# Calculus, vol 2 - Multi Variable Calculus and Linear Algebra With Applications to Differen

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**Calculus, vol 2 - Multi Variable Calculus and Linear Algebra With Applications to Differential Equation and Probability**

by Tom M. Apostol

by Tom M. Apostol

This book is a continuation of the author’s Calculus, Volume I, Second Edition. The

present volume has been written with the same underlying philosophy that prevailed in the

first. 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. As in Volume I, historical remarks are included to give the

student a sense of participation in the evolution of ideas.

The second volume is divided into three parts, entitled Linear Analysis, Nonlinear

Ana!ysis, and Special Topics. The last two chapters of Volume I have been repeated as the

first two chapters of Volume II so that all the material on linear algebra will be complete

in one volume.

Part 1 contains an introduction to linear algebra, including linear transformations,

matrices, determinants, eigenvalues, and quadratic forms. Applications are given to

analysis, in particular to the study of linear differential equations. Systems of differential

equations are treated with the help of matrix calculus. Existence and uniqueness theorems

are proved by Picard’s method of successive approximations, which is also cast in the

language of contraction operators.

Part 2 discusses the calculus of functions of several variables. Differential calculus is

unified and simplified with the aid of linear algebra. It includes chain rules for scalar and

vector fields, and applications to partial differential equations and extremum problems.

Integral calculus includes line integrals, multiple integrals, and surface integrals, with

applications to vector analysis. Here the treatment is along more or less classical lines and

does not include a formal development of differential forms.

The special topics treated in Part 3 are Probability and Numerical Analysis. The material

on probability is divided into two chapters, one dealing with finite or countably infinite

sample spaces; the other with uncountable sample spaces, random variables, and distribution

functions. The use of the calculus is illustrated in the study of both one- and

two-dimensional random variables.

The last chapter contains an introduction to numerical analysis, the chief emphasis

being on different kinds of polynomial approximation. Here again the ideas are unified

by the notation and terminology of linear algebra. The book concludes with a treatment of

approximate integration formulas, such as Simpson’s rule, and a discussion of Euler’s

summation formula.

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