Half Title -- Title Page -- Copyright -- Dedication -- Contents -- Preface -- 1 Problem Solving and Numerical Mathematics -- 1.1 Problem Solving -- 1.2 Numbers and Measurements -- 1.3 Numerical Mathematical Operations -- 1.3.1 Binary Arithmetic Operations -- 1.3.2 Additional Numerical Operations -- 1.4 Units of Measurement -- 1.5 The Factor-Label Method -- 1.6 Measurements, Accuracy, and Significant Digits -- 1.6.1 Scientific Notation -- 1.6.2 Rounding -- 1.6.3 Significant Digits in a Calculated Quantity -- 2 Mathematical Functions -- 2.1 Mathematical Functions in Physical Chemistry -- 2.1.1 Functions in Thermodynamics -- 2.1.2 Functions in Quantum Mechanics -- 2.1.3 Function Notation -- 2.1.4 Continuity -- 2.1.5 Graphs of Functions -- 2.2 Important Families of Functions -- 2.2.1 Linear Functions -- 2.2.2 Quadratic Functions -- 2.2.3 Cubic Functions -- 2.2.4 Logarithms -- 2.2.5 Exponentials -- 2.2.6 Trigonometric Functions -- 2.2.7 Inverse Trigonometric Functions -- 2.2.8 Hyperbolic Trigonometric Functions -- 2.2.9 Significant Digits in Logarithms, Exponentials, and Trigonometric Functions -- 2.3 Generating Approximate Graphs -- 3 Problem Solving and Symbolic Mathematics: Algebra -- 3.1 The Algebra of Real Scalar Variables -- 3.2 Coordinate Systems in Two Dimensions -- 3.3 Coordinate Systems in Three Dimensions -- 3.3.1 Cartesian Coordinates -- 3.3.2 Spherical Polar Coordinates -- 3.3.3 Cylindrical Polar Coordinates -- 3.4 Imaginary and Complex Numbers -- 3.4.1 Mathematical Operations with Complex Numbers -- 3.4.2 The Argand Diagram -- 3.4.3 The Complex Conjugate -- 3.4.4 The Magnitude of a Complex Quantity -- 3.4.5 Roots of a Complex Number -- 3.5 Problem Solving and Symbolic Mathematics -- 4 Vectors and Vector Algebra -- 4.1 Vectors in Two Dimensions -- 4.1.1 The Sum and Difference of Two Vectors -- 4.1.2 The Product of a Vector and a Scalar.

4.1.3 Unit Vectors -- 4.1.4 The Scalar Product of Two Vectors -- 4.1.5 The Magnitude of a Vector -- 4.2 Vectors in Three Dimensions -- 4.2.1 Unit Vectors in Three Dimensions -- 4.2.2 The Magnitude of a Vector -- 4.2.3 The Sum and Difference of Two Vectors -- 4.2.4 The Product of a Scalar and a Vector -- 4.2.5 The Scalar Product of Two Vectors -- 4.2.6 The Vector Product of Two Vectors -- 4.3 Physical Examples of Vector Products -- 4.3.1 Magnetic Force -- 4.3.2 Electrostatic Force -- 4.3.3 Angular Momentum -- 5 Problem Solving and the Solution of Algebraic Equations -- 5.1 Algebraic Methods for Solving One Equation with One Unknown -- 5.1.1 Polynomial Equations -- 5.1.2 Approximate Solutions to Equations -- 5.2 Numerical Solution of Algebraic Equations -- 5.2.1 Graphical Solution of Algebraic Equations -- 5.2.2 Trial and Error -- 5.2.3 The Method of Bisection -- 5.2.4 Solving Equations Numerically with Excel -- 5.3 A Brief Introduction to Mathematica -- 5.3.1 Numerical Calculations with Mathematica -- 5.3.2 Symbolic Algebra with Mathematica -- 5.3.3 Solving Equations with Mathematica -- 5.3.4 Graphing with Mathematica -- 5.4 Simultaneous Equations: Two Equations with Two Unknowns -- 5.4.1 The Method of Substitution -- 5.4.2 The Method of Elimination -- 5.4.3 Consistency and Independence in Simultaneous Equations -- 5.4.4 Homogeneous Linear Equations -- 5.4.5 Using Mathematica to Solve Simultaneous Equations -- 6 Differential Calculus -- 6.1 The Tangent Line and the Derivative of a Function -- 6.1.1 The Derivative -- 6.1.2 Derivatives of Specific Functions -- 6.2 Differentials -- 6.3 Some Useful Derivative Identities -- 6.3.1 The Derivative of a Constant -- 6.3.2 The Derivative of a Function Times a Constant -- 6.3.3 The Derivative of a Product of Two Functions -- 6.3.4 The Derivative of the Sum of Two Functions.

