Mathematical Analysis

• 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.

From the reviews: "Many teachers and students could benefit from thinking about its unique perspectives." ---SIAM Review "Giaquinta ... and Modica ... present the differential and integral calculus for real functions of a single real variable from a sophisticated point of view. Their book is far better suited as a source of unique perspectives on the logical development of calculus ideas and their physical applications ... [so that] it will better serve the experienced calculus instructor seeking new approaches to familiar material, or the student who has already mastered the basic ideas and techniques of calculus but wishes to see it all again, 'done right,' ... Highly recommended." ---CHOICE "This self-contained book introduces the main ideas and fundamental methods of mathematical analysis without loosing sight of the context in which it was developed and the role played in science, particularly in physics... Each chapter has a short summary where the most important facts discussed are collected. There is also a large number of exercises inserted at various points into the text... The book is meant principally for students in mathematics, physics, engineering, and computer science, but it can be used at technological and scientific faculties by anyone who wants to approach these topics." ---ZENTRALBLATT MATH "The presentation of the theory is clearly arranged, all theorems have rigorous proofs and every chapter closes with a summing up of the results and exericeses with different requirements. The aim of the authors was very successful[ly] realized, and this book is excellently suitable for students in mathematics, physics, engineering, computer science and all students of technological and scientific faculties." ---Journal of Analysis and its Applications "This is the first of several volumes on analysis by these authors. ... It is essentially a calculus book at the advanced undergraduate level, or an analysis textbook for someone who has never seen calculus before. ... I like this book ... ." (Warren Johnson, The Mathematical Association of America, October, 2009)