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VectorCalculus

  

DotProduct

  

computes the dot product of Vectors and differential operators

 

Calling Sequence

Parameters

Description

Examples

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)

v1Vector1,1,2,coordinates=cartesianx,y,z

v1:=exey+2ez

(2)

v2Vector0,1,1,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.

vsVectorSpace1,π3,π3,sphericalr,p,t

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

(5)

v1RootedVectorroot=vs,1,1,1

v1:=111

(6)

v2RootedVectorroot=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.

v1RootedVectorroot=1,π4,1,1,1,1,cylindricalr,p,h

v1:=111

(9)

v2Vector0,0,1,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.

vf1VectorFieldr,φ,θ,sphericalr,φ,θ

vf1:=re_r+φe_φ+θe_θ

(12)

vf2VectorFieldr2,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.

vf1VectorFieldx,yz,z,cartesianx,y,z

vf1:=xe_xyze_y+ze_z

(15)

Del.vf1

2z

(16)

vf2VectorFieldrt,φ,t,cylindricalr,φ,t

vf2:=rte_r+φe_φ+te_t

(17)

Del.vf2

2rt+r+1r

(18)

Construct a directional derivative operator.

VVectorFieldx,yz,z,cartesianx,y,z

V:=xe_xyze_y+ze_z

(19)

WVectorFieldyz,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.

pv1PositionVectorp,p,polarr,t

pv1:=pcosppsinp

(22)

pv2PositionVector1,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.

vf3VectorFieldyz,xz,xy,cartesianx,y,z

vf3:=yze_x+xze_y+xye_z

(25)

v3Vector1,2,1,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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