compute the curl by using the nabla differential operator - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Physics : Physics Package : Vectors : Physics/Vectors/differential_operators

Physics[Vectors][Curl] - compute the curl by using the nabla differential operator

Physics[Vectors][Divergence] - compute the divergence by using the nabla differential operator

Physics[Vectors][Gradient] - compute the gradient by using the nabla differential operator

Physics[Vectors][Laplacian] - compute the Laplacian by using the nabla differential operator

Calling Sequence

Curl(A)

Divergence(A)

Gradient(A)

Laplacian(A)

Remark: these calling sequences are also valid with the inert %Curl, %Divergence, %Gradient, %Laplacian commands

Parameters

A

-

any algebraic (vector or scalar) expression

Description

• 

Curl, Divergence, Gradient and Laplacian, respectively return the curl, the divergence, the gradient and the Laplacian of a given vectorial or scalar function. When the command's name is prefixed by %, an unevaluated representation for these operations is returned.

• 

The %Curl, %Divergence, %Gradient and %Laplacian are the inert forms of Curl, Divergence, Gradient and Laplacian, that is: they represent the same mathematical operations while holding the operations unperformed. To activate the operations use value.

• 

Curl, Divergence and Gradient check their arguments (for consistency) before sending the task to Nabla. So, if A_ is a vector, then Gradient(A_) will interrupt the computation with an error message, as well as Divergence(A) and Curl(A) when A is a scalar (not a vector). All these differential operations are realized just w.r.t the geometrical coordinates phi,r,rho,theta,x,y,z. Therefore, if A does not depend on these global geometrical coordinates, these commands (as well as Nabla) return 0.

• 

For the conventions about the geometrical coordinates and vectors see ?conventions

Examples

withPhysics[Vectors]

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

(1)

Setupmathematicalnotation=true

mathematicalnotation=true

(2)

The Gradient of a function

∇fx,y,z

xfx,y,zi+yfx,y,zj+zfx,y,zk

(3)

The Divergence of a Gradient is equal to the Laplacian

∇·∇fx,y,z

2x2fx,y,z+2y2fx,y,z+2z2fx,y,z

(4)

Laplacianfx,y,z

2x2fx,y,z+2y2fx,y,z+2z2fx,y,z

(5)

The Laplacian of a vector function in spherical coordinates

LaplacianA_r,θ,φ

2r2Ar,θ,φr+2rAr,θ,φr+2θ2Ar,θ,φsinθ+θAr,θ,φcosθr2sinθ+2φ2Ar,θ,φr2sinθ2

(6)

The Curl of a Gradient is identically zero

eq:=∇Ar,θ,φ

eqrAr,θ,φr+θAr,θ,φθr+φAr,θ,φφrsinθ

(7)

∇×eq

0

(8)

Depending on the context the inert representations of these commands, obtained by prefixing the command's name with %, serve better the purpose of representing the mathematical objects

%GradientAr,θ,φ

Ar,θ,φ

(9)

∇×

0

(10)

The Curl of non-projected vector function (note the underscore in 'V_' to represent vectors)

eq:=∇×V_r,θ,φ

eq×Vr,θ,φ

(11)

The Divergence of a Curl is identically zero

∇·eq

0

(12)

The Divergence and Curl of a projected vector function (projected vectors don't need an "arrow"  - the underscore "_" mentioned in the previous example to be represented)

V:=_r+f1θ,φ_θ+f2θ,φ_φ

Vr+f1θ,φθ+f2θ,φφ

(13)

∇·V

2r+θf1θ,φsinθ+f1θ,φcosθrsinθ+φf2θ,φrsinθ

(14)

∇×V

θf2θ,φsinθ+f2θ,φcosθφf1θ,φrrsinθf2θ,φθr+f1θ,φφr

(15)

See Also

operations, Physics, Physics conventions, Physics examples, Physics/Vectors, Setup


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam