In this chapter we investigate the fundamental operators of vector analysis and their applications to the study of vector fields from L2 (Ω). Since such fields are not required to be smooth, the Riesz theorem on the representation of linear functionals is going to play a significant role. We begin by introducing the spaces of velocity fields of an ideal fluid in a closed or open container. Our goal will be to investigate the velocity fields that describe the movement of a viscous incompressible fluid. We introduce the form that defines the velocity of energy dissipation and derive the corresponding fundamental Korn inequality. The Stokes operator, which plays a major role in all the investigations of the movement of a viscous fluid, will also be defined. In the last part of this chapter we discuss and solve the basic boundary value problems pertaining to the movement of a viscous fluid in an open container.