Laplace operator for curvilinear coordinate systems. TAU. Laplace operator and transfer functions. Placement operator new() and operator delete()

Vono є okremim vypadkom equal Helmholtz. You can look at the tri-world (1), two-world (2), one-world and n-peaceful spaces:

The operator is called the Laplace operator (The Laplace operator is equivalent to the successive capture of the gradient and divergence operations.).

Solution of Laplace

Laplace's solution to harmonic functions.

The level of Laplace rises to the elliptical lines. The heterogeneous equivalence of Laplace becomes the equivalence of Poisson.

The skin solution of Laplace's alignment in the boundary region G is unambiguously seen by regional minds, which are superimposed on the behavior of the solution (or even similar ones) in the boundary region G. If the solution is located in the left space, the boundary mind is brought up to the order of the asymptotics for f at a. The task of knowing such decisions is called the regional task. Most often, the Dirichle control is used, if the value of the function f itself is set on the cordon, and the Neman control, if the value f of the normal to the cordon is set.

Laplace alignment in spherical, polar and cylindrical coordinates

Laplace's alignment can be written in Cartesian coordinates.

For spherical coordinates (Laplace's alignment may look like this:

In polar coordinates (the coordinate system can be equal:

In cylindrical coordinates (equal may look:

Before Laplace's level, it is necessary to induce a rich problem of physics and mechanics, for which the physical quantity is a function of less than the coordinates of a point. So, the Laplace equation describes the potential in the region, which does not avenge heavy masses, the potential of the electrostatic field - in the region, which does not avenge charges, the temperature during stationary processes, etc. around the hull of the ship, stationary filtering of underground waters, the winding up of the field near the electromagnet, as well as the stationary electric field in the vicinity of the porcelain insulator, or the electric cable buried in the ground, the replacement of the transverse cross section, to be built up to the completion of the paralysis. The Laplace operator is very important in quantum mechanics.

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The Laplace operator is a differential operator that operates in a linear space of smooth functions and a symbol. F functions

The Laplace operator is equivalent to the successive capture of the gradient and divergence operations.

Gradієnt - a vector that shows directly the most obvious growth of a given value, the value of which changes from one point to another (scalar field). For example, if you take the height of the surface of the Earth above the level of the sea, then the gradient at the skin point of the surface will show "a straight-up steep slope." Rosemіr (module) of the vector gradієnta dorіvnyuє svidkostі rostannya tsomu prіnemі. For a trivial space, a vector function with components is called a gradient, and a scalar function of the coordinates x, y, z is called a deacer.

Since it is a function of n variables, then the n-world vector is called a gradient.

Components of which are based on private arguments that are similar to all. The gradient is denoted by grad, or by the variation of the nabla operator,

From the point of view of the gradient, you can see that:

Gradient sense of whether a scalar function f in that which has a scalar body with an infinitely small displacement vector gives the latest differential of the function for a variable change of coordinates in space, on which f is assigned, then linear (in time shackled camp won’t be the head) part of the change f when shifted. Using the same letter for denoting a function in the form of a vector and a second function in the form of a coordinate, you can write:

Varto here respect that the scales of the formula of the total differential cannot be found in the form of coordinates xi, then in the nature of the parameters x are taken, then the differential is subtracted by the invariant, then it is a scalar, with any transformations of coordinates, and the scales dx are the vector, then the gradient, calculations are sizable rank, vyyavlyaєєєєєєєєєєєєєєяєєєєєєєєєєєєєєєєєєєє єєєє 's vector (contravariant), then the vector, we represent at the dual basis, which only and can give a scalar with a simple summation of the coordinates of the stellar (contravariant), then the vector, we write down at the sizable basis.

In this order, viraz (well apparently - for more curvilinear coordinates) can be completely correct and invariantly written as:

Abo omitting the sumi sign behind the Einstein rule,

Divergence is a differential operator that displays a vector field on a scalar (this is the operation of differentiation, as a result of which a scalar field goes to the vector field), which determines (for a skin point), "how much to diverge in and out of a small area of ​​a given point field" ( more precisely - there are differences between the incoming and outgoing flows).

If you believe that a flow can be assigned a sign of algebra, then there is no need to check the incoming and outgoing flows OKremo, everything will be automatically checked when the sign is improved. To this we can give a more brief description of the divergence:

divergence is a differential operator on a vector field, which characterizes the flow of a given field through the surface of a small area near the skin inner point of the area of ​​the designated field.

