 #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,zy)
[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^{3} the Lie bracket is the same as the cross product using the vector: X = (y,x,0); Y = (0,zy)
[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 xcomponent is different. Any help would be appreciated.
[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 xcomponent is different. Any help would be appreciated.