24.2 Vector Fields[1PW]

E447

[1PX]Note:exercise 4, written exam 20 June 2017.

Let \(F\) be a continuous vector field on \(ℝ^ n ⧵ \{ 0\} \), such that, for every \(x ≠ 0\), \(F (x)\) is a scalar multiple of \(x\). For \(r {\gt} 0\), we denote with \(S_ r\) the sphere of radius \(r\) centered in \(0\).

• Prove that, for each regular arc \(γ\) with support contained in a sphere \(S_ r\) , we have \(∫_γ F = 0\).

• Prove that, if such a field \(F\) is conservative, then \(|F (x)|\) is constant on every sphere \(S_ r\), and therefore that \(F(x)=x𝜌(|x|)\) with \(𝜌:ℝ^ n ⧵ \{ 0\} →ℝ\) continuous.

Hidden solution: [UNACCESSIBLE UUID ’1PY’]