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Student[VectorCalculus]

  

Curl

  

compute the curl of a vector field in R^3

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Curl(F)

Curl(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 Curl(F) calling sequence computes the curl of the vector field F in R^3.  This is equivalent to Del &x F and CrossProduct(Del, F).

• 

If F is a Vector-valued procedure, the default coordinate system is used, and it must be indexed by the coordinate names.  Otherwise, F must be a vector field.

• 

If F is a procedure, the result is a procedure.  Otherwise, the result is a vector field.

• 

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

Examples

withStudent[VectorCalculus]:

FVectorFieldy,x,0

Fye_xxe_y

(1)

CurlF

2e_z

(2)

To display the differential form of the curl operator:

Curl

yVF [3]x,y,zzVF [2]x,y,ze_x+zVF [1]x,y,zxVF [3]x,y,ze_y+xVF [2]x,y,zyVF [1]x,y,ze_z

(3)

SetCoordinatescylindricalr,θ,z:

Curl

θVF [3]r,θ,zzrVF [2]r,θ,zre_r+zVF [1]r,θ,zrVF [3]r,θ,ze_θ+rrVF [2]r,θ,zθVF [1]r,θ,zre_z

(4)

Curls,φ,w

φVF [3]s,φ,wwsVF [2]s,φ,wse_s+wVF [1]s,φ,wsVF [3]s,φ,we_φ+ssVF [2]s,φ,wφVF [1]s,φ,wse_w

(5)

Curlspherical

φrsinφVF [3]r,φ,θθrVF [2]r,φ,θr2sinφe_r+θVF [1]r,φ,θrrsinφVF [3]r,φ,θrsinφe_φ+rrVF [2]r,φ,θφVF [1]r,φ,θre_θ

(6)

Curlsphericalα,ψ,γ

ψαsinψVF [3]α,ψ,γγαVF [2]α,ψ,γα2sinψe_α+γVF [1]α,ψ,γααsinψVF [3]α,ψ,γαsinψe_ψ+ααVF [2]α,ψ,γψVF [1]α,ψ,γαe_γ

(7)

Nabla is a synonym for Del.

SetCoordinatescartesian

cartesian

(8)

Del &x F

2e_z

(9)

Nabla &x F

2e_z

(10)

CrossProductDel,F

2e_z

(11)

Curlx,y,z→x2,y2,z2

x,y,z→VectorCalculus:-Vector0,0,0,attributes=vectorfield,coords=cartesianx,y,z

(12)

SetCoordinatescylindricalr,θ,z

cylindricalr,θ,z

(13)

Curlr,θ,z→fr,θ,z,gr,θ,z,hr,θ,z

r,θ,z→VectorCalculus:-Vectorθhr,θ,zrzgr,θ,zr,zfr,θ,zrhr,θ,z,gr,θ,z+rrgr,θ,zθfr,θ,zr,attributes=vectorfield,coords=cylindricalr,θ,z

(14)

See Also

Student[VectorCalculus]

Student[VectorCalculus][CrossProduct]

Student[VectorCalculus][Del]

Student[VectorCalculus][Divergence]

Student[VectorCalculus][Laplacian]

Student[VectorCalculus][Nabla]

Student[VectorCalculus][SetCoordinates]

Student[VectorCalculus][Vector]

Student[VectorCalculus][VectorField]

 


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