**Course Classification**

**Core Courses (CC)**

Core/Compulsory Courses are courses students must take and pass before graduation. However, a student may be allowed to proceed to the next level of study if he/she failed some Core-Courses, but have earned enough credit units to qualify for the next level.

The failed Core -Course will always reflect in the remark column of his/her results as outstanding.

**Elective Courses (EL)**

Elective courses are courses students take base on their areas of specialization and interest. In addition, to the Core- Courses, students are allowed to take these courses to enable them to register for the minimum allowable per semester.

However, there are restrictive and un-restrictive electives, depending on areas of specialization and Faculty. Students may consult their academic advisers for guidance in selecting Elective Courses.

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**Pre-Requisite Courses**

Pre-requisite courses are those that students must take and pass before registering higher level course(s). However, a student in a graduating class may register for a higher and lower course concurrently, provided that he/she had attempted the lower course once and that the total number of credit units registered in a semester, does not exceed 24.

**General Studies Courses (GS)**

All undergraduate students are required to pass all the General Studies Courses for a total of 8 Credits Units. This must be passed before a degree can be awarded.

**Industrial Training**

Industrial attachment is very crucial for all students in Sub-Department of Mathematics in Nnamdi Azikiwe University Awka. The minimum duration of this training is 6 months (24 weeks, semester plus long vacation).

The Students Industrial Work Experience Scheme (SIWES) will make our undergraduate training effective, as students will be exposed to various equipment, manufacturing processes and quality control procedures during the attachment. The SIWES is to be conducted under strict supervision and on successful completion of the scheme, it is graded and no students shall graduate without passing it.

**Course Contents and Descriptions**

**100-LEVEL**

**MAT. 100:** **BASIC MATHEMATICS (FOR NON-SCIENCES)**

**CREDIT ****3: SEMESTER 1 ^{ST} PRE-REQUESITE: Credit pass in O’ level MATHEMATICS**

Number systems natural, integer, rational, irrational and real numbers. Set theory. Permutations and combinations. Elementary probability including mathematical expectation, Baye’s theorem for conditional probability and total probability. Indices, logarithms and surds. The Binomial theorem for positive integers. Progressions: arithmetic, geometric, harmonic, arithmetic-geometric; including arithmetic, geometric and harmonic means of a sequence of numbers and the n-means-arithmetic, geometric and harmonic – between two numbers. Series as infinite sums of the progression. Elementary functions.

**MAT 101:** **ELEMENTARY MATHEMATICS 1**

**CREDIT: 3 SEMESTER 1 ^{ST}PRE-REQUISITE: Credit Pass In “’Level Mathematics**

Number systems – natural, integer, rational, irrational, real and complex number. Elementary set theory. Indices, surds and logarithms. Quadratic equations. Polynomials and their factorizations:- the remainder and factor theorems. Rational functions and partial fractions. The PMI (Principle of mathematical induction). Permutations and combinations. The binomial theorem for rational index. Progressions – arithmetic, geometric. Harmonic, arithmetic-geometric. Series. Solution of inequalities. The algebra of complex numbers:- addition, subtraction, multiplication and divisions; Argand diagrams and the geometry of complex numbers; modulus; arguments and polar coordinates; the de Movire’s theorem. Complex roots of unity and complex solution to a = z^{n} . trigonometry – circular measure, elementary properties of trigonometric functions, radian measure, addition formulae and other trigonometric identities, sine and cosine laws; solution of triangles, heights and distances.

**MAT 102: ELEMENTARY MATHERMATICS II**

**CREDIT – 3 SEMESTER 2 ^{ND }PRE-REQUISITE: MAT 101**

Functions:-concept and definition, examples – polynomial, exponential, logarithmic and trigonometric functions. Graphs and their properties. Plan analytic geometry:-equations of a straight line, circle, parabola, ellipse and hyperbola. Tangents and normals. Limits and continuity. Differentiation from first principles of some polynomial and ergonometric functions. Techniques of differentiation – sum, product, quotient and chain rules including implicit differentiation. Differentiation of simple algebraic, trigonometric, exponential, logarithmic and composite functions. Higher order derivatives. Applications to extremum and simple rate problems. L’Hospital rule, simple Taylor/Maclaurin expansion. Curve sketching Integration as anti-differentiation. The fundamental theorem of integral calculus. Applications to areas and volumes.

