##### MATA1614 (16 credits) – Engineering Statics

Three lectures and two hours practical per week in the first semester.

One three-hour paper.

**Contents:** Vector operations, resultants of forces, moments of forces about points and axes, equilibrium of forces acting on a point or a rigid body, friction, center of gravity and centroids, moments of inertia.

**Outcomes:** Add and subtract forces. Calculate moments of forces. Calculate projections of forces along given lines. Analyse the equilibrium of given force systems. Calculate centroids and centers of gravity. Calculate certain moments of inertia.

##### MATA1624 (16 credits) – Engineering Dynamics

Three lectures and two hours practical per week in the second semester.

One three-hour paper.

**Contents:** Particle kinematics, including continuous, erratic, rectilinear, curvilinear and relative motion. Particle kinetics, including equations of motion for particles and systems of particles in several types of coordinate systems, work and energy, impulse and momentum.

**Outcomes:** Analyse the motion of particles acted upon by given force systems. Apply the principles of work and energy, as well as conservation of energy. Calculate power and efficiency. Apply the principles of momentum and conservation of momentum to collisions and other relevant mechanical situations.

##### MATM1614 (16 credits) – Calculus

Four lectures and three hours practical per week in the first semester.

One three-hour paper.

**Contents:** Single variable calculus, including: the real numbers, functions, limits, continuity, differentiation (theory, techniques and applications), the mean value theorem, curve sketching, inverse functions, transcendental functions, integration (theory, techniques and applications).

**Outcomes:** Find the domain and range of given functions including complicated functions. Find the inverse of a given function including complicated functions. Investigate the continuity and differentiability of given functions. Calculate limits of given functions using L’Hospital’s rule and other techniques. Calculate the derivative of a given function, including transcendental functions. Find the tangent line to a given curve at a point. State and prove the mean value theorem. Apply the mean value theorem. Calculate rates of change. Calculate local and global minima and maxima as well as inflection points of given functions. Investigate the concavity of a given function. Sketch the graph of a given function.

Evaluate definite and indefinite integrals using partial fractions, integration by parts, and substitution techniques. Calculate areas with curved boundaries

##### MATM1624 (16 credits) – Algebra and Logic

Four lectures and three hours practical per week in the second semester.

One three-hour paper.

**Contents:** Logic. Algebra, including the binomial theorem, complex numbers, conical sections, vectors in R2 and R3, systems of linear equations, matrices, and determinants.

**Outcomes:** Use the principals of logic to prove results. Prove and use the binomial theorem Perform algebraic operations on complex numbers. State and prove de Moivre’s theorem and related results Solve algebraic equations involving complex numbers. Derive various vector identities. Perform algebraic operations on vectors in R^2 and R^3. Calculate inner products, norms and vector products. Use vectors to describe the geometrical properties of lines and planes. Calculate areas and volumes by means of vector methods. Perform algebraic operations on matrices. Calculate the determinant and inverse of a matrix. Use projections of vectors on lines. Use elementary matrices to perform Gaussian elimination. Prove various theorems involving elementary matrices. Solve systems of linear equations using Gaussian elimination. Investigate the properties of given conical sections

##### MATM1534 (16 credits) – Calculus

Three lectures and three hours practical per week in the first semester.

One two-hour paper.

**Contents:** Functions, graphs, limits, continuity and the derivative. Polynomial, trigonometric, exponential, and logarithmic functions. Differentiation. Critical points and local maxima and minima. Introduction to modelling. The definite integral. Integration techniques.

**Outcomes:** Find the domain and range of a given function. Find the inverse of an invertible function. Shift and stretch a given function. Solve simple problems involving exponential functions, including population growth and radioactive decay. Solve equations using logarithmic functions. Solve problems involving sinusoidal and tangent functions. Find the horisontal and vertical asymptotes of rational functions. Identify parts of a function which are continuous, and points at which it is not. Calculate limits, including left and right limits of a function. Identify the inner and outer functions of a composite function. Construct a composite function from given functions. Calculate the derivative of polynomial functions using the definition of the derivative at a point and as a function. Use the rules of differentiation to calculate derivative functions for polynomial, exponential, logarithmic, trigonometric and inverse trigonometric functions. Find the derivatives of implicit functions. Finding and identifying local maxima and minima and inflection points of functions. Find the global maximum and minimum of a given function and apply this to simple optimisation problems. Calculate indefinite integrals using some simple rules.