6.3.5 The Derivative of the Difference of Two Functions -- 6.3.6 The Derivative of the Quotient of Two Functions -- 6.3.7 The Derivative of a Function of a Function (the Chain Rule) -- 6.4 Newton's Method -- 6.5 Higher-Order Derivatives -- 6.5.1 The Curvature of a Function -- 6.6 Maximum-Minimum Problems -- 6.7 Limiting Values of Functions -- 6.8 l'Hôpital's Rule -- 7 Integral Calculus -- 7.1 The Antiderivative of a Function -- 7.1.1 Position, Velocity, and Acceleration -- 7.2 The Process of Integration -- 7.2.1 The Definite Integral as an Area -- 7.2.2 Facts about Integrals -- 7.2.3 Derivatives of Definite Integrals -- 7.3 Tables of Indefinite Integrals -- 7.4 Improper Integrals -- 7.5 Techniques of Integration -- 7.5.1 The Method of Substitution -- 7.5.2 Integration by Parts -- 7.5.3 The Method of Partial Fractions -- 7.5.4 Integration with Mathematica -- 7.6 Numerical Integration -- 7.6.1 The Bar-Graph Approximation -- 7.6.2 The Trapezoidal Approximation -- 7.6.3 Simpson's Rule -- 7.6.4 Numerical Integration with Mathematica -- 8 Differential Calculus with Several Independent Variables -- 8.1 Functions of Several Independent Variables -- 8.2 Changes in a Function of Several Variables, Partial Derivatives -- 8.2.1 Differentials -- 8.3 Change of Variables -- 8.4 Useful Partial Derivative Identities -- 8.4.1 The Variable-Change Identity -- 8.4.2 The Reciprocal Identity -- 8.4.3 The Euler Reciprocity Relation -- 8.4.4 The Maxwell Relations -- 8.4.5 The Cycle Rule -- 8.4.6 The Chain Rule -- 8.5 Thermodynamic Variables Related to Partial Derivatives -- 8.6 Exact and Inexact Differentials -- 8.6.1 Integrating Factors -- 8.7 Maximum and Minimum Values of Functions of Several Variables -- 8.7.1 Constrained Maximum/Minimum Problems.

8.7.2 Lagrange's Method of Undetermined MultipliersNamed for Joseph Louis Lagrange (born Guisepps Lodovico Lagrangia), 1736-1813, French-Italian physicist and mathematician. -- 8.8 Vector Derivative Operators -- 8.8.1 Vector Derivatives in Cartesian Coordinates -- 8.8.2 Vector Derivatives in Other Coordinate Systems -- 9 Integral Calculus with Several Independent Variables -- 9.1 Line Integrals -- 9.1.1 Line Integrals of Exact Differentials -- 9.1.2 Line Integrals of Inexact Differentials -- 9.1.3 Line Integrals with Three Integration Variables -- 9.1.4 Line Integrals in Thermodynamics -- 9.2 Multiple Integrals -- 9.2.1 Double Integrals -- 9.2.2 The Double Integral Representing a Volume -- 9.2.3 Triple Integrals -- 9.2.4 Changing Variables in Multiple Integrals -- 10 Mathematical Series -- 10.1 Constant Series -- 10.1.1 Some Convergent Constant Series -- 10.1.2 The Geometric Series -- 10.1.3 The Harmonic Series -- 10.1.4 Tests for Convergence -- 10.2 Power Series -- 10.2.1 Maclaurin Series -- 10.2.2 Taylor Series -- 10.2.3 The Convergence of Power Series -- 10.2.4 Power Series in Physical Chemistry -- 10.3 Mathematical Operations on Series -- 10.4 Power Series with More than One Independent Variable -- 11 Functional Series and Integral Transforms -- 11.1 Fourier Series -- 11.1.1 Finding the Coefficients of a Fourier Series-Orthogonality -- 11.1.2 Fourier Series with Complex Exponential Basis Functions -- 11.2 Other Functional Series with Orthogonal Basis Sets -- 11.2.1 Hilbert Space -- 11.2.2 Determining the Expansion Coefficients -- 11.3 Integral Transforms -- 11.3.1 Fourier Transforms (Fourier Integrals) -- 11.3.2 Laplace Transforms -- 12 Differential Equations -- 12.1 Differential Equations and Newton's Laws of Motion -- 12.2 Homogeneous Linear Differential Equations with Constant Coefficients -- 12.2.1 The Harmonic Oscillator.

12.2.2 The Damped Harmonic Oscillator-A Nonconservative System -- 12.3 Inhomogeneous Linear Differential Equations: The Forced Harmonic Oscillator -- 12.3.1 Variation of Parameters Method -- 12.4 Differential Equations with Separable Variables -- 12.5 Exact Differential Equations -- 12.6 Solution of Inexact Differential Equations Using Integrating Factors -- 12.7 Partial Differential Equations -- 12.7.1 Waves in a String -- 12.7.2 Solution by Separation of Variables -- 12.7.3 The Schrödinger Equation -- 12.8 Solution of Differential Equations using Laplace Transforms -- 12.9 Numerical Solution of Differential Equations -- 12.9.1 Euler's Method -- 12.9.2 The Runge-Kutta Method -- 12.9.3 Solution of Differential Equations with Mathematica -- 13 Operators, Matrices, and Group Theory -- 13.1 Mathematical Operators -- 13.1.1 Eigenfunctions and Eigenvalues -- 13.1.2 Operator Algebra -- 13.1.3 Operators in Quantum Mechanics -- 13.2 Symmetry Operators -- 13.3 The Operation of Symmetry Operators on Functions -- 13.4 Matrix Algebra -- 13.4.1 The Equality of Two Matrices -- 13.4.2 The Sum of Two Matrices -- 13.4.3 The Product of a Scalar and a Matrix -- 13.4.4 The Product of Two Matrices -- 13.4.5 The Identity Matrix -- 13.4.6 The Inverse of a Matrix -- 13.4.7 Matrix Terminology -- 13.5 Determinants -- 13.6 Matrix Algebra with Mathematica -- 13.7 An Elementary Introduction to Group Theory -- 13.8 Symmetry Operators and Matrix Representations -- 14 The Solution of Simultaneous Algebraic Equations with More than Two Unknowns -- 14.1 Cramer's Rule -- 14.2 Linear Dependence and Inconsistency -- 14.3 Solution by Matrix Inversion -- 14.4 Gauss-Jordan Elimination -- 14.5 Linear Homogeneous Equations -- 14.6 Matrix Eigenvalues and Eigenvectors -- 14.7 The Use of Mathematica to Solve Simultaneous Equations.

14.8 The Use of Mathematica to Find Matrix Eigenvalues and Eigenvectors.