As a second-order differential operator, the Laplace operator maps Ck functions to Ck−2 functions for k ≥ 2. It is a linear operator Δ : Ck(Rn) → Ck−2(Rn), or more generally, an operator Δ : Ck(Ω) → Ck−2(Ω) for any
open setΩ ⊆ Rn.
Motivation
Diffusion
In the
physical theory of
diffusion, the Laplace operator arises naturally in the mathematical description of
equilibrium.[1] Specifically, if u is the density at equilibrium of some quantity such as a chemical concentration, then the
net flux of u through the boundary ∂V of any smooth region V is zero, provided there is no source or sink within V:
Since this holds for all smooth regions V, one can show that it implies:
The left-hand side of this equation is the Laplace operator, and the entire equation Δu = 0 is known as
Laplace's equation. Solutions of the Laplace equation, i.e. functions whose Laplacian is identically zero, thus represent possible equilibrium densities under diffusion.
The Laplace operator itself has a physical interpretation for non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in a sense made precise by the
diffusion equation. This interpretation of the Laplacian is also explained by the following fact about averages.
Averages
Given a twice continuously differentiable function and a point .
Then, the average value of over the ball with radius centered at is:[2]
Similarly, the average value of over the sphere (the boundary of a ball) with radius centered at is:
This is a consequence of
Gauss's law. Indeed, if V is any smooth region with boundary ∂V, then by Gauss's law the flux of the electrostatic field E across the boundary is proportional to the charge enclosed:
where the first equality is due to the
divergence theorem. Since the electrostatic field is the (negative) gradient of the potential, this gives:
Since this holds for all regions V, we must have
The same approach implies that the negative of the Laplacian of the
gravitational potential is the
mass distribution. Often the charge (or mass) distribution are given, and the associated potential is unknown. Finding the potential function subject to suitable boundary conditions is equivalent to solving
Poisson's equation.
Energy minimization
Another motivation for the Laplacian appearing in physics is that solutions to Δf = 0 in a region U are functions that make the
Dirichlet energyfunctionalstationary:
To see this, suppose f : U → R is a function, and u : U → R is a function that vanishes on the boundary of U. Then:
In spherical coordinates in N dimensions, with the parametrization x = rθ ∈ RN with r representing a positive real radius and θ an element of the
unit sphereSN−1,
where ΔSN−1 is the
Laplace–Beltrami operator on the (N − 1)-sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as:
As a consequence, the spherical Laplacian of a function defined on SN−1 ⊂ RN can be computed as the ordinary Laplacian of the function extended to RN∖{0} so that it is constant along rays, i.e.,
homogeneous of degree zero.
In fact, the algebra of all scalar linear differential operators, with constant coefficients, that commute with all Euclidean transformations, is the polynomial algebra generated by the Laplace operator.
The vector Laplace operator, also denoted by , is a
differential operator defined over a
vector field.[6] 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. When computed in
orthonormalCartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.
The vector Laplacian of a
vector field is defined as
where , , and are the components of the vector field , and just on the left of each vector field component is the (scalar) Laplace operator. This can be seen to be a special case of Lagrange's formula; see
Vector triple product.
The Laplacian of any
tensor field ("tensor" includes scalar and vector) is defined as the
divergence of the
gradient of the tensor:
For the special case where is a
scalar (a tensor of degree zero), the
Laplacian takes on the familiar form.
If is a vector (a tensor of first degree), the gradient is a
covariant derivative which results in a tensor of second degree, and the divergence of this is again a vector. The formula for the vector Laplacian above may be used to avoid tensor math and may be shown to be equivalent to the divergence of the
Jacobian matrix shown below for the gradient of a vector:
And, in the same manner, a
dot product, which evaluates to a vector, of a vector by the gradient of another vector (a tensor of 2nd degree) can be seen as a product of matrices:
This identity is a coordinate dependent result, and is not general.
A version of the Laplacian can be defined wherever the
Dirichlet energy functional makes sense, which is the theory of
Dirichlet forms. For spaces with additional structure, one can give more explicit descriptions of the Laplacian, as follows.
The Laplacian also can be generalized to an elliptic operator called the Laplace–Beltrami operator defined on a
Riemannian manifold. The Laplace–Beltrami operator, when applied to a function, is the
trace (tr) of the function's
Hessian:
where the trace is taken with respect to the inverse of the
metric tensor. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on
tensor fields, by a similar formula.
Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the
exterior derivative, in terms of which the "geometer's Laplacian" is expressed as
Here δ is the
codifferential, which can also be expressed in terms of the
Hodge star and the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian is defined on
differential formsα by
It is the generalization of the Laplace operator in the sense that it is the differential operator which is invariant under the
isometry group of the underlying space and it reduces to the Laplace operator if restricted to time-independent functions. The overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high-energy
particle physics. The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the
wave equations, and it is also part of the
Klein–Gordon equation, which reduces to the wave equation in the massless case.
The additional factor of c in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the x direction were measured in meters while the y direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that c = 1 in order to simplify the equation.