Search results
Results from the WOW.Com Content Network
In mathematics and physics, 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 = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
The free-space circular cylindrical Green's function (see below) is given in terms of the reciprocal distance between two points. The expression is derived in Jackson's Classical Electrodynamics. [1] Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in
A solution to Laplace's equation defined on an annulus.The Laplace operator is the most famous example of an elliptic operator.. In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator.
In this equation, we used sup and inf instead of max and min because the graph (,) does not have to be locally finite (i.e., to have finite degrees): a key example is when () is the set of points in a domain in , and (,) if their Euclidean distance is at most . The importance of this example lies in the following.
The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions: (,,) = | | (,) (,) where the () are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary ...
It can be further verified that the above identity also applies when ψ is a solution to the Helmholtz equation or wave equation and G is the appropriate Green's function. In such a context, this identity is the mathematical expression of the Huygens principle , and leads to Kirchhoff's diffraction formula and other approximations.
In potential theory, an area of mathematics, a double layer potential is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface S in three-dimensions.
However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector field, which can be a point of confusion. For example, the vector field v = ( x y , y z , z x ) {\displaystyle {\bf {v}}=(xy,yz,zx)} satisfies Laplace's equation, but it has both nonzero divergence and nonzero curl and is not a ...