Differential Calculus

Type of physical science: Mathematical methods

Field of study: Calculus

Differential calculus is a mixture of mathematical theory and practice that concentrates on the question of how dependent quantities vary as the quantities on which they depend change in various ways. Differential calculus has become an indispensable part of the mathematical background of every physical scientist and engineer.

Overview

Differential calculus is an amalgam of theory and practice that focuses on the behavior of functions, or dependent quantities, by asking how these quantities vary with the independent quantity or quantities on which they depend. To be more specific, let us concentrate on the case of functions of one variable. Suppose that the dependent variable, y, depends on only one independent variable, x. Supposing y depends on x in a functional way, we give the very relation between them a name, f, and write y = f(x). (In many elementary examples, f(x) is some sort of formula or expression involving x, and it is these relations with which differential calculus is particularly powerful in dealing; but it has its uses as well in dealing with the more mysterious functions that are not readily expressible.)

What does it mean to ask how y varies with x? For example, one might ask whether y increases or decreases as x increases. It may happen that y will sometimes increase as x increases, and sometimes decrease as x increases, so one refines the question by asking for those ranges for the independent variable x over which y behaves one way or the other. If it is determined that y increases, say, as x increases over an interval I, then one might be interested in how fast y increases.

One can go even deeper in the study of how y varies with x by comparing the function f, on I, with other functions on I or parts of I.

For example, suppose y = f(x) = x/(x² + 1) (for simplicity, x is confined to non-negative values). Using the methods of differential calculus, it is possible to show that y increases from 0 to 1/2 as x increases from 0 to 1. As x increases further, y decreases, tending to zero without ever reaching it as x increases without bound.

If one is concerned about how y increases as x increases over the interval I consisting of zero, one, and all values in between, one might be led to compare y with y₁ = g(x) = x/2, which also increases steadily from 0 to 1/2 as x increases from 0 to 1.

When is y greater than y₁, and when smaller? When is y increasing faster than y₁, and when more slowly? Differential calculus gives meaning to the phrase "increasing faster than" and provides the means to answer these questions. It would be extremely difficult or impossible to pose, and to answer, such questions outside the framework of differential calculus.

The great simplifying concept that has made possible the remarkable successes of differential calculus is that of the derivative, introduced independently by the co-inventors of differential calculus, Sir Isaac Newton and Gottfried Wilhelm Leibniz. If y = f(x) is a dependent variable, a function of the independent variable x, the derivative of y, or of f, at a particular value x₀ of x, is the limit, if any, to which the Newton quotients

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tend, as x tends toward x₀ (that is, as h tends to zero). Each Newton quotient is the result of dividing the difference in values of the dependent variable at different values of the independent variable, by the corresponding difference in values of the independent variable.

Think of these differences as the net changes in the values of the variables, with x₀, f(x₀) as the starting values. The quotient

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can then be interpreted as an average, or net, rate of change of y with respect to x, for the change in the value of x from x₀ to x₀ + h. (This interpretation can be swallowed with less strain if y happens to be the distance along a directed line from some starting point and x represents amount of time elapsed from some starting time.) From this interpretation of the Newton quotients arises the interpretation of the derivative of f at x₀, denoted f(x₀), as the instantaneous rate of change of the dependent variable y, at x₀ (or when x = x₀). This is the "physical" interpretation of the derivative upon which much application is based. Apropos, notice that, if y = f(x) and y₁ = g(x) depend on the same independent variable x, it makes sense to say that y is increasing faster than y₁, when x = x₀, if f(x₀) is larger than g(x₀), provided both derivatives exist.

The derivative function, or function derived from f, is a new dependent variable y = f(x), whose value, at each point x where defined, is the derivative-at-the-point-x as defined above. This derived dependent variable is also commonly denoted dy/dx; this notation is attributable to Leibniz and has been both a blessing and a curse in the development of calculus. The notation suggests that the derivative is a ratio. The symbols dy, dx, called differentials, are thought to represent figmentary quantities, infinitesimal changes in the variable values. The curse imparted by this notation arises from the fact that these infinitesimal quantities do not really exist as quantities, and the derivative is not really a ratio; it is a limit of ratios of noninfinitesimal quantities.

