Search results
Results from the WOW.Com Content Network
In spherical coordinates in N dimensions, with the parametrization x = rθ ∈ R N with r representing a positive real radius and θ an element of the unit sphere S N−1, = + + where Δ S N−1 is the Laplace–Beltrami operator on the (N − 1)-sphere, known as the spherical Laplacian.
The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator.
6.10 Laplacian on functions. ... Download as PDF; Printable version; ... An orthonormal inertial frame is a coordinate chart such that, ...
The spherical Laplacian is the Laplace–Beltrami operator on the (n − 1)-sphere with its canonical metric of constant sectional curvature 1. It is convenient to regard the sphere as isometrically embedded into R n as the unit sphere centred at the origin. Then for a function f on S n−1, the spherical Laplacian is defined by
Starting from homogeneous initial conditions with only one point-like perturbation, the correct growth process would yield a circle (left). If growth is faster along the coordinate axes, which is caused by the anisotropic effect of central difference discretization, the circle would turn into star-shaped structure, as the errors propagate (right):.
Because is a linear differential operator, the solution () to a general system of this type can be written as an integral over a distribution of source given by (): = (, ′) (′) ′ where the Green's function for Laplacian in three variables (, ′) describes the response of the system at the point to a point source located at ...
The negative of the Laplacian in R d given by = = is a uniformly elliptic operator. The Laplace operator occurs frequently in electrostatics. The Laplace operator occurs frequently in electrostatics. If ρ is the charge density within some region Ω, the potential Φ must satisfy the equation − Δ Φ = 4 π ρ . {\displaystyle -\Delta \Phi =4 ...
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid.For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matrix.