Two-Dimensional Impulse Functions

Just as in one dimension we need entities that are concentrated at a point in time, so in two dimensions we find it useful to introduce corresponding entities that are concentrated at a point in the (x, y)-plane. But in two dimensions there are two generalizations, the second being concentration on a line. One class of examples is furnished by the familiar mechanical notion of pressure; in mechanics, the oldest branch of physics, the concept that we require is well established in connection with the theoretical idea of a point force or point load applied to a surface. We recall that in one dimension the term impulse itself derives from mechanics, where it signifies the time integral of a force that is applied for a time that is more or less short. It is therefore useful, in introducing the theory of impulse functions on the plane, to appeal to the subject of mechanics, where the essential concepts are already familiar. First we deal with the two-dimensional point impulse or dot and then with a variety of impulse arrays and with impulsive entities that are not restricted to a point but are distributed along straight or curved lines. The δ-notation of P. A. M. Dirac (1902–1987) generalizes to allow convenient representation of the two-dimensional concepts.