Differential and Integral Calculus Vol. II

Description

PREFACE CONTENTS CHAPTER I PRELIMINARY REMARKS ON ANALYTICAL GEOMETRY AND VECTOR ANALYSIS 1. Rectangular Co-ordinates and Vectors 2. The Area of a Triangle, the Volume of a Tetrahedron, the Vector Multiplication of Vectors 3. Simple Theorems on Determinants of the Second and Third Order 4. Affine Transformations and the Multiplication of Determinants CHAPTER II FUNCTIONS OF SEVERAL VARIABLES AND THEIR DERIVATIVES 1. The Concept of Function in the Case of Several Variables 2. Continuity 3. The Derivatives of a Function 4. The Total Differential of a Function and its Geometrical Meaning 5. Functions of Functions (Compound Functions) and the Introduction of New Independent Variables 6. The Mean Value Theorem and Taylor's Theorem for Functions of Several Variables 7. The Application of Vector Methods APPENDIX 1. The Principle of the Point of Accumulation in Several Dimensions and its Applications 2. The Concept of Limit for Functions of Several Variables 3. Homogeneous Functions CHAPTER III DEVELOPMENTS AND APPLICATIONS OF THE DIFFERENTIAL CALCULUS 1. Implicit Functions 2. Curves and Surfaces in Implicit Form 3. Systems of Functions, Transformations, and Mappings 4. Applications 5. Families of Curves, Families of Surfaces, and their Envelopes 6. Maxima and Minima APPENDIX 1. Sufficient Conditions for Extreme Values 2. Singular Points of Plane Curves 3. Singular Points of Surfaces 4. Connexion between Euler's and Lagrange's Representations of the Motion of a Fluid 5. Tangential Representation of a Closed Curve CHAPTER IV MULTIPLE INTEGRALS 1. Ordinary Integrals as Functions of a Parameter 2. The Integral of a Continuous Function over a Region of the Plane or of Space 3. Reduction of the Multiple Integral to Repeated Single Integrals 4. Transformation of Multiple Integrals 5. Improper Integrals 6. Geometrical Applications 7. Physical Applications APPENDIX 1. The Existence of the Multiple Integral 2. General Formula for the Area (or Volume) of a Region bounded by Segments of Straight Lines or Plane Areas (Guldin's Formula). The Polar Planimeter 3. Volumes and Areas in Space of any Number of Dimensions 4. Improper Integrals as Functions of a Parameter 5. The Fourier Integral 6. The Eulerian Integrals (Gamma Function) 7. Differentiation and Integration to Fractional Order. Abel's Integral Equation 8. Note on the Definition of the Area of a Curved Surface CHAPTER V INTEGRATION OVER REGIONS IN SEVERAL DIMENSIONS 1. Line Integrals 2. Connexion between Line Integrals and Double Integrals in the Plane. (The Integral Theorems of Gauss, Stokes, and Green) 3. Interpretation and Applications of the Integral Theorems for the Plane 4. Surface Integrals 5. Gauss's Theorem and Green's Theorem in Space 6. Stokes's Theorem in Space 7. The Connexion between Differentiation and Integration for Several Variables APPENDIX 1. Remarks on Gauss's Theorem and Stokes's Theorem 2. Representation of a Source-free Vector Field as a Curl CHAPTER VI DIFFERENTIAL EQUATIONS 1. The Differential Equations of the Motion of a Particle in Three Dimensions 2. Examples on the Mechanics of a Particle 3. Further Examples of Differential Equations 4. Linear Differential Equations 5. General Remarks on Differential Equations 6. The Potential of Attracting Charges 7. Further Examples of Partial Differential Equations CHAPTER VII CALCULUS OF VARIATIONS 1. Introduction 2. Euler's Differential Equation in the Simplest Case 3. Generalizations CHAPTER VIII FUNCTIONS OF A COMPLEX VARIABLE 1. Introduction 2. Foundations of the Theory of Functions of a Complex Variable 3. The Integration of Analytic Functions 4. Cauchy's Formula and its Applications 5. Applications to Complex Integration (Contour Integration) 6. Many-valued Functions and Analytic Extension SUPPLEMENT Real Numbers and the Concept of Limit Miscellaneous Examples Summary of Important Theorems and Formulae Answers and Hints Index