In the past few decades, the evaporation of a sessile droplet on a solid substrate is of fundamental importance in a wide range of situations ranging from technical applications such as DNA mapping, disease diagnosis, and ink-jet printing. To solve the droplet evaporation problem numerically, a steady-state multiphysics model has been found and developed by a large various of researchers. But the applicability of the steady-state approximation is still unclear. And a complete theory for the transients in the evaporation of drying sessile droplets is still lacking. In this paper, the initial transients in the droplet evaporation (i.e., the transient processes immediately after the droplet is placed on the substrate) has been investigated first both numerically and experimentally. The results show that, during the initial stage of droplet evaporation, the vapor concentration changes monotonically when approaching its steady-state while the temperature changes nonmonotonically. The initial transient times needed for the vapor concentration and for the temperature to change from their initial values to the steady-state values are almost the same, and the ratio between the initial transient time and the droplet drying time is mainly determined by the properties of the liquid, not by the underlying substrate or by the pressure of the surrounding atmosphere. The duration of the initial transients was found much longer than the duration of the process transients which denotes the adjustment of the vapor concentration and the temperature to the change in the droplet shape as the evaporation proceeds. Finally, a quantitative criterion for evaluating the applicability of the steady-state approximation in the droplet evaporation is obtained, indicating that the approximation is more reasonable for liquids with a lower saturated vapor concentration. The criterion is corroborated experimentally and may contribute to the body of knowledge concerning the droplet evaporation.