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laplacian

  

Laplacian of an expression

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

laplacian(f, v)

laplacian(f, v, co)

Parameters

f

-

scalar expression

v

-

vector or list of variables

co

-

(optional), is either of type `=` or a list of three elements. This option is used to compute the Laplacian in orthogonally curvilinear coordinate systems.

Description

• 

Important: The linalg package has been deprecated. Use the superseding packages VectorCalculus[Laplacian], instead.

  

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

• 

laplacian(f, v) computes the Laplacian of f with respect to v.

• 

The Laplacian is defined to be the sum of the second derivatives 2x2f for x in v.

• 

In the case of three dimensions, where f is a scalar expression of three variables and v is a list or a vector of three variables:

  

If the optional third argument co is of the form coords = coords_name or coords = coords_name({[const]}), laplacian will operate on commonly used orthogonally curvilinear coordinate systems. See ?coords for the list of coordinate systems supported by Maple.

For orthogonally curvilinear coordinates v[1], v[2], v[3]

with unit vectors a[1], a[2], a[3], and scale factors

h[1], h[2], h[3].

Let the rectangular coordinates x, y, z be defined in terms of the

specified orthogonally curvilinear coordinates. We have:

   h[n]^2 = [diff(x,v[n])^2 + diff(y,v[n])^2 + diff(z,v[n])^2], n=1,2,3.

The formula for the laplacian of f is:

   laplacian(f) = 1/(h[1]*h[2]*h[3])*sum(diff(h[1]*h[2]*h[3]/h[n]^2*

                  diff(f,v[n]),v[n]),n=1..3)

  

If the optional third argument co is a list of three elements which specify the scale factors, laplacian will operate on orthogonally curvilinear coordinate systems.

• 

To compute the Laplacian in other orthogonally curvilinear coordinate systems, use the addcoords routine.

• 

The two dimensional case is similar to the three dimensional one.

• 

The command with(linalg,laplacian) allows the use of the abbreviated form of this command.

Examples

Important: The linalg package has been deprecated. Use the superseding packages VectorCalculus[Laplacian], instead.

withlinalg:

laplacianx2yz,x,y,z

2yz

(1)

frsinθz2:vr,θ,z:

laplacianf,v,coords=cylindrical

2rsinθ

(2)

define the scale factors in cylindrical coordinates

h1,r,1:

laplacianf,v,h

2rsinθ

(3)

gr2sinθcosφ:vr,θ,φ:

laplaciang,v,coords=spherical

5r2sinθ2cosφ+r2cosθ2cosφr2cosφr2sinθ

(4)

See Also

addcoords

coords

linalg(deprecated)[curl]

linalg(deprecated)[diverge]

linalg(deprecated)[grad]

LinearAlgebra

VectorCalculus[Laplacian]