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VectorCalculus

  

Laplacian

  

compute the Laplacian of a function from R^n to R, or of a vector field

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Laplacian(f, c)

Laplacian(F)

Parameters

f

-

algebraic expression

c

-

(optional) list(name) or name[name, name, ...]; specify the list of variable names or coordinate system indexed by coordinate names

F

-

(optional) Vector or Vector-valued procedure; specify a vector field

Description

• 

The Laplacian(f, c) command computes the Laplacian of the function f.  This is a synonym for (Del . Del)(f) or Del . Del(f).

• 

If c is a list of names, the Laplacian is taken in the current default coordinate system by using the names in c as the coordinate names.  If the number of given names is not compatible with this coordinate system, an error is raised.

  

If c is a name indexed by other names, the Laplacian is computed in this coordinate system by using the indices as the coordinate names.  If the number of names is not compatible with the coordinate system, an error is raised.

  

If c is not specified, the current default coordinates are used. The default coordinates must be indexed by coordinate names, otherwise an error is raised.

• 

The Laplacian(F) command, where F is either a vector field or a Vector-valued procedure (which is interpreted as a vector field), computes the Laplacian of the vector field as follows:

– 

If the coordinate system of F is cartesian, then map the algebraic Laplacian onto the component functions.

– 

Otherwise, if F is a 3-D vector field, use the formula DelDel.FDel &x Del &x F.

– 

Otherwise, map F to cartesian coordinates, apply the algebraic Laplacian to the component functions, and then map the result back to the original coordinate system of F.

• 

The Laplacian() command returns the differential form of the Laplacian operator in the current coordinate system.  For more information, see SetCoordinates.

Examples

withVectorCalculus:

Laplacianx2+y2+z2,x,y,z

6

(1)

Laplacianfr,θ,'polar'r,θ

rfr,θ+r2r2fr,θ+2θ2fr,θrr

(2)

SetCoordinates'cylindrical'r,θ,z

cylindricalr,θ,z

(3)

Laplacianfr,θ,z

rfr,θ,z+r2r2fr,θ,z+2θ2fr,θ,zr+r2z2fr,θ,zr

(4)

Del.Delfr,θ,z

rfr,θ,z+r2r2fr,θ,z+2θ2fr,θ,zr+r2z2fr,θ,zr

(5)

Laplacian

rrrSF r,θ,z+θθSF r,θ,zr+zrzSF r,θ,zr

(6)

FVectorFieldr3,zθ,z

F:=r3e_r+zθe_θ+ze_z

(7)

simplifyLaplacianF

24r3θ2+zθ2r2e_rzθ22r2θ3e_θ14z3/2e_z

(8)

See Also

VectorCalculus

VectorCalculus[Curl]

VectorCalculus[Del]

VectorCalculus[Divergence]

VectorCalculus[DotProduct]

VectorCalculus[Gradient]

VectorCalculus[Nabla]

VectorCalculus[SetCoordinates]

VectorCalculus[VectorField]

Vectors/Laplacian

 


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