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• Copyright
• Preface
• To the Student
• Contents
• CHAPTER 1 Infinite Series, Power Series
• 1. THE GEOMETRIC SERIES
• 2. DEFINITIONS AND NOTATION
• 3. APPLICATIONS OF SERIES
• 4. CONVERGENT AND DIVERGENT SERIES
• 5. TESTING SERIES FOR CONVERGENCE; THE PRELIMINARY TEST
• 6. CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
• A. The Comparison Test
• B. The Integral Test
• C. The Ratio Test
• D. A Special Comparison Test
• 7. ALTERNATING SERIES
• 8. CONDITIONALLY CONVERGENT SERIES
• 9. USEFUL FACTS ABOUT SERIES
• 10. POWER SERIES; INTERVAL OF CONVERGENCE
• 11. THEOREMS ABOUT POWER SERIES
• 12. EXPANDING FUNCTIONS IN POWER SERIES
• 13. TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
• A. Multiplying a Series by a Polynomial or by Another Series
• B. Division of Two Series or of a Series by a Polynomial
• C. Binomial Series
• D. Substitution of a Polynomial or a Series for the Variable in Another Series
• E. Combination of Methods
• F. Taylor Series Using the Basic Maclaurin Series
• G. Using a Computer
• 14. ACCURACY OF SERIES APPROXIMATIONS
• 15. SOME USES OF SERIES
• 16. MISCELLANEOUS PROBLEMS
• CHAPTER 2 Complex Numbers
• 1. INTRODUCTION
• 2. REAL AND IMAGINARY PARTS OF A COMPLEX NUMBER
• 3. THE COMPLEX PLANE
• 4. TERMINOLOGY AND NOTATION
• 5. COMPLEX ALGEBRA
• A. Simplifying to x+iy form
• B. Complex Conjugate of a Complex Expression
• C. Finding the Absolute Value of z
• D. Complex Equations
• E. Graphs
• F. Physical Applications
• 6. COMPLEX INFINITE SERIES
• 7. COMPLEX POWER SERIES; DISK OF CONVERGENCE
• 8. ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
• 9. EULER’S FORMULA
• 10. POWERS AND ROOTS OF COMPLEX NUMBERS
• 11. THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
• 12. HYPERBOLIC FUNCTIONS
• 13. LOGARITHMS
• 14. COMPLEX ROOTS AND POWERS
• 15. INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS
• 16. SOME APPLICATIONS
• 17. MISCELLANEOUS PROBLEMS
• CHAPTER 3 Linear Algebra
• 1. INTRODUCTION
• 2. MATRICES; ROW REDUCTION
• 3. DETERMINANTS; CRAMER’S RULE
• 4. VECTORS
• 5. LINES AND PLANES
• 6. MATRIX OPERATIONS
• 7. LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
• 8. LINEAR DEPENDENCE AND INDEPENDENCE
• 9. SPECIAL MATRICES AND FORMULAS
• 10. LINEAR VECTOR SPACES
• 11. EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
• 12. APPLICATIONS OF DIAGONALIZATION
• 13. A BRIEF INTRODUCTION TO GROUPS
• 14. GENERAL VECTOR SPACES
• 15. MISCELLANEOUS PROBLEMS
• CHAPTER 4 Partial Differentiation
• 1. INTRODUCTION AND NOTATION
• 2. POWER SERIES IN TWO VARIABLES
• 3. TOTAL DIFFERENTIALS
• 4. APPROXIMATIONS USING DIFFERENTIALS
• 5. CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
• 6. IMPLICIT DIFFERENTIATION
• 7. MORE CHAIN RULE
• 8. APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
• 9. MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
• 10. ENDPOINT OR BOUNDARY POINT PROBLEMS
• 11. CHANGE OF VARIABLES
• 12. DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
• 13. MISCELLANEOUS PROBLEMS
• CHAPTER 5 Multiple Integrals; Applications of Integration
• 1. INTRODUCTION
• 2. DOUBLE AND TRIPLE INTEGRALS
• 3. APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
• 4. CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
• 5. SURFACE INTEGRALS
• 6. MISCELLANEOUS PROBLEMS
• CHAPTER 6 Vector Analysis
• 1. INTRODUCTION
• 2. APPLICATIONS OF VECTOR MULTIPLICATION
• 3. TRIPLE PRODUCTS
• 4. DIFFERENTIATION OF VECTORS
• 5. FIELDS
• 6. DIRECTIONAL DERIVATIVE; GRADIENT
• 7. SOME OTHER EXPRESSIONS INVOLVING ∇
• 8. LINE INTEGRALS
• 9. GREEN’S THEOREM IN THE PLANE
• 10. THE DIVERGENCE AND THE DIVERGENCE THEOREM
• 11. THE CURL AND STOKES’ THEOREM
• 12. MISCELLANEOUS PROBLEMS
• CHAPTER 7 Fourier Series and Transforms
• 1. INTRODUCTION
• 2. SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
• 3. APPLICATIONS OF FOURIER SERIES
• 4. AVERAGE VALUE OF A FUNCTION
• 5. FOURIER COEFFICIENTS
• 6. DIRICHLET CONDITIONS
• 8. OTHER INTERVALS
• 9. EVEN AND ODD FUNCTIONS
• 10. AN APPLICATION TO SOUND
