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Student[VectorCalculus]

  

Divergence

  

compute the divergence of a vector field

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Divergence(F)

Divergence(c)

Parameters

F

-

(optional) vector field or Vector-valued procedure; specify the components of the vector field

c

-

(optional) specify the coordinate system

Description

• 

The Divergence(F) calling sequence computes the divergence of the vector field F.  This calling sequence is equivalent to Del.F and DotProduct(Del, F).

• 

If F is a Vector-valued procedure, the default coordinate system is used. The default coordinate system must be indexed by the coordinate names.

  

Otherwise, F must be a Vector with the vectorfield attribute set, and it must have a coordinate system attribute that is indexed by the coordinate names.

• 

If F is a procedure, the returned object is a procedure. Otherwise, the returned object is an expression.

• 

The Divergence(c) calling sequence returns the differential form of the divergence operator in the coordinate system specified by c, which can be given as:

  

* an indexed name, e.g., sphericalr,φ,θ

  

* a name, e.g., spherical; default coordinate names will be used

  

* a list of names, e.g., r,φ,θ; the current coordinate system will be used, with these as the coordinate names

• 

The Divergence() calling sequence returns the differential form of the divergence operator in the current coordinate system.  For more information, see SetCoordinates.

Examples

withStudent[VectorCalculus]:

To create a vector field, use the Student[VectorCalculus][VectorField] command.

FVectorFieldx2,y2,z2

F:=x2e_x+y2e_y+z2e_z

(1)

DivergenceF

2x+2y+2z

(2)

Del.F

2x+2y+2z

(3)

Nabla.F

2x+2y+2z

(4)

DotProductDel,F

2x+2y+2z

(5)

Divergencex,y,z→sinx,cosy,tanz

x,y,z→cosxsiny+1+tanz2

(6)

To display the differential form of the divergence operator:

Divergence

xVF 1x,y,z+yVF 2x,y,z+zVF 3x,y,z

(7)

SetCoordinatescylindricalr,θ,z:

Divergence

rrVF 1r,θ,z+θVF 2r,θ,z+zrVF 3r,θ,zr

(8)

Divergences,φ,w

ssVF 1s,φ,w+φVF 2s,φ,w+wsVF 3s,φ,ws

(9)

Divergencespherical

rr2sinφVF 1r,φ,θ+φrsinφVF 2r,φ,θ+θrVF 3r,φ,θr2sinφ

(10)

Divergencesphericalα,ψ,γ

αα2sinψVF 1α,ψ,γ+ψαsinψVF 2α,ψ,γ+γαVF 3α,ψ,γα2sinψ

(11)

To display the divergence of an arbitrary vector-valued function (r,theta) -> <f(r,theta),g(r,theta)> in the polar coordinate system:

SetCoordinatespolarr&comma;&theta;

polarr&comma;&theta;

(12)

Divergencer&comma;&theta;&rarr;fr&comma;&theta;&comma;gr&comma;&theta;

r&comma;&theta;&rarr;fr&comma;&theta;&plus;rrfr&comma;&theta;&plus;&theta;gr&comma;&theta;r

(13)

See Also

Student[VectorCalculus]

Student[VectorCalculus][Curl]

Student[VectorCalculus][Del]

Student[VectorCalculus][DotProduct]

Student[VectorCalculus][Laplacian]

Student[VectorCalculus][Nabla]

Student[VectorCalculus][SetCoordinates]

Student[VectorCalculus][Vector]

Student[VectorCalculus][VectorField]

 


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