## Otto Haupt's calculus textbooks

 Otto Haup published the 3-volume work Differential und Integralrechnung in 1938. This is a jointly authored work but it would appear that it was mainly due to Haupt since it is described as written by 0tto Haupt, assisted by Georg Aumann. A second edition, now entitled Differential- und Integralrechnung unter besonderer Berücksichtigung neueren Ergebnisse, lists three authors, Haupt, Aumann and Christian Pauc. It is a revision by Haupt and Pauc of the original text and the three volumes where published in 1948, 1950 and 1955. A third edition, considerably rewritten, was entitled Einführung in die reelle Analysis and had the two original authors, Haupt and Aumann. The three volumes were published in 1974, 1979 and 1983. We give below some extracts from reviews of these various editions.

1. Differential- und Integralrechnung Volumes 1, 2 and 3 (1938), by 0tto Haupt, assisted by Georg Aumann.
1.1. Review by: P J Daniell.
The Mathematical Gazette 23 (255) (1939), 331-332.

This book, or set of books, falls partly into each of two classes. Its extreme generality and deep analysis place it in the class of fundamental treatises. On the other hand it gives weight here and there more particularly to less advanced aspects of the subjects treated and in other respects it tends more towards the textbook type. There are even occasionally a few examples set at the ends of chapters. There appears to be some transition occurring in the attitude in German universities towards mathematics. At the beginning, for example, it is stated that geometrical ideas are drawn upon to make a more intuitive demonstration and it is true that some propositions, even in handling such general objects of thought as derived numbers of measurable functions, are expressed geometrically. The actual proofs, however, are all, very properly, arithmetical. In the first three chapters there are clear signs of the teacher, and the account of convergent sequences is clear and very full. Very early (p. 38) there appears the Heine-Borel-Lebesgue covering theorem. The first volume is about sequences, sets of points and functions of a real variable or combination of variables. There is a long discussion of monotone and convex functions. The second volume concerns itself with differentiation and there is a brief aside on the inverse operation. ... The third volume handles integration and follows a part of the procedure of Caratheodory's 'Reelle Funktionen', but although it includes the Lebesgue integral a great many pages are devoted to Jordan content and the Riemann integral.

1.2. Review by: Seymour Sherman.
Amer. Math. Monthly 50 (2) (1943), 116-117.

Differential- und Integralrechnung is designed for an analysis course intermediate between the traditional advanced calculus and the modern super-real variables, which begins with boolean algebras, lattices, topology and continues with Banach Spaces and integrals with values therein. Emphasis is placed on intuitive development rather than on ultra-formal argument. Nevertheless, some novel features are included. In the first half of the first volume, Introduction to Real Analysis, the continuum is introduced rapidly (long before the reader gets tired) and the usual limit properties of number sequences and sets are defined and explored. In the second half real functions of one and more real variables are considered. At the end of this half the author has a section on inequalities and convex functions, a subject not generally found in a book of this type. The second volume, Differential Calculus, again, is divided into two parts. In the first part the notion of a derivative is introduced and discussed. After the rules of differentiation are developed, the mean value theorem, Taylor series, what some texts call evaluation of indeterminates, the indefinite integral, and circular and hyperbolic functions are considered. These are then illustrated mainly with geometric examples and then the properties of monotone functions are given further attention. In the second half functions of many real variables are considered, with emphasis, at the end, on systems of equations and Jacobians. The last volume, Integration, is the most abstract of the three, which is of course pedagogically justified when one considers the additional sophistication the reader has supposedly acquired on his way to the end. Measure and content, for the most part in general fields of sets are dealt with in the first half. Here we find more emphasis on the notion of content than is customary. In the second half there is considered first, integration associated with content and second, integration associated with measure. The final section on examples and applications begins with properties of the Lebesgue integral and ends with a consideration of bounded k-dimensional surfaces in E_n and generalizations to E_n of the theorems of Gauss and Stokes.

2. Differential- und Integralrechnung unter besonderer Berücksichtigung neueren Ergebnisse Vol. I, Einführung in die reele Analysis (1948), by Otto Haupt, Georg Aumann and Christian Pauc.
2.1. Review by: Orrin Frink.
Mathematical Reviews, MR0036280 (12,84c).

