- #1
mmwave
- 647
- 2
I have a scalar function of position only, V(r) where r is the position vector (x, y,z). Since V depends only on position, is it true to say that
dV/dx = dV/dy = dV/dz ?
(these should be partial derivatives)
I am trying to show L = r x [nab] commutes with any radial function V(r) meaning that for any function f
r x [nab] ( V(r) * f) - V(r) * r x [nab] f = 0
most of the terms cancel out but I am left with
(y dV/dz - x dV/dz -zdV/dy + xdV/dy + z dVdx - y dV/dx) * f
or (y (dV/dz - dV/dx) + x( dV/dy - dVdz ) +z( dV/dx - dV/dy ) ) * f
If the partial derivatives cancel then I am done. If not I have no clue how to continue but I do know the final answer must be zero.
dV/dx = dV/dy = dV/dz ?
(these should be partial derivatives)
I am trying to show L = r x [nab] commutes with any radial function V(r) meaning that for any function f
r x [nab] ( V(r) * f) - V(r) * r x [nab] f = 0
most of the terms cancel out but I am left with
(y dV/dz - x dV/dz -zdV/dy + xdV/dy + z dVdx - y dV/dx) * f
or (y (dV/dz - dV/dx) + x( dV/dy - dVdz ) +z( dV/dx - dV/dy ) ) * f
If the partial derivatives cancel then I am done. If not I have no clue how to continue but I do know the final answer must be zero.