The Elements of Calculus

Calculus is the foundation of the modern mathematics of physics and engineering

Calculus was invented by Newton and Leibnitz at the end of the 17th century. While algebra had been an important addition to mathematics in the immediately preceding centuries, it provided mainly a notational convenience. A very significant convenience, to be sure, but it did not provide much additional power to mathematical investigations. Calculus, however, changed mathematics radically, and possessed great power and range, in practical matters as well as theoretical. Calculus involves a convenient representation of certain limiting processes, which were hinted at in earlier mathematics, but were very difficult to use, and then only in particular problems.

In the educational curriculum for science and engineering, calculus forms a bridge between elementary mathematics, such as geometry, algebra and trigonometry, and advanced mathematics, such as differential equations, vector analysis and complex variables. In this position, it has other duties to perform other than simply introducing its elements. To begin the study of calculus, concepts of function, continuous function and limits are necessary, as well as some idea of the nature of mathematical proof. During the course, the student also should be introduced to the theory of curves, infinite series, power series, elementary functions, convergence, Fourier series, and other topics, as examples to which calculus can be applied. Finally, the course should foster manipulative skills, and even introduce numerical methods, such as differencing and quadrature. In all this, the central idea of calculus tends to be obscured.

The essence of calculus are the definitions of the derivative and integral as limit processes. The derivative of a continuous function y(x) is y'(x) = limit Δx→0 {[y(x+Δx) - y(x)]/Δx}, and its integral between a and b is the limit as the largest interval Δx→0 of the sum Y(a,b) = Σ y(ξi)Δxi, where the Δx are a subdivision of the interval a ≥ x ≥ b, and ξi is some x in the interval Δi. The Fundamental Theorem of Calculus states that the derivative of the function Y(x) = Y(a,x), which is the integral of y(x) from a to x (the indefinite integral), is Y'(x) = y(x), or, going the other way, that the indefinite integral of y'(x) is y(x). Aside from the details, that is all there is to calculus. The use of these concepts is called analysis.

It used to be the custom to divide the Calculus course into a term of Differential Calculus followed by a term of Integral Calculus. This tenacious error appealed to the orderly paedagogue who viewed calculus as a set of rules, but was not efficient in presenting the subject to the reasoning mind. Richard Courant perceived this clearly, and promoted a unified development based on the reciprocal relations between differential and integral calculus. His textbook, first published in English in 1934 (see References), and popular into the 1950's, reflects this idea. Although I did not encounter his book until later, it still appears to be the best calculus text ever likely to be written. The selection of topics, and the methods of treatment, are excellent. Among other things, it has good exercises with answers and hints.

The differential calculus exploits the definition of the derivative, and freely uses concepts like that of the differential, dx, which is different from the finite difference Δx. The derivative y'(x) can now be written dy/dx. The somewhat hidden limiting process makes a rigorous mathematician nervous, but the power in applications is great. Famous direct applications are to the finding of extrema (maxima and minima) of a function f(x) by solving f '(x) = 0, and the evaluation of the limit of f(x)/g(x) at a point x where both f and g vanish; that is, to the form 0/0. In this case, we simply differentiate f and g iteratively until the form is defined. One studies the derivatives of polynomials, and finds that dex/dx = ex, so that tables of the derivatives of the elementary functions, including trigonometric, hyperbolic, logarithmic and exponential functions can be created. An important rule is the Chain Rule, d(xy) = xdy + ydx, in differential notation. There is also the Mean Value theorem for a continuous function, that [y(b) - y(a)]/(b - a) = y'(ξ), where ξ is some value in the interval [b,a]. While studying differential calculus, one can also go into the matter of finite differences and their uses.

The integral calculus makes great use of the Fundamental Theorem, mentioned above, and shows that for every property of the derivative, there is a corresponding property of the integral. For example, the Chain Rule becomes integration by parts: ∫d(xy) = xy = ∫xdy + ∫ydx, or ∫xdy = xy - ∫ydx. In particular, the table of derivatives can also be interpreted as a table of integrals. The indefinite integral is always accompanied by an arbitrary constant addend, since the derivative of a constant is zero ( limit 0/Δx = 0). There is great fun in finding the integrals of elementary functions. By making a substitution, such as v = v(x), which makes dv = v'dx, an integral may be thrown into a recognizable form. If ∫xdy can't be done, perhaps ∫ydx can, so that integration by parts can do the job. The Mean Value theorem, ∫(a,b) f(x)dx = (b - a)f(ξ), shows how to estimate integrals numerically. Of course, it is the mirror image of the Mean Value theorem for the derivative. Since many problems can be analyzed using elementary functions, integral calculus is quite powerful in applications such as the determination of areas, volumes, centroids, moments of inertia, as well as in the solution of problems of motion. A very convenient tool is a handbook of integrals. The mental and algebraic agility acquired by "doing integrals" can be obtained in no other way.

