The Laplace operator is a differential operator acting in a linear space of smooth functions and is denoted by the symbol. He associates the function F with the function
The Laplace operator is equivalent to taking the gradient and divergence operations sequentially.
The gradient is a vector showing the direction of the fastest increase of a certain quantity, the value of which changes from one point in space to another (scalar field). For example, if we take the height of the Earth's surface above sea level as the height, then its gradient at each point on the surface will show the “direction of the steepest rise.” The magnitude (modulus) of the gradient vector is equal to the growth rate in this direction. For the case of three-dimensional space, the gradient is a vector function with components, where is some scalar function of the coordinates x, y, z.
If is a function of n variables, then its gradient is an n-dimensional vector
The components of which are equal to the partial derivatives of all its arguments. The gradient is denoted by grad, or using the nabla operator,
From the definition of gradient it follows that:
The meaning of the gradient of any scalar function f is that its scalar product with an infinitesimal displacement vector gives the total differential of this function with a corresponding change in coordinates in the space on which f is defined, that is, linear (in the case general position it is also the main) part of the change in f when shifted by. Using the same letter to denote a function of a vector and the corresponding function of its coordinates, we can write:
It is worth noting here that since the formula for the total differential does not depend on the type of coordinates x i, that is, on the nature of the parameters x in general, the resulting differential is an invariant, that is, a scalar, under any coordinate transformations, and since dx is a vector, then the gradient calculated in the usual way, turns out to be a covariant vector, that is, a vector represented in a dual basis, which is the only scalar that can be given by simply summing the products of the coordinates of an ordinary (contravariant) one, that is, a vector written in a regular basis.
Thus, the expression (generally speaking for arbitrary curvilinear coordinates) can be quite correctly and invariantly written as:
Or, according to Einstein’s rule, omitting the sum sign,
Divergence is a differential operator that maps a vector field onto a scalar one (that is, the differentiation operation that results in a scalar field when applied to a vector field), which determines (for each point) “how much the field incoming and outgoing from a small neighborhood of a given point diverges” (more precisely, how much the incoming and outgoing flows diverge).
If we take into account that an algebraic sign can be assigned to a flow, then there is no need to take into account the incoming and outgoing flows separately; everything will be automatically taken into account when summing taking into account the sign. Therefore, we can give a shorter definition of divergence:
divergence is a differential operator on a vector field that characterizes the flow of this field through the surface of a small neighborhood of each internal point of the field definition domain.
The divergence operator applied to the field F is denoted as or
The definition of divergence looks like this:
where ФF is the flow of the vector field F through a spherical surface of area S, limiting the volume V. Even more general, and therefore convenient to use, is the definition when the shape of a region with surface S and volume V is allowed to be any. The only requirement is that it be inside a sphere with a radius tending to zero. This definition, unlike the one given below, is not tied to specific coordinates, for example, to Cartesian ones, which can be an additional convenience in certain cases. (For example, if you choose a neighborhood in the shape of a cube or parallelepiped, the formulas for Cartesian coordinates given in the next paragraph are easily obtained).
thus, the value of the Laplace operator at a point can be interpreted as the density of sources (sinks) of the potential vector field gradF at this point. In the Cartesian coordinate system, the Laplace operator is often denoted as follows, that is, in the form of a scalar product of the Nabla operator with itself.
It is a special case of the Helmholtz equation. Can be considered in three-dimensional (1), two-dimensional (2), one-dimensional and n-dimensional spaces:
The operator is called the Laplace operator (The Laplace operator is equivalent to taking the gradient and divergence operations sequentially.).
Solution of Laplace's equation
The solutions to Laplace's equation are harmonic functions.
Laplace's equation belongs to elliptic equations. Laplace's inhomogeneous equation becomes Poisson's equation.
Each solution of the Laplace equation in a bounded domain G is uniquely identified by boundary conditions imposed on the behavior of the solution (or its derivatives) on the boundary of the domain G. If the solution is sought in the entire space , the boundary conditions are reduced to the prescription of some asymptotic behavior for f at . The problem of finding such solutions is called a boundary value problem. The most common are the Dirichlet problem, when the value of the function f itself is given on the boundary, and the Neman problem, when the value of f is given along the normal to the boundary.
Laplace's equation in spherical, polar and cylindrical coordinates
Laplace's equation can be written not only in Cartesian coordinates.
