This book is aimed at students who encounter mathematical models in other disciplines.

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Description

Content

This book is aimed at students who encounter mathematical models in other disciplines. It assumes some knowledge of calculus, and explains the tools and concepts for analysing models involving sets of either algebraic or 1st order differential equations. The text emphasises commonalities between these modelling approaches.

The approach is practical, aiming at insight to understand the mathematical principles, but recognising that real world modelling uses computer algebra software. Hands on exploration is supported by giving software commands interspersed in the text, as well as their output.

- Introduction Mathematical Modelling
- What is a mathematical model?
- Using mathematical models
- Types of models
- How is this book useful for modelling?

- Simultaneous Linear Equations
- Introduction
- Matrices
- Applying matrices to simultaneous equations
- Determinants
- Inverting a Matrix by Elementary Row Operations
- Solving Equations by Elementary Row Operations
- Homogeneous and Non-homogeneous equations

- Matrices in Geometry
- Reflection
- Shear
- Plane Rotation
- Orthogonal and orthonormal vectors
- Geometric addition of vectors
- Matrices and vectors as objects

- Eigenvalues and Diagonalization
- Linear superpositions of vectors
- Calculating Eigenvalues and Eigenvectors
- Similar matrices and diagonalisation
- How eigenvalues relate to determinants
- Using diagonalisation to decouple linear equations
- Orthonormal eigenvectors
- Summary: eigenvalues, eigenvectors and diagonalisation.

- Revision: Calculus Results
- Differentiation formulas
- Rules of Differentiation
- Integration Formulas
- Integration Methods

- First Order Differential Equations
- Introduction
- Initial value problems
- Classifying First Order Differential Equations
- Separation of variables
- General Method for solving LFODE’s.
- Applications to modelling real world problems
- Characterising Solutions Using a Phase Line
- Variation of Parameters method
- The Main Points Again – A stepwise strategy for solving FODE’s.

- General Properties of Solutions to Differential Equations
- Introduction
- Homogenous Linear Equations

- Systems of Linear Differential Equations
- Introduction
- Homogenous Systems
- The Fundamental Matrix
- Repeated Eigenvalues
- Non-homogenous systems

- Appendix: Complex Numbers
- Representing complex numbers
- Algebraic operations
- Euler’s formula
- Log, Exponential and Hyperbolic functions
- Differentiation Formulae