**MAT 112:** **INTRODUCTION TO ACTUARIAL MATHEMATICS I**

**CREDIT: 3 SEMESTER 2 ^{ND} PRE-REQUISITE: MAT 100, MAT 101**

Mathematical theory and practical problems as regards simple and compound interest, annuities, debt extinction by

(i) amortization and (ii) sinking funds, depreciation, investment in bonds, capitalization, endowment funds, perpetuities, shares and stocks. Introduction to life insurance mathematics (calculus of life contingency).

**MAT 161: ELEMENTARY MECHANICS I (STATICS)**

**CREDIT 3 SEMESTER 1 ^{ST} CO-REQUISITE: MAT 101, MAT 102**

Geometric representation of vectors in 1-3 dimensions. Components, direction cosines. Addition ,scalar multiplication of vectors and linear dependence. Dot (scalar) and cross (vector) product of vectors. Line vectors and division of a line in a given ratio. Areal vectors, volume of a parallelepiped. Sine and cosine laws. Forces as vectors; resolution of forces-parallelogram, polygon and triangle of forces. Lami’s theorem. Resultants of a system of forces acting at a point. Laws of friction. Equilibrium of forces. Particles rough horizontal and inclined planes. Tension in a string. Forces in a plane acting on a rigid body. Like and unlike forces, mechanical advantage (MA), velocity ratio (VR), efficiency (E) and systems of pulleys. Center of gravity.

**MAT 162: ELEMENTARY MECHANICS II (DYNAMICS)**

**CREDIT 3 SEMESTER 2 ^{ND}**

**PRE-REQUISITE MAT 161**

Differentiate and integration of vectors with respect to a scalar variable. Components of a velocity and acceleration of a particle moving in a plane. Forces, momentum, laws of motion, motion under gravity, projectiles, resisted vertical motion. Elastic string, simple pendulum. Impulse. Impact of two smooth sphere on a smooth sphere. Relative velocity. General motion of a particle in two dimensions, motion in horizontal and vertical circle; simple harmonic motion of a rigid body about a fixed axis; moment of inertia calculations.

200-Level.

**MAT 200: BASIC MATHEMATICS (FOR NON-SCIENCE) II**

**CREDIT 3 SEMESTER 2 ^{ND} PRE-REQUISITE MAT 100**

Vectors- column, row, addition, cross and dot products. Matrices and determinants. Systems of linear equations. Introduction to linear programming- graphical method and elementary simplex method. Limits and continuity. Differentiation and integration with applications to Economics; marginal cost and revenues, elasticity, total revenue etc.

**MAT 201 LINEAR ALGEBRA 1**

**CREDIT 3 SEMESTER 1 ^{ST}**

**PRE-REQUISITE MAT 101, MAT 102**

Vectors and vector algebra. Vector space over the real field. Linear dependence and independence; basis and dimension. The dot and cross products in 3-dimensions. Equations of lines and planes in free space. Linear transformation and their representation by matrices. Matrix algebra. Operations on matrices; rank, range, null space, nullity. Determinants and inverses of matrices. Singular and non-singular transformations.

**MAT 202: ELEMENTARY DIFFERENTIAL EQUATIONS**

**CREDIT 3 SEMESTER 2 ^{ND}**

**PRE-REQUISITE MAT102**

Methods of integration. Introduction to differential equations- classification, order, degree. Ordinary differential equations of the first order. Examples to illustrate the sources of differential equations from the physical and biological sciences-growth, decay, cooling problems and the law of mass action. Linear differential equations of second order. Application of first and second order linear differential equations to falling problems and simple circuits. Replace transformation.

