If I understand correctly, you have a 3D vector field defined by:
{-u v + (u^2 *0.5)/(1 + u^2) - 1/120 u Log[2],
(u^2 *0.5)/(1 + u^2) - 1/120 w Log[2],
(2*2*0.5)/(1 + w^2) - 1/30 v Log[2]}
In that case you would plot a vector field of this with the commad:
vectors= VectorPlot3D[{-u v + (u^2*0.5)/(1 + u^2) -
1/120 u Log[2], (u^2*0.5)/(1 + u^2) -
1/120 w Log[2], (2*2*0.5)/(1 + w^2) - 1/30 v Log[2]}, {u, 0,
10}, {w, 0, 100}, {v, 0, 2}, VectorScale -> 0.003]
You can combine this with your surface plot:
surface =
ContourPlot3D[{0 == (2*2*0.5)/(1 + w^2) - 1/30 v Log[2],
0 == (u^2*0.5)/(1 + u^2) - 1/120 w Log[2]}, {u, 0, 30}, {w, 0,
100}, {v, 0, 2}, AxesLabel -> Automatic,
ContourStyle -> {Lighter@Cyan, Lighter@Red}];
Show[surface, vectors]
