## UNISA Applied Mathematics Course Module

Major combinations:

NQF Level: 5: MAT1503, MAT1512 and any TWO of the following: APM1513, APM1514, APM1612, PHY1505

NQF Level: 6: APM2611, MAT1613, MAT2615 and at least two other APM modules on NQF Level: 6

NQF Level: 7: FIVE of the following: (a) APM3701 (b) APM3711 (c) APM3712 (d) APM3713 (e) MAT3706 (f) MAT3707

Mechanics and Calculus of Variations – APM3712 | |||
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Under Graduate Degree | Semester module | NQF level: 7 | Credits: 12 |

Module presented in English |
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Pre-requisite: Any 2 APM or MAT modules on second level |
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Purpose: To enable students to demonstrate a basic understanding of generalised coordinates, Hamilton’s principle, calculus of variations and the Euler-Lagrange equations, the problem of Lagrange and the isoperimetric problem, Hamilton-Jacobi theory and Poisson brackets, Equivalent Lagrangians, canonical transformations and Noether’s theorem and application of the variational principles in mechanics. |

Special Relativity and Riemannian Geometry – APM3713 | |||
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Under Graduate Degree | Semester module | NQF level: 7 | Credits: 12 |

Module presented in English |
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Pre-requisite: Any 2 APM or MAT modules on second level |
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Purpose: To introduce students to special relativity and the basics of general relativity. Introductory geometry in non-Euclidian spaces and tensor algebra will also be covered. |

Cosmology – APM4801 | |||
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Honours | NQF level: 8 | Credits: 12 | |

Module presented in English |
Module presented online | ||

Purpose: Cosmology is the study of the physical universe. The module first introduces properties of the visible universe, including concepts such as distance scales, redshift, isotropy and homogeneity. This is followed by a brief survey of the structure and evolution of galaxies and stars. An empirical basis is used to show that the physical universe in its entirety has structure and evolves. The module focuses mainly on big-bang models of the universe and gives a description of both Newtonian cosmology and general relativistic cosmology. The big-bang type of evolution of the universe is followed from its early stages, including neutrino decoupling and the radiation dominated era. This is then pursued through decoupling and the origin of the cosmic microwave background radiation, and into the matter dominated era. The module is concluded with a fairly introductory discussion of observational cosmology. The latter looks at a variety of cosmological observations, all using discreet sources of radiation, to test the validity of models. The module is aimed at students who majored in applied mathematics, physics, or astronomy. |

Continuous Time Stochastic Processes – APM4802 | |||
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NQF level: 8 | Credits: 12 | ||

Module presented in English |
Module presented online | ||

Purpose: The purpose of this module is for students to gain insight into the use of continuous time processes as a tool of applied mathematics. |

Introduction to General Relativity – APM4804 | |||
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Honours | NQF level: 8 | Credits: 12 | |

Module presented in English |
Module presented online | ||

Purpose: General relativity is the relativistic theory of gravitation, and is widely regarded as one of the major achievements of 20th century physical science. The detailed syllabus is: Review of special relativity. The equivalence principle and the physical ideas that lead to general relativity. The Einstein field equations. The linearised field equations: Comparison with Newtonian theory, and gravitational waves. The Schwarzschild solution: Derivation and properties. Introduction to black holes. The Friedmann-Robertson-Walker solution. |

Mathematics of Optimization Theory – APM4805 | |||
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Honours | NQF level: 8 | Credits: 12 | |

Module presented in English |
Module presented online | ||

Purpose: The concept of optimization, in its various forms, is a very fundamental one with an important role to play in various branches of mathematics and of course also in the application of mathematics in other disciplines such as economics and engineering. The infinite dimensional case of optimization is studied in the calculus of variations and in optimal control theory. This module presents the classical theory of optimization in the finite dimensional situation. The emphasis is on the development of the mathematical theory and techniques of optimization (convex analysis, Lagrange multiplier rules) rather than computational or numerical techniques for finding optimal points. |

Riemannian Geometry and Tensor Calculus – APM4806 | |||
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Honours | Year module | NQF level: 8 | Credits: 12 |

Module presented in English |
Module presented online | ||

Purpose: Vectors and tensors in general coordinate systems. Covariant differentiation. The Riemann curvature tensor and associated tensors. The Weyl tensor and conformal metrics. Lie derivatives. Description of hypersurfaces. This module may be taken independently of APM4804, but it has been set up to provide the necessary mathematical background for a proper study of general relativity. It is concerned with the description of an N-dimensional non-Euclidean space referred to arbitrary coordinates. |

