MATH 603 Homework Set VI
Due Friday, 5 December 2014
1. A vector field F(x) on Rn is called a Central Field provided it has the form F(x) = φ(x) x
where φ(x) is a scalar valued function.
(a) Assume φ(x) = α(r) where r = |x|, i.e. φ(x) is constant on spheres centered at the
origin. Show that F(x) is conservative.
(b) Prove the converse of part (a), i.e. If a central vector field F(x) = φ(x) x is conservative,
then φ(x) = α(r) for some scalar valued function α(r).
(c) Find all conservative central fields in Rn that satisfy Div(F) = 0.
2. This problem is expressed using Polar Coordinates in R2 . Consider a vector field of the form
F(r, θ) = φ(r, θ) j2 (θ), where j1 (θ) = cos(θ)i1 + sin(θ)i2 and j2 (θ) = − sin(θ)i1 + cos(θ)i2 .
(a) Show that Curl(F)(r, θ) = 0 for r > 0 if and only if φ(r, θ) = ψ(θ)/r for some scalar
valued function ψ(θ).
R 2π
(b) If in part (a), 0 ψ(θ) dθ 6= 0, show that the vector field F is not conservative. Explain
why this does not contradict the theorem proved in class that says “Curl free” implies
conservative.
3. Consider the mapping f (·) : R3 −→ R3 given in Spherical Coordinates as (R, Θ, Φ) −→
(ρ(R, Φ), θ(Θ, Φ), φ(Φ)) with ρ(R, Φ) = R f (Φ), θ(Θ, Φ) = Θ + ωΦ, φ(Φ) = g(Φ), ω being
a given constant. Geometrically, this mapping takes a cone with apex at the origin and
central axis coincident with the positive Z-axis into another such cone with different opening
angle and with a torsional twist. Using the notation from problem (2), one has f (R, Θ, Φ) =
ρ(R, Φ)k1 (θ(Θ, Φ), φ(Φ)). Let F denote the second order tensor ∇f .
(a) Show that F = amn (Φ) km (θ(Θ, Φ), φ(Φ))⊗kn (Θ, Φ) with a11 = f (Φ), a13 = f 0 (Φ), a22 =
f (Φ) sin(g(Φ))/ sin(Φ), a23 = ωf (Φ) sin(g(Φ)), a33 = f (Φ)g 0 (Φ) with all other akl equal
zero. (f 0 (Φ) denotes differentiation with respect to Φ.)
(b) Show that det(F ) = f (Φ)3 g 0 (Φ) sin(g(Φ))/ sin(Φ).
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