This is an introduction to the theory of real functions, with some modern improvements. In spite of the title, it contains no differential calculus, and very little integral calculus. A first edition by Haupt and Aumann appeared in 1938; the present revision is by Haupt and Pauc, and is the first volume of a projected larger work. Chapter titles are: real numbers, limits of sequences, the function concept, real functions of a real variable, of several real variables, the general theory of limits.

3. Differential- und Integralrechnung unter besonderer Berücksichtigung neuerer Ergebnisse. Vol. II, Differentialrechnung (1950), by Otto Haupt, Georg Aumann and Christian Pauc.
3.1. Review by: Orrin Frink.
Mathematical Reviews, MR0040360 (12,681c).

Some topics included are: continuity and differentiability, tangents, formal rules of differentiation, derivatives of the elementary functions, higher derivatives, the mean value theorem, Taylor's formula with remainder, Taylor's series, indeterminate forms, existence and uniqueness of primitive functions, elementary integration formulas, divided differences, Newton and Lagrange interpolation polynomials, derivatives (generalized derivatives) derivatives of higher order as limits of divided differences, convex and convexoid functions of nth order, osculating circle and curvature, limit sets of functions of one or more real variables, derivate sets and contingents, nowhere differentiable functions, differentiability properties of rectifiable arcs, contingents and generalized differentials of functions of several variables, differentials of higher order, partial derivatives of first and higher order, Taylor's formula with remainder for functions of several variables, implicit functions, functional determinants, maxima and minima of functions of several variables, multiplier rule, differentiable mappings, functional dependence and independence.

4. Differential- und Integralrechnung unter besonderer Berücksichtigung neuerer Ergebnisse. Vol. III, Integralrechnung (1955), by Otto Haupt, Georg Aumann and Christian Pauc.
4.1. Review by: Orrin Frink.
Mathematical Reviews, MR0077613 (17,1066a).

This third volume of the new edition is a thoroughly modern textbook on measure theory and integration. The authors strive always for the greatest generality and adopt where possible the abstract viewpoint. The five parts, which include a total of eleven chapters, deal with, first, content, measure, and their extensions; second, partition integrals, s-additive functions, and linear functionals; third, measure and integration in topological spaces; fourth, primitive functions and the indefinite integral; and fifth, applications.

5. Einführung in die reelle Analysis Vol. I, Funktionen einer reellen Veränderlichen (1974), by 0tto Haupt and Georg Aumann.
5.1. Review by: Editors.
Mathematical Reviews, MR0352349 (50 #4836).

Apart from the revision of details the main difference [from the second edition] is a substantial rearrangement of the material: Vol. I is devoted to functions of one real variable, including the Riemann-Stieltjes integral; Vol. II will cover functions of several variables, including Lebesgue integration (i.e., will cover much of the material of the old Vol. III), and Vol. III will be devoted to topics in functional analysis.

6. Einführung in die reelle Analysis Vol. II, Differentialrechnung der Funktionen mehrerer Veränderlicher (1979), by 0tto Haupt and Georg Aumann.
6.1. Review by: Editors.
Mathematical Reviews, MR0565653 (83e:26001).

In the present, third edition this well-known text has undergone a through revision. In order to keep down the size of Volume II, the authors decided to move the theory of integration into Vol. III, which "is expected to appear soon''. Vol. II deals mainly with functions of several real variables. It consists of three parts: 1. Algebraic and geometric foundations, 2. Functions and mappings, 3. Differential calculus.

7. Einführung in die reelle Analysis Vol. III, Integralrechnung der Funktionen mehrerer Veränderlicher (1983), by 0tto Haupt and Georg Aumann.
7.1. Review by: Editors.
Mathematical Reviews, MR0699454 (84e:26001).

This edition has been "completely reworked''. Mathematical Reviews has not covered the previous editions of Volume III. ... Chapter headings: 1. Set systems. Spaces of measurement; 2. Content and measure; 3. Integral and real functions with respect to a measure; 4. s-additive functions; 5. Bases of derivation. Derivatives of set functions; 6. Supplements; 7. Product measures. Fubini's theorem; 8. Theorems of Gauss and Stokes; 9. Linear positive continuous functionals.

JOC/EFR April 2016