In laser wake field acceleration, it is relatively easy to achieve and control electron injection by adopting a plasma density gradient. This scheme of plasma density gradient injection has been studied in recent years both theoretically and experimentally, but thus far theoretical studies have been done mostly by particle-in-cell simulations. In this paper the density gradient injection and acceleration of electrons are studied with a newly developed analytic approach. The energy threshold for electron injection versus plasma density gradient scale length is given. It is shown that in the plasma density gradient region, the energy threshold of electron injection becomes lower at later times after the driving laser pusle or when the gradient becomes sharper. Evolution of plasma wave's phase velocity and motion of the background electrons in the plasma density gradient are worked out in the linear plasma wave regime, i.e. the normalized laser intensity is a0～1. The energy, the location, and the timing of the injected electrons are obtained. Separatrices of test electrons in the gradient region are obtained by Hamiltonian theory. The influence of injection timing in the density gradient region on the succeeding acceleration in the homogeneous plasma density region is also discussed. It is indicated that whether the injected electrons may be accelerated efficiently or not in the homogeneous region depends on both the energy of the electrons and the phase of the plasma wave at the gradient-to-homogeneous turning point. The analytic results are confirmed by particle-in-cell simulations.