(Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on A , not on v ; thus, there is no need of using Feynman's subscript notation in the equation above). Using the chain rule, the total derivative of A is: d A d t = ∂ ∂ A ∂ ∂ t + ( v ⋅ ⋅ ∇ ∇ ) A so that the above expression becomes: F = q [ − − ∇ ∇ ( ϕ ϕ − − v ⋅ ⋅ A ) − − d A d t ] .