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The Chemistry Maths Book, Second Edition

(Grace) #1

  • 1 Numbers, variables, and units

    • 1.1 Concepts

    • 1.2 Real numbers

    • 1.3 Factorization, factors, and factorials

    • 1.4 Decimal representation of numbers

    • 1.5 Variables

    • 1.6 The algebra of real numbers

    • 1.7 Complex numbers

    • 1.8 Units

    • 1.9 Exercises



  • 2 Algebraic functions

    • 2.1 Concepts

    • 2.2 Graphical representation of functions

    • 2.3 Factorization and simplification of expressions

    • 2.4 Inverse functions

    • 2.5 Polynomials

    • 2.6 Rational functions

    • 2.7 Partial fractions

    • 2.8 Solution of simultaneous equations

    • 2.9 Exercises



  • 3 Transcendental functions

    • 3.1 Concepts

    • 3.2 Trigonometric functions

    • 3.3 Inverse trigonometric functions

    • 3.4 Trigonometric relations

    • 3.5 Polar coordinates

    • 3.6 The exponential function

    • 3.7 The logarithmic function

    • 3.8 Values of exponential and logarithmic functions

    • 3.9 Hyperbolic functions

    • 3.10 Exercises



  • 4 Differentiation

    • 4.1 Concepts

    • 4.2 The process of differentiation

    • 4.3 Continuity

    • 4.4 Limits

    • 4.5 Differentiation from first principles

    • 4.6 Differentiation by rule

    • 4.7 Implicit functions

    • 4.8 Logarithmic differentiation viii Contents

    • 4.9 Successive differentiation

    • 4.10 Stationary points

    • 4.11 Linear and angular motion

    • 4.12 The differential

    • 4.13 Exercises



  • 5 Integration

    • 5.1 Concepts

    • 5.2 The indefinite integral

    • 5.3 The definite integral

    • 5.4 The integral calculus

    • 5.5 Uses of the integral calculus

    • 5.6 Static properties of matter

    • 5.7 Dynamics

    • 5.8 Pressure–volume work

    • 5.9 Exercises



  • 6 Methods of integration

    • 6.1 Concepts

    • 6.2 The use of trigonometric relations

    • 6.3 The method of substitution

    • 6.4 Integration by parts

    • 6.5 Reduction formulas

    • 6.6 Rational integrands. The method of partial fractions

    • 6.7 Parametric differentiation of integrals

    • 6.8 Exercises



  • 7 Sequences and series

    • 7.1 Concepts

    • 7.2 Sequences

    • 7.3 Finite series

    • 7.4 Infinite series

    • 7.5 Tests of convergence

    • 7.6 MacLaurin and Taylor series

    • 7.7 Approximate values and limits

    • 7.8 Operations with power series

    • 7.9 Exercises



  • 8 Complex numbers

    • 8.1 Concepts

    • 8.2 Algebra of complex numbers

    • 8.3 Graphical representation

    • 8.4 Complex functions

    • 8.5 Euler’s formula

    • 8.6 Periodicity

      • 8.7 Evaluation of integrals Contents ix

      • 8.8 Exercises



    • 9 Functions of several variables

      • 9.1 Concepts

      • 9.2 Graphical representation

      • 9.3 Partial differentiation

      • 9.4 Stationary points

      • 9.5 The total differential

      • 9.6 Some differential properties

      • 9.7 Exact differentials

      • 9.8 Line integrals

      • 9.9 Multiple integrals

      • 9.10 The double integral

      • 9.11 Change of variables

      • 9.12 Exercises





  • 10 Functions in 3 dimensions

    • 10.1 Concepts

    • 10.2 Spherical polar coordinates

    • 10.3 Functions of position

    • 10.4 Volume integrals

    • 10.5 The Laplacian operator

    • 10.6 Other coordinate systems

    • 10.7 Exercises



  • 11 First-order differential equations

    • 11.1 Concepts

    • 11.2 Solution of a differential equation

    • 11.3 Separable equations

    • 11.4 Separable equations in chemical kinetics

    • 11.5 First-order linear equations

    • 11.6 An example of linear equations in chemical kinetics

    • 11.7 Electric circuits

    • 11.8 Exercises



  • 12 Second-order differential equations. Constant coefficients

    • 12.1 Concepts

    • 12.2 Homogeneous linear equations

    • 12.3 The general solution

    • 12.4 Particular solutions

    • 12.5 The harmonic oscillator

    • 12.6 The particle in a one-dimensional box

    • 12.7 The particle in a ring

    • 12.8 Inhomogeneous linear equations

    • 12.9 Forced oscillations

    • 12.10 Exercises

    • special functions 13 Second-order differential equations. Some

    • 13.1 Concepts

    • 13.2 The power-series method

    • 13.3 The Frobenius method

    • 13.4 The Legendre equation

    • 13.5 The Hermite equation

    • 13.6 The Laguerre equation

    • 13.7 Bessel functions

    • 13.8 Exercises



  • 14 Partial differential equations

    • 14.1 Concepts

    • 14.2 General solutions

    • 14.3 Separation of variables

    • 14.4 The particle in a rectangular box

    • 14.5 The particle in a circular box

    • 14.6 The hydrogen atom

    • 14.7 The vibrating string

    • 14.8 Exercises



  • 15 Orthogonal expansions. Fourier analysis

    • 15.1 Concepts

    • 15.2 Orthogonal expansions

    • 15.3 Two expansions in Legendre polynomials

    • 15.4 Fourier series

    • 15.5 The vibrating string

    • 15.6 Fourier transforms

    • 15.7 Exercises



  • 16 Vectors

    • 16.1 Concepts

    • 16.2 Vector algebra

    • 16.3 Components of vectors

    • 16.4 Scalar differentiation of a vector

    • 16.5 The scalar (dot) product

    • 16.6 The vector (cross) product

    • 16.7 Scalar and vector fields

    • 16.8 The gradient of a scalar field

    • 16.9 Divergence and curl of a vector field

    • 16.10 Vector spaces

    • 16.11 Exercises



  • 17 Determinants

    • 17.1 Concepts

    • 17.2 Determinants of order

    • 17.3 The general case

    • 17.4 The solution of linear equations Contents xi

    • 17.5 Properties of determinants

    • 17.6 Reduction to triangular form

    • 17.7 Alternating functions

    • 17.8 Exercises



  • 18 Matrices and linear transformations

    • 18.1 Concepts

    • 18.2 Some special matrices

    • 18.3 Matrix algebra

    • 18.4 The inverse matrix

    • 18.5 Linear transformations

    • 18.6 Orthogonal matrices and orthogonal transformations

    • 18.7 Symmetry operations

    • 18.8 Exercises



  • 19 The matrix eigenvalue problem

    • 19.1 Concepts

    • 19.2 The eigenvalue problem

    • 19.3 Properties of the eigenvectors

    • 19.4 Matrix diagonalization

    • 19.5 Quadratic forms

    • 19.6 Complex matrices

    • 19.7 Exercises



  • 20 Numerical methods

    • 20.1 Concepts

    • 20.2 Errors

    • 20.3 Solution of ordinary equations

    • 20.4 Interpolation

    • 20.5 Numerical integration

    • 20.6 Methods in linear algebra

    • 20.7 Gauss elimination for the solution of linear equations

    • 20.8 Gauss–Jordan elimination for the inverse of a matrix

    • 20.9 First-order differential equations

    • 20.10 Systems of differential equations

    • 20.11 Exercises



  • 21 Probability and statistics

    • 21.1 Concepts

    • 21.2 Descriptive statistics

    • 21.3 Frequency and probability

    • 21.4 Combinations of probabilities

    • 21.5 The binomial distribution

    • 21.6 Permutations and combinations

    • 21.7 Continuous distributions

    • 21.8 The Gaussian distribution



  • 21.9 More than one variable xii Contents

  • 21.10 Least squares

  • 21.11 Sample statistics

  • 21.12 Exercises

  • Appendix. Standard integrals

  • Solutions to exercises

  • Index

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