that is allowed to range over all non-negative integer values. The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of partial differential equations, it can sometimes be more fruitful to work instead with the Sobolev spaces. Continuity The terms parametric continuity (Ck) and geometric continuity (Gn) were introduced by Brian Barsky, to show that the smoothness of a curve could be measured by removing restrictions on the speed, with which the parameter traces out the curve. Parametric continuity Parametric continuity (Ck) is a concept applied to parametric curves, which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve is said to be of class Ck, if exists and is continuous on , where derivatives at the end-points are taken to be one sided derivatives (i.e., at from the right, and at from the left). As a practical application of this concept, a curve describing the motion of an object with a parameter of time must have C1 continuity and its first derivative is differentiable—for the object to have finite acceleration. For smoother motion, such as that of a camera's path while making a film, higher orders of parametric continuity are required. Order of parametric continuity The various order of parametric continuity can be described as follows: : zeroth derivative is continuous (curves are continuous) : zeroth and first derivatives are continuous : zeroth, first and second derivatives are continuous : 0-th through -th derivatives are continuous Geometric continuity The concept of geometrical continuity or geometric continuity (Gn) was primarily applied to the conic sections (and related shapes) by mathematicians such as Leibniz, Kepler, and Poncelet. The concept was an early attempt at describing, through geometry rather than algebra, the concept of continuity as expressed through a parametric function. The basic idea behind geometric continuity was that the five conic sections were really five different versions of the same shape. An ellipse tends to a circle as the eccentricity approaches zero, or to a parabola as it approaches one; and a hyperbola tends to a parabola as the eccentricity drops toward one; it can also tend to intersecting lines. Thus, there was continuity between the conic sections. These ideas led to other concepts of continuity. For instance, if a circle and a straight line were two expressions of the same shape, perhaps a