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The vector Laplace operator, also denoted by , is a differential operator defined over a vector field. [7] The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field , returning a vector quantity.
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.
Conversely, characterizes the Laplace–Beltrami operator completely, in the sense that it is the only operator with this property. As a consequence, the Laplace–Beltrami operator is negative and formally self-adjoint, meaning that for compactly supported functions f {\displaystyle f} and h {\displaystyle h} ,
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.
In potential theory (a branch of mathematics), the Laplacian of the indicator is obtained by letting the Laplace operator work on the indicator function of some domain D. It is a generalisation of the derivative (or "prime function") of the Dirac delta function to higher dimensions; it is non-zero only on the surface of D.
Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation .
In mathematics, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator , where p {\displaystyle p} is allowed to range over 1 < p < ∞ {\displaystyle 1<p<\infty } .
In mathematics, the infinity Laplace (or -Laplace) operator is a 2nd-order partial differential operator, commonly abbreviated .It is alternately defined, for a function : of the variables = (, …,), by