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Introduction to Calculus and Analysis II/1 de Richard Courant

Introduction to Calculus and Analysis II/1: Classics in Mathematics

Autor Richard Courant, Fritz John
en Limba Engleză Paperback – 14 dec 1999

Acest manual universitar, publicat în prestigioasa serie Classics in Mathematics de către Springer, reprezintă prima parte a celui de-al doilea volum dintr-o lucrare monumentală dedicată analizei matematice. Suntem de părere că ediția din 2000 păstrează intactă rigoarea pedagogică ce a format generații de matematicieni, oferind o tranziție fluidă de la calculul diferențial de bază către studiul avansat al funcțiilor de mai multe variabile. Apreciem în mod deosebit echilibrul pe care Richard Courant și Fritz John îl mențin între demonstrațiile matematice stricte și aplicațiile practice, o trăsătură care face textul accesibil nu doar studenților la matematică, ci și cercetătorilor din fizică sau inginerie.

Structura volumului este organizată meticulos, începând cu topologia mulțimilor de puncte și continuând cu derivatele parțiale, diferențiala unei funcții și semnificația sa geometrică. Un capitol esențial este dedicat calculului vectorial și matricial, pregătind terenul pentru transformările liniare. Spre deosebire de Multidimensional Differential and Integral Calculus de Giorgio Riccardi, care adoptă un stil conversațional și informal, lucrarea de față rămâne fidelă formalismului clasic, fiind mult mai densă în conținut teoretic. Totodată, Introduction to Calculus and Analysis II/1 completează perspectiva oferită de Basic Analysis II de James K. Peterson, adăugând o profunzime istorică și o rigoare a demonstrațiilor care lipsesc deseori din manualele moderne simplificate.

Subliniem importanța secțiunilor de apendice, unde sunt tratate riguros noțiuni precum principiul punctului de acumulare în mai multe dimensiuni. Această abordare reflectă filozofia autorului, întâlnită și în lucrarea sa Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, unde problemele specifice sunt utilizate pentru a introduce metode matematice generale și durabile.

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Specificații

ISBN-13: 9783540665694
ISBN-10: 3540665692
Pagini: 588
Ilustrații: XXV, 556 p. 157 illus.
Dimensiuni: 155 x 235 x 31 mm
Greutate: 0.87 kg
Ediția:2000
Editura: Springer Berlin, Heidelberg
Colecția Springer
Seria Classics in Mathematics

Locul publicării:Berlin, Heidelberg, Germany

Public țintă

Research

De ce să citești această carte

Recomandăm acest volum studenților la facultățile de profil tehnic și științific care doresc o bază solidă în analiza multivariată. Cititorul câștigă o înțelegere profundă a calculului diferențial în $R^n$, sprijinită de exerciții cu grade diferite de dificultate și figuri iluminatoare. Este o investiție în educația matematică pe termen lung, fiind o lucrare de referință care nu se demodează odată cu schimbarea curriculei.

Despre autor

Richard Courant (1888–1972) a fost unul dintre cei mai influenți matematicieni ai secolului XX, conducând Departamentul de Matematică de la New York University și punând bazele institutului care astăzi îi poartă numele (Courant Institute of Mathematical Sciences). Recunoscut pentru capacitatea sa de a unifica matematica pură cu cea aplicată, Courant a lăsat moștenire lucrări fundamentale precum „Methoden der mathematischen Physik”. Stilul său pedagogic se caracterizează prin claritate și prin convingerea că matematica trebuie să pornească de la probleme concrete pentru a ajunge la generalizări teoretice, viziune care strălucește în întreaga serie „Introduction to Calculus and Analysis”.