Calculate definite integrals using the fundamental theorem of calculus. Use simple substitutions to calculate definite and indefinite integrals. Use integration by parts to calculate definite and indefinite integrals.

##### MATM1544 (16 credits) – Calculus & Algebra

Three lectures and three hours practical per week in the second semester.

One two-hour paper.

**Contents:** This module contains some theory and applications of Calculus and Algebra, including: calculation of definite and indefinite integrals by substitution and partial fractions, solving separable ordinary differential equations, complex numbers, vectors in 2 and 3 dimensions, vector equations of lines and planes, solving systems of linear equations, introduction to matrix algebra.

**Outcomes:** Recognise and calculate indefinite and definite integrals which can be calculated by algebraic, sin, and cos substitutions. Recognise and calculate both definite and indefinite integrals which can be solved by partial fractions. Recognise a separable ordinary differential equation and solve it. Calculate the absolute value and conjugate of a complex number. Add, subtract, multiply and divide complex numbers and write the result in standard form. Convert a complex number to polar form and back. Calculate an integer power of a complex number. Calculate all the roots of a complex number for a given integer root. Convert a vector from its geometrical definition to component form and back. Add and subtract vectors. Calculate the dot and vector product of two vectors. Use the vector product to calculate the areas of triangles and parallelograms. Calculate the box product of three vectors. Use the box product to calculate the volume of a parallelepiped. Write the equation of a line in vector and parametric form in two and three dimensions. Write the equation of a plane in vector and parametric form in three dimensions. Calculate relationships between lines and lines and planes using vector methods. Add, subtract and multiply matrices. Calculate the determinant of a matrix. Calculate the inverse of an invertible matrix by both the cofactor and Gauss-Jordan method.

##### MATM1542 (8 credits) – Introductory calculus and statics

Two lectures and one hour practical per week during the second semester.

One two-hour paper.

**Contents:** Calculus: polynomial, trigonometric, exponential and logarithmic functions, curve sketching, the function concept, an outline of differentiation and integration. Statics: forces and moments, stress and strain, shear force and bending moment, trusses.

**Outcomes:** Students master basic differentiation, integration and strength of materials, and can use calculus to solve construction problems.

*Note: This module is meant for Quantity Surveying and Construction Management students . *

##### MATM1574 (16 credits) – Precalculus I

Three lectures and three hours practical per week in the first semester.

One two-hour paper.

**Contents:** Number systems, Properties of real numbers. Notations. Exponents and radicals. Special product formulas. Factorizing. Distance and midpoint. Simplify algebraic expressions. Solve equations. Modelling. Applications. Interest; speed; distance; time; percentage; depreciation; inflation; ratio and proportion. Exponential and logarithmic laws. Functions. Domain and Range. Graphs: linear, quadratic, circles, half-circles and hyperbolas, exponential and logarithmic, absolute value. Elimination and substitution. Principles of geometry. Perimeter, circumference, area, volume and total area. Principles of trigonometry and solving triangles; applications and modeling. Arithmetic and Geometric series.

**Outcomes:** Have a good understanding of natural numbers; integers; rational and real numbers. : Simplify and Factorize Algebraic and Rational expressions. Solve equations. Sketch graphs of functions and determine their domains and ranges. Use the concepts of ratio and proportion to solve practical problems. Have a thorough understanding of the basic geometry of triangles, circles, quadrilaterals and prisms. Use the various trigonometric functions to solve triangles. Derive the sum formulas for geometric and arithmetic series, and apply these, as well as induction to solve financial problems involving simple and compound interest, mortgages, depreciation and inflation.

##### MATM1584 (16 credits) – Precalculus II

Three lectures and three hours practical per week in the second semester.

One two-hour paper.

