#  ## Introduction to calculus and analysis  v. 1 ～ v. II/2

Richard Courant, Fritz John

From the reviews: "...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

「Nielsen BookData」より

From the reviews: "...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

「Nielsen BookData」より

From the Preface: (...) The book is addressed to students on various levels, to mathematicians, scientists, engineers. It does not pretend to make the subject easy by glossing over difficulties, but rather tries to help the genuinely interested reader by throwing light on the interconnections and purposes of the whole. Instead of obstructing the access to the wealth of facts by lengthy discussions of a fundamental nature we have sometimes postponed such discussions to appendices in the various chapters. Numerous examples and problems are given at the end of various chapters. Some are challenging, some are even difficult; most of them supplement the material in the text.

「Nielsen BookData」より

[目次]

• Functions of Several Variables and Their Derivatives: Points and Points Sets in the Plane and in Space
• Functions of Several Independent Variables
• Continuity
• The Partial Derivatives of a Function
• The Differential of a Function and Its Geometrical Meaning
• Functions of Functions (Compound Functions) and the Introduction of New Independent Variables
• The mean Value Theorem and Taylor's Theorem for Functions of Several Variables
• Integrals of a Function Depending on a Parameter
• Differentials and Line Integrals
• The Fundamental Theorem on Integrability of Linear Differential Forms
• Appendix.- Vectors, Matrices, Linear Transformations: Operatios with Vectors
• Matrices and Linear Transformations
• Determinants
• Geometrical Interpretation of Determinants
• Vector Notions in Analysis.- Developments and Applications of the Differential Calculus: Implicit Functions
• Curves and Surfaces in Implicit Form
• Systems of Functions, Transformations, and Mappings
• Applications
• Families of Curves, Families of Surfaces, and Their Envelopes
• Alternating Differential Forms
• Maxima and Minima
• Appendix.- Multiple Integrals: Areas in the Plane
• Double Integrals
• Integrals over Regions in three and more Dimensions
• Space Differentiation. Mass and Density
• Reduction of the Multiple Integral to Repeated Single Integrals
• Transformation of Multiple Integrals
• Improper Multiple Integrals
• Geometrical Applications
• Physical Applications
• Multiple Integrals in Curvilinear Coordinates
• Volumes and Surface Areas in Any Number of Dimensions
• Improper Single Integrals as Functions of a Parameter
• The Fourier Integral
• The Eulerian Integrals (Gamma Function)
• Appendix

「Nielsen BookData」より

[目次]

• Relations Between Surface and Volume Integrals: Connection Between Line Integrals and Double Integrals in the Plane
• Vector Form of the Divergence Theorem. Stokes's Theorem
• Formula for Integration by Parts in Two Dimensions: Green's Theorem
• The Divergence Theorem Applied to the Transformation of Double Integrals
• Area Differentiation
• Interpretation of the Formulae of Gauss and Stokes by Two-Dimensional Flows
• Orientation of Surfaces
• Integrals of Differential Forms and of Scalars over Surfaces
• Gauss's and Green's Theorems in Space
• Appendix: General Theory of Surfaces and of Surface Integrals.- Differential Equations: The Differential Equations for the Motion of a Particle in Three Dimensions
• The General Linear Differential Equation of the First Order
• Linear Differential Equations of Higher Order
• General Differential Equations of the First Order
• Systems of Differential Equations and Differential Equations of Higher Order
• Integration by the Method of Undermined Coefficients
• The Potential of Attracting Charges and Laplace's Equation
• Further Examples of Partial Differential Equations from Mathematical Physics .- Calculus of Variations: Functions and Their Extreme Values of a Functional
• Generalizations
• Problems Involving Subsidiary Conditions. Lagrange Multipliers.- Functions of a Complex Variable: Complex Functions Represented by Power Series
• Foundations of the General Theory of Functions of a Complex Variable
• The Integration of Analytic Functions
• Cauchy's Formula and Its Applications
• Applications to Complex Integration (Contour Integration)
• Many-Valued Functions and Analytic Extension.- List of Biographical Dates Index

「Nielsen BookData」より

[目次]

• 1 Introduction.- 2 The Fundamental Ideas of the Integral and Differential Calculus.- 3 The Techniques of Calculus.- 4 Applications in Physics and Geometry.- 5 Taylor's Expansion.- 6 Numerical Methods.- 7 Infinite Sums and Products.- 8 Trigonometric Series.- 9 Differential Equations for the Simplest Types of Vibration.- List of Biographical Dates.

「Nielsen BookData」より

### 書名 Introduction to calculus and analysis Courant, Richard John, Fritz Fritz John John F. Courant R. Classics in mathematics v. 1 v. II/1 v. II/2 Springer-Verlag c1999-c2000 Reprint of the 1989 ed 3 v. 24-25 cm 3540665706 3540665692 9783540650584 BA39430861 ※クリックでCiNii Booksを表示 英語 ドイツ

##  ##  