By: Mary L. Boas
Edition: 3rd
Year: 2005
Now in its third edition, Mathematical Concepts in the Physical Sciences, 3rd Edition provides a comprehensive introduction to the areas of mathematical physics. It combines all the essential math concepts into one compact, clearly written reference.
This book is intended for students who have had a two-semester or three-semester introductory calculus course. Its purpose is to help students develop, in a short time, a basic competence in each of the many areas of mathematics needed in advanced courses in physics, chemistry, and engineering. Students are given sufficient depth to gain a solid foundation (this is not a recipe book). At the same time, they are not overwhelmed with detailed proofs that are more appropriate for students of mathematics. The emphasis is on mathematical methods rather than applications, but students are given some idea of how the methods will be used along with some simple applications.
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
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
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
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