Applied Linear Algebra – APM1513 | |||
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Under Graduate Degree | Semester module | NQF level: 5 | Credits: 12 |

Module presented in English |
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Purpose: To enable students to master and apply the following aspects of the numerical solution of systems of linear equations: the method of least squares; linear programming (simplex method); eigenvalues, eigenvectors, diagonalisation as well as some miscellaneous applications. |

Numerical Solutions to Partial Differential Equations – APM4808 | |||
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Honours | Year module | NQF level: 8 | Credits: 12 |

Module presented in English |
Module presented online | ||

Purpose: Partial differential equations (PDE’s) have formed a basis of many mathematical models of chemical, physical and biological problems. More recently, the use of PDE’s has also extended to include the fields of economics and financial forecasting. In this module we study various finite difference methods for the numerical solution of these PDE’s. The efficiency of these methods is then examined by means of theoretical analysis of their consistency, convergence and stability. |

Mathematical Modelling – APM1514 | |||
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Under Graduate Degree | Semester module | NQF level: 5 | Credits: 12 |

Module presented in English |
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Purpose: To enable students to demonstrate a basic understanding of solution, equilibrium points and stability of difference equations and firstorder differential equations; applications to population models; harvesting strategies; epidemics; economics and other situations; simple optimisation and applications. |

Optimal Control – APM4809 | |||
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Honours | Year module | NQF level: 8 | Credits: 12 |

Module presented in English |
Module presented online | ||

Purpose: Systems that evolve in time occur naturally in various fields such as dynamics, economics, medicine and ecology, and modelling the behaviour of such systems provides an important application of mathematics. These systems can be completely deterministic, but often it may be possible to control their behaviour through the application of external controls. The theory of optimal control is concerned with finding the controls which, at minimum cost, either direct the system along a given trajectory or enable it to reach a desired target state. This module introduces some of the basic tools of optimal control. Many applications from various fields are also included, to show how the various ‘maximum principles’ help us find the optimal controls in practice. Contents: Terminology and classifications of optimal control problems. Controllability and reachability. Linear time optimal problem. The Time Optimal Principle. The Pontryagin Maximum Principle. Linear equations with quadratic costs. |

Applied Linear Algebra (Extended) – XPM1513 | |||
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Under Graduate Degree | Year module | NQF level: 5 | Credits: 12 |

Module presented in English |
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Purpose: To enable students to master and apply the following aspects of the numerical solution of systems of linear equations: the method of least squares; linear programming (simplex method); eigenvalues, eigenvectors, diagonalisation as well as some miscellaneous applications. |

An Introduction to the Finite Element Method – APM4810 | |||
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Honours | Year module | NQF level: 8 | Credits: 12 |

Module presented in English |
Module presented online | ||

Purpose: I want to introduce you to a module on Finite Elements. This module will develop the basic mathematical theory of the Finite Element Method (FEM).This method is the most widely used technique for engineering design and mathematical physics. In studying this module the student will obtain a clear knowledge of what the Finite Element Method is, how it works and how to use it to solve boundary-value problems. The Finite Element Method is a general technique for constructing approximate solutions to boundary-value problems. The method involves dividing the domain of the solution into a finite number of sub domains, the finite elements, and using variational concepts to construct an approximation of the solution over the collection of finite elements. |

Mathematical Modelling (Extended) – XPM1514 | |||
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Under Graduate Degree | Year module | NQF level: 5 | Credits: 12 |

Module presented in English |
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Purpose: To enable students to demonstrate a basic understanding of solution, equilibrium points and stability of difference equations and firstorder differential equations; applications to population models; harvesting strategies; epidemics; economics and other situations; simple optimisation and applications. |

Applied Functional Analysis – APM4811 | |||
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Honours | Year module | NQF level: 8 | Credits: 12 |

Module presented in English |
Module presented online | ||

Purpose: Research in the solvability of PDE’s leads us to a much wider scope. In the realization that alternative methods can be used for proving the existence and uniqueness of solutions of linear and nonlinear PDE’s, a new research area in Applied Mathematics was introduced. The theory of Sobolev Spaces was developed, which turned out to be a suitable setting in which to apply ideas of functional analysis to glean information concerning PDE’s. For this reason I want to introduce you to a module on Applied Functional Analysis. |

Mechanics II – APM1612 | |||
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Under Graduate Degree | Semester module | NQF level: 6 | Credits: 12 |

