Laplacian

Laplacian[f,{x1,,xn}]

gives the Laplacian .

Laplacian[f,{x1,,xn},chart]

gives the Laplacian in the given coordinates chart.

Details

  • Laplacian[f,x] can be input as f. The character can be typed as del or \[Del]. The list of variables x and the 2 are entered as a subscript and superscript, respectively.
  • An empty template can be entered as del2, and moves the cursor from the subscript to the main body.
  • All quantities that do not explicitly depend on the variables given are taken to have zero partial derivative.
  • Laplacian[f,{x1,x2,}] yields a result with the same dimensions as f.
  • In Laplacian[f,{x1,,xn},chart], if f is an array, it must have dimensions {n,,n}. The components of f are interpreted as being in the orthonormal basis associated to chart.
  • Coordinate charts in the third argument of Laplacian can be specified as triples {coordsys,metric,dim} in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.
  • Laplacian works with SparseArray and structured array objects.

Examples

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Basic Examples  (4)

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:

Scope  (5)

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 squared norm in n dimensions:

Activate the sum to get the simple result:

Applications  (3)

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 potentialthe electric potential of a point chargeto 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:

Properties & Relations  (6)

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:

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

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:

Interactive Examples  (1)

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

Introduced in 2012
 (9.0)
 |
Updated in 2014
 (10.0)