**Contents:** Inequalities; Absolute Value; Definition of a function, Graphs of functions; piecewise graphs; domain and range; symmetry; even and odd functions; translating and combining functions; composite functions; inverse functions; linear and quadratic functions; power functions and polynomials; rational functions and their properties; exponential and logarithmic functions; the exponential and logarithmic laws and applications; the trigonometric functions; trigonometric identities; basic statistics

**Outcome:** Students are familiar with the elementary functions and their graphs and have a good basis for a calculus module.

*Note: This module can give access to MATM1614 .*

##### MATA2614 (16 credits) – Dynamics of rigid bodies

Three lectures and two hours practical per week in the first semester.

One three-hour paper.

**Contents:** Planar kinematics of a rigid body, including translation, rotation about a fixed axis, absolute and relative motion analysis, rotating axes. Planar kinetics of a rigid body, including moments of inertia, equations of motion for translation, rotation about a fixed axis and general planar motion; Work and energy; Impulse and momentum; vibrations

**Outcomes:** Analyse the motion of a rigid body subject to a given system of planar forces. Calculate power and energy, and apply the principles of energy and the conservation of energy to the motion of rigid bodies where suitable. Calculate the momentum of a rigid body, and apply the principles of momentum and impulse to the motion of rigid bodies under suitable circumstances. Analyse vibrating systems.

##### MATM2614 (16 credits) – Vector analysis

Two lectures and two hours practical per week in the first semester.

One three-hour paper.

**Contents:** Vector functions: limits, derivatives, and integrals. Curves: parametrization, tangent vectors, arc length. Multivariable functions: quadratic surfaces, partial derivatives, limits, continuity, differentiability, gradients, and directional derivatives, the Mean Value theorem, the chain rule for partial derivatives, tangent planes. Multiple and line integrals: Theory and applications.

**Outcomes:** Students understand the theory and applications of more advanced calculus, including vector calculus, multivariable functions, line integrals, and surface integrals.

##### MATM2624 (16 credits) – Linear algebra

Two lectures and two hours practical in the second semester.

One three-hour paper.

**Contents:** Real vectors/spaces, subspaces, basis, dimension, rank, nullity, matrix transformations. Eigenvectors and diagonalisation. Inner products and Gram-Schmidt process. Orthogonal matrices and orthogonal diagonalisation. General linear transformation and isomorphism.

**Outcomes:** Understand the notions of linear algebra. Solve problems of linear algebra using the concepts of matrices.

##### MATA2634 (16 credits) – Mathematical modelling

Two lectures and two hours practical per week in the first semester.

One three-hour paper.

**Contents:** Principles of modelling. Optimisation models. Dimensional analysis. Physical, chemical, biological and financial models. Decision and game theory.

**Outcomes:** Apply modelling techniques, such as difference equations, proportionality, curve fitting and interpolation techniques, and elementary optimisation techniques. Apply decision and game theory techniques based on probabilities. Be able to use the basic steps to build a model, in conjunction with the necessary techniques to construct a simple model individually, or as part of a small team.

##### MATA2644 (16 credits) – Ordinary differential equations

Two lectures and three hours practical per week in the second semester.

One three-hour paper.

**Contents:** Non-linear first order differential equations: substitution techniques, exact equations, integration factors. Non-homogeneous higher order differential equations with constant coefficients. Series methods. Systems of linear differential equations. Applications such as mixtures, orthogonal trajectories and the logistic equation.

**Outcomes:** Solve various non-linear first order differential equations, linear second order differential equations with constant coefficients, as well as some with non-constant coefficients.

Apply ordinary differential equations to solve some basic scientific problems from various disciplines.

##### MATM2654 (16 credits) – Scientific Computing

Two lectures and two hours practical per week in the first semester.

One three-hour paper.

**Contents:** Programming with Matlab. Scientific computing. Introductory numerical techniques.

**Outcomes:** Students can implement mathematical formulas, computations, and algorithms on a computer and use it to solve problems numerically.

##### MATM2664 (16 credits) – Sequences and series

Two lectures and two hours practical per week in the second semester.

One three-hour paper.

**Contents:** Sequences of real numbers: convergence, limits, boundedness, indeterminate forms, L'Hospital's rule. Improper integrals. Infinite series: tests for convergence, absolute and conditional convergence. Taylor series. Power series: intervals of convergence. Fourier analysis.