Integrating f(x) is really solving the differential equation y' = f(x), and integral calculus goes over smoothly into the study of differential equations. If the student is going no farther in analysis, the calculus course should end with a short treatment of the most useful differential equations (e.g., linear first order, and linear of any order with constant coefficients). Integration also creates new functions that can be studied, such as the gamma function, and elliptic integrals.

Another important concept introduced in calculus courses is the Taylor series, where the derivatives of a function at a point determine its behavior in the neighborhood. Indeed, the Taylor series is y(x + h) = y(x) + y'(x)h + y"(x)h2/2! + ..., which is an example of a power series in h. The properties of infinite series, such as convergence and uniform convergence, are conveniently introduced at the same time, concentrating on power series. These concepts are useful for further progress in understanding mathematics.

The first course in calculus leads on to many extensions and applications, both theoretical and practical. An obvious extension is to dependence on several independent variables, which introduces partial derivatives, ∂f/∂x and ∂f/∂y, where f = f(x,y), and multiple integrals, such as ∫∫f(x,y)dydx. Extensions to more than two variables, and to line integrals, ∫f(x,y)ds are easy. The extension to three dimensions is aided by vector concepts, and the results can be applied to differential geometry. An important result is the equality of cross partials, ∂2/∂x∂y = ∂2/∂y∂x, which is used in differential equations and thermodynamics, for example. Limits and infinte series deserve further attention, and are applied to improper integrals, those with infinite limits. The gamma function and related matters, such as Stirling's approximation to n!, are useful, and attention can be paid to Fourier series and integral transforms, such as the Laplace transform. These topics in differential and integral calculus may form a course in Advanced Calculus, such as is presented by the second volume of Courant, or the excellent text by Widder.

The first course in calculus can be begun as soon as sufficent algebraic skills have been learned, best accomplished by good courses in algebra and trigonometry, and mathematical curiosity has been excited. This could happen in high school, but usually is delayed until the first year in college, when calculus is presented to the poorly prepared, mathematically immature students produced in today's high schools. Calculus should immediately follow a good course in algebra and analytic geometry for best results. A course in differential equations can immediately follow calculus for students of physics and engineering, since it is applied early in these curricula. An Advanced Calculus course requires mathematical maturity in addition to manipulative skills. Physics students may well have been introduced to many of its topics earlier in their studies. For example, electromagnetism involves vectors, surface integrals and partial derivatives. Thermodynamics makes use of partial derivatives and concepts from differential geometry. Mechanics involves simple differential equations. All these subjects call upon the mathematical maturity developed in Calculus. Advanced calculus is a good place to tie all these things together, one final survey of the field, and a proper goal for the undergraduate.

A good algebra course is essential for success in calculus. Like calculus courses, algebra courses are mainly applications, the fundamentals blending into the background. Unfortunately, most courses are very poor because of the lure of presenting them as a system of rules and the lack of competent instructors. Manipulation is important, but understanding is golden. Preparation in algebra should include complex numbers, introduced as roots of algebraic equations, and simple vectors. These concepts should continually be exercised at all levels, and can be applied, for example, to alternating-current circuits (as well as to optics and other areas where phase is important). The combination of complex numbers and analysis must wait until postgraduate work. Calculus is essential preparation for this. In any case, much the same ground must be covered in iterative passes, going deeper each time. This is the only way a good understanding of anything worthwhile can be acquired.


  1. R. Courant, Differential and Integral Calculus, 2nd ed. (London: Blackie, 1937), 2 vols. The first volume contains the introductory course in calculus, with optional advanced topics.
  2. H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (New York: Macmillan, 1961). I understand that this book is now out of print, which is a great shame. The alternatives are not nearly as convenient and useful.
  3. D. V. Widder, Advanced Calculus, 2nd ed. (New York: Dover, 1989) 1st ed. 1947.

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Composed by J. B. Calvert
Created 4 March 2001
Last revised