Differential and Integral Calculus, Volume I

by R. Courant

Hardcover, 1966

Status

Available

Call number

515

Library's review

Indeholder "Introductory Remarks", "Chapter I. Introduction", "1. The Continuum of Numbers", "2. The Concept of Function", "3. More Detailed Study of the Elementary Functions", "4. Functions of an Integral Variable. Sequences of Numbers", "5. The Concept of the Limit of a Sequence", "6. Further
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Discussion of the Concept of Limit", "7. The Concept of Limit where the Variable is Continuous", "8. The Concept of Continuity", "Appendix I. Preliminary Remarks", "1. The Principle of the Point of Accumulation and its Applications", "2. Theorems on Continuous Functions", "3. Some Remarks on the Elementary Functions", "Appendix II.", "1. Polar Co-ordinates", "2. Remarks on Complex Numbers", "Chapter II. The Fundamental Ideas of the Integral and Differential Calculus", "1. The Definite Integral", "2. Examples", "3. The Derivative", "4. The Indefinite Integral, the Primitive Function, and the Fundamental Theorems of the Differential and Integral Calculus", "5. Simple Methods of Graphical Integration", "6. Further Remarks on the Connexion between the Integral and the Derivative", "7. The Estimation of Integrals and the Mean Value Theorem of the Integral Calculus", "Appendix", "1. The Existence of the Definite Integral of a Continuous Function", "2. The Relation between the Mean Value Theorem of the Differential Calculus and the Mean Value Theorem of the Integral Calculus", "Chapter III. Differentiation and Integration of the Elementary Functions", "1. The Simplest Rules for Differentiation and their Applications", "2. The Corresponding Integral Formulæ", "3. The Inverse Function and its Derivative", "4. Differentiation of a Function of a Function", "5. Maxima and Minima", "6. The Logarithm and the Exponential Function", "7. Some Applications of the Exponential Function", "8. The Hyperbolic Functions", "9. The Order of Magnitude of Functions", "Appendix", "1. Some Special Functions", "2. Remarks on the Differentiability of Functions", "3. Some Special Formulæ", "Chapter IV. Further development of the Differential Calculus", "1. Elementary Integrals", "2. The Method of Substitution", "3. Further Examples of the Substitution Method", "4. Integration by Parts", "5. Integration of Rational Functions", "6. Integration of Some Other Classes of Functions", "7. Remarks on Functions which are not Integrable in Terms of Elementary Functions", "8. Extension of the Concept of Integral. Improper Integrals.", "Appendix. The Second Mean Value Theorem of the Integral Calculus", "Chapter V. Applications", "1. Representation of Curves", "2. Applications to the Theory of Plane Curves", "3. Examples", "4. Some very Simple Problems in the Mechanics of a Particle", "5. Further Applications: Particle sliding down a Curve", "6. Work", "Appendix", "1. Properties of the Evolute", "2. Areas bounded by Closed Curves", "Chapter VI. Taylor's Theorem and the Approximate Expressions of Functions by Polynomials", "1. The Logarithm and the Inverse Tangent", "2. Taylor's Theorem", "3. Applications. Expansions of the Elementary Functions", "4. Geometrical Applications", "Appendix", "1. Example of a Function which cannot be expanded in a Taylor Series", "2. Proof that e is Irrational", "3. Proof that the Binomial Series Converges", "4. Zeroes and Infinities of Functions, and So-called Indeterminate Expressions", "Chapter VII. Numerical Methods", "Preliminary Remarks", "1. Numerical Integration", "2. Applications of the Mean Value Theorem and of Taylor's Theorem. The Calculus of Errors", "3. Numerical Solution of Equations", "Appendix. Stirling"s Formula", "Chapter VIII. Infinite Series and Other Limiting Processes", "Preliminary Remarks", "1. The Concepts of Convergence and Divergence", "2. Tests for Convergence and Divergence", "3. Sequences and Series of Functions", "4. Uniform and Non-uniform Convergence", "5. Power Series", "6. Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples.", "7. Power Series with Complex Terms", "Appendix", "1. Multiplication and Division of Series", "2. Infinite Series and Improper Integrals", "3. Infinite Products", "4. Series involving Bernoulli"s Numbers", "Chapter IX. Fourier Series", "1. Periodic Functions", "2. Use of Complex Notation", "3. Fourier Series", "4. Examples of Fourier Series", "5. The Convergence of Fourier Series", "Appendix. Integration of Fourier Series", "Chapter X. A Sketch of the Theory of Functions of Several Variables", "1. The Concept of Function in the Case of Several Variables", "2. Continuity", "3. The Derivatives of a Function of Several Variables", "4. The Chain Rule and the Differentiation of Inverse Functions", "5. Implicit Functions", "6. Multiple and Repeated Integrals", "Chapter XI. The Differential Equations for the Simplest Types of Vibrations.", "1. Vibration Problems of Mechanics and Physics", "2. Solution of the Homogeneous Equation. Free Oscillations", "3. The Non-homogeneous Equation. Forced Oscillations", "4. Additional Remarks on Differential Equations", "Summary of Important Theorems and Formulæ", "Miscellaneous Examples", "Answers and Hints", "Index".

Virkelig nydelig lærebog for lærenemme og lærevillige matematikstuderende. Grundliggende matematisk analyse med masser af gennemarbejdede eksempler og en klar rød tråd. Selv om Richard Courant skrev den i 30'ernes Tyskland, er den stadig værd at læse.
Fx finder vi i dette bind på side 299 og frem en smuk gennemgang af en friktionsfri partikels skæbne, når den er sluppet løs på en kurve. Side 223 og frem bruges på Wallis's produkt og man finder en formel for sqrt(pi) som passer som hånd i handske med andelen af bitstrenge med lige mange ettere og nuller, hvis man er datalog
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Publication

Blackie (1966), Hardcover, 616 pages

Description

The classic introduction to the fundamentals of calculus Richard Courant's classic text Differential and Integral Calculus is an essential text for those preparing for a career in physics or applied math. Volume 1 introduces the foundational concepts of "function" and "limit", and offers detailed explanations that illustrate the "why" as well as the "how". Comprehensive coverage of the basics of integrals and differentials includes their applications as well as clearly-defined techniques and essential theorems. Multiple appendices provide supplementary explanation and author notes, as well as solutions and hints for all in-text problems.

User reviews

LibraryThing member nealjking
This is a classic book on the calculus: It covers the foundations of real analysis as well as the practical calculational techniques. This is calculus for people that enjoy mathematics.

Subjects

Language

Original language

German

Physical description

616 p.; 22.2 cm

Local notes

Omslag: Ikke angivet
Omslaget viser en saddelkurve u = x^2 + y^2
Indskannet omslag - N650U - 150 dpi
Fra forordet til den tyske udgave:
"The beginner should note that I have avoided blocking the entrance to the concrete facts of the differential and integral calculus by discussions of fundamental matters, for which he is not yet ready. Instead, these are collected in appendices to the chapters, and the student whose main purpose is to acquire the facts rapidly or to proceed to practical applications may postpone reading these until he feels the need for them. The appendices also contain some additions to the subject-matter; they have been made relatively concise. The reader will notice, too, that the general style of presentation, at first detailed, is more condensed towards the end of the book. He should not, however, let himself be disheartened by isolated difficulties which he may find in the concluding chapters. Such gaps in understanding, if not too frequent, usually fill up of their own accord."

Pages

616

Library's rating

Rating

(4 ratings; 4)

DDC/MDS

515
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