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

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VectorCalculus[DotProduct] - computes the dot product of Vectors and differential operators

Calling Sequence

DotProduct(v1, v2)

v1 . v2

Parameters

v1

-

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

v2

-

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

Description

• 

The DotProduct(v1, v2) command ( scalar product ) computes the dot product of v1 and v2, where v1 and v2 can be free Vectors, rooted Vectors, position Vectors, vector fields, Del or Nabla.

• 

The function can be accessed through  . or DotProduct exports.

• 

If v2 is a VectorField, the divergence  of v2 can be computed as DotProduct(Del, v2), or .v2.

• 

If v1 is a VectorField, an operator representing the directional derivative  in the direction of v1 is obtained as DotProduct(v1, Del), or v1..

• 

The behavior of the dot product of two Vectors is described by the following table:

 

v1

coord (v1)

v2

coord (v2)

v1.v2

1

free Vector

cartesian

free Vector

cartesian

scalar

 

free Vector

curved

free Vector

any

error

2

free Vector

cartesian

rooted Vector (root2)

coord2

scalar

 

free Vector

curved

rooted Vector (root2)

coord2

error

3

free Vector

cartesian

vector field

cartesian

scalar

 

free Vector

cartesian

vector field

curved

error

 

free Vector

curved

vector field

any

error

4

free Vector

cartesian

position Vector

cartesian

scalar

5

rooted Vector (root1)

coord1

rooted Vector (root1)

coord1

scalar

 

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.v2root1

7

rooted Vector (root1)

cartesian

position Vector

cartesian

scalar

8

vector field

coord1

vector field

coord1

scalar field

 

vector field

coord1

vector field

coord2

error

9

vector field

coord1

position Vector

cartesian

error

10

position Vector

cartesian

position Vector

cartesian

scalar

Examples

restart

withVectorCalculus:

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

1,1,1.1,1,1

1

(1)

v1:=Vector1,1,2,coordinates=cartesianx,y,z

v1:=exey+2ez

(2)

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

v2:=ey+ez

(3)

DotProductv1,v2

1

(4)

Take the dot 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

(5)

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

v1:=111

(6)

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

v2:=110

(7)

DotProductv1,v2

0

(8)

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

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

v1:=111

(9)

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

v2:=ez

(10)

v2.v1

1

(11)

The dot 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

(12)

vf2:=VectorFieldr2,φ+θ,0,sphericalr,φ,θ

vf2:=r2e_r+φ+θe_φ

(13)

DotProductvf1,vf2

r3+φφ+θ

(14)

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

vf1:=VectorFieldx,yz,z,cartesianx,y,z

vf1:=xe_xyze_y+ze_z

(15)

Del.vf1

2z

(16)

vf2:=VectorFieldrt,φ,t,cylindricalr,φ,t

vf2:=rte_r+φe_φ+te_t

(17)

Del.vf2

2rt+r+1r

(18)

Construct a directional derivative operator.

V:=VectorFieldx,yz,z,cartesianx,y,z

V:=xe_xyze_y+ze_z

(19)

W:=VectorFieldyz,xz,xy,cartesianx,y,z

W:=yze_x+xze_y+xye_z

(20)

V.DelW

yz2+yze_x+2xze_y+xyz+xye_z

(21)

The dot product of two position vectors is defined.

pv1:=PositionVectorp,p,polarr,t

pv1:=pcosppsinp

(22)

pv2:=PositionVector1,p,parabolicu,v

pv2:=1212p2p

(23)

pv1.pv2

pcosp1212p2+p2sinp

(24)

The dot product of a cartesian free Vector and a cartesian vector field is defined.

vf3:=VectorFieldyz,xz,xy,cartesianx,y,z

vf3:=yze_x+xze_y+xye_z

(25)

v3:=Vector1,2,1,coordinates=cartesianx,y,z

v3:=ex+2ey+ez

(26)

vf3.v3

xy+2xz+yz

(27)

See Also

LinearAlgebra[DotProduct], VectorCalculus, VectorCalculus[Del], VectorCalculus[Divergence], VectorCalculus[Divergence], VectorCalculus[Laplacian], VectorCalculus[PositionVector], VectorCalculus[RootedVector], VectorCalculus[Vector], VectorCalculus[VectorField]


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