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VectorCalculus

  

DirectionalDiff

  

computes the directional derivative of a scalar field in the direction given by a vector

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

DirectionalDiff(F,v,c)

DirectionalDiff(F,p,dir,c)

Parameters

F

-

the scalar or vector field to differentiate

v

-

Vector(algebraic); the direction Vector or vector field

p

-

point=list(algebraic) or point=Vector(algebraic); point where the derivative will be evaluated

dir

-

list(algebraic) or Vector(algebraic); components specifying the direction of the directional derivative in a specified coordinate system

c

-

(optional) list(name) or symbol[name, name, ...]; list of names or name of the coordinate system indexed by the coordinate names

Description

• 

The DirectionalDiff(F,v,c) command, where F is a scalar function, computes the directional derivative of F at the location and direction specified by v.  The expression F is interpreted in the coordinate system specified by c, if provided, and otherwise in the current coordinate system.

• 

The DirectionalDiff(F,v,c) command, where F is a VectorField, computes the VectorField of directional derivatives of each component of F with respect to v.

• 

The argument v can be a free Vector in Cartesian coordinates, a position Vector, a vector field or a rooted Vector.  If v is one of the first three, the result will be a scalar field of all directional derivatives in Rn in the directions specified by v; this scalar field will be given in the same coordinate system as is used to interpret expression F.  If v is a rooted Vector, the result is the value of the directional derivative of F in the direction of v taken at the root point of v.

• 

If F is a scalar function, the Vector v is normalized. If F is a VectorField, the Vector v is not normalized.

• 

The DirectionalDiff(F,p,dir,c) command computes the directional derivative of F at the point p in the direction dir, where F is interpreted in the coordinate system specified by c, if provided, and otherwise in the current coordinate system.  The point p can be a list, a free Vector in Cartesian coordinates or a position Vector. The direction dir can be a free Vector in Cartesian coordinates, a position Vector or a vector field.  The result is the value of DirectionalDiff(F,dir,c) evaluated at the point p.

– 

If c is a list of names, the directional derivative of F is taken with respect to these names in the current coordinate system.

– 

If c is an indexed coordinate system, F is interpreted in the combination of that coordinate system and coordinate names.

– 

If c is not specified, F is interpreted in the current coordinate system, whose coordinate name indices define the function's variables.

Note that c has no influence on the interpretation of the direction vector v.

• 

An operator implementing the directional derivative with respect to a VectorField can be obtained using the dot operator with Del, as in V.Del.

Examples

withVectorCalculus:

Introductory examples where a coordinate system is specified

SetCoordinatescartesianx,y

cartesianx,y

(1)

v11,2:

DirectionalDiffr2,v1,polarr,t

25rcost5+45rsint5

(2)

WVectorFieldu+v,v,cartesianu,v

W:=u+ve_u+ve_v

(3)

DirectionalDiffr2,point=1,π,W,polarr,t

2

(4)

ddDirectionalDiffr2,W,polarr,t:

simplifyevaldd,r=1,t=π

2

(5)

ddDirectionalDiffVectorFieldxy,xy,W

dd:=x+yyxye_x+x+yy+yxe_y

(6)

Examples where a list of variable names is provided

DirectionalDiffpq,1,2,p,q

15q5+25p5

(7)

v21,0:

SetCoordinatespolar

polar

(8)

ddDirectionalDiffrcosθ,v2,r,θ:

simplifydd

1

(9)

Examples where the information is given in the form of a Rooted Vector

SetCoordinatespolarr,t

polarr,t

(10)

vsVectorSpace1,π2,polarr,t:

v3vs:-Vector1,1

v3:=11

(11)

v4vs:-Vector0,1

v4:=01

(12)

DirectionalDiffr2,v3

2

(13)

DirectionalDiffr2,v4

0

(14)

SetCoordinatescartesianx,y

cartesianx,y

(15)

DirectionalDiffy2x2,point=1,2,0,1,cartesianx,y

4

(16)

DirectionalDiffy2x2,RootedVectorroot=1,2,0,1,cartesianx,y

4

(17)

DirectionalDiffy2x2,RootedVectorroot=1,π2,1,1,polarr,t,cartesianx,y

0

(18)

Examples using the dot operator to construct a directional derivative operator

SetCoordinatescartesianx,y,z

cartesianx,y,z

(19)

VVectorFieldyz,xz,xy

V:=yze_x+xze_y+yxe_z

(20)

normalV.Delxyz

x2y2+x2z2+y2z2

(21)

V.DelVectorField1x,1y,1z

yzx2e_xxzy2e_yyxz2e_z

(22)

See Also

Physics[Vectors][DirectionalDiff]

Student[MultivariateCalculus][DirectionalDerivative]

tensor[directional_diff]

VectorCalculus

VectorCalculus[diff]

VectorCalculus[DotProduct]

VectorCalculus[Gradient]

VectorCalculus[SetCoordinates]

 


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