**MAT 204 LINEAR ALGEBRA II**

**CREDIT 3 SEMESTER 2 ^{ND}**

**PRE-REQUISITE MAT 201**

Systems of linear equation, change of basis, equivalence and similarity, eigenvalues and eigenvectors. Minimal and characteristic polynomials of a linear transformation (matrix). Cayley-Hamilton theorem. Bilinear and quadratic forms. Orthogonal diagonalisation. Canonical forms.

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**MAT 211 INTRODUCTION TO ACTUARIAL MATHEMATICS II**

**CREDIT 2 SEMESTER 1 ^{ST} PRE-REQUISITE MAT 112**

Application of mathematical methods in Economics. Supply and demand curves. Elasticities. Relation between average and marginal costs. Relationship between average and marginal. Budgeting.

**MAT 222 NUMERIC METHODS**

**CREDIT 3 SEMESTER 2 ^{ND} PRE-REQUISITE MAT101, MAT 102**

Error analysis, solution of algebraic and transcendental equations- Newton, Newton-Raphson, regular falsi, chord or secant, tangents, bisection and basic iteration methods. Curve-fitting. Interpolation and approximation. (zeros of nonlinear equations of one variable.) systems of linear equations. Gauss-Siedel and Jacobi iterative methods. 111-conditioned systems. Numerical differential and integration (quadrature). Trapezoidal and Simpson’s rules for quadrature. Romberg integration.

**MAT 231 CALCULUS**

**CREDIT 3 SEMESTER ^{1ST} PRE-REQUISITE MAT102**

Vector function and their derivatives. Partial derivatives. Directional derivative. Tangent plane and normal line. Gradient, curl and divergence. The chain rule. Maxima and minima problems. Optimization and Lagrange multiplier method. Rolle’s and mean- value theorems. Taylor’s theorem. Multiple integrals. Applications to areas, volumes, centers of mass. Moment of inertia etc.

**MAT 242 NUMBER THEORIES**

**CREDIT 3 SEMESTER 2 ^{ND} PRE-REQUISITE MAT 251**

Basic set theory. Symbolic logic. Methods of mathematical proof. Relations- partial ordering, equivalence, upper and lower bounds, maximal, minimal, maximum and minimum elements of sets of real numbers. Elementary treatment of the well ordering principle. Zorn’s lemma and Axiom of choice. Prime numbers- infinitude of, divisibility and modulo systems. GCDs and LCMs. Euclid’s division algorithm. The fundamental theorem of Arithmetic or unique factorization theorem. Continued fractions and the solvability of linear congruence. Transversals and the solvability of polynomial congruences (elementary treatment only).

**MAT251** **INTRODUCTION TO REAL ANALYSIS**

**CREDIT 3 SEMESTER 1 ^{ST} PRE-REQUISITE MAT 102**

Limit (more rigorous treatise using epsilon-delta) – sums, products and quotients of limits. Bounds for real numbers. Sequences of real numbers- definition, types (monotone, etc), bounds, convergence. Cauchy sequences. Theorem of nested intervals. Series of real numbers- definition, tests for convergence of series of non-negative terms, absolute and conditional convergence, alternating series and rearrangement. Continuity and uniform continuity (epsilon-delta approach). Monotone functions. Differentiability. Rolle’s and mean-value theorems for differentiable function. Taylor and maclaurin series.

**MAT 252** **INTRODUCTION TO COMPLEX ANALYSIS**

**CREDIT 3 SEMESTER 2 ^{ND} PRE-REQUISITE MAT 251**

Polar representation of complex numbers include in a review of complex numbers. geometric and analytic interpretation of regions in the complex plane-discs, domains, annuli, spheres, circles, parabolas, ellipses, etc. limits of sequences of complex numbers. Definition and examples of complex valued functions of a complex variable Cauchy-Riemann equations. Analytic functions and Taylor series. Contour integrals including Cauchy’s and Cauchy-Goursat integral theorems (elementary treatise only).

