Laplacian

Laplacian[f,{x1,,xn}]

gives the Laplacian .

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

gives the Laplacian in the given coordinates chart.

Details

• Laplacian is also known as LaplaceBeltrami operator. When applied to vector fields, it is also known as vector Laplacian.
• 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.
• For coordinate charts on Euclidean space, Laplacian[f,{x1,,xn},chart] can be computed by transforming f to Cartesian coordinates, computing the ordinary Laplacian and transforming back to chart. »
• A property of Laplacian is that if chart is defined with metric g, expressed in the orthonormal basis, then Laplacian[g,{x1,,xn]},chart] gives zero. »
• 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

open allclose all

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(8)

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,,xn},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,,xn},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:

Interactive Examples(1)

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

Wolfram Research (2012), Laplacian, Wolfram Language function, https://reference.wolfram.com/language/ref/Laplacian.html (updated 2014).

Text

Wolfram Research (2012), Laplacian, Wolfram Language function, https://reference.wolfram.com/language/ref/Laplacian.html (updated 2014).

CMS

Wolfram Language. 2012. "Laplacian." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/Laplacian.html.

APA

Wolfram Language. (2012). Laplacian. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Laplacian.html

BibTeX

@misc{reference.wolfram_2022_laplacian, author="Wolfram Research", title="{Laplacian}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Laplacian.html}", note=[Accessed: 20-March-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_laplacian, organization={Wolfram Research}, title={Laplacian}, year={2014}, url={https://reference.wolfram.com/language/ref/Laplacian.html}, note=[Accessed: 20-March-2023 ]}