On the plane with the origin removed

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Group theory and symmetry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

On the plane with the origin removed, consider the two vector fields

\begin{aligned}X &= x\,\partial_x + y\,\partial_y &&(\text{uniform radial expansion}),\\ Y &= x\,\partial_y - y\,\partial_x &&(\text{rotation}).\end{aligned}

((a)) Find the flow

\varphi_t

X

. ((b)) Compute the Lie bracket

[X, Y]

and say what this implies about whether the flows of

X

and

Y

commute.