Calculus

Calculus is at once the most important and most difficult subject encountered early by

students of mathematics; introductory courses often succeed only in turning students

away from mathematics, and from the many subjects in which the calculus plays a major

role.

The present text introduces calculus in the informal manner adopted in my Arithmetic [1],

a manner endorsed by Lakatos [2], and by the following words of Lanczos from his

preface to [3]:

Furthermore, the author has the notion that mathematical formulas have their “secret

life” behind their Golem-like appearance. To bring out the “secret life” of

mathematical relations by an occasional narrative digression does not appear to him

a profanation of the sacred rituals of formal analysis but merely an attempt to a more

integrated way of understanding. The reader who has to struggle through a maze of

“lemmas”, “corollaries”, and “theorems”, can easily get lost in formalistic details, to

the detriment of the essential elements of the results obtained. By keeping his mind

on the principal points he gains in depth, although he may lose in details. The loss is

not serious, however, since any reader equipped with the elementary tools of algebra

and calculus can easily interpolate the missing details. It is a well-known experience

that the only truly enjoyable and profitable way of studying mathematics is the

method of “filling in the details” by one’s own efforts.

The scope is broader than is usual in an introduction, embracing not only the differential

and integral calculus, but also the difference calculus so useful in approximations, and

the partial derivatives and the fractional calculus usually met only in advanced courses.

Such breadth is achievable in small compass not only because of the adoption of

informality, but also because of the executable notation employed. In particular, the array

character of the notation makes possible an elementary treatment of partial derivatives in

the manner used in tensor analysis.

The text is paced for a reader familiar with polynomials, matrix products, linear

functions, and other notions of elementary algebra; nevertheless, full definitions of such

matters are also provided.