Two motions of a point of are generated by the vector fields X = _x + y , _z, Y = _y…

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

Two motions of a point

(x,y,z)

\mathbb{R}^3

are generated by the vector fields

X = \partial_x + y\,\partial_z, \qquad Y = \partial_y - x\,\partial_z.

Along either motion

z

changes by

y\,\mathrm{d}x - x\,\mathrm{d}y

, which is minus twice the signed area swept about the origin by

(x,y)

. ((a)) Compute the Lie bracket

[X,Y]

, giving its components

([X,Y]^x, [X,Y]^y, [X,Y]^z)

. ((b)) Start at the origin and follow

X

for a small parameter

\epsilon

and then

Y

for

\epsilon

; separately, follow

Y

then

X

. Say which order ends with the larger

z

, and how the sign of

[X,Y]^z

shows this.