Hi, I was woking on a problem from Taylor Mechanics.(4.27) It reads: Suppose that the force F(r , t) depends on the time t but stillsatisfies ∇ × F = 0. It is a mathematical fact that the work integral ∫12F(r , t) ⋅ dr (evaluated at any one time t) is independent of the path taken between the points 1 and 2. Use this to show that the time-dependent PE defined by (4.48) -∫r0rF(r' , t) ⋅ dr', for any fixed time t, has the claimed property that F(r , t) = -∇U(r , t). Can you see what goes wrong with the argument leading to Equation (4.19), that is, conservation of energy? Equation (4.19) is nothing but Δ(T + U) = 0. I tried to write down the gradient of (4.48) directly, but I failed because I do not know how to handle ∇∫r0rF(r' , t) ⋅ dr'. And, I think, thus I cannot see what goes wrong with the argument leading to (4.19). (I am aware of the fact that mechanical energy is no longer conserved since potential energy U depends on time) Can somebody help me with that gradient part, or is there any other way to do this?