• 11. PARSEVAL’S THEOREM
• 12. FOURIER TRANSFORMS
• 13. MISCELLANEOUS PROBLEMS
• CHAPTER 8 Ordinary Differential Equations
• 1. INTRODUCTION
• 2. SEPARABLE EQUATIONS
• 3. LINEAR FIRST-ORDER EQUATIONS
• 4. OTHER METHODS FOR FIRST-ORDER EQUATIONS
• 5. SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
• 6. SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
• 7. OTHER SECOND-ORDER EQUATIONS
• 8. THE LAPLACE TRANSFORM
• 9. SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
• 10. CONVOLUTION
• 11. THE DIRAC DELTA FUNCTION
• 12. A BRIEF INTRODUCTION TO GREEN FUNCTIONS
• 13. MISCELLANEOUS PROBLEMS
• CHAPTER 9 Calculus of Variations
• 1. INTRODUCTION
• 2. THE EULER EQUATION
• 3. USING THE EULER EQUATION
• 4. THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
• 5. SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
• 6. ISOPERIMETRIC PROBLEMS
• 7. VARIATIONAL NOTATION
• 8. MISCELLANEOUS PROBLEMS
• CHAPTER 10 Tensor Analysis
• 1. INTRODUCTION
• 2. CARTESIAN TENSORS
• 3. TENSOR NOTATION AND OPERATIONS
• 4. INERTIA TENSOR
• 5. KRONECKER DELTA AND LEVI-CIVITA SYMBOL
• 6. PSEUDOVECTORS AND PSEUDOTENSORS
• 7. MORE ABOUT APPLICATIONS
• 8. CURVILINEAR COORDINATES
• 9. VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
• 10. NON-CARTESIAN TENSORS
• 11. MISCELLANEOUS PROBLEMS
• CHAPTER 11 Special Functions
• 1. INTRODUCTION
• 2. THE FACTORIAL FUNCTION
• 3. DEFINITION OF THE GAMMA FUNCTION; RECURSION RELATION
• 4. THE GAMMA FUNCTION OF NEGATIVE NUMBERS
• 5. SOME IMPORTANT FORMULAS INVOLVING GAMMA FUNCTIONS
• 6. BETA FUNCTIONS
• 7. BETA FUNCTIONS IN TERMS OF GAMMA FUNCTIONS
• 8. THE SIMPLE PENDULUM
• 9. THE ERROR FUNCTION
• 10. ASYMPTOTIC SERIES
• 11. STIRLING’S FORMULA
• 12. ELLIPTIC INTEGRALS AND FUNCTIONS
• 13. MISCELLANEOUS PROBLEMS
• CHAPTER 12 Series Solutions of Differential Equations; Legendre, Bessel, Hermite, and Laguerre Funct
• 1. INTRODUCTION
• 2. LEGENDRE’S EQUATION
• 3. LEIBNIZ’ RULE FOR DIFFERENTIATING PRODUCTS
• 4. RODRIGUES’ FORMULA
• 5. GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
• 6. COMPLETE SETS OF ORTHOGONAL FUNCTIONS
• 7. ORTHOGONALITY OF THE LEGENDRE POLYNOMIALS
• 8. NORMALIZATION OF THE LEGENDRE POLYNOMIALS
• 9. LEGENDRE SERIES
• 10. THE ASSOCIATED LEGENDRE FUNCTIONS
• 11. GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
• 12. BESSEL’S EQUATION
• 13. THE SECOND SOLUTION OF BESSEL’S EQUATION
• 14. GRAPHS AND ZEROS OF BESSEL FUNCTIONS
• 15. RECURSION RELATIONS
• 16. DIFFERENTIAL EQUATIONS WITH BESSEL FUNCTION SOLUTIONS
• 17. OTHER KINDS OF BESSEL FUNCTIONS
• 18. THE LENGTHENING PENDULUM
• 19. ORTHOGONALITY OF BESSEL FUNCTIONS
• 20. APPROXIMATE FORMULAS FOR BESSEL FUNCTIONS
• 21. SERIES SOLUTIONS; FUCHS’S THEOREM
• 22. HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
• 23. MISCELLANEOUS PROBLEMS
• CHAPTER 13 Partial Differential Equations
• 1. INTRODUCTION
• 2. LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
• 3. THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHRODINGER EQUATION
• 4. THEWAVE EQUATION; THE VIBRATING STRING
• 5. STEADY-STATE TEMPERATURE IN A CYLINDER
• 6. VIBRATION OF A CIRCULAR MEMBRANE
• 7. STEADY-STATE TEMPERATURE IN A SPHERE
• 8. POISSON’S EQUATION
• 9. INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
• 10. MISCELLANEOUS PROBLEMS
• CHAPTER 14 Functions of a Complex Variable
• 1. INTRODUCTION
• 2. ANALYTIC FUNCTIONS
• 3. CONTOUR INTEGRALS
• 4. LAURENT SERIES
• 5. THE RESIDUE THEOREM
• 6. METHODS OF FINDING RESIDUES
• 7. EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
• 8. THE POINT AT INFINITY; RESIDUES AT INFINITY
• 9. MAPPING
• 10. SOME APPLICATIONS OF CONFORMAL MAPPING
• 11. MISCELLANEOUS PROBLEMS
• CHAPTER 15 Probability and Statistics
• 1. INTRODUCTION
• 2. SAMPLE SPACE
• 3. PROBABILITY THEOREMS
• 4. METHODS OF COUNTING
• 5. RANDOM VARIABLES
• 6. CONTINUOUS DISTRIBUTIONS
• 7. BINOMIAL DISTRIBUTION
• 8. THE NORMAL OR GAUSSIAN DISTRIBUTION
• 9. THE POISSON DISTRIBUTION
• 10. STATISTICS AND EXPERIMENTAL MEASUREMENTS
• 11. MISCELLANEOUS PROBLEMS
• References
• Answers to Selected Problems
• Index