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AuthorCourant, Richard. author
TitleIntroduction to Calculus and Analysis II/1 [electronic resource] / by Richard Courant, Fritz John
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2000
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Descript XXV, 556 p. 157 illus. online resource


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 Gรถttingen, 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 Gรถttingen 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 the space 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)


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 Taylorโ{128}{153}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 Eulerโ{128}{153}s and Lagrangeโ{128}{153}s Representation of the motion of a Fluid -- A.6 Tangential Representation of a Closed Curve and the Isoperi-metric Inequality -- 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. Stokesโ{128}{153}s Theorem -- 5.3 Formula for Integration by Parts in Two Dimensions. Greenโ{128}{153}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 Gaussโ{128}{153}s and Greenโ{128}{153}s Theorems in Space -- 5.10 Stokesโ{128}{153}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.3 Stokesโ{128}{153}s Theorem -- A.4 Surfaces and Surface Integrals in Euclidean Spaces of Higher Dimensions -- A.5 Integrals over Simple Surfaces, Gaussโ{128}{153}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 Laplaceโ{128}{153}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 Cauchyโ{128}{153}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

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