Descriere scurtă

From the reviews: "These books (Introduction to Calculus and Analysis Vol. I/II) are very well written. The mathematics are rigorous but the many examples that are given and the applications that are treated make the books extremely readable and the arguments easy to understand. These books are ideally suited for an undergraduate calculus course. Each chapter is followed by a number of interesting exercises. More difficult parts are marked with an asterisk. There are many illuminating figures...Of interest to students, mathematicians, scientists and engineers. Even more than that."
Newsletter on Computational and Applied Mathematics, 1991
"...one of the best textbooks introducing several generations of mathematicians to higher mathematics. ... This excellent book is highly recommended both to instructors and students."
Acta Scientiarum Mathematicarum, 1991

Cuprins

1 Functions of Several Variables and Their Derivatives.- 1.1 Points and Points Sets in the Plane and in Space.- 1.2 Functions of Several Independent Variables.- 1.3 Continuity.- 1.4 The Partial Derivatives of a Function.- 1.5 The Differential of a Function and Its Geometrical Meaning.- 1.6 Functions of Functions (Compound Functions) and the Introduction of New In-dependent Variables.- 1.7 The Mean Value Theorem and Taylor’s Theorem for Functions of Several Variables.- 1.8 Integrals of a Function Depending on a Parameter.- 1.9 Differentials and Line Integrals.- 1.10 The Fundamental Theorem on Integrability of Linear Differential Forms.- Appendix A.1. The Principle of the Point of Accumulation in Several Dimensions and Its Applications.- A.2. Basic Properties of Continuous Functions.- A.3. Basic Notions of the Theory of Point Sets.- A.4. Homogeneous functions..- 2 Vectors, Matrices, Linear Transformations.- 2.1 Operations with Vectors.- 2.2 Matrices and Linear Transformations.- 2.3 Determinants.- 2.4 Geometrical Interpretation of Determinants.- 2.5 Vector Notions in Analysis.- 3 Developments and Applications of the Differential Calculus.- 3.1 Implicit Functions.- 3.2 Curves and Surfaces in Implicit Form.- 3.3 Systems of Functions, Transformations, and Mappings.- 3.4 Applications.- 3.5 Families of Curves, Families of Surfaces, and Their Envelopes.- 3.6 Alternating Differential Forms.- 3.7 Maxima and Minima.- Appendix A.1 Sufficient Conditions for Extreme Values.- A.2 Numbers of Critical Points Related to Indices of a Vector Field.- A.3 Singular Points of Plane Curves 360 A.4 Singular Points of Surfaces.- A.5 Connection Between Euler’s and Lagrange’s Representation of the motion of a Fluid.- A.6 Tangential Representation of a Closed Curve and the Isoperi-metricInequality.- 4 Multiple Integrals.- 4.1 Areas in the Plane.- 4.2 Double Integrals.- 4.3 Integrals over Regions in three and more Dimensions.- 4.4 Space Differentiation. Mass and Density.- 4.5 Reduction of the Multiple Integral to Repeated Single Integrals.- 4.6 Transformation of Multiple Integrals.- 4.7 Improper Multiple Integrals.- 4.8 Geometrical Applications.- 4.9 Physical Applications.- 4.10 Multiple Integrals in Curvilinear Coordinates.- 4.11 Volumes and Surface Areas in Any Number of Dimensions.- 4.12 Improper Single Integrals as Functions of a Parameter.- 4.13 The Fourier Integral.- 4.14 The Eulerian Integrals (Gamma Function).- Appendix: Detailed Analysis of the Process Of Integration A.1 Area.- A.2 Integrals of Functions of Several Variables.- A.3 Transformation of Areas and Integrals.- A.4 Note on the Definition of the Area of a Curved Surface.- 5 Relations Between Surface and Volume Integrals.- 5.1 Connection Between Line Integrals and Double Integrals in the Plane (The Integral Theorems of Gauss, Stokes, and Green).- 5.2 Vector Form of the Divergence Theorem. Stokes’s Theorem.- 5.3 Formula for Integration by Parts in Two Dimensions. Green’s Theorem.- 5.4 The Divergence Theorem Applied to the Transformation of Double Integrals.- 5.5 Area Differentiation. Transformation of Au to Polar Coordinates.- 5.6 Interpretation of the Formulae of Gauss and Stokes by Two-Dimensional Flows.- 5.7 Orientation of Surfaces.- 5.8 Integrals of Differential Forms and of Scalars over Surfaces.- 5.9 Gauss’s and Green’s Theorems in Space.- 5.10 Stokes’s Theorem in Space.- 5.11 Integral Identities in Higher Dimensions.- Appendix: General Theory Of Surfaces And Of Surface Integals A.I Surfaces and Surface Integrals in Three dimensions.- A.2 The Divergence Theorem.- A.3Stokes’s Theorem.- A.4 Surfaces and Surface Integrals in Euclidean Spaces of Higher Dimensions.- A.5 Integrals over Simple Surfaces, Gauss’s Divergence Theorem, and the General Stokes Formula in Higher Dimensions.- 6 Differential Equations.- 6.1 The Differential Equations for the Motion of a Particle in Three Dimensions.- 6.2 The General Linear Differential Equation of the First Order.- 6.3 Linear Differential Equations of Higher Order.- 6.4 General Differential Equations of the First Order.- 6.5 Systems of Differential Equations and Differential Equations of Higher Order.- 6.6 Integration by the Method of Undermined Coefficients.- 6.7 The Potential of Attracting Charges and Laplace’s Equation.- 6.8 Further Examples of Partial Differential Equations from Mathematical Physics.- 7 Calculus of Variations.- 7.1 Functions and Their Extrema.- 7.2 Necessary conditions for Extreme Values of a Functional.- 7.3 Generalizations.- 7.4 Problems Involving Subsidiary Conditions. Lagrange Multipliers.- 8 Functions of a Complex Variable.- 8.1 Complex Functions Represented by Power Series.- 8.2 Foundations of the General Theory of Functions of a Complex Variable.- 8.3 The Integration of Analytic Functions.- 8.4 Cauchy’s Formula and Its Applications.- 8.5 Applications to Complex Integration (Contour Integration).- 8.6 Many-Valued Functions and Analytic Extension.- List of Biographical Dates.