In spherical coordinates (Laplace's equation has the following form:
In polar coordinates (coordinate system), the equation is:
In cylindrical coordinates (the equation is:
Many problems in physics and mechanics lead to the Laplace equation, in which a physical quantity is a function only of the coordinates of a point. Thus, Laplace’s equation describes the potential in a region that does not contain gravitating masses, the potential of an electrostatic field in a region that does not contain charges, temperature during stationary processes, etc. A large number of engineering problems associated, in particular, with slow stationary flow around a ship’s hull , stationary filtration of groundwater, the emergence of a field around the electromagnet, as well as stationary electric field in the vicinity of a porcelain insulator or an electrical cable of variable cross-section buried in the ground, comes down to solving the three-dimensional Laplace or Poisson equations. Great value The Laplace operator plays quantum mechanics.
Examples of problem solving
EXAMPLE 1
| Exercise | Find the field between two coaxial cylinders with radii and , the potential difference between which is equal to |
| Solution | Let us write Laplace's equation in cylindrical coordinates, taking into account axial symmetry:
It has a solution Hence We get:
As a result we have: |
| Answer | The field between two coaxial cylinders is given by the function
|
+B. Let's choose the zero potential on the outer cylinder, find it, and get:
EXAMPLE 2
| Exercise | Investigate the stability of the equilibrium of a positively charged particle in an electric field (Earnshaw’s theorem). |
| Solution | Let us place the origin of coordinates at the equilibrium position of the particle. In this case, we can assume that the potential is represented in the form: |
We considered three main operations of vector analysis: calculating gradtx for a scalar field a and rot a for a vector field a = a(x, y, z). These operations can be written in more in simple form using the symbolic operator V (“nabla”): The operator V (Hamilton operator) has both differential and vector properties. Formal multiplication, for example, multiplication ^ by the function u(x, y), will be understood as partial differentiation: Within the framework of vector algebra, formal operations on the operator V will be carried out as if it were a vector. Using this formalism, we obtain the following basic formulas: 1. If is a scalar differentiable function, then by the rule of multiplying a vector by a scalar we obtain where P, Q, R are differentiable functions, then by the formula for finding the scalar product we obtain Hamilton operator Second order differential operations Operator Laplace Concept of curvilinear coordinates Spherical coordinates 3. Calculating the vector product, we obtain For a constant function and = c we obtain and for a constant vector c we have From the distribution property for the scalar and vector products we obtain Remark 1. Formulas (5) and (6) can be interpreted Tamka as a manifestation of the differential properties of the “nabla” operator (V is a linear differential operator). We agreed that the operator V acts on all quantities written after it. In this sense, for example, is a scalar differential operator. When applying the operator V to the product of any quantities, one must keep in mind the usual rule for differentiating the product. Example 1. Prove that According to formula (2), taking into account Remark 1, we obtain or To note the fact that “obs a” does not act on any value included in the complex formula, this value is marked with the index c (“const” ), which is omitted in the final result. Example 2. Let u(xty,z) be a scalar differentiable function, and (x,y,z) be a vector differentiable function. Prove that 4 Rewrite the left side of (8) in symbolic form Taking into account the differential nature of the operator V, we obtain. Since u is a constant scalar, it can be taken out of the sign of the scalar product, so that a (at the last step we omitted the index e). In the expression (V, iac), the operator V acts only on a scalar function and, therefore, As a result, we obtain Remark 2. Using the formalism of acting with the operator V as a vector, we must remember that V is not an ordinary vector - it has neither length, no direction, so. for example, vector, Where Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): H_i\- Lamé coefficients.
Cylindrical coordinates
In cylindrical coordinates outside the line Unable to parse expression (Executable file texvc not found; See math/README for setup help.):\r=0
:
texvc not found; See math/README for setup help.): \Delta f = (1 \over r) (\partial \over \partial r) \left(r (\partial f \over \partial r) \right) + ( \partial^2f \over \partial z^2) + (1 \over r^2) (\partial^2 f \over \partial \varphi^2)
Spherical coordinates
In spherical coordinates outside the origin (in three-dimensional space):
Unable to parse expression (Executable filetexvc not found; See math/README for help with setting up.: \Delta f = (1 \over r^2) (\partial \over \partial r) \left(r^2 (\partial f \over \partial r) \ right) + (1 \over r^2 \sin \theta) (\partial \over \partial \theta) \left(\sin \theta (\partial f \over \partial \theta) \right) + (1 \ over r^2\sin^2 \theta) (\partial^2 f \over \partial \varphi^2)
Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \Delta f = (1 \over r) (\partial^2 \over \partial r^2) \left(rf \right) + (1 \over r^2 \sin \theta) (\partial \over \partial \theta) \left(\sin \theta (\partial f \over \partial \theta) \right) + (1 \over r^2 \sin^2 \theta ) (\partial^2 f \over \partial \varphi^2).
