Freedom, Constraint And Control In Multivariable Calculus

Certainly, everyone interested in technology should possess an understanding of the models of deterministic, continuous multivariable control. The study of multivariable calculus can be viewed as a natural extension of the unfortunately named "single variable" calculus. Ordinary "single variable" calculus is the study of equations in two variables, F(x, y) = 0. Equations in two variables provide mechanisms for studying: a) continuous curves in the two dimensional plane, b) continuous models of control (where one variable controls a second variable), c) continuous models of time variation (signals, trends or evolution) or d) situations where two variables track together continuously. These functions and curves of ordinary calculus are real entities having observable and predictable properties. What happens when a single variable depends on or is controlled by several other variables? How do we visualize and treat the situation when several variables depend on or are controlled by a single variable? What if several variables control several other variables? How will small changes in the controlling variables affect the controlled variables? This paper begins with a discussion of the importance of counting variables at the beginning of multivariable problems, proceeds to a classification of the entities of multivariable calculus and continues to describe the concept of the derivative as it pertains to each of the entities of multivariable calculus. Similar to the functions of calculus, the objects of multivariable calculus, including space curves, warped surfaces and models of multivariable control are also real and also have observable and predictable properties. Every additional variable provides an additional degree of freedom by adding a dimension to the space spanned by the collection of variables. Every additional equation provides a constraint, which removes a dimension from the space spanned by the variables. P ge 608.1 Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition Copyright ! 2001, American Society for Engineering Education Counting Variables A study of the numeric properties of rectangles might begin by listing the important quantitative properties of rectangles and noting relationships, which hold among these properties. These quantitative properties would be listed as the variables: L the length of a rectangle, W the width of the rectangle, A the area of the rectangle, P the perimeter of the rectangle and D the diagonal of the rectangle. The values for some of these variables can be arbitrarily specified but not all of the variables can be chosen arbitrarily. What are the rules that determine the available possibilities? Relationships between the variables, which are true for all rectangles must hold. In this case these relationships are the equations: A = L*W, P = 2(L + W) and D = L + W. Each equation permits one variable to be evaluated and therefore the three equations permit three of the variables to be evaluated. In this situation any two variables can be considered as independent and if properly chosen, the remaining three variables will be determined. If n is the total number of variables and m, the number of equations, the number of independent, free variables is n – m. The following cases should be recognized: A. If the number of equations equals the number of variables the solution might consist of a point or maybe several isolated points. B. If the number, n, of variables is one more than the number of equations, the solution will be a curve that is, a one-dimensional manifold, in n-space. In three-dimensional space two canonical forms for a curve should be recognized: i. two equations, each in three variables: each equation represents a surface and the curve is the intersection of the two surfaces: F(x, y, z) = 0 G(x, y, z) = 0 ii. three equations, each displaying how a point, P(x, y, z) is controlled by a parameter: x = f(u) y = g(u) z = h(u) Here if u represents time, t, the three equations describe how the point, P(x, y, z) moves with time. The parameter, u, sometimes is chosen to represent the distance s along the curve from a fixed point P0(x0, y0, z0) to the moving point P(x, y, z) . Page 608.2 Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition Copyright ! 2001, American Society for Engineering Education C. If the number, n, of variables is two more than the number of equations the solution will be a surface that is, a two-dimensional manifold, in n space, etc. Here too, in three-dimensional space two canonical forms for the surface should be recognized: i. F(x, y, z) = 0 , x and y determine z. ii. x = f(u, v) y = g(u, v) z = h(u, v) In case ii, u and v provide co-ordinatizations for the surface, each choice of the parameters u and v determines the location of a point P(x, y, z) on the surface. D. If the number, n, of variables exceeds the number of equations by three we might try to visualize the resulting manifold as a one parameter family of surfaces or if the parameter is time, we may see it as a surface moving as a function of time. In what follows the equations will be assumed to be independent and if the number of variables is limited to three, three-dimensional spatial visualization can be applied.