The blessing arises from this: that while it is inaccurate to regard the derivative at a point as a ratio, it has turned out to be useful to do so; and although the differentials dy, dx are fictional quantities, they have proved to be useful fictions. Their uses reside in certain symbolic manipulations that are beyond the scope of this article; let it suffice to say that these manipulations of the differentials, rightly derided and objected to in the eighteenth and nineteenth centuries, have, in the twentieth century, been put on a rigorous footing.

Besides its interpretation as instantaneous rate of change and, more dubiously, as a ratio of infinitesimal changes, the derivative bears a geometric interpretation that leads to certain applications. The derivative f'(x₀) is the slope of the line tangent to the graph of f (the graph of the equation y = f(x) in the Cartesian plane) at the point (x₀, f(x₀)) on that graph. A pictorial grasp of this fact makes plausible the three basic theorems about the derivative upon which all the practices and applications of differential calculus are based:

A function f is constant on an interval (that is, the dependent variable y = f(x) takes only one value as x roams that interval) if and only if f(x) = 0 for all x in that interval.

If f'(x) greater than or equal to 0 for all x in an interval, then y = f(x) is nondecreasing as x increases over the interval.

If f achieves its greatest, or least, value on an open interval I at a point x0 ∊ I, and if f(x₀) exists, then f'(x₀) = 0.

Applications

Theorem 2 above (and its corollary, in which f is nonincreasing and f(x) less than or equal to 0 on the interval) is of great use in investigating the global behavior of functions. In fact, this elementary theorem is so much a part of the standard toolkit of applied mathematicians and physical scientists that its applications are hardly noticed. It would be impossible to single out a typical application, but here is a rather mathematical example of this theorem in action. Fix p between 0 and 1. The application concerns the problem of comparing the quantities (a + b)^p and a^p + b^p, as a and b range over the positive real numbers. Applying a trick that will become clear later, we focus on the expressions (x + 1)^p and

First consider f(x) = (x + 1)^p/(x^p + 1). Standard techniques for taking the derivative, and the assumption that 0 less than p less than 1, show that f'(x) is negative for 0 less than x less than 1 and positive for x greater than 1. It follows that f(x) decreases as x increases from 0 to 1, and then increases as x increases further. Thus, f = 2p^-1 is the unique lowest of the values f(x), x greater than or equal to 0. That is, f(x) = (x + 1)^p/p ≥ 2^p-1 for all x greater than or equal to 0, with equality if and only if x = 1. Multiply this inequality through by x^p + 1, replace x by a/b, and multiply through by b to the power of p to find: (a + b)^p greater than or equal to 2p^-1 (a^p + b^p) for all a, b greater than 0, with equality if and only if a = b.

Further inspection of the function of above, or of g(x) = x^p + 1 - (x + 1)^p (noting that g(0) = 0 and finding that g'(x) greater than 0 for x greater than 0) yields the additional information that (a + b)^p ≤ a^p + b^p for all a, b greater than or equal to 0, with equality if and only if one of a, b is zero. Estimates, or comparisons, of this sort may seem crude, but they are extremely useful in a number of contexts that crop up in the physical sciences.

In the preceding example, it was found that the quantity f(x), with x greater than or equal to 0, achieves its least value, 2p^-1, when x = 1.

This is an example of an optimization result, and, in this case, one arrives at the result using only the basic theorem 2 of the preceding section (besides some derivative-taking and elementary formula-wringing that are beyond the scope of this article). In fact, basic theorem 3 is more commonly brought to bear on optimization problems than is theorem 2, although a skilled optimizer will often use both, together with more refined, "higher-order" results that we need not bother with here. As it stands, theorem 3 suggests the following general strategy for solving the problem of where a function y = f(x) of one independent variable x takes its greatest, or least value, if anywhere, as x roams over a closed interval I: Make a collection consisting of the endpoint(s) of I, the points inside I where f has no derivative, and the solutions inside I of the equation f'(x) = 0; the solution of the optimization problem, if there is a solution, will be found among the points collected. If, as is often the case, there is a good reason why f(x) must achieve a highest, or lowest, value on I, then the place where the optimum value is achieved may be selected from the candidates in the collection by simply comparing the values f(x), as x ranges over the collection. This simple strategy is surprisingly powerful and useful.

Its power lies in this: that it often replaces the problem of comparing the values f(x) as x ranges over an infinite set by the problem of comparing those values with x confined to a finite set.