In case Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \ f=f(r) V n-dimensional space:
texvc not found; See math/README for help with setting up.): \Delta f = (d^2 f\over dr^2) + (n-1 \over r ) (df\over dr).
Parabolic coordinates
In parabolic coordinates (in three-dimensional space) outside the origin:
Unable to parse expression (Executable filetexvc not found; See math/README for help with setup.): \Delta f= \frac(1)(\sigma^(2) + \tau^(2)) \left[ \frac(1)(\sigma) \frac (\partial )(\partial \sigma) \left(\sigma \frac(\partial f)(\partial \sigma) \right) + \frac(1)(\tau) \frac(\partial )(\partial \tau) \left(\tau \frac(\partial f)(\partial \tau) \right)\right] + \frac(1)(\sigma^2\tau^2)\frac(\partial^2 f)(\partial \varphi^2)
Cylindrical parabolic coordinates
In the coordinates of a parabolic cylinder outside the origin:
Unable to parse expression (Executable filetexvc not found; See math/README - help with setup.): \Delta F(u,v,z) = \frac(1)(c^2(u^2+v^2)) \left[ \frac(\partial ^2 F )(\partial u^2)+ \frac(\partial^2 F )(\partial v^2)\right] + \frac(\partial^2 F )(\partial z^2).
General curvilinear coordinates and Riemannian spaces
Let on a smooth manifold Unable to parse expression (Executable file texvc a local coordinate system is specified and Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): g_(ij)- Riemannian metric tensor on Unable to parse expression (Executable file texvc not found; See math/README for setup help.): X, that is, the metric has the form
texvc not found; See math/README - help with setup.): ds^2 =\sum^n_(i,j=1)g_(ij) dx^idx^j
.
Let us denote by Unable to parse expression (Executable file texvc not found; See math/README for setup help.): g^(ij) matrix elements Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): (g_(ij))^(-1) And
texvc not found; See math/README for setup help.): g = \operatorname(det) g_(ij) = (\operatorname(det) g^(ij))^(-1)
.
Vector field divergence Unable to parse expression (Executable file texvc not found; See math/README for setup help.): F, specified by the coordinates Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): F^i(and representing the first order differential operator Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \sum_i F^i\frac(\partial)(\partial x^i)) on the manifold X calculated by the formula
texvc not found; See math/README for setup help.): \operatorname(div) F = \frac(1)(\sqrt(g))\sum^n_(i=1)\frac(\partial)(\partial x ^i)(\sqrt(g)F^i)
,
and the gradient components of the function f- according to the formula
Unable to parse expression (Executable filetexvc not found; See math/README - help with setup.): (\nabla f)^j =\sum^n_(i=1)g^(ij) \frac(\partial f)(\partial x^i).
Laplace operator - Beltrami on Unable to parse expression (Executable file texvc not found; See math/README for setup help.): X
:
texvc not found; See math/README - help with setup.): \Delta f = \operatorname(div) (\nabla f)= \frac(1)(\sqrt(g))\sum^n_(i=1)\frac (\partial)(\partial x^i)\Big(\sqrt(g) \sum^n_(k=1)g^(ik) \frac(\partial f)(\partial x^k)\Big) .
Meaning Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \Delta f is a scalar, that is, it does not change when transforming coordinates.
Application
Using this operator it is convenient to write Laplace's, Poisson's and wave equations. In physics, the Laplace operator is applicable in electrostatics and electrodynamics, quantum mechanics, in many equations of continuum physics, as well as in the study of the equilibrium of membranes, films or interfaces with surface tension (see Laplace pressure), in stationary problems of diffusion and thermal conductivity, which reduce, in the continuous limit, to the usual equations of Laplace or Poisson or to some of their generalizations.
Variations and generalizations
- The D'Alembert operator is a generalization of the Laplace operator for hyperbolic equations. Includes the second derivative with respect to time.
- The vector Laplace operator is a generalization of the Laplace operator to the case of a vector argument.
See also
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