This compact textbook provides a foundation in mathematics for STEM students entering university. The book helps students from different disciplines and backgrounds make the transition to university. Based on the author’s teaching for many years, the book can be used as a textbook and a resource for lecturers and professors. Its accessibility is such that it is can also be used by students in their final year in school before university and help them continue their mathematical studies at college. The book is designed so that students will return to the book repeatedly as their undergraduate careers progress. Although compact and concise, it loses no rigour. All the topics are carefully explained meaningfully, not just presented as a set of rules or rote-learned procedures.
Author(s): Philip Prewett
Edition: 1
Publisher: Springer
Year: 2022
Language: English
Pages: 125
Tags: Trigonometry; Real and Complex Numbers; Vector Algebra; Matrices; Differentiation; Integration; Functional Analysis; Conic Sections; 1st Order Differential Equations; 2nd Order Differential Equations; Gaussian Statistics
Preface
Acknowledgements
Contents
1 Trigonometry
Philip Prewett
1.1 Trigonometric Relationships
1.2 Pythagoras' Theorem
1.3 The Sine and Cosine Rules
1.3.1 Sine Rule
1.3.2 Cosine Rule
1.4 Compound Angles
1.5 Derivatives of the Circular Functions
2 Real and Complex Numbers
Philip Prewett
2.1 Surds
2.2 Solution of Quadratic Equations
2.2.1 Equations Which Can Be Factorized
2.2.2 Equations Which Cannot Be Factorized
2.3 Complex Numbers
2.4 Euler's Theorem
2.4.1 De Moivre's Theorem
2.5 Phase Ambiguity
3 Vector Algebra
Philip Prewett
3.1 Addition of Vectors
3.2 Subtraction of Vectors
3.3 Multiplication of Vectors
3.3.1 Mutually perp Vectors
3.4 Direction Cosines
3.5 Position Vector r
3.6 Vector/Cross Product
3.6.1 Cartesian Form of Vector Product
3.7 Applications of Vectors
3.7.1 Work Done by a Force on a Body
3.7.2 Moment of Force (Torque)
4 Matrices
Philip Prewett
4.1 Basic Matrices
4.1.1 Addition and Subtraction of Matrices
4.1.2 Multiplying a Matrix by a Scalar
4.1.3 Multiplication of Matrices
4.1.4 Square Matrices
4.1.5 Transpose of a Matrix AT
4.2 Origins of Matrices
4.2.1 Some More 2D Matrix Operations
4.3 Inverse of a Matrix A-1
4.3.1 Minor of a Matrix Element
4.3.2 Cofactor of a Matrix Element
4.3.3 An Application of Minors and Cofactors
4.3.4 Adjugate or Adjoint Matrix
4.4 Inverse of a 3times3 Matrix
4.5 Matrices for Simultaneous Equations
4.6 Eigenvalues and Eigenvectors
4.7 Matrices in System Stability
4.7.1 Note on |A-sI|=0
5 Differentiation
Philip Prewett
5.1 The Chain Rule
5.2 Logarithmic Functions
5.3 The Exponential Function
5.4 Function of a Function
6 Integration
Philip Prewett
6.1 The Meaning of Integration
6.2 Integration as the Inverse of Differentiation
6.3 Integration of the Circular Functions
6.4 Integration by Parts
6.5 Integration by Substitution
6.6 Method of Partial Fractions
6.7 Some Special Integrals
6.8 Calculation of Areas
7 Functional Analysis
Philip Prewett
7.1 Maxima, Minima and Point of Inflection
7.2 Discontinuities
7.3 Tangent, Normal and Curvature
7.4 The Binomial Theorem
7.4.1 Understanding the Binomial Theorem
7.5 Taylor and McLaurin Series
7.5.1 McLaurin Series for Key Functions
7.6 Example of Series Approximation
7.7 Series Method for intcosx2dx
7.8 The Hyperbolic Functions
7.8.1 Hyperbolic Functions in Integration
8 The Conic Sections
Philip Prewett
8.1 Straight Line
8.2 Circle
8.3 Ellipse
8.4 Parabola
8.5 Hyperbola
8.6 Rectangular Hyperbola
8.7 Summary of Canonical Forms
8.8 Perimeter, Area and Volume
8.8.1 Right Angled Triangle
8.8.2 Circle
8.8.3 Sphere
8.8.4 Ellipse
9 1st Order Differential Equations
Philip Prewett
9.1 Summary of General Methods
9.2 Capacitor Discharge
9.3 Understanding Exponential Growth
10 2nd Order Differential Equations
Philip Prewett
10.1 General Solution
10.2 Case of Repeated Roots
10.3 Case of Complex Roots
10.4 2nd Order ODEs in Applications
11 Gaussian Statistics
Philip Prewett
11.1 Basic Statistics
11.2 Continuous Distribution
11.3 Standard Normal Form
11.4 Confidence Intervals
11.5 The Error Function erf z
Glossary
Glossary
