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the classical partial differential equations for Euclidean space (Topic)

Consider some partial differential equations in which the number of independent variables is greater than two, we note here that the most important equations.

Laplace's equation

$\displaystyle \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = 0$

The wave equation

$\displaystyle \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial ... ...{\partial^2 V}{\partial z^2} = \frac{1}{c^2} \frac{\partial^2 V}{\partial t^2} $

The equation of the conduction of heat

$\displaystyle \frac{\partial^2 V}{\partial t^2} = \kappa \left( \frac{\partial^... ...\frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} \right ) $

The equation for the conduction of electricity

$\displaystyle c^2 \left( \frac{\partial^2 E}{\partial x^2} + \frac{\partial^2 E... ...u \frac{\partial^2 E}{\partial t^2} + \sigma \mu \frac{\partial E}{\partial t} $

The wave equation of Schrodinger's theory of wave mechanics.

This last equation takes many different forms and we shall mention here only the simple form of the equation in which the dependence of $\psi$ on the time has already been taken into consideration. The reduced equation is then

$\displaystyle \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\pa... ...ial^2 \psi}{\partial z^2} + \frac{8 \pi^2}{h^2} \left ( E - V \right ) \psi = 0$

where $V$ is a function of $x$, $y$ and $z$ and $E$ is a constant to be determined.

In these equations $\kappa$ represents the diffusivity of thermometric conductivity of the medium, $K$ the specific inductive capacity, $\mu$ the permeability, and $\sigma$ the electric conductivity of the medium. The quantities $c$ and $h$ are universal constants, $c$ being the velocity of light in vacuum and $h$ being Plank's constant which occurs in his theory of radiation.

Laplace's equation, which for brevity may be written in the form

$\displaystyle \nabla^2 V = 0 $

may be obtained in various ways from a set of linear equations of the first order. One ,set,

$\displaystyle X = \frac{\partial V}{\partial x}, \,\,\, Y = \frac{\partial V}{\... ...partial x} + \frac{\partial V}{\partial y} + \frac{\partial V}{\partial z} = 0,$

occurs naturally in the theory of attractions, $V$ being the gravitational potential and $X$, $Y$, $Z$ the components of force per unit mass. The last equation is then a consequence of Gauss's theorem that the surface integral of the normal force is zero for any closed surface not containing any attracting matter.

The same equations occur also in hydrodynamics, the potential $V$ being replaced by the velocity potential $\phi$ and the quantities $X$, $Y$, $Z$ by the component velocities $u$, $v$, $w$. The equation is then the equation of continuity of an incompressible fluid.

The electric and magnetic interpretations of $X$, $Y$, $Z$ and $V$ are similar to the gravitational except that the electric (or magnetic) potential is usually taken to be - $V$ when $X$, $Y$, $Z$ are the force intensities.

As in the two-dimensional theory, Laplace's equation is satisfied by the potential $V$ because by the principle of superposition $V$ is expressed as the sum of a number of elementary potentials each of which happens to be a solution of Laplace's equation, the elementary potential being of type

$\displaystyle V = \left[ \left(x - x' \right)^2 + \left(y - y' \right)^2 + \left(z - z' \right)^2 \right]^{-\frac{1}{2}} = 1 / R $

When $V$ is interpreted as the electrostatic potential this elementary potential is regarded as that of a unit point charge at the point $\left( x', y', z' \right)$; when $V$ is interpreted as a magnetic potential the elementary potential is that of a unit magnetic pole. In the theory of gravitation the elementary potential is that of unit mass concentrated as the point $\left( x, y, z \right)$. A more general expression for a potential is

$\displaystyle V = \sum m_s \left[ \left(x - x' \right)^2 + \left(y - y' \right)^2 + \left(z - z' \right)^2 \right]^{-\frac{1}{2}} $

where the coefficient $m_s$ is a measure of the strength of the charge, pole or mass concentrated at the point $\left( x_s, y_s, z_s \right)$. If we wrote $\phi$ in place of $V$, where $\phi$ is a velocity potential for a fluid motion in three dimensions, the elementary potential is that of a source and the coefficient $m_s$ can be interpreted as the strength of the source at $\left( x_s, y_s, z_s \right)$. Sources and sinks are useful in hydrodynamics as they give a convenient representation of the disturbance produced by a body when it is placed in a steady stream.

This article is a derivative work of the public domain in [1].

Bibliography

[1] Bateman, H., "Partial Differential Equations of Mathematical Physics" Cambridge University Press, 1923.



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Cross-references: domain, work, representation, motion, charge, type, two-dimensional, theorem, mass, radiation, velocity, universal constants, conductivity, function, mechanics, wave, heat, conduction, wave equation, partial differential equations

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Physics Classification: 02.30.Jr (Partial differential equations)
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