Basic counting principal; permutations and combinations. Random phenomena; sample spaces and events; axioms of probability; random selection; rules of probability. Conditional probability; Bayes' rule; stochastic independence. Discrete random variables; expected value and variance; discrete distributions: binomial, Poisson, geometric, hypergeometric and negative-binomial. Basiese telbeginsel; permutasies en kombinasies. Stogastiese verskynsels; steekproefruimtes en gebeurtenisse; waarskynlikheidsaksiomas; ewekansige seleksie; waarskynlikheidsreëls. Voorwaardelike waarskynlikheid; Bayes se reël; stogastiese onafhanklikheid. Diskrete kansveranderlikes; verwagte waarde en variansie; diskrete verdelings: binomiaal, Poisson, geometries, hipergeometries en negatief-binomiaal.

Development of the skilled use of vector, differential and integral calculus in the modelling of dynamics of simple physical systems, including the analysis of force fields, motion and modelling assumptions. Ontwikkeling van die vaardige gebruik van vektor-, differensiaal- en integraalrekening in die modellering van dinamika van eenvoudige fisiese sisteme, insluitend analise van kragtevelde, beweging en modellerings-aannames.

Linear systems. Orthogonality: application to curve fitting. Eigenvalues and -vectors: application to systems of differential equations. Singular values: application to image processing. Numerical computations such as LU, QR and SVD factorisation and the computation of eigenvalues and -vectors. Condition numbers: sensitivity of linear systems. Lineêre stelsels. Ortogonaliteit: toepassing op krommepassing. Eiewaardes en -vektore: toepassing op stelsels van differensiaalvergelykings. Singulierwaardes: toepassing op beeldverwerking. Numeriese bewerkings soos LU-, QR- en SVD-ontbinding en die berekening van eiewaardes en -vektore. Kondisiegetalle: sensitiwiteit van lineêre stelsels.

Modelling of a wide variety of applications using ordinary differential equations (DEs). Linear, non-linear, separable and homogeneous DEs as well as systems of DEs. Analytical techniques (including Laplace transforms) as well as numerical methods for solving models. Emphasis on the various steps of the classic modelling process. Modellering van 'n verskeidenheid toepassings met behulp van gewone differensiaalvergelykings (DVs). Lineêre, nie-lineêre, skeibare en homogene DVs sowel as stelsels van DVs. Analitiese tegnieke (insluitend Laplace transforms) sowel as numeriese metodes vir die oplos van modelle. Klem op die verskillende stappe van die klassieke modelleringsproses.

Applications of prime factorisation, divisibility, greatest common divisors, the Euler phi function, modular arithmetic, multiplicative inverses, algebraic groups and elementary combinatorics in cryptology (the protection of information) and coding theory (the integrity of information). Introductory graph theory: planarity, colourings, Hamiltonian and Euler graphs. Toepassings van priemfaktorisasie, deelbaarheid, grootste gemene delers, die Euler-phi-funksie, modulêre rekenkunde, multiplikatiewe inverses, algebraïse groepe en elementêre kombinatorika in kriptologie (die beveiliging van inligting) en kodeerteorie (die integriteit van inligting). Inleidende grafiektoerie: planêriteit, kleurings, Hamilton- en Eulergrafieke.

Numerical stability, and conditioning. Methods for solving non-linear equations; convergence analysis. Interpolation with polynomials and spline functions; error analysis. Numerical differentiation and integration. Numerical methods for solving initial value problems. The use of software like Matlab or Python for numerical calculations. Numeriese stabiliteit, en sleggeaarheid. Metodes vir die oplos van nie-lineêre vergelykings; analise van konvergensie. Interpolasie met polinome en latfunksies; foutanalise. Numeriese differensiasie en integrasie. Numeriese metodes vir die oplos van aanvangswaardeprobleme. Die gebruik van sagteware soos Matlab of Python vir numeriese berekeninge.

Modelling of the dynamics of continuous systems; convective and diffusive transport as special cases of the general transport theorem; stress dyadic; energy and heat transport; constitutive equations for fluids; derivation and solution of the Navier-Stokes equation; ideal flow; potential flow; computational simulation of fluid dynamics. Modellering van die dinamika van kontinue sisteme; konvektiewe en diffusiewe oordrag aan die hand van die algemene transportteorema; spanningsdiade; energie- en warmte-oordrag; gedragvergelykings vir vloeistowwe; aflei en oplos van die Navier-Stokes vergelyking; ideale vloei; potensiaalvloei; numeriese vloeisimulasie.

Fourier series, continuous and discrete Fourier transforms, convolution, Laplace transform, Sturm-Liouville theory, orthogonal functions. Applications in signal and image processing, as well as in the solution of ordinary and partial differential equations. Numerical Fourier analysis and the famous FFT (fast Fourier transform). Fourier-reekse, kontinue en diskrete Fourier-transforms, konvolusie, Laplace-transform, Sturm-Liouville-teorie, ortogonale funksies. Toepassings in sein- en beeldverwerking, sowel as in die oplossing van gewone en parsiële differensiaalvergelykings. Numeriese Fourier-analise en die beroemde FFT (vinnige Fourier-transform).