TAU. Laplace operator and transfer functions. Laplace equation Laplace operator in a curvilinear coordinate system
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 a simpler 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 setup help.: \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 - help with setup.): \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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