Recenzii

From the reviews: "These books (Introduction to Calculus and Analysis Vol. I/II) are very well written. The mathematics are rigorous but the many examples that are given and the applications that are treated make the books extremely readable and the arguments easy to understand. These books are ideally suited for an undergraduate calculus course. Each chapter is followed by a number of interesting exercises. More difficult parts are marked with an asterisk. There are many illuminating figures...Of interest to students, mathematicians, scientists and engineers. Even more than that." Newsletter on Computational and Applied Mathematics, 1991 "...one of the best textbooks introducing several generations of mathematicians to higher mathematics. ... This excellent book is highly recommended both to instructors and students." Acta Scientiarum Mathematicarum, 1991

Textul de pe ultima copertă

Biography of Richard Courant
Richard Courant was born in 1888 in a small town of what is now Poland, and died in New Rochelle, N.Y. in 1972. He received his doctorate from the legendary David Hilbert in Göttingen, where later he founded and directed its famed mathematics Institute, a Mecca for mathematicians in the twenties. In 1933 the Nazi government dismissed Courant for being Jewish, and he emigrated to the United States. He found, in New York, what he called "a reservoir of talent" to be tapped. He built, at New York University, a new mathematical Sciences Institute that shares the philosophy of its illustrious predecessor and rivals it in worldwide influence.
For Courant mathematics was an adventure, with applications forming a vital part. This spirit is reflected in his books, in particular in his influential calculus text, revised in collaboration with his brilliant younger colleague, Fritz John.
(P.D. Lax)

Biography of Fritz John

Fritz John was born on June 14, 1910, in Berlin. After his school years in Danzig (now Gdansk, Poland), he studied in Göttingen and received his doctorate in 1933, just when the Nazi regime came to power. As he was half-Jewish and his bride Aryan, he had to flee Germany in 1934. After a year in Cambridge, UK, he accepted a position at the University of Kentucky, and in 1946 joined Courant, Friedrichs and Stoker in building up New York University the institute that later became the Courant Institute of Mathematical Sciences. He remained there until his death in New Rochelle on February 10, 1994.
John's research and the books he wrote had a strong impact on the development of many fields of mathematics, foremost in partial differential equations. He also worked on Radon transforms, illposed problems, convex geometry, numerical analysis, elasticity theory. In connection with his work in latter field, he and Nirenberg introduced thespace of the BMO-functions (bounded mean oscillations). Fritz John's work exemplifies the unity of mathematics as well as its elegance and its beauty.
(J. Moser)