The strategy is adaptable to multivariable optimization problems, in which the dependent quantity is a function of more than one variable. Partial derivatives replace the derivative, and the endpoints of the interval are replaced by the boundary of a region; these problems are difficult, but the theorem that the derivatives at an optimal point either must be zero or must not exist serves once again to eliminate infinitely many possibilities.

Optimization problems, such as the closely related global behavior problems, are so thoroughly embedded in the discourse in the physical sciences that it would be quite fallacious to point to a single sort of such problem as typical. There are, of course, practical optimization problems in applied mathematics and engineering that are quite impressive, but it is not at all clear that these constitute a more important type of application of differential optimization techniques than their use in pure scientific discussion.

Finally, there is a third broad area of application of differential calculus, which intertwines with studies in optimization and global behavior and which, like such studies, is integral to the fundamental stock of ideas and techniques of every physical scientist: differential equations. Differential equations involve derivatives by their definition. The importance of differential equations in much (but not all) of physical science arises from Newton's definition of force, according to which the total force on a particle of constant mass is that mass times the acceleration of the particle. The accleration of a particle is the instantaneous rate of change--that is, the derivative with respect to time, of the velocity of the particle, which is itself the time derivative of the position of the particle, a vector quantity described by an arrow that goes from some reference point to the particle. In many situations, the force on the particle depends on its position and its velocity and time itself--or at least on some of these; consequently, analysis of these situations gives rise to systems of differential equations, relating the unknown coordinates of the moving particle to their derivatives, and the derivatives of their derivatives, with respect to time.

Differential equations arise in many other contexts besides that of particle motion; they are inevitable in the "modeling" of any physical situation in which some relation can be seen between the rate of change of some quantity of interest, and the quantity itself and the independent variables.

Theorem 1 of the preceding section is modest but venerable; it is the primordial ancestor of all existence-and-uniqueness theorems in the realm of differential equations. Such theorems say something about the sets of all solutions of differential equations of various forms.

Theorem 1 describes all possible solutions, on intervals, of the simplest possible differential equation, dy/dx = 0. It is not merely the spiritual ancestor of more impressive existence and uniqueness theorems; it can almost always be discovered at the fundament from which such theorems are deduced.

Context

In the fifteenth, sixteenth, and seventeenth centuries, in Europe, a subtle intellectual revolution took place in the world of physical science. The point of view that had dominated at the beginning of the Renaissance (often attributed to Aristotle, but probably inseparable from the intellectual climate of ancient Greek society) was that physical events took place in the imperfect real world; one described what happened, and then one sought to go beyond the superficial description only by inquiring into the essential nature of the things, the objects, involved in the event.

The central feature of the revolution was a tacit despair of ever penetrating to the essence of things; the quest for essence was replaced, little by little, by a focus on (supposedly) observable quantities associated with the event. For example, in Galileo's famous treatise on falling objects, he cares little whether the object be granite or marble or a porcelain dish (as long as the object is not a feather or a piece of parchment); he is concerned with the relations among the mass of the object, how far it has fallen, how fast it is falling, and how much time has elapsed since it was dropped.

The ancient Greek concept of idealization was reborn in this new way of thinking about physics, with the difference that it was not the objects involved that were idealized, but the "whole picture," for want of a better term. For example, from his dismissal of the case of a falling feather, or similar such object, and from other remarks he made, it seems that Galileo understood that what he was bearing down on was not a comprehensive description of what can happen when one drops any object whatever near the surface of the earth, but rather an analysis of an idealized, hypothetical situation, in which a hypothetical object is acted upon by a hypothetical constant force, with no messy impediments, such as air resistance.

Another great instance of idealization lies in Newton's mental picture of an object moving through space with no force pushing on it whatsoever; no human ever has or ever will verifiably observe such a wonder, but this picture is fundamental to Newton's definition of force as the rate of change (time derivative) of momentum.

Physics became, by the time of Newton, the study of the relations among supposedly observable quantities associated with idealized physical events. Once this point is grasped, the development of differential calculus, viewed as the art of dealing with rates of change of quantities that depend on other quantities, seems inevitable. In fact, differential calculus was invented independently by Newton and Leibniz at about the same time--in the second half of the seventeenth century, after the new physics had percolated for a time in the minds of natural philosophers.