The operator of divergence, stopovers up to the field F, means as or

The definition of divergence looks like this:

de FF - the flow of the vector field F through the spherical surface of the area S, which surrounds V. The only help is to change in the middle of the sphere with a radius that is not equal to zero. The designation, on the view pointing below, is not tied to the sing coordinates, for example, Cartesian, which can represent additional stability in the sing views. (For example, if you choose around the shape of a cube or a parallelepiped, it is easy to find formulas for Cartesian coordinates, hover in the next paragraph).

In this way, the value of the Laplace operator at the point can be vitlumachene as the value of the core (stock) of the potential vector field gradF at the ts_y point. In the Cartesian coordinate system, the Laplace operator is often designated as such in the view of the scalar creation of the operator nabla on itself.

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The Laplace operator is equivalent to the successive capture of the gradient and divergence operations: texvc NOT knowledge; Math/README - grading.): \Delta=\operatorname(div)\,\operatorname(grad) In this way, the value of the Laplace operator at a point can be vitlumated as the strength of the cores (stocks) of the potential vector field It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - update about customization.): \ \operatorname(grad)F at this point. In Cartesian coordinate systems, the Laplace operator is often denoted as follows It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README - cdot update.): \Delta=\nabla\cdot\nabla=\nabla^2, then create a nabla operator on yourself like a scalar operator. The Laplace operator is symmetric.

Further definition of the Laplace operator

Laplace operator є natural zagalnennyam functions ї dekіlkoh zminnyh zminnyh other pokhodnії functions ї odnієї zminnoї. True, as a function It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - math/readme: \ f (x) may be in the vicinity of the point It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - update about the fix.): \ x_0 without interruption to a friend It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - proofreading about tweaking.): \ f""(x) those who see the Taylor formulas

another friend is ugly

Like, passing to the function It is impossible to open the virus (winning file texvc view It is impossible to open the virus (winning file texvc change, fix it the same way, then for a given point It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - finalizing the adjustment.): M_0(x_1^0,x_2^0, ... ,x_k^0) look at її It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - proofreading about tweaking.): \ k-peaceful Kulova outskirts It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - completion of the adjustment.): \ Q_r radius It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - a statement about the adjustment.): \ r and difference between the arithmetic mean

functions It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - a statement about the alignment.): \ F at the cordon It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - proof of the fix.): \ S_r such a neighborhood with a cordon area It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - proofreading about tweaking.): \ \sigma(S_r) that meaning It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - completion of the setup.): \ F(M_0) near the center and around It is impossible to open the virus (winning file texvc , then the time of the continuity of other private similar functions It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - a statement about the alignment.): \ F on the outskirts of the point It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - update about the fix.): \ M_0 Laplacian value It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - finalization.): \ \Delta F at this point there is a boundary

One hour ahead of time for the Laplace operator of the function It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - a statement about the alignment.): \ F, which can be uninterrupted other pokhіdnі, the formula is valid

Tsya formula shows a non-intermediate connection of the Laplacian function with the volume average on the outskirts of the center of the point.

The proof of these formulas can be known, for example, from .

Vishchevikladenі interі, in all vipadkah, if stinks іsnuyut, can be assigned to the Laplace operator of the function It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - an update on the alignment.): \ F. This is more important than the sizable stance of the Laplacian, which transfers the use of other similar functions, which are looked at, and zbіgaєtsya z sizable sizable values ​​at the same time without interruption of these seperate ones.

Virazi for the Laplace operator in various curvilinear coordinate systems

At fairly orthogonal curvilinear coordinates at the trivial space It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - tweaking.): q_1, \ q_2, \ q_3 :

Cylindrical coordinates

For cylindrical coordinates, the posture is straight It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - completion of the adjustment.): \ r=0 :

Spherical coordinates

For spherical coordinates, posture on the cob in front of (near the trivi-world space):

At the same time It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - completion of the improvement.): \ f=f(r) in n- peaceful space:

Parabolic coordinates

At parabolic coordinates (near the trivi-world space) posture on the cob in front of:

Cylindrical parabolic coordinates

At the coordinates of the parabolic cylinder, the position of the cob is:

Zagalny curvilinear coordinates of the Riemannian space

Let's go on a smooth rіznomanіttі It is impossible to open the virus (winning file texvc given local system coordinates ta It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - finalization about tweaking.): g_(ij)- Riemannian metric tensor on It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - finalization about tweaking.): X, then the metric can be seen

Significantly through It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - finalization about nailing.): g^(ij) matrix elements It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - proof of the nailing.): (g_(ij))^(-1)і

Vector field divergence It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - proof of the alignment.): F, given by coordinates It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - an update about the fix.): F^i(І represents the first-order differential operator It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - correction about the alignment.): \sum_i F^i\frac(\partial)(\partial x^i)) on a different scale X calculate by formula

and the components of the function gradient f- behind the formula

Operator Laplace - Beltrami on It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - finalization about tweaking.): X :

Value It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - correction about the adjustment.): \Delta fє scalar, so it does not change the hour of the coordinate transformation.

