In this paper, effective Eulerian algorithms are introduced for the computation of the forward finite time Lyapunov exponent (FTLE) of smooth flow fields. The advantages of the proposed algorithms mainly manifest in two aspects. First, previous Eulerian approaches for computing the FTLE field are improved so that the Jacobian of the flow map can be obtained by directly solving a corresponding system of partial differential equations, rather than by implementing certain finite difference upon the flow map, which can significantly improve the accuracy of the numerical solution especially near the FTLE ridges. Second, in the proposed algorithms, all computations are done on the fly, that is, all required partial differential equations are solved forward in time, which is practically more natural. The new algorithms still maintain the optimal computational complexity as well as the second order accuracy. Numerical examples demonstrate the effectiveness of the proposed algorithms.