To present an exploration of vector spaces and matrices with particular reference to the development of computational techniques and application skills.
Vectors and vector spaces; subspaces, linear independence, bases, physical applications. Inner products, norm and distance, orthogonality, orthonormal basis.
Matrices; matrix operations, echelon matrices, algebra of square matrices, classical adjoint, matrix inversion. Games of strategy; matrix games, applications to optimal decision making.
Linear equations; methods of solution, Gaussian elimination, use of determinants, applications to electrical networks, traffic flow. Linear programming; graphical methods, simplex method, applications in management science.
Linear mappings; kernel and image, vector space isomorphisms, space of linear mappings. Linear transformations; matrix representation, change of basis. Eigenvalues and eigenvectors, characteristic polynomial.
GROSSMAN, S.I., Multivariable Calculus, Linear Algebra and Differential Equations, (Harcourt, Brace, Jovanovich), 1986.
ANTON, H., Elementary Linear Algebra, (Wiley), 1973.
KOLMAN, B., Introductory Linear Algebra with Applications, (Macmillan), 1980.
MORRIS, A.O., Linear Algebra An Introduction, (Van Nostrand Reinhold), 1983.
WHITELAW, T.A., An Introduction to Linear Algebra, (Blackie), 1983.
To study functions of a real variable and of sequences and series of real numbers. To explore applications of advanced calculus.
Real numbers; field and order properties, least upper bounds, greatest lower bounds, completeness axiom, limits of sequences and series, infinite decimals.
Real valued functions of real numbers; limits, continuity, intermediate value theorem, boundedness properties of continuous functions.
Differentiation; chain rule, relative maxima and minima, differentiation of inverse functions, applications.
Rolle's theorem, mean value theorem, applications to optimization problems and approximation methods.
Taylor's theorem, applications.
GROSSMAN, S.I., Multivariable Calculus, Linear Algebra and Differential Equations, (Harcourt, Brace, Jovanovich), 1986.
FLANDERS, H., Calculus, (Freeman), 1985.
LANG, S., Analysis 1, (Addison Wesley), 1968.
ROSS, K.A., Elementary Analysis: The Theory of Calculus, (Springer Verlag), 1980.
To foster an appreciation of information technology and to explore applications in mathematics. To introduce programming and to develop practical skills in the use of computers.
Introduction to computers; hardware, software, main components.
Data storage and retrieval; input and output, memory, backup store.
Development of computers; calculating devices, modern PCs.
Introduction to computer programming; problem analysis, program design, features of high level language, program writing for mathematics.
Use of software packages; mathematical applications. Spreadsheets; data tabulation, graphical representation.
BISHOP, P., Comprehensive Computer Studies, (Arnold), 1987.
HUBBARD, John, Programming with C++, Schaum's Outline Series, McGraw Hill (1996).
SEDGEWICK, Robert Algorithms in C++, Addison Wesley, (1992).
LIPPMAN, Stanley, C++ Primer, Addison Wesley, (1995).
To develop further the exploration of advanced calculus through a range of topics not covered elsewhere and to include an introduction to differential equations and applications.
Real vector spaces of dimension n, norm, inner product and cross product. Lines and planes in 3-dimensional space, curves and surfaces. Cylindrical and spherical coordinates.
Calculus of several variables, continuity and derivative for multivariable functions. Partial derivatives. Maxima and minima, constrained and unconstrained optimisation. Directional derivative, gradient, divergence and curl. Chain rule. Higher derivatives. Taylor’s formula in several variables.
Inverse functions and implicit functions. Tangent planes and normal lines.
Definition of multiple integral, Fubini’s theorem. Double and line integrals, surface and volume integrals.
Introduction to differential equations, ordinary and partial differential equations. The Heat Equation, the Wave Equation.
McCALLUM, William G, et al., Calculus: Multivariable, (John Wiley & Sons) 2005
SPIVAK, Michael, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, (Addison-Wesley), 1968.
STEWART, James, Calculus: Concepts and Contexts, (London: Pacific Grove), 1998.
KLINE, Morris, Calculus: An Intuitive and Physical Approach, (Wiley), 1977.
COURANT, R., John, F., Introduction to Calculus and Analysis, Volume 2, (Wiley) 1974.