Zastosuvannya

For the help of this operator, manually write down the equalization of Laplace, Poisson and the equal equalization. У фізиці оператор Лапласа застосуємо в електростатиці та електродинаміці, квантовій механіці, у багатьох рівняннях фізики суцільних середовищ, а також при вивченні рівноваги мембран, плівок або поверхонь розділу фаз з поверхневим натягом (див. Лапласово тиск), в стаціонарних задачах дифузії та теплопровідності, зводяться , at the uninterrupted boundary, to the great equals of Laplace and Poisson and some of them zagalnennyam.

Variations and zagalnennya

• The D'Alembert operator is a generalization of the Laplace operator for hyperbolic equalities. Turn on a friend for an hour.
• The vector Laplace operator is a generalization of the Laplace operator to the point of view of the vector argument.

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Literature

Posilannya

Laplacian, - differential operator, which is defined by the formula

(here - coordinates in), as well as deakі yogo zagalnennya. L. o. (1) є simplest elіptich. differential operator of the 2nd order. L. o. played an important role in the mathematical analysis, mathematical physics and geometry (div., for example, Laplace equal, Laplace - Beltrami equal, Harmony function, Harmony form).

Let Pomsta n-peaceful Roman space with metrics

Nehai - matrix, back to matrix Todi L. o. (or the Laplace operator - Beltrami) of the Riemannian metric (2)

de - local coordinates on M. The operator (1) is recognized by the sign of L. o. standard Euclidean metric

Introduction to the operator (3) є L. o. on differential forms. To the very expanse of all known differential forms on the ML. about. may look

de d- operator of ovn_shny differentiation form, d*- formally speaking to the dooperator, who is appointed to help the offensive creation on smooth financial forms:

de * - Hodge operator, generations by the metric (2) and translating p-forms ( etc)-form. In formula (5), the forms a and b are considered descriptive; on complex forms, it is necessary to follow the Hermitian continuation of the scalar addition (5). The sounding of the operator (4) on the O-form (that is, functions) is given by formula (3). On p-forms with a large number of L. o. in local coordinates it is written as

Here - covariant trips for

The curvature tensor is the Ricci tensor. Let it be given a dovilny eliptich. complex

de E r - dіysnі or komplektnі razsharuvannya on raznomanіttі M, G (E r) - the expanse of their smooth cuts E r Hermitian metric, as well as having added a volume element to M, it is possible to distinguish the Hermitian-scalar twir at the expanses of smooth finitnyh pereriziv roses Єr. Todi appointed operators d*, formally associated with operators d. Behind formula (3) will be L. o. on the skin space G( E r). If we take the de Rham complex as a complex (6), then with a natural choice of metrics in p-forms and an obligatory element generated by the metric (2), it will appear as a L. o. de Rham's complex of descriptions of L. o. on forms.

On a complex rіznomanіttі z complex de Rama є elіptich. complexes

de - expanse of smooth forms type ( p, q).on M. Introducing the Hermitian structure in M, you can induce L. o. (4) the de Rham and L. o. complexes (7), (8):

Kozhen z tsikh operators translate in itself space Yakshcho M - Kählerian diversity, and the Hermitian structure on Mindukovan Kählerian metric, then

An important fact that determines the role of L. o. elliptical. complex, є іsnuvannya іn razі compact raznomanіtya Mortogonal razkladanі Khoja:

At tsimu razkladі de - L. o. complex (6), so - expanse of "harmonious" redistributions E r(At the same time, the de Rham complex has a lot of common harmonic forms of the step p). The direct sum of the first two additions at the right side of the formula (9) is more and the sum of the two remaining dodankivs is directly Zocrema, layout (9) sets the isomorphism of the space of the cohomology complex (6) at the term of that space of harmonies. redistribution

Lit.: Ram J. de, Differentiation of Raznomanittya, prov. from fr., M., 1956; Zhen Shen-Shen, Comprehensive Raznomanittya, prov. from English, M., 1961; Wells R., Differential calculation on complex variances, prov. from English, M., 1976. M. A. Shubin.

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