computes the cross product of Vectors and differential operators - Maple Help

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VectorCalculus[CrossProduct] - computes the cross product of Vectors and differential operators

Calling Sequence

CrossProduct(v1,v2)

v1 &x v2

Parameters

v1

-

Vector(algebraic); Vector, Vector-valued procedure, or differential operator

v2

-

Vector(algebraic); Vector, Vector-valued procedure, or differential operator

Description

• 

The CrossProduct(v1, v2) function ( vector product ) computes the cross product of v1 and v2, where v1 and v2 can be either three dimensional free Vectors, rooted Vectors, position Vectors, vector fields, Del or Nabla.

• 

The function  and can be accessed through  &x or CrossProduct exports.

• 

The behavior of cross product is contained in the following table

 

v1

coord (v1)

v2

coord (v2)

v1 &x v2

coord (v1 &x v2)

1

free Vector

cartesian

free Vector

cartesian

free Vector

cartesian

 

free Vector

curved

free Vector

any

error

 

2

free Vector

cartesian

rooted Vector (root2)

coord2

rooted Vector (root2)

coord2

3

free Vector

any

vector field

any

error

 

4

free Vector

cartesian

position Vector

cartesian

free Vector

cartesian

 

free Vector

curved

position Vector

cartesian

error

 

5

rooted Vector (root1)

coord1

rooted Vector (root1)

coord1

rooted Vector

coord1

 

rooted Vector (root1)

coord1

rooted Vector (root2)

coord1

error

 

 

rooted Vector (any)

coord1

rooted Vector (any)

coord2

error

 

6

rooted Vector (root1)

coord1

vector field

coord2

v1 &x v2root1

coord2

7

rooted Vector (root1)

cartesian

position Vector

cartesian

rooted Vector (root1)

cartesian

8

vector field

coord1

vector field

coord1

vector field

coord1

 

vector field

coord1

vector field

coord2

error

 

9

vector field

coord1

position Vector

cartesian

error

 

10

position Vector

cartesian

position Vector

cartesian

position Vector

cartesian

Examples

restart

withVectorCalculus:

Take the cross product of two free Vectors in cartesian coordinates.

1,0,0 &x 0,1,0

ez

(1)

v1:=Vector1,4,0,coordinates=cartesianx,y,z

v1:=ex+4ey

(2)

v2:=Vector1,1,1,coordinates=cartesianx,y,z

v2:=ex+ey+ez

(3)

CrossProductv1,v2

4exey3ez

(4)

GetCoordinatesCrossProductv1,v2

cartesianx,y,z

(5)

Take the cross product of two rooted vectors if they have the same coordinate system and root point.

vs:=VectorSpace1,π3,π3,sphericalr,p,t

vs:=modulelocal_origin,_coords,_coords_dim;exportGetCoordinates,GetRootPoint,Vector;end module

(6)

v1:=RootedVectorroot=vs,1,1,1

v1:=111

(7)

v2:=RootedVectorroot=vs,1,0,1

v2:=101

(8)

v1 &x v2

101

(9)

GetRootPointv1 &x v2

er+13πep+13πet

(10)

The cross product of a cartesian free Vector and a rooted Vector is valid. The resulting Vector is rooted.

v1:=RootedVectorroot=1,π4,1,1,1,1,cylindricalr,p,h

v1:=111

(11)

v2:=Vector0,0,1,coordinates=cartesianx,y,z

v2:=ez

(12)

v2 &x v1

110

(13)

GetRootPointv2 &x v1

er+14πep+eh

(14)

GetCoordinatesv2 &x v1

cylindricalr,p,h

(15)

The cross product of two vector fields is defined if they are in the same coordinate system.

vf1:=VectorFieldr,φ,θ,sphericalr,φ,θ

vf1:=re_r+φe_φ+θe_θ

(16)

vf2:=VectorFieldr2,φ,φ,sphericalr,φ,θ

vf2:=r2e_r+φe_φ+φe_θ

(17)

CrossProductvf1,vf2

φ2φθe_r+r2θφre_φ+φr2+φre_θ

(18)

Use differential operators to compute the Curl of a vector field.

F:=VectorField1,x,y,cartesianx,y,z

F:=e_xxe_y+ye_z

(19)

Del &x F

e_xe_z

(20)

G:=VectorFieldr,t,p,sphericalr,p,t

G:=re_r+te_p+pe_t

(21)

Del &x G

rcospp+rsinprr2sinpe_rpre_p+tre_t

(22)

The cross product of two position Vectors is defined. The result is a position Vector.

pv1:=PositionVector1,p,q,sphericalr,φ,θ

pv1:=sinpcosqsinpsinqcosp

(23)

pv2:=PositionVector1,p,p,cylindricalr,φ,h

pv2:=cospsinpp

(24)

CrossProductpv1,pv2

sinpsinqpcospsinpsinpcosqp+cosp2sinp2cosqsinpsinqcosp

(25)

AboutCrossProductpv1,pv2

Type: Position VectorComponents: sinpsinqpcospsinp,sinpcosqp+cosp2,sinp2cosqsinpsinqcospCoordinates: cartesianRoot Point: 0,0,0

(26)

See Also

LinearAlgebra[CrossProduct], VectorCalculus, VectorCalculus[Curl], VectorCalculus[Del], VectorCalculus[PositionVector], VectorCalculus[RootedVector], VectorCalculus[Vector], VectorCalculus[VectorField]


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