Author | Courant, Richard. author |
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Title | Introduction to Calculus and Analysis [electronic resource] : Volume I / by Richard Courant, Fritz John |
Imprint | New York, NY : Springer New York, 1989 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-8955-2 |
Descript | XXIII, 661 p. online resource |
1 Introduction -- 1.1 The Continuum of Numbers -- 1.2 The Concept of Function -- 1.3 The Elementary Functions -- 1.4 Sequences -- 1.5 Mathematical Induction -- 1.6 The Limit of a Sequence -- 1.7 Further Discussion of the Concept of Limit -- 1.8 The Concept of Limit for Functions of a Continuous Variable -- Supplements -- Problems -- 2 The Fundamental Ideas of the Integral and Differential Calculus -- 2.1 The Integral -- 2.2 Elementary Examples of Integration -- 2.3 Fundamental Rules of Integration -- 2.4 The Integral as a Function of the Upper Limit (Indefinite Integral) -- 2.5 Logarithm Defined by an Integral -- 2.6 Exponential Function and Powers -- 2.7 The Integral of an Arbitrary Power of x -- 2.8 The Derivative -- 2.9 The Integral, the Primitive Function, and theFundamental Theorems of the Calculus -- Supplement The Existence of the Definite Integral of a Continuous Function -- Problems -- 3 The Techniques of Calculus -- A Differentiation and Integration of the Elementary Functions -- B Techniques of Integration -- C Further Steps in the Theory of Integral Calculus -- 4 Applications in Physics and Geometry -- 4.1 Theory of Plane Curves -- 4.2 Examples -- 4.3 Vectors in Two Dimensions -- 4.4 Motion of a Particle under Given Forces -- 4.5 Free Fall of a Body Resisted by Air -- 4.6 The Simplest Type of Elastic Vibration -- 4.7 Motion on a Given Curve -- 4.8 Motion in a Gravitational Field -- 4.9 Work and Energy -- Problems -- 5 Taylorโs Expansion -- 5.1 Introduction: Power Series -- 5.2 Expansion of the Logarithm and the Inverse Tangent -- 5.3 Taylorโs Theorem -- 5.4 Expression and Estimates for the Remainder -- 5.5 Expansions of the Elementary Functions -- 5.6 Geometrical Applications -- Appendix I -- Appendix II Interpolation -- Problems -- 6 Numerical Methods -- 6.1 Computation of Integrals -- 6.2 Other Examples of Numerical Methods -- 6.3 Numerical Solution of Equations -- Problems -- 7 Infinite Sums and Products -- 7.1 The Concepts of Convergence and Divergence -- 7.2 Tests for Absolute Convergence and Divergence -- 7.3 Sequences of Functions -- 7.4 Uniform and Nonuniform Convergence -- 7.5 Power Series -- 7.6 Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples -- 7.7 Power Series with Complex Terms -- Problems -- 8 Trigonometrie Series -- 8.1 Periodic Functions -- 8.2 Superposition of Harmonic Vibrations -- 8.3 Complex Notation -- 8.4 Fourier Series -- 8.5 Examples of Fourier Series -- 8.6 Further Discussion of Convergence -- 8.7 Approximation by Trigonometric and RationalPolynomials -- Appendix I -- Appendix II -- Problems -- 9 Differential Equations jor the SimplestTypes of Vibration -- 9.1 Vibration Problems of Mechanics and Physics -- 9.2 Solution of the Homogeneous Equation. Free Oscillations -- 9.3 The Nonhomogeneous Equation. Forced Oscillations -- List of Biographical Dates