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**MAT 261 MECHANICS**

**CREDIT 3 SEMESTER 1 ^{ST} PRE-REQUISITE MAT 162**

Kinematics and rectilinear motion of particle. Free motion of a rigid body in two dimensions and stability of equilibrium. General motion of a rigid body as a translation plus a rotation. Moments and products of inertia in three dimensions. Parallel and perpendicular axes theorems. Principal axes, momentum, kinetic energy of a rigid body impulsive motion. Examples involving one and two dimensional motion of simple systems. Moving frames of reference, rotating and translating frames of reference. Coriolis force.

**300-Level**

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**MAT 311 OPTIMIZATION THEORY AND METHODS**

**CREDIT 3 SEMESTER 1ST PRE-REQUISITE MAT 201**

Linear programming models. The simplex method:- formulation and theory, quality integer programming. Transportation and scheduling problems. Two-person-zero- sum games. Nonlinear programming-quadratic programming, Kuhn-Tucker methods, optimality criteria. Single variable optimization. Multivariable optimization techniques. Gradient methods.

**MAT 321 NUMERICAL ANALYSIS 1**

**CREDIT 3 SEMESTER 1 ^{ST} PRE-REQUISITE MAT 222**

Finite differences and difference calculus. Newton’s forward and backward interpolation formula. Numerical differentiation and integration. Newton-Cotes quadrature formulae. Numerical solution of differential equations. Monte-Carlo methods.

**MAT 331 ADVANCED CALCULUS**

**CREDIT 3 SEMESTER 1 ^{ST} PRE-REQUISITE MAT 231**

Leibnitz rule for successive differentiation and its extension. Functions in R^{n} continuity and differentiability. Partial derivatives, the tangent plane, the chain rule, total differential. Scalar and vector fields. The gradient and directional derivatives. Curl and divergence. Green’s Stoke’s and Gauss’s (divergence) theorems. Jacobians and curvilinear coordinates. Change of variables in multiple integrals. Functions defined by integrals; gamma and beta functions and their elementary properties.

**MAT 333** **DIFFERENTIAL EQUATIONS**

CREDIT 3 SEMESTER 1^{ST} PRE-REQUISITE MAT 202

Linear equation of second order, properties of their solutions. Series solution of second order linear equations about ordinary and singular points- including the solution of Bessel, Legendre and Gauss hypergeometric equations. Sturm- Liouville problems. Orthogonal functions and polynomials. Fourier, Fourier-Bessel and Fourier-Legendre series. Fourier transform. Solution of heat, wave and Laplace’s equations by Fourier (separation of variable) method.

**MAT 335** **DIFFERENTIAL GEOMETRY**

**CREDIT 3 SEMESTER 1 ^{ST}**

**PRE-REQUISITE MAT 231, MAT 202**

Vector functions of real variable. Boundedness, limits, continuity and differentiability. Functions of class C^{m} . Taylor’s formula. Analytic functions. Curves: regular, differentiable and smooth. Curvature and torsion. Tangent line and normal planes. Vector functions and vector variables. Linearity, Directional derivatives of functions of class C^{m}. Taylor’s theorem and the inverse function theorem. Concept of a surface: Parametric representation, tangent plane and normal lines. Topological properties of simple surfaces.

**MAT 341** **ABSTRACT ALGEBRA I (GROUP THEORY)**

**CREDIT 3** **SEMESTER 1 ^{ST} PRE-REQUISITE MAT 242**

Basic definitions and examples of algebraic structure: Semigroups, groupoids, monoids, groups, rings, and fields. Groups: subgroups and cosets. Lagrange’s theorem and applications, Permutation groups, Cyclic groups, Normal subgroups and quotient (factor) groups. Homomorphism and isomorphism theorems. Cayley’s theorem. Authomorphisms Aut(G) and Inner authormophisms Inn(G). Direct products of groups. Groups of small order. Groups acting on sets. Sylow theorems.

