Thanks for your replies. My confusion I think lies with the derivation I watched from a Yale video which went like (please notice dot product and vectors)
K = mv.v/2
dK/dt = m dv/dt.v
dK = ma.v = ma.dr/dt so
∫dK = ∫F.dr and then the lhs was set to K2-K1 to get
K2-K1 =...
Since KE depends only on the magnitude of the vector, any path I take having the same starting and ending velocities will of course have the same work.
Since the change of kinetic energy, K2-K1, ALWAYS equals the integral of F.dr along any path, how can that integral depend upon the path? I realize that the integral is ONLY equal to the change in potential energy (F is the derivative of the potential) at the end points when F is a conservative...