Chapter 9: Vector Calculus
Section 9.4: Differential Identities
Show that for sufficiently well-behaved functions fρ,φ,θ in spherical coordinates, the gradient of f is curl-free.
In spherical coordinates, the gradient of f is given by ∇f=fρfφρfθρ sin(φ)≡uvw, as per Table 9.3.2.
By Table 9.3.3, the curl in spherical coordinates is
∇×∇f=eρρ2sinφeφρ sinφeθρ∂ρ∂φ∂θuρ vρ w sinθ=w cos(φ)ρ sin(φ)+wφρ−vθρ sin(φ)uθρ sin(φ)−wρ−wρvρ+vρ−uφρ
=(fθρ sin(φ))ρcos(φ)sin(φ)+∂φ(fθρ sin(φ))ρ−∂θ(fφρ)ρ sin(φ)∂θ(fρ)ρ sin(φ)−(fθρ sin(φ))ρ−∂ρ(fθρ sin(φ))(fφρ)ρ+∂ρ(fφρ)−∂φ(fρ)ρ =fθ cos(φ)ρ2sin2(φ)+fθ φsin(φ)−fθ cos(φ)sin2(φ) ρ2−fφ θρ2sin(φ)fρ θρ sin(φ)−fθρ2sin(φ)−ρ fθ ρ−fθρ2sin(φ)fφρ2+ρ fφ ρ−fφρ2−fρ φρ
=fθ cos(φ)ρ2sin2(φ)+fθ φρ2sin(φ)−fθ cos(φ)ρ2sin2(φ)−fφ θρ2sin(φ)fρ θρ sin(φ)−fθρ2sin(φ)−fθ ρρ sin(φ)+fθρ2sin(φ)fφρ2+fφ ρρ−fφρ2−fρ φρ=fθ φρ2sin(φ)−fφ θρ2sin(φ)fρ θρ sin(φ)−fθ ρρ sin(φ)fφ ρρ−fρ φρ=000
where the final simplification to the zero vector hinges on the equality of the mixed partial derivatives. That is why an assumption on the behavior of the function f must be made.
Maple Solution - Interactive
Tools≻Load Package: Student Vector Calculus
Tools≻Tasks≻Browse: Calculus - Vector≻
Vector Algebra and Settings≻
Display Format for Vectors
Press the Access Settings button and select
"Display as Column Vector"
Display Format for Vectors
Obtain the curl of the gradient of f in spherical coordinates
Write the scalar fρ,φ,θ.
Context Panel: Student Vector Calculus≻Differentiate≻Gradient
(Complete the dialog as per the figure to the right.)
Context Panel: Student Vector Calculus≻Curl
The scalar f does not carry (as an attribute) its coordinate system. Hence, it is necessary to use the Context Panel, not typeset math notation, to obtain the gradient. However, the gradient vector does carry its coordinate system, so the curl operator could be applied either by typeset math notation or by the Context Panel. It is clearly simpler to chain the calculations and continue using the Context Panel in this example.
Maple Solution - Coded
Load the Student VectorCalculus package and execute the BasisFormat command.
Apply the Curl and Gradient commands from the Student VectorCalculus package.
Because scalars do not carry a coordinate system, Maple's Gradient command must be told to act in spherical coordinates.
CurlGradientfr,θ,z,sphericalρ,φ,θ = 000
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