**MAT 351** **REAL ANALYSIS**

**CREDIT 3** **SEMESTER 1 ^{ST}**

**PRE-REQUISITE MAT 251**

The set of real numbers. Rational and irrational numbers. Open interval, open sets. Cantor set. Limits, derived sets. The Bolzano-Weierstrasse and Heine-Borel theorems. Limits superior and inferior of sequence of real numbers. Supremum (l.u.b.) and infimum (g.l.b.) of sets of real numbers. Completeness of the reals and incompleteness of the rationals. Convergence of sequence and series of real numbers and functions. Uniform convergence. Continuous functions of real variable. Uniform continuity including equi-continuity(uniform continuity of a family of functions). Riemann integral of real valued functions with real domains. Continuous monopositive functions. Functions of bounded variation. The Riemann-Stieltje’s integral. Effects on limits and sums when the functions are continuous, differentiable or Riemann integrable. Power series.

**MAT 353** **METRIC SPACE TOPOLOGY**

**CREDIT 3** **SEMESTER 1 ^{ST}**

**CO-REQUISITE MAT 351**

Sets. Metrics and examples. Open balls, closed balls, spheres. Open sets and neighborhoods. Closed sets. Interior, exterior, boundary (frontier), limit points and closure of a set. Dense subsets and separable metric spaces. Convergence in metric spaces. Homeomorphisms, continuity, compactness (including countable and sequential compactness, and Lindelof property), connectedness.

**MAT 361** **TENSOR ANALYSIS**

**CREDIT 3** **SEMESTER** **1 ^{ST}**

**CO-REQUISITE MAT 333**

Vector algebra, Vector, dot and cross products. Equations of curves and surfaces. Vector differentiation and applications. Gradient, divergence and curl. Vector integrals – line, surface and volume integrals. Green’s, Stoke’s and divergence theorems. Tensor products of vector spaces. Tensor algebra. Symmetry. Cartesian Tensor. Application of tensors in geometry and mathematical physics.

**MAT 363** **ANALYTICAL DYNAMICS I**

**CREDIT 3** **SEMESTER 1 ^{ST}**

**PRE-REQUESITE**

**MAT 261**

Degrees of freedom, Holonomic and non-holonomic systems (with constraints). Generalized coordinates. Lagrange’s equation for holonomic systems; forces as vector fields (forces dependent on coordinates only). Conservative fields (forces obtainable from potentials). Impulsive force. Motion of rigid bodies: moving frames of reference. Corilis force. Motion near the Earth’s surface. The Foucault’s pendulum. Euler’s dynamical equations for the motion of a rigid body with one point fixed. The symmetrical top.

**MAT 365** **QUATUUM MECHANICS**

**CREDIT 3** **SEMESTER 1 ^{ST}**

**CO-REQUISITE MAT 363**

Stress, strain and deformation. Rate of deformation tensor. Finite strain and deformation. Eulerian and Lagragian formulation. General principles. Constitutive equations.

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**400-Level**

**MAT 411** **ACTUARIAL MATHEMATICS**

**CREDIT 3** **SEMESTER 1 ^{ST}**

**PRE-REQUISITE**

**MAT 211**

Probability and decision making. Mathematical Expectation. Decision rules and trees. Redundancy. Replacement theory. Optimum replacement age; replacing at convenient moment. Stock control under certainty and under uncertainty.

**MAT 413** **OPERATIONS RESEARCH I**

**CREDIT 3** **SEMESTER 1 ^{ST}**

**PRE-REQUISITE**

**MAT 311**

Phases of operations research study. Classification of operations research models: linear, dynamic and integer programming. Decision theory. Inventory models, critical path analysis and project control.

**MAT 412** **OPERATIONS RESEARCH II**

**CREDIT: 3** **SEMESTER: 2 ^{ND}**

**