Occurrence and Derivation of Laplace’s Equation

Abstract

If a vector v can be associated with each point in a given space, then v is said to be a vector field. Two everyday examples are an electric field and a velocity field in a moving fluid. In an electric field, the electric intensity E at any point is a vector whose magnitude and direction are equal to the magnitude and direction of the force which would be exerted on a unit charge if it were placed at that point ; generally the vector E varies from point to point. In a fluid the velocity v at any point is the velocity of the particle instantaneously situated at that point. Although in general vector fields are functions of time as well as of space, in this book we shall be concerned only with space dependence and we shall assume that all vector fields with which we deal are steady, i.e. they do not change with time.