- #1

nigelscott

- 135

- 4

- Homework Statement
- Prove that for a 2 sphere in R R[SUP]3[/SUP] the Lie bracket is the same as to cross product.

- Relevant Equations
- Vector: X = (y,-x,0); Y = (0,z-y)

[X,Y] = J[SUB]Y[/SUB]X - J[SUB]X[/SUB]Y where the J's are the Jacobean matrices.

Prove that for a 2 sphere in R

[X,Y] = J

I computed J

^{3}the Lie bracket is the same as the cross product using the vector: X = (y,-x,0); Y = (0,z-y)[X,Y] = J

_{Y}X - J_{X}Y where the J's are the Jacobean matrices.I computed J

_{Y}X - J_{X}Y to get (-z,0,x). I computed (y,-x,0) ^ (0,z,-y) and obtained (xy,y^{2},yz) = (z,0,x) using Wolfram but I can't figure out why the sign of the x-component is different. Any help would be appreciated.