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Physics[Vectors][DirectionalDiff] - the directional derivative

Calling Sequence

DirectionalDiff(A, B_)

Parameters

A

-

any algebraic (vectorial or scalar) expression

B_

-

a vector

Description

• 

DirectionalDiff(A, B_) computes the directional derivative of A in the direction of B_, that is, the scalar product of a unitary vector in the direction of B_ times Nabla - the differential operator - applied to the function A. Two cases can happen:

1. 

A is not a vector. Hence DirectionalDiffA,B_=B_NormB_.NablaA

2. 

A_ is a vector. Hence DirectionalDiffA_,B_=B_.Nabla.A_NormB_

• 

The %DirectionalDiff is the inert form of DirectionalDiff, that is: it represents the same mathematical operation while holding the operation unperformed. To activate the operation use value.

Examples

withPhysics[Vectors]

&x,`+`,`.`,ChangeBasis,ChangeCoordinates,Component,Curl,DirectionalDiff,Divergence,Gradient,Identify,Laplacian,Nabla,Norm,Setup,diff

(1)

Setupmathematicalnotation=true

mathematicalnotation=true

(2)

The definition of directional derivative

DirectionalDiffax,y,z,_i=_i.Nablaax,y,z

xax,y,z=xax,y,z

(3)

Directional derivative in spherical coordinates

DirectionalDiffar,θ,φ,_θ=_θ.Nablaar,θ,φ

θar,θ,φr=θar,θ,φr

(4)

Directional derivative of a vector function

R:=ax,y,z_i+bx,y,z_j+cx,y,z_k

Rax,y,zi+bx,y,zj+cx,y,zk

(5)

DirectionalDiffR,_i=_i.NablaR

xax,y,zi+xbx,y,zj+xcx,y,zk=xax,y,zi+xbx,y,zj+xcx,y,zk

(6)

Note that, when the vector which defines the direction (the second argument) is projected over one coordinate system, the function being differentiated is expected to be expressed using the same coordinate system; otherwise an error interruption happens and a corresponding message is displayed

DirectionalDiffax,y,z,_r

Error, (in Physics:-Vectors:-DirectionalDiff) vectors must be projected over one and the same base

For this example, correct input could be

DirectionalDiffax,y,z,ChangeBasis_r,1

xax,y,zsinθcosφ+yax,y,zsinθsinφ+zax,y,zcosθcosφ2sinθ2+sinφ2sinθ2+cosθ2

(7)

See Also

Identify, Nabla, operations, Physics, Physics conventions, Physics examples, Student[MultivariateCalculus][DirectionalDerivative], tensor/directional_diff, VectorCalculus[DirectionalDiff]., Vectors, Vectors[`.`]


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