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Laplace’s Equation – W.B.C.S. Examination Notes – On Chemistry.
ল্যাপ্লেসের সমীকরণ – WBCS পরীক্ষার নোট – রসায়নের উপর।
Laplace’s equation is a second-order partial differential equation named after Pierre-Simon Laplace
who first studied its properties. This is often written as, Continue Reading Laplace’s Equation – W.B.C.S. Examination Notes – On Chemistry.
where ∆ = ∇
2 is the Laplace operator and φ is a scalar function. In general, ∆ = ∇
2 is the Laplace–Beltrami or
Laplace–de Rham operator.
Laplace’s equation and Poisson’s equation are the simplest examples of elliptic partial differential equations. The
general theory of solutions to Laplace’s equation is known as potential theory. The solutions of Laplace’s equation
are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism,
astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric,
gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat
equation.
Definition
In three dimensions, the problem is to find twice-differentiable real-valued functions f, of real variables x, y, and z,
such that
In Cartesian coordinates
In cylindrical coordinates,
In spherical coordinates,
In Curvilinear coordinates,
or
This is often written as
or, especially in more general contexts,
where ∆ = ∇
2 is the Laplace operator or “Laplacian”
where ∇ • is the divergence operator (also symbolized “div”) which maps vectors to scalars, and ∇ is the gradient
operator (also symbolized “grad”) which maps vectors to vectors. (hence, the Laplacian Δf ≝ div grad f, maps the
vector function f to a scalar magnitude; specifically it maps the grad (the partial derivatives ) of f to a scalar
(function).)
The Laplace equation can be considered the prototypical elliptic PDE. At this point we supplement the discussion motivated by the method of separation of variables with some additional observations. The importance of Laplace’s equation for electrostatics has stimulated the development of a great variety of methods for its solution in the presence of boundary conditions ranging from simple and symmetrical to complicated and convoluted. Techniques for present-day engineering problems tend to rely heavily on computational methods. The thrust of this section, however, will be on general properties of the Laplace equation and its solutions.
The basic properties of the Laplace equation are independent of the coordinate system in which it is expressed; we assume for the moment that we will use Cartesian coordinates. Then, because the PDE sets the sum of the second derivatives, ∂2ψ/∂x2i, to zero, it is obvious that if any of the second derivatives has a positive sign, at least one of the others must be negative. This point is illustrated in Example 9.4.1, where the x and y dependence of a solution to the Laplace equation was sinusoidal, and as a result, the zdependence was exponential (corresponding to different signs for the second derivative). Since the second derivative is a measure of curvature, we conclude that if ψ has positive curvature in any coordinate direction, it must have negative curvature in some other coordinate direction. That observation, in turn, means that all the stationary points of ψ (points where its first derivatives in all directions vanish) must be saddle points, not maxima or minima. Since the Laplace equation describes the static electric potential in charge-free regions, we conclude that the potential cannot have an extremum at a point where there is no charge. A corollary to this observation is that the extrema of the electrostatic potential in a charge-free region must be on the boundary of the region.
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