate systems

The class of curvilinear coordinates, compared to the class of rectilinear coordinates, is extensive and much more diverse and, from an analytical point of view, is the most universal, since it expands the capabilities of the method of rectilinear coordinates. The use of curvilinear coordinates can sometimes greatly simplify the solution of many problems, especially problems solved directly on the surface of rotation. For example, when solving a problem on a surface of revolution associated with finding a certain function, it is possible, in the area where this function is specified on a given surface, to select a system of curvilinear coordinates that will allow this function to be endowed with a new property - to be constant in a given coordinate system, which cannot always be done using rectilinear coordinate systems.

A system of curvilinear coordinates, defined in a certain region of three-dimensional Euclidean space, associates each point of this space with an ordered triple of real numbers - φ, λ, r (curvilinear coordinates of the point).

If the system of curvilinear coordinates is located directly on some surface (surface of revolution), then in this case, two real numbers are assigned to each point on the surface - φ, λ, which uniquely determine the position of the point on this surface.

There must be a mathematical connection between the system of curvilinear coordinates φ, λ, r and the rectilinear Cartesian coordinate system (X, Y, Z). Indeed, let the system of curvilinear coordinates be specified in a certain region of space. Each point of this space corresponds to a single triple of curvilinear coordinates – φ, λ, r. On the other hand, the only triple of rectilinear Cartesian coordinates corresponds to the same point - X, Y, Z. Then it can be argued that in general form

ϕ = ϕ (X,Y,Z);

λ = λ (,); (2.58)

X Y Z

r = r (X, Y, Z).

There is both a direct (2.58) and an inverse mathematical connection between these SCs.

From the analysis of formulas (2.58) it follows that with a constant value of one of the spatial curvilinear coordinates φ, λ, r, for example,

ϕ =ϕ(Х,У,Z)= const,

And variable values ​​of the other two (λ, r ), we obtain in general a surface, which is called a coordinate surface. Coordinate surfaces corresponding to the same coordinate do not intersect each other. However, two coordinate surfaces corresponding to different coordinates intersect and produce a coordinate line corresponding to a third coordinate.

2.3.2. Curvilinear coordinates on a surface

For geodesy, surface curvilinear coordinates are of greatest interest.

Let the surface equation be a function of Cartesian coordinates in

By directing the unit vectors i, j, l along the coordinate axes (Fig. 2.11), the surface equation can be written in vector form

r = X i + Y j + Z l . (2.60)

Let us introduce two new independent variables φ and λ, such that the functions

satisfy equation (2.59). Equalities (2.61) are parametric equations of the surface.

λ1 =const

Rice. 2.11. Curvilinear surface coordinate system

Each pair of numbers φ and λ corresponds to a certain (single) point on the surface, and these variables can be taken as the coordinates of surface points.

If we give φ different constant values ​​φ = φ1, φ = φ2, ..., then we get a family of curves on the surface corresponding to these constants. Similarly, by giving constant values ​​for λ we will have

second family of curves. Thus, a network of coordinate lines φ = const and λ = const is formed on the surface. Coordinate lines in general

are curved lines. Therefore, the numbers φ, λ are called

curvilinear coordinates points on the surface.

Curvilinear coordinates can be either linear or angular quantities. The simplest example of a system of curvilinear coordinates, in which one coordinate is a linear quantity and the other is an angular quantity, can be polar coordinates on a plane.

The choice of curvilinear coordinates does not necessarily have to precede the formation of coordinate lines. In some cases, it is more expedient to establish a network of coordinate lines that is most convenient for solving certain problems on the surface, and then select for these lines such parameters (coordinates) that would have a constant value for each coordinate line.

A certain system of parameters corresponds to a completely definite network of coordinate lines, but for each given family of coordinate lines it is possible to select many other parameters that are continuous and unambiguous functions of a given parameter. In the general case, the angles between the coordinate lines of the family φ = const and the lines of the family λ = const can have different values.

We will consider only orthogonal systems of curvilinear coordinates in which each coordinate line φ = const intersects any other coordinate line λ = const at a right angle.

When solving many problems on a surface, especially problems related to calculating the curvilinear coordinates of surface points, it is necessary to have differential equations for changing the curvilinear coordinates φ and λ depending on the change in the length S of the surface curve.

The connection between the differentials dS, dφ, dλ can be established by introducing a new variable α, i.e. angle

α dS

λ = const

λ+d λ = const

positive direction of line λ = const to positive

direction of this curve (Fig. 2.12). This angle, as it were, sets the direction (orientation) of the line in

a given point on the surface. Then (without output):

Rice. 2.12. The geometry of the connection between the differential of an arc of a curve on a surface and the changes (differentials) of curvilinear

coordinates

+ ∂ Z 2 ;

∂ϕ

+ ∂ Z 2 . ∂λ

IN geodesy angle α corresponds to geodetic azimuth: α = A.

