Laplacian
✖
Laplacian

Details

- Laplacian is also known as Laplace–Beltrami 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[f,VectorSymbol[…]] computes the Laplacian with respect to the vector symbol. »
- Laplacian works with SparseArray and structured array objects.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
The Laplacian in three-dimensional Cartesian coordinates:

https://wolfram.com/xid/09b58m8lm-jo1t8y

The Laplacian in three-dimensional cylindrical coordinates:

https://wolfram.com/xid/09b58m8lm-hhtha2

The Laplacian in two-dimensional polar coordinates:

https://wolfram.com/xid/09b58m8lm-80ahcj

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

https://wolfram.com/xid/09b58m8lm-jhdup

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

https://wolfram.com/xid/09b58m8lm-fyv1xr

Scope (6)Survey of the scope of standard use cases
Laplacian applies to arrays of arbitrary rank:

https://wolfram.com/xid/09b58m8lm-i2o0z6


https://wolfram.com/xid/09b58m8lm-6uz0pu

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

https://wolfram.com/xid/09b58m8lm-foqved

A Laplacian specifying metric, coordinate system, and parameters:

https://wolfram.com/xid/09b58m8lm-bxhu6q

Laplacian works on curved spaces:

https://wolfram.com/xid/09b58m8lm-q0dzv2

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

https://wolfram.com/xid/09b58m8lm-3dxwcl

https://wolfram.com/xid/09b58m8lm-1xc556

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

https://wolfram.com/xid/09b58m8lm-ld958

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

https://wolfram.com/xid/09b58m8lm-rdmmlv

The Laplacian of the squared norm in n dimensions:

https://wolfram.com/xid/09b58m8lm-3pi6az

Activate the sum to get the simple result:

https://wolfram.com/xid/09b58m8lm-le2c2b

Applications (3)Sample problems that can be solved with this function
Poisson's equation in spherical coordinates:

https://wolfram.com/xid/09b58m8lm-fazez8

Solve for a radially symmetric charge distribution :

https://wolfram.com/xid/09b58m8lm-t7jw7k

The Laplacian on the unit sphere:

https://wolfram.com/xid/09b58m8lm-bjoj6h

The spherical harmonics are eigenfunctions of this operator with eigenvalue
:

https://wolfram.com/xid/09b58m8lm-dwptuj


https://wolfram.com/xid/09b58m8lm-nnh8zk

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

https://wolfram.com/xid/09b58m8lm-shiqtq

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

https://wolfram.com/xid/09b58m8lm-mhae9s


https://wolfram.com/xid/09b58m8lm-oahxvt

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

https://wolfram.com/xid/09b58m8lm-yi1hbh

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

https://wolfram.com/xid/09b58m8lm-0yoqhg

Properties & Relations (8)Properties of the function, and connections to other functions
Laplacian preserves the shape of an array:

https://wolfram.com/xid/09b58m8lm-2g67ex


https://wolfram.com/xid/09b58m8lm-w0sh4c

The Laplacian is equal to the divergence of the gradient:

https://wolfram.com/xid/09b58m8lm-9qp481


https://wolfram.com/xid/09b58m8lm-cmgslw

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

https://wolfram.com/xid/09b58m8lm-c3fhyc


https://wolfram.com/xid/09b58m8lm-lsep4m

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

https://wolfram.com/xid/09b58m8lm-4hrf73


https://wolfram.com/xid/09b58m8lm-gj0fov


https://wolfram.com/xid/09b58m8lm-gmk3he

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

https://wolfram.com/xid/09b58m8lm-rwnq60


https://wolfram.com/xid/09b58m8lm-bzw6df


https://wolfram.com/xid/09b58m8lm-qkiv4d

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

https://wolfram.com/xid/09b58m8lm-c517r8

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

https://wolfram.com/xid/09b58m8lm-7j4pt7


https://wolfram.com/xid/09b58m8lm-dzm10o

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

https://wolfram.com/xid/09b58m8lm-0q5baa

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

https://wolfram.com/xid/09b58m8lm-cnvlaf


https://wolfram.com/xid/09b58m8lm-lvctfi


https://wolfram.com/xid/09b58m8lm-ritu9j


https://wolfram.com/xid/09b58m8lm-6o2lni

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

https://wolfram.com/xid/09b58m8lm-xckotd

https://wolfram.com/xid/09b58m8lm-wn0xv3

https://wolfram.com/xid/09b58m8lm-t9dzxl

https://wolfram.com/xid/09b58m8lm-szlfyh

Laplacian preserves the symmetry structure of SymmetrizedArray objects:

https://wolfram.com/xid/09b58m8lm-7eudwf

The Laplacian has the same symmetry as the input:

https://wolfram.com/xid/09b58m8lm-gyijiw

Wolfram Research (2012), Laplacian, Wolfram Language function, https://reference.wolfram.com/language/ref/Laplacian.html (updated 2024).
Text
Wolfram Research (2012), Laplacian, Wolfram Language function, https://reference.wolfram.com/language/ref/Laplacian.html (updated 2024).
Wolfram Research (2012), Laplacian, Wolfram Language function, https://reference.wolfram.com/language/ref/Laplacian.html (updated 2024).
CMS
Wolfram Language. 2012. "Laplacian." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Laplacian.html.
Wolfram Language. 2012. "Laplacian." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. 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
Wolfram Language. (2012). Laplacian. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Laplacian.html
BibTeX
@misc{reference.wolfram_2025_laplacian, author="Wolfram Research", title="{Laplacian}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Laplacian.html}", note=[Accessed: 02-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_laplacian, organization={Wolfram Research}, title={Laplacian}, year={2024}, url={https://reference.wolfram.com/language/ref/Laplacian.html}, note=[Accessed: 02-May-2025
]}