Mathematics for Scientists and Engineers
Preface
Acknowledgements
1
Foundations
1.1
Numbers
1.2
Algebra
1.2.1
BEDMAS
1.2.2
Notation for Multiplication and Division
1.2.3
Rules of Exponents
1.2.4
More on exponents
1.2.5
Brackets
1.2.6
Simplifiying Expressions
1.3
Equations
1.3.1
Formulas
1.3.2
Conditional equations
1.4
Inequalities
1.5
Common mistakes!
2
Functions and Graphs
2.1
Linear Functions
2.1.1
Parallel lines
2.1.2
Perpendicular lines
2.2
Polynomials
2.3
Rational functions
2.4
Trigonometric functions
2.5
Exponentials
2.6
Hyperbolic functions
2.7
Inverse Functions
2.7.1
Finding the inverse of a function
2.7.2
Root functions
2.7.3
Inverse Trigonometric Functions
2.7.4
Logarithms
2.7.5
Logarthmic plots
3
Solving Equations and Inequalities
3.1
Quadratic equations
3.1.1
Factorisation by inspection
3.1.2
Completing the square
3.1.3
The quadratic formula
3.2
Higher order polynomials
3.3
Simultaneous equations
3.3.1
Simultaneous linear equations
3.3.2
Non-linear simultaneous equations
3.4
Exponentials and Logarithms
3.4.1
Standard Bases
3.4.2
Change of Base for Exponentials
3.5
Inequalities
4
Trigonometry
4.1
Pythagoras
4.2
Degrees and radians
4.3
Trigonometric ratios
4.4
Sine and cosine rules
4.5
Angles in Cartesian Coordinates
4.6
Trigonometric waveforms
4.7
Trigonometric identities
4.7.1
Pythagorean identities
4.7.2
Compound angle formulae
4.7.3
Double angle formulae
4.7.4
Product to sum formulae
4.7.5
Sum to product formulae
5
Complex Numbers
5.1
\(i\)
and complex numbers
5.2
Complex arithmetic
5.3
The argument, polar and exponential form for complex numbers
5.4
Roots of complex numbers
5.5
\(e^{i\theta}\)
and trigonometric identities
6
Vectors
6.1
Vector addition
6.2
Scalar multiplication
6.3
Vectors in Cartesian coordinates
6.4
Vector products
6.4.1
Dot product
6.4.2
Cross product
7
Systems of Linear Equations
7.1
Lines and planes
7.2
Solving linear systems - Gaussian elimination
7.3
Echelon Form
8
Matrices
8.1
Solving linear systems revisited
8.2
Determinants
9
Linear Transformations
9.1
Matrix Algebra
9.1.1
Addition and Subtraction
9.1.2
Multiplication by a scalar
9.1.3
Matrix multiplication
9.1.4
Functional interpretations
9.1.5
Two special matrices
9.1.6
Some further properties
9.2
Matrix inverse
9.2.1
Finding the inverse of a
\(2\times 2\)
matrix
9.2.2
Finding the inverse of an
\(n\times n\)
matrix
10
Eigenvalues and Eigenvectors
10.1
Matrix Diagonalisation
10.2
Finding Eigenvalues
10.3
Finding Eigenvectors
10.4
Powers of diagonalisable matrices
11
Differentiation
11.1
Concept
11.2
Standard derivatives
11.3
Rules
11.4
Higher order derivatives
11.5
Further Techniques
11.5.1
Implicit Differentiation
11.5.2
Logarithmic Differentiation
11.5.3
Parametric Differentiation
12
Applications of Differentiation
12.1
Maxima and Minima
12.2
Points of Inflection
12.3
Higher Order Derivative Test
12.4
Graph Sketching
12.5
Optimisation
13
Sequences
13.1
Limits of Sequences
14
Series
14.1
Sigma Notation
14.2
Infinite Series
14.3
Power Series
14.4
Taylor Series
14.4.1
Linearisation
14.4.2
Taylor Polynomials
14.4.3
Taylor Series
15
Integration
15.1
Indefinite Integrals
15.1.1
Standard Integrals
15.2
Definite Integrals
16
Further Integration Techniques
16.1
Integration by Substitution
16.2
Integrals of trigonometric and hyperbolic functions
16.3
Integrals of rational functions using partial fractions
16.4
Integration by Parts
17
Differential Equations
17.1
Linear or Nonlinear ODEs
17.2
Separable First Order ODEs
17.3
Linear first order ODEs
17.4
Linear Second Order ODEs
17.4.1
Homogeneous equation with constant coefficients
17.4.2
Inhomogeneous equation with constant coefficients
18
Numerical Methods
18.1
Numerical Root Finding
18.2
Numerical differentiation
18.3
Numerical Integration
18.4
Numerical Solutions to ODEs
19
Probability Fundamentals
19.1
Sample space and events
19.2
Counting
19.3
Permutations and Combinations
19.3.1
Permutations
19.3.2
Combinations
19.4
Probabilities
19.5
Conditional Probability
19.6
Independence
19.7
Law of Total Probability
19.8
Bayes’ Theorem
20
Discrete Random Variables
20.1
Bernoulli trials
20.2
Binomial Distribution
20.3
Cumulative Distribution Functions
20.4
Poisson distribution
20.5
Expectation and Variance
20.5.1
Expectation
20.5.2
Variance and Standard Deviation
20.6
Summary of Discrete Distributions
21
Continuous Random Variables
21.1
Uniform distribution
21.2
Cumulative Distribution Function
21.3
Expectation and Variance
21.4
Normal Distribution
21.5
Standard Normal Distribution Tables
21.6
The Law of Large Numbers
21.7
Central Limit Theorem
22
Statistics
22.1
Frequency
22.1.1
Tables
22.1.2
Graphs
22.2
Measures of central tendency
22.2.1
Mean
22.2.2
Mode
22.2.3
Median
22.3
Measures of Dispersion
22.3.1
Range
22.3.2
Interquartile Range and Percentiles
22.3.3
Variance and Standard Deviation
Exercise Set 1
Exercise Set 2
Exercise Set 3
Exercise Set 4
Exercise Set 5
Exercise Set 6
Exercise Set 7
Exercise Set 8
Exercise Set 9
Exercise Set 10
Exercise Set 11
Exercise Set 12
Exercise Set 13
Exercise Set 14
Exercise Set 15
Exercise Set 16
Exercise Set 17
Exercise Set 18
Notation
References
Published with bookdown
Mathematics for Scientists and Engineers
References