**Outcomes:** Students understand the basic theory of sequences and series of real numbers. They can apply the theory by determining the power series expansion and intervals of convergence of functions.

##### MATM3714 (16 credits) – Complex analysis

Two lectures and two hours practical per week in the first semester.

One three-hour paper.

**Contents:** The complex numbers. Functions of a complex variable. Limits, continuity, and differentiability. The Cauchy-Riemann equations. Power series. Analytic functions. Cauchy's theorem. Residue theory and applications.

**Outcomes:** Students understand the basic theory of complex functions (which includes residue theory and applications).

##### MATM3724 (16 credits) – Real analysis

Two lectures and two hours practical per week in the second semester.

One three-hour paper.

**Contents:** Axiomatic construction of the real numbers. Sequences of real numbers. The Weierstrass-Bolzano theorem. Limits and continuity. The intermediate value theorem. The Riemann integral.

**Outcomes:** Students understand the basic theory of the field of real numbers. Continuity, differentiability, and Riemann integrability of real functions form part of this module.

##### MATM3734 (16 credits) – Discrete Mathematics

Two lectures and two hours practical per week in the first semester.

One two-hour paper.

**Contents:** Logic, method of proof, set theory, functions and relations, elementary number theory, induction, recursion, effectivity of algorithms.

**Outcomes:** Students understand the foundation of mathematics and know when sentences are logically equivalent. Notions such as countability and infinity are mastered. Students will have enough background to study and understand the theory of algorithms.

##### MATM3744 (16 credits) – Algebra

Two lectures and two hours practical per week in the second semester.

One two-hour paper.

**Contents:** Integers: Induction, greatest common divisors, well-ordering principle, equivalence relations, arithmetic modulo n. Groups: Finite and infinite groups, subgroups, cyclic groups, dihedral groups, permutation groups, Lagrange’s theorem, cosets, conjunction, homomorphisms, isomorphism theorems. Rings: Commutative rings, rings with unity, integral domains, polynomial rings, fields, principle ideal domains, ideals, homomorphisms, fields of fractions of an integral domain, isomorphism theorems

**Outcomes:** Describe notions around certain algebraic structures such as groups, rings and fields.

Apply these notions.

##### MATA3764 (16 credits) – Industrial Mathematics

Two lectures and two hours practical per week in the second semester.

One two-hour paper.

**Contents:** Introduction to linear programming. Actual problems from industry with the necessary mathematics to model it mathematically and solve the models. Communication of results. Project.

**Outcomes:** Students can solve simple programming problems. They are familiar with several actual problems from industry and are able to solve similar simple problems themselves and communicate results.

##### MATA3774 (16 credits) – Numerical analysis

Two lectures and two hours practical per week in the first semester.

One two-hour paper.

**Contents:** Non-linear equations in one variable: iterative methods, error analysis. Polynomial interpolation: Lagrange, barycentric, Newton, Chebyshev and Hermite interpolation ; splines; error estimation. Numerical differentiation and integration. Initial-value problems in ordinary differential equations: elementary theory, high-order Taylor, Runge-Kutta and multistep methods, stability..

**Outcomes:** Implement the theory of numerical techniques such as the iterative solution of non-linear equations, interpolation, numerical differentiation and integration, and the numerical solution of ordinary differential equations on a computer. Perform accuracy and reliability tests.

##### MATA3784 (16 credits) – Dynamical systems

Two lectures and two hours practical per week in the second semester.

One three-hour paper.

**Contents:** Elementary stability considerations in systems of linear first order ordinary differential equations: chemical, medical, biological, and other applications. Systems of non-linear first order ordinary differential equations. Local stability and the classification of fixed points: Applications to biological and medical models. Global stability and limit cycles: Forced non-linear oscillations. First order perturbation techniques. Applications of ordinary differential equations.

**Outcomes:** Students can use phase diagrams to analyse equilibrium points and trajectories of non-linear ordinary differential equations. Students can use techniques from asymptotic analysis to obtain approximate solutions of such differential equations. Students can apply these techniques to manipulate models in Chemistry, Physics, Medical Science, and Biology.

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T: +27 51 401 3199

E: damonsle@ufs.ac.za

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T: +27 51 401 2531

E: lottere@ufs.ac.za