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]