Mathematical Analysis

For more than two thousand years some familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today the traditional place of mathematics in education is in grave danger. Unfortunately, professional representatives of mathematics share in the reponsibiIity. The teaching of mathematics has sometimes degen­ erated into empty drill in problem solving, which may develop formal ability but does not lead to real understanding or to greater intellectual indepen­ dence. Mathematical research has shown a tendency toward overspecialization and over-emphasis on abstraction. Applications and connections with other fields have been neglected . . . But . . . understanding of mathematics cannot be transmitted by painless entertainment any more than education in music can be brought by the most brilliant journalism to those who never have lis­ tened intensively. Actual contact with the content of living mathematics is necessary. Nevertheless technicalities and detours should be avoided, and the presentation of mathematics should be just as free from emphasis on routine as from forbidding dogmatism which refuses to disclose motive or goal and which is an unfair obstacle to honest effort. (From the preface to the first edition of What is Mathematics? by Richard Courant and Herbert Robbins, 1941.

1. Numbers, Functions and their Graphs.- 1.1 Real Numbers: a Description.- 1.2 The Cartesian Plane.- 1.3 Elementary Functions.- 1.4 Remarks on Common Language and the Language of Mathematics.- 1.5 Exercises.- 2. Limits and Continuity.- 2.1 Limits.- 2.2. Continuous Functions.- 2.3. Continuous functions on an interval.- 2.4 Weierstrass's Theorem.- 2.5 Summing Up.- 2.6 Exercises.- 3. The Fundamental Ideas of the Differential and Integral Calculus.- 3.1 Differential Calculus.- 3.2 Integral Calculus.- 3.3 The Fundamental Theorem of Calculus.- 3.4 Calculus: Some Historical Remarks.- 3.5 Summing Up.- 3.6 Exercises.- 4. The Calculus of Derivatives and of Integrals.- 4.1 Computation of Derivatives.- 4.2 Integrals and Primitives.- 4.3 A Definition of the Trigonometric, Logarithmic and Exponential Functions.- 4.4 Some Differential Equations.- 4.5 Generalized Integrals.- 4.6 Summing Up.- 4.7 Exercises.- 5. Further Developments in Calculus.- 5.1 Taylor's Formula.- 5.2 The Calculus of Limits.- 5.3 Convex Functions.- 5.4 Some Inequalities.- 5.5 Graphing a Function.- 5.6 Summing Up.- 5.7 Exercises.- 6. Toward Differential Equations and Minimum Principles.- 6.1 Linear Ordinary Differential Equations.- 6.2 First Order ODEs.- 6.3 One-Dimensional Motions.- 6.4 Optimization Problems.- 6.5 Summing Up.- 6.6 Exercises.- A. Matematicians and Other Scientists.- B. Bibliographical Notes.- C. Index.