To physicists and engineers, differential calculus has remained what it set out to be, a sort of technical art of use in analyzing and understanding physical events, and in solving practical problems. For mathematicians, differential calculus was not put on a rigorous footing until the late nineteenth century. The practical profit from the rigorization of the theory has been the sure-footed extension of the formal methods of differential calculus to exotic contexts undreamed of by Newton: One can now do optimization problems in which the dependent quantity to be optimized is not a real quantity at all, but something much more glamorous, such as an "operator" on a "function space." The context has changed, but the methods, in broad outline, have not. The derivative still reigns as the great simplifying concept.

Principal terms:

DEPENDENT VARIABLE: a variable quantity whose value depends on (is a function of) another quantity or other quantities, the independent variable or variables

DERIVATIVE: a dependent variable or function derived from another function by taking a limit of a Newton quotient; the instantaneous rate of change of the dependent variable per unit change of the independent variable

DIFFERENTIAL: a symbol associated to a variable with uses in formula manipulation, thought of as representing an infinitesimal increment or change in the value of the variable

DIFFERENTIAL EQUATION: an equation in which the unknown is a function, or dependent variable quantity, and in which appear the derivative, or some partial derivatives, or subsequent derivatives of the derivative(s) of the unknown function

FUNCTION: a dependent variable quantity whose value depends on one or more independent variable quantities

INDEPENDENT VARIABLE: a variable quantity thought of as roaming freely, without constraint or dependence on other quantities, over a certain set of possible values

NEWTON QUOTIENT: an expression of the form [f(x) - f(x₀)]/(x - x₀), or [f(x₀ + h) - f(x₀)]/h, where f(x) is a dependent quantity whose value depends on the independent variable x, and x₀ is a number in the set of possible values over which x is allowed to roam

OPTIMIZATION PROBLEM: a problem of which the solution is a set of values of the independent variables at which the dependent variable is as large, or as small, as possible

PARTIAL DERIVATIVE: a derivative of a function of several variables (a dependent quantity which depends on more than one independent quantity) obtained by fixing all but one of the independent variables and taking the derivative of the resulting function of one variable

Bibliography

Apostol, T.M., et al., eds. SELECTED PAPERS ON CALCULUS. Belmont, Calif.: Dickerson, 1969. This is a collection of articles from the AMERICAN MATHEMATICAL MONTHLY and MATHEMATICS magazine, two MAA publications. Teachers of calculus will find much of interest in these articles. Most of the articles are accessible to calculus students, who may find their patience rewarded here.

Chaundy, Theodore. THE DIFFERENTIAL CALCULUS. Oxford, England: Clarendon Press, 1935. Intended for the English university students of its era, this prosy, intelligent work reads more like a treatise, an extended essay, than a modern textbook. Recommended for earnest study to the independent student-readers who want to learn calculus on their own, and for browsing to everyone else.

Grabiner, Judith V. THE ORIGINS OF CAUCHY'S RIGOROUS CALCULUS. Cambridge, Mass.: MIT Press, 1981. An account of the attempts in the nineteenth century to put calculus on a rigorous foundation, and of the developments leading up to that attempt, from the beginning of differential calculus 150 years before. This is serious math history by a major math historian. A knowledge of calculus is a prerequisite for enjoying this book.

Hall, A. Rupert. PHILOSOPHERS AT WAR: THE QUARREL BETWEEN NEWTON AND LEIBNIZ. New York: Cambridge University Press, 1980. Accessible to all; the author judiciously avoids dwelling on or belaboring the mathematical issues and developments involved in the dispute over who gets the credit for inventing differential calculus. Perhaps in consequence, the mathematics is quite clear and interesting, while the material in which the author is interested, who said or wrote what to whom, and when, and what they might have been thinking at the time, becomes a bit murky. Worth a look, though.

Newman, James R. THE WORLD OF MATHEMATICS. 2 vols. New York: Simon & Schuster, 1956. Part 2 is on history and biography in mathematics, a selection of excerpts interspersed with Newman's commentary. The items on Newton, and the excerpt from Bishop Berkeley's essay, "The Analyst," are recommended. Part 5 is entitled "Mathematics and the Physical World." The items on Galileo and the Bernoullis are especially of interest.

Thomas, G. B., Jr. CALCULUS AND ANALYTIC GEOMETRY. 3d ed. Reading, Mass.: Addison-Wesley, 1951. The very model of the comprehensive "modern" textbook aimed at American physical science and engineering students, this one has the distinction of being the most popular calculus text in North America in the 1950's and 1960's, and perhaps for all time.