Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra, With Applications to Differential Equations and Probability

Description

An introduction to the calculus, with an excellent balance between theory and technique. Integration is treated before differentiation -- this is a departure from most modern texts, but it is historically correct, and it is the best way to establish the true connection between the integral and the derivative. Proofs of all the important theorems are given, generally preceded by geometric or intuitive discussion. This Second Edition introduces the mean-value theorems and their applications earlier in the text, incorporates a treatment of linear algebra, and contains many new and easier exercises. As in the first edition, an interesting historical introduction precedes each important new concept. **

§11.31 "Extensions to higher dimensions" (DjVu pp. 430-2) mentioned here regarding Fubini's theorem applied to integrals of order ≥3

MR0248290

This volume completes the second edition [first edition, Vol. I: Introduction with vectors and analytic geometry, 1961; Vol. II: Calculus of several variables with applications to probability and vector analysis, 1962] of the author's unusually thorough Calculus, the first volume of the second edition having appeared in 1967 [Vol. I: One-variable calculus, with an introduction to linear algebra, 1967; MR0214705]. Two sentences from the preface characterize the work: "Sound training in technique is combined with a strong theoretical development. Every effort has been made to convey the spirit of modern mathematics without undue emphasis on formalization.'' The second edition provides more exercises than the first, mainly easier ones, some reordering and rewriting of material of the first edition, but mainly it adds a new section on linear algebra. The first part of this new section appeared as the last two chapters in the second edition Volume I, and these are repeated as the first two chapters of Volume II to make the entire new section available in one volume, where full use is made thereafter. Chapter headings in Volume II are: Linear spaces; Linear transformations and matrices; Determinants; Eigenvalues and eigenvectors; Eigenvalues of operators acting on Euclidean spaces; Linear differential equations; Systems of differential equations; Differential calculus of scalar and vector fields; Applications of the differential calculus; Line integrals; Multiple integrals; Surface integrals; Set functions and elementary probability; Calculus of probabilities; Introduction to numerical analysis. Reviewed by F. Brooks