2.3.3. Polar coordinate systems and their generalizations

2.3.4. Spatial polar coordinate system

To specify a spatial polar coordinate system, you must first select a plane (hereinafter we will call it the main one). A certain point O is selected on this plane

measurements

segments

space, then

position

any point in space will be

definitely

be determined

quantities: r, φ, λ, where r –

polar

straight distance from pole

O to point Q (Fig. 2.13); λ –

polar angle - the angle between

polar

Rice. 2.13. Spatial system

orthogonal

projection

polar radius to main

polar coordinates and its modifications

plane

changes

(polar radius) and its

0 ≤ λ < 2π); φ – угол между

vector

projection

OQ0 on

main

plane, considered positive (0 ≤ φ ≤ π/2) for points of the positive half-space and negative (-π/2 ≤ φ ≤ 0) for points of the negative half-space.

Any spatial polar CS can be easily associated (transformed) with a spatial Cartesian rectangular CS.

If we take the scale and origin of the polar system as the scale and origin of coordinates in a spatial rectangular system, the polar axis OR as the semi-abscissa axis OX, the line OZ drawn from the pole O perpendicular to the main plane in the positive direction of the polar system as the semi-axis OZ of the rectangular Cartesian system, and take the semi-axis – OU to be the axis into which the abscissa axis goes when it is rotated by an angle π/2 in the positive direction in the main plane of the polar system, then from Fig. 2.13

Formulas (2.64) allow us to express X, Y, Z in terms of r, φ, λ and vice versa

On any surface you can establish a coordinate system, determining the position of a point on it, again with two numbers. To do this, we will somehow cover the entire surface with two families of lines so that through each of its points (perhaps with a small number of exceptions) one, and only one, line from each family passes. Now you just need to provide the lines of each family with numerical marks according to some solid rule that allows you to find the desired line of the family using the numerical mark (Fig. 22).

Point coordinates M numbers serve as surfaces u, v, Where u-- numerical marking of the line of the first family passing through M, And v-- marking lines of the second family. We will continue to write: M(u; v), numbers And, v are called curvilinear coordinates of a point M. What has been said will become completely clear if we turn to the sphere for an example. It can all be covered with meridians (first family); each of them corresponds to a numerical mark, namely the longitude value u(or c). All parallels form a second family; each of them is associated with a numerical mark - latitude v(or and). Through each point on the sphere (excluding the poles) there is only one meridian and one parallel.

As another example, consider the lateral surface of a right circular cylinder of height N, radius a(Fig. 23). For the first family we take the system of its generators, one of them is taken as the initial one. We assign a mark to each generator u, equal to the length of the arc on the base circle between the initial generatrix and the given one (we will count the arc, for example, counterclockwise). For the second family we take the system of horizontal sections of the surface; numerical mark v We will consider the height at which the section is drawn above the base. With proper selection of axes x, y, z in space we will have for any point M(x;y; z) our surface:

(Here the arguments for cosine and sine are not in degrees, but in radians.) These equations can be considered as parametric equations for the surface of a cylinder.

Problem 9. Along what curve should a piece of sheet metal be cut to make a drainpipe elbow, so that after proper bending a cylinder of radius is obtained? A, truncated by a plane at an angle of 45° to the plane of the base?

Solution. Let's use the parametric equations of the cylinder surface:

We draw a cutting plane through the axis Oh, her equation z=y. Combining this with the equations we just wrote, we get the equation

intersection lines in curvilinear coordinates. After unwrapping the surface onto a plane, the curvilinear coordinates And And v will turn into Cartesian coordinates.

So, a piece of tin should be outlined on top along a sinusoid

Here u And v already Cartesian coordinates on the plane (Fig. 24).

Both in the case of a sphere and a cylindrical surface, and in the general case, defining a surface by parametric equations entails establishing a curvilinear coordinate system on the surface. Indeed, the expression for Cartesian coordinates x, y, z arbitrary point M(x;y;z) surfaces through two parameters u, v(this is generally written like this: X=ts ( u; v), y= ts (u;v), z=š (u;v), ts, w, sh - functions of two arguments) makes it possible, knowing a pair of numbers u, v, find the corresponding coordinates x, y, z, which means the position of the point M on a surface; numbers u, v serve as its coordinates. By giving one of them a constant value, for example u=u 0, we get the expression x, y, z through one parameter v, i.e., the parametric equation of the curve. This is the coordinate line of one family, its equation u=u 0 . Exactly the same line v=v 0 -- coordinate line of another family.

coordinate cartesian radius vector