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Summary: Now in its third edition, Mathematical Concepts in the Physical Sciences provides a comprehensive introduction to the areas of mathematical physics. It combines all the essential math concepts into one compact, clearly written reference.Edition/Copyright: 3RD 06
Chapter 1: Infinite Series, Power Series.
The Geometric Series.
Definitions and Notation.
Applications of Series.
Convergent and Divergent Series.
Convergence Tests for Series of Positive Terms.
Conditionally Convergent Series.
Useful Facts about Series.
Power Series; Interval of Convergence.
Theorems about Power Series.
Expanding Functions in Power Series.
Accuracy of Series Approximations.
Some Uses of Series.
Chapter 2: Complex Numbers.
Real and Imaginary Parts of a Complex Number.
The Complex Plane.
Terminology and Notation.
Complex Infinite Series.
Complex Power Series; Disk of Convergence.
Elementary Functions of Complex Numbers.
Powers and Roots of Complex Numbers.
The Exponential and Trigonometric Functions.
Complex Roots and Powers.
Inverse Trigonometric and Hyperbolic Functions.
Chapter 3: Linear Algebra.
Matrices; Row Reduction.
Determinants; Cramer's Rule.
Lines and Planes.
Linear Combinations, Functions, Operators.
Linear Dependence and Independence.
Special Matrices and Formulas.
Linear Vector Spaces.
Eigenvalues and Eigenvectors.
Applications of Diagonalization.
A Brief Introduction to Groups.
General Vector Spaces.
Chapter 4: Partial Differentiation.
Introduction and Notation.
Power Series in Two Variables.
Approximations using Differentials.
More Chain Rule.
Maximum and Minimum Problems.
Constraints; Lagrange Multipliers.
Endpoint or Boundary Point Problems.
Change of Variables.
Differentiation of Integrals.
Chapter 5: Multiple Integrals.
Double and Triple Integrals.
Applications of Integration.
Change of Variables in Integrals; Jacobians.
Chapter 6: Vector Analysis.
Applications of Vector Multiplication.
Differentiation of Vectors.
Directional Derivative; Gradient.
Some Other Expressions Involving V.
Green's Theorems in the Plane.
The Divergence and the Divergence Theorem.
The Curl and Stokes' Theorem.
Chapter 7: Fourier Series and Transforms.
Simple Harmonic Motion and Wave Motion; Periodic Functions.
Applications of Fourier Series.
Average Value of a Function.
Complex Form of Fourier Series.
Even and Odd Functions.
An Application to Sound.
Chapter 8: Ordinary Differential Equations.
Linear First-Order Equations.
Other Methods for First-Order Equations.
Linear Equations (Zero Right-Hand Side).
Linear Equations (Nonzero Right-Hand Side).
Other Second-Order Equations.
The Laplace Transform.
Laplace Transform Solutions.
The Dirac Delta Function.
A Brief Introduction to Green's Functions.
Chapter 9: Calculus of Variations.
The Euler Equation.
Using the Euler Equation.
The Brachistochrone Problem; Cycloids.
Several Dependent Variables; Lagrange's Equations.
Chapter 10: Tensor Analysis.
Tensor Notation and Operations.
Kronecker Delta and Levi-Civita Symbol.
Pseudovectors and Pseudotensors.
More about Applications.
Chapter 11: Special Functions.
The Factorial Function.
Gamma Function; Recursion Relation.
The Gamma Function of Negative Numbers.
Formulas Involving Gamma Functions.
Beta Functions in Terms of Gamma Functions.
The Simple Pendulum.
The Error Function.
Elliptic Integrals and Functions.
Chapter 12: Legendre, Bessel, Hermite, and Laguerre functions.
Leibniz' Rule for Differentiating Products.
Generating Function for Legendre Polynomials.
Complete Sets of Orthogonal Functions.
Orthogonality of Legendre Polynomials.
Normalization of Legendre Polynomials.
The Associated Legendre Polynomials.
Generalized Power Series or the Method of Frobenius.
The Second Solutions of Bessel's Equation.
Graphs and Zeros of Bessel Functions.
Differential Equations with Bessel Function Solutions.
Other Kinds of Bessel Functions.
The Lengthening Pendulum.
Orthogonality of Bessel Functions.
Approximate Formulas of Bessel Functions.
Series Solutions; Fuch's Theorem.
Hermite and Laguerre Functions; Ladder Operators.
Chapter 13: Partial Differential Equations.
Laplace's Equation; Steady-State Temperature.
The Diffusion of Heat Flow Equation; the Schrodinger Equation.
The Wave Equation; the Vibrating String.
Steady-State Temperature in a Cylinder.
Vibration of a Circular Membrane.
Steady-State Temperature in a Sphere.
Integral Transform Solutions of Partial Differential Equations.
Chapter 14: Functions of a Complex Variable.
The Residue Theorem.
Methods of Finding Residues.
Evaluation of Definite Integrals.
The Point at Infinity; Residues of Infinity.
Some Applications of Conformal Mapping.
Chapter 15: Probability and Statistics.
Methods of Counting.
The Normal or Gaussian Distribution.
The Poisson Distribution.
Statistics and Experimental Measurements.
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