The Laplacian in three-dimensional Cartesian coordinates:

The Laplacian in three-dimensional cylindrical coordinates:

The Laplacian in two-dimensional polar coordinates:

Use del to enter ∇, for the list of subscripted variables, and to enter the 2:

Use del2 to enter the template , fill in the variables, press , and fill in the function:

Laplacian applies to arrays of arbitrary rank:

In a curvilinear coordinate system, a vector with constant components may have a nonzero Laplacian:

A Laplacian specifying metric, coordinate system, and parameters:

Laplacian works on curved spaces:

The Laplacian of the coordinate vector is SymbolicZerosArray[{n}]:

The Laplacian of the squared norm is expressed in terms of SymbolicIdentityArray[{n}]:

Use TensorExpand to simplify to the expected result, namely twice the dimension:

The Laplacian of the squared norm in n dimensions:

Activate the sum to get the simple result:

Poisson's equation in spherical coordinates:

Solve for a radially symmetric charge distribution :

The Laplacian on the unit sphere:

The spherical harmonics are eigenfunctions of this operator with eigenvalue :

The generalization of the Coulomb potential—the electric potential of a point charge—to n dimensions is:

Since the charge density is only nonzero at the origin, the Laplacian must be equal to zero everywhere else:

Simplify the result:

Activating the result in specific dimensions and combining denominators shows the result is zero:

This result can also be obtained in each dimension using spherical coordinates:

Laplacian preserves the shape of an array:

The Laplacian is equal to the divergence of the gradient:

Since Grad uses an orthonormal basis, the Laplacian of a scalar equals the trace of the double gradient:

For higher-rank arrays, this is the contraction of the last two indices of the double gradient:

Compute Laplacian in a Euclidean coordinate chart c by transforming to and then back from Cartesian coordinates:

The result is the same as directly computing Laplacian[f,{x₁,…,x_n},c]:

The Laplacian of an array equals the Laplacian of its components only in Cartesian coordinates:

If chart is defined with metric g, expressed in the orthonormal basis, Laplacian[g,{x₁,…,x_n},chart] is zero:

For a vector field in three-dimensional flat space, the Laplacian is equal to :

In a flat space of dimension , the Laplacian of a vector field equals . For :

Laplacian preserves the symmetry structure of SymmetrizedArray objects:

The Laplacian has the same symmetry as the input:

View expressions for the Laplacian of a scalar function in different coordinate systems: