Maths for Chemists 2nd Edition by Martin Cockett, Graham Doggett ISBN 9781782624950 1782624953

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ISBN 10: 1782624953
ISBN 13: 9781782624950
Author: Martin Cockett, Graham Doggett

Maths for Chemists 2nd Edition Table of contents:

1 Numbers and Algebra

1.1 Real Numbers

1.1.1 Integers

1.1.2 Rational Numbers

1.1.3 Irrational Numbers

1.1.4 Decimal Numbers

1.1.4.1 Working with Decimal Numbers

1.1.4.2 Observations on Rounding

1.1.4.3 Rounding Errors

1.1.5 Combining Numbers

1.1.5.1 Powers or Indices

1.1.5.2 Rational Powers

1.1.5.3 Further Properties of Indices

1.1.6 Scientific Notation

1.1.6.1 Combining Numbers Given in Scientific Form

1.1.6.2 Names and Abbreviations for Powers of Ten

1.1.7 Relationships Between Numbers

1.1.7.1 Negative Numbers

1.1.7.2 Very Large and Very Small Numbers

1.1.7.3 Infinity

1.1.7.4 The Magnitude

1.2 Algebra

1.2.1 Generating a Formula for the Sum of the First n Positive Integers

1.2.2 Algebraic Manipulation

1.2.2.1 Dealing with Negative and Positive Signs

1.2.2.2 Working with Rational Expressions

1.2.3 Polynomials

1.2.3.1 Factorising a Polynomial

1.2.3.2 Forming a Common Denominator

1.2.4 Coping with Units

Summary of Key Points

2 Functions and Equations: Their Form and Use

2.1 Defining Functions

2.1.1 Functions in a Chemical Context

2.1.1.1 Understanding the Role of Units – The Mathematically Correct Approach

2.1.1.2 Creating a Function from a Formula

2.1.1.3 Understanding the Role of Units – The Pragmatic Approach

2.2 Representation of Functions

2.2.1 Tabular Representations of Functions of a Single Variable

2.2.2 Graphical Representations of Functions of a Single Variable

2.2.2.1 A Chemical Example of a Point Plot

2.2.2.2 Improving on the Boyle Model

2.2.3 Representing a Function in Terms of a Prescription

2.2.3.1 Prescription Functions in Chemistry

2.3 Some Special Mathematical Functions

2.3.1 Exponential Functions

2.3.1.1 Two Chemical Examples

2.3.2 Logarithm Functions

2.3.2.1 Defining the Logarithm Function

2.3.2.2 Properties of Logarithms

2.3.3 Trigonometric Functions

2.3.3.1 The Question of Angle

2.3.3.2 Angle Measure in S.I. Units

2.3.3.3 Sign Conventions for Angles and Trigonometric Functions

2.3.3.4 Special Values for Trigonometric Functions

2.3.3.5 Reciprocal Trigonometric Functions

2.3.3.6 Domains and Periodic Nature of Trigonometric Functions

2.3.3.7 Important Identities Involving Trigonometric Functions

2.3.3.8 Further Important Properties of Trigonometric Functions

2.3.4 Exponential Functions with Base e Revisited

2.3.4.1 Symmetric and Antisymmetric Functions

2.3.4.2 The Product Function x2e x

2.3.4.3 Product of a Polynomial or Trigonometric Function with an Exponential Function

2.3.4.4 A Chemical Example: The 3s Atomic Radial Wave Function for the Hydrogen Atom

2.3.5 Explicit and Implicit Functions

2.4 Equations

2.4.1 An Algebraic Method for Finding Roots of Polynomial Equations

2.4.2 Solving Polynomial Equations in a Chemical Context

2.4.2.1 Polynomial Equations of Higher Degree in Chemistry

2.4.2.2 Examples of Cubic Polynomial Equations in Physical Chemistry

Summary of Key Points

3 Limits

3.1 Mathematical and Chemical Examples

3.1.1 Point Discontinuities

3.1.2 Limiting Behaviour for Increasingly Large Positive or Negative Values of the Independent Variable

3.1.3 Limiting Behaviour for Increasingly Small Values of the Independent Variable

3.2 Defining the Limiting Process

3.2.1 Finding the Limit Intuitively

3.2.2 An Algebraic Method for Evaluating Limits

3.2.3 Evaluating Limits for Functions Whose Values Become Indeterminate

3.2.4 The Limiting Form of Functions of More Than One Variable

Summary of Key Points

4 Differentiation

4.1 The Average Rate of Change

4.2 The Instantaneous Rate of Change

4.2.1 Differentiation from First Principles

4.2.2 Differentiation by Rule

4.2.2.1 Some Standard Derivatives

4.2.2.2 An Introduction to the Concept of the Operator

4.2.3 Basic Rules for Differentiation

4.2.4 Chain Rule

4.3 Higher Order Derivatives

4.3.1 Operators Revisited: An Introduction to the Eigenvalue Problem

4.3.1.1 The Eigenvalue Problem

4.4 Maxima, Minima and Points of Inflection

4.4.1 Finding and Defining Stationary Points

4.4.1.1 Turning Points (Maxima and Minima)

4.4.1.2 Points of Inflection

4.5 The Differentiation of Functions of Two or More Variables

Summary of Key Points

5 Differentials

5.1 The Effects of Incremental Change

5.1.1 The Concept of Infinitesimal Change

5.1.1.1 The Origins of the Infinitesimal

5.1.2 Differentials in Action

5.2 The Differential of a Function of Two or More Variables

5.3 The Propagation of Errors

Summary of Key Points

6 Integration

6.1 Reversing the Effects of Differentiation

6.2 The Definite Integral

6.2.1 Finding the Area Under A Curve The Origin of Integral Calculus

6.2.1.1 Negative ‘Areas’

6.2.2 A Chemical Example: Where is the Electron in the Hydrogen Atom?

6.3 The Indefinite Integral

6.4 General Strategies for Solving More Complicated Integrals

6.4.1 Integration by Parts

6.4.2 Integration Using the Substitution Method

6.4.2.1 Use of Trigonometrical Substitutions

6.4.2.2 General Comment

6.5 The Connection Between the Definite and Indefinite Integral

Summary of Key Points

7 Differential Equations

7.1 Using the Derivative of a Function to Create a Differential Equation

7.2 Some Examples of Differential Equations Arising in Classical and Chemical Contexts

7.3 First Order Differential Equations

7.3.1 F(x, y) is Independent of y

7.3.2 Boundary Conditions

7.3.3 F(x, y) is in the Form F(x)G(y)

7.3.3.1 Separation of Variables Method

7.3.4 Separable First Order Differential Equations in Chemical Kinetics

7.3.5 First Order Differential Equations Linear in y

7.3.5.1 The Constant of Integration

7.3.6 First Order Differential Equations in Radioactive Decay Processes

7.3.7 First Order Differential Equations in Chemical Kinetics Processes

7.4 Second Order Differential Equations

7.4.1 Simple Harmonic Motion

7.4.2 Second Order Differential Equations with Constant Coefficients

7.4.3 How is an Eigenvalue Problem Recognised?

7.4.3.1 The Search for Eigenfunctions

Summary of Key Points

8 Power Series

8.1 Sequences

8.1.1 Finite Sequences

8.1.1.1 The Geometric Progression

8.1.1.2 Arithmetic Progression

8.1.2 Sequences of Indefinite Length

8.1.3 Functions Revisited

8.2 Finite Series

8.3 Infinite Series

8.3.1 π Revisited – The Rate of Convergence of an Infinite Series

8.3.2 Testing a Series for Convergence

8.3.2.1 The Ratio Test

8.3.2.2 The Infinite Geometric Series

8.4 Power Series as Representations of Functions

8.4.1 The Maclaurin Series: Expansion about the Point x = 0

8.4.1.1 The Maclaurin Series Expansion for ex

8.4.1.2 Truncating the Exponential Power Series

8.4.1.3 The Maclaurin Expansions of Trigononometric Functions

8.4.1.4 The Problem with Guessing the General Term: A Chemical Counter Example

8.4.2 The Taylor Series: Expansion about Points other than Zero

8.4.3 Manipulating Power Series

8.4.3.1 Combining Power Series

8.4.3.2 A Shortcut for Generating Maclaurin Series

8.4.4 The Relationship Between Domain and Interval of Convergence

8.4.5 Limits Revisited: Limiting Forms of Exponential and Trigonometric Functions

8.4.5.1 Exponential Functions

8.4.5.2 Trigonometric Functions

Summary of Key Points

9 Numbers Revisited: Complex Numbers

9.1 The Imaginary Number i

9.2 The General Form of Complex Numbers

9.3 Manipulation of Complex Numbers

9.3.1 Addition, Subtraction and Multiplication

9.3.2 The Complex Conjugate

9.3.3 Division of Complex Numbers

9.4 The Argand Diagram

9.4.1 The Modulus and Argument of z

9.5 The Polar Form for Complex Numbers

9.5.1 Euler’s Formula

9.5.1.1 The Number eiπ

9.5.2 Powers of Complex Numbers

9.5.3 The De Moivre Theorem

9.5.4 Complex Functions

9.5.4.1 The Periodicity of the Exponential Function

9.5.4.2 The Eigenvalue Problem Revisited

9.5.4.3 Structure Factors in Crystallography

9.5.5 Roots of Complex Numbers

Summary of Key Points

10 Working with Arrays I: Determinants

10.1 Origins – The Solution of Simultaneous Linear Equations

10.2 Expanding Determinants

10.3 Properties of Determinants

10.4 Strategies for Expanding Determinants Where n > 3

Summary of Key Points

11 Working with Arrays II: Matrices and Matrix Algebra

11.1 Introduction: Some Definitions

11.2 Rules for Combining Matrices

11.2.1 Multiplication of a Matrix by a Constant

11.2.2 Addition and Subtraction of Matrices

11.2.3 Matrix Multiplication

11.2.3.1 Properties of Matrix Multiplication

11.3 Origins of Matrices

11.3.1 Coordinate Transformations

11.3.1.1 Sequential Linear Transformations

11.3.2 Coordinate Transformations in Three Dimensions: A Chemical Example

11.4 Operations On Matrices

11.4.1 The Transpose of a Matrix

11.4.2 The Complex Conjugate Matrix

11.4.3 The Complex Conjugate Transposed Matrix

11.4.4 The Trace of a Square Matrix

11.4.5 The Matrix of Cofactors

11.5 The Determinant of a Product of Square Matrices

11.6 Special Matrices

11.6.1 The Null Matrix

11.6.2 The Unit Matrix

11.7 Matrices with Special Properties

11.7.1 Symmetric Matrices

11.7.2 Orthogonal Matrices

11.7.3 Singular Matrices

11.7.4 Hermitian Matrices

11.7.5 Unitary Matrices

11.7.6 The Inverse Matrix

11.8 Solving Sets of Linear Equations

11.8.1 Solution of Linear Equations: A Chemical Example

11.9 Molecular Symmetry and Group Theory

11.9.1 An Introduction to Group Theory

11.9.2 Groups of Matrices

11.9.3 Group Theory in Chemistry

Summary of Key Points

12 Vectors

12.1 The Geometric Construction of Vectors

12.1.1 Vectors in Two Dimensions

12.1.2 Conventions

12.2 Addition and Subtraction of Vectors

12.2.1 Vector Addition

12.2.2 Vector Subtraction

12.3 Base Vectors

12.3.1 The Magnitude of a Vector in Three Dimensional Space

12.3.2 Vector Addition, Subtraction and Scalar Multiplication Using Algebra

12.4 Multiplication of Vectors

12.4.1 Scalar Product of Two Vectors

12.4.1.1 Specifying the Angle θ

12.4.1.2 The Scalar Product in the Chemical Context

12.4.1.3 Scalar Products of Vectors Expressed in Terms of Base Vectors

12.4.1.4 Finding the Angle Between Two Vectors

12.4.1.5 Simple Application of the Scalar Product: the Cosine Law for a Triangle

12.4.2 Vector Product of Two Vectors

12.4.2.1 Vector Products in a Chemical Context

12.4.3 Area of a Parallelogram

12.5 Matrices and Determinants Revisited: Alternative Routes to Evaluating Scalar and Vector Products

12.5.1 The Scalar Product

12.5.2 The Vector Product

12.5.3 The Scalar Triple Product

Summary of Key Points

13 Simple Statistics and Error Analysis

13.1 Errors

13.1.1 Systematic Errors

13.1.2 Random Errors

13.2 The Statistics of Repeated Measurement

13.2.1 The Binomial Distribution

13.2.2 The Gaussian Distribution

13.2.2.1 The Mean, Variance and Standard Deviation

13.2.2.2 The Sample Variance and Sample Standard Deviation

13.2.2.3 Confidence Intervals

13.2.2.4 Standard Error of the Mean and Standard Error of the Sample Mean

13.2.2.5 Student t-Factors and the 95% Confidence Limit

13.3 Linear Regression Analysis

13.3.1 The Least Squares Method

13.3.1.1 Finding the Best Gradient and y Intercept

13.3.1.2 Finding the Uncertainties Associated with the Gradient and y Intercept

13.4 Propagation of Errors

13.4.1 Uncertainty in a Single Variable

13.4.2 Combining Uncertainties in More Than One Variable

Summary of Key Points

Worked Answers to Problems

Glossary

Subject Index

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