Module presented in English |
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Pre-requisite: MAT1512 (or XAT1512) & PHY1505 (or XHY1505) |
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Purpose: To enable students to demonstrate a basic understanding of definite integrals, line integrals and the vector product; dynamics of systems of particles and rigid bodies; in particular mass centres, moments of forces, moments of inertia and angular momentum. |

Introduction to Mechanics of Fluids – APM4812 | |||
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Honours | Year module | NQF level: 8 | Credits: 12 |

Module presented in English |
Module presented online | ||

Purpose: The purpose of this module is to make students aware of some fundamental aspects of fluid motion, including important fluid properties, regime flow, pressure variations in fluids at rest and in motion, fluid kinematics, methods of flow description and analysis and the Bernoulli equation. This module conveys the essential elements of kinematics, including Eulerian and Lagrangian mathematical description of flow phenomena, and indicates the vital relationship between the two views. The basic analysis methods generally used to solve or to begin solving fluid mechanics problems (linear motion and deformation, angular motion and deformation, conservation of mass, conservation of linear momentum, viscous flow) are also introduced. Emphasis is placed on understanding how flow phenomena are described mathematically and on when and how to use infinitesimal and finite control volumes. Important notions such as boundary layers, transition from laminar to turbulent flow will also be introduced. |

Differential Equations – APM2611 | |||
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Under Graduate Degree | Semester module | NQF level: 7 | Credits: 12 |

Module presented in English |
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Pre-requisite: MAT1512 (or XAT1512) & MAT1613 |
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Purpose: To enable students to obtain knowledge of first-order ordinary differential equations, linear differential equations of higher order, series solutions of differential equations (method of Frobenius), Laplace transform and partial differential equations (only an introduction). |

Honours Research Project in Applied Mathematics – HRAPM81 | |||
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Honours | NQF level: 8 | Credits: 24 | |

Module presented in English |
Module presented online | ||

Co-requisite: HMAPM80 |
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Purpose: The purpose of this module is to learn how to do research in applied mathematics, and how to present the research. This is demonstrated during a research project under supervision of an academic supervisor, and usually involves the preparation of a document describing the research according to the norms and expectations in applied mathematics. |

Applied Dynamical Systems – APM2614 | |||
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Under Graduate Degree | Semester module | NQF level: 7 | Credits: 12 |

Module presented in English |
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Pre-requisite: APM1513 (or XPM1513) & MAT1503 (or XAT1503) , MAT1512 (or XAT1512) & MAT1613 |
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Purpose: To enable students to master and apply fundamental aspects of discrete and continuous systems including linear systems; phase portraits: equilibrium points, stability, limit cycles; Liapunov stability; elementary control theory as well as applications to mechanics, ecology, economics and elsewhere. |

Honours Research in Applied Mathematics – HRAPM82 | |||
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Honours | NQF level: 8 | Credits: 36 | |

Module presented in English |
Module presented online | ||

Purpose: The purpose of this module is to introduce the student to academic writing skills for mathematics using LATEX and BibTEX, the process of conducting mathematics literature searches and the preparation of mathematics research documents, and to use these skills to engage in a research project in mathematics, under supervision of an academic supervisor. |

Computer Algebra – APM2616 | |||
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Under Graduate Degree | Semester module | NQF level: 7 | Credits: 12 |

Module presented in English |
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Pre-requisite: APM1513 (or XPM1513)& MAT1503 (or XAT1503), MAT1512 (or XAT1512) & MAT1613 |
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Purpose: To give students an understanding of the power of modern computer algebra systems, and specifically to enable students to use computer algebra to solve analytically a variety of mathematical problems including the algebraic equations (both linear and nonlinear), differentiation, integration, differential equations, matrix manipulation, series expansions, and limits; and to represent mathematical functions graphically, 2D and 3D, and to produce mathematical reports. |

Partial Differential Equations – APM3701 | |||
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Under Graduate Degree | Semester module | NQF level: 7 | Credits: 12 |

Module presented in English,Afrikaans |
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Pre-requisite: Any 2 APM or MAT modules on second level |
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Purpose: To introduce students to the following topics in partial differential equations; the equation of Laplace, the heat equation and the wave equation treated as typical examples of elliptic, parabolic and hyperbolic partial differential equations respectively, and methods of solution of the corresponding boundary value problems are also discussed. |

Numerical Methods II – APM3711 | |||
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Under Graduate Degree | Semester module | NQF level: 7 | Credits: 12 |

Module presented in English |
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Pre-requisite: COS2633 |
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Purpose: To equip students with numerical techniques for the approximate solution of initial and boundary value problems of differential equations and function approximations. |