Mathematics 1

Overview

Complex Arithmetic:
1. Complex numbers: fundamentals, modulus and argument, Argand diagrams.
2. Complex forms: cartesian, exponential, conversions between forms, conjugation
3. Arithmetic: addition/subtraction, multiplication, division, exponentiation
4. DeMoivre’s theorem

Linear Algebra:
1. Vector arithmetic: concept, high-dimensional objects and arithmetic operations.
2. Matrices: fundamentals, notation, determinants, transposition
3. Matrix arithmetic: addition/subtraction, multiplication, division, inversion, triangularisation.
4. Linear equations: solution by Gaussian Elimination, Cramer’s Rule, Matrix Inversion.

Differentiation:
Fundamentals; Curve Sketching; Product and Chain Rules; Parametric Differentiation; Logarithmic Differentiation; Parital Differentiation

Differential Equations:
Fundamentals; 1st Order Methods; 2nd Order Methods

Integration:
Fundamentals; Integrating functions of functions; Integration functions of linear functions; Integration by parts; Integration by substitution; Integration by Reduction Formula; Applications

Sequences and Series:
Fundamentals; Convergence and Limits; Tests of Convergence; Power Series Properties; Limits for Indeterminate Solutions; L’Hopitals’s Rule;

Function Approximation:
Fundamentals; imiting Indeterminate Analytical Functions; Taylor’s and Maclaurin’s Series; Compositie Series Approximations; Accuracy Limitations

Learning Objectives

• Understanding of the concept and forms of, and motivation for complex numbers.
• The ability to represent complex numbers in Cartesian, exponential and graphical forms.
• The ability to perform fundamental arithmetic operations on complex numbers.
• The ability to measure the modulus and argument of a complex number.
• The ability to use complex arithmetic to represent the roots of any number.
• Understanding of the concept of vector arithmetic.
• The ability to manipulate high-dimensional mathematical objects and apply fundamental arithmetic operations thereon.
• An understanding of the form and concepts behind manipulation of matrices.
• The ability to perform fundamental arithmetic operations on matrices.
• The ability to transform matrices.
• The ability to exploit matrices for the solution of linear algebraic equations.
• The ability to perform matrix triangularisation and inversion.
• The ability to use matrix triangularisation, matrix inverse and matrix determinants to solve systems of simultaneous equations.
• Differentiation of simple, parameteric and logarithmic functions
• 1st and 2nd order differential equations
• Integration of functions of functions, functions of linear functions, by parts, substitution or reduction
• Sequences and series
• Functional approximation

Skills

• Formulation and analysis of arithmetic problems including complex numbers.
• The ability to derive the roots of any number.
• Formulation and manipulation of high-dimensional mathematical objects.
• Formulation and solution of high-dimensional linear algebraic problems using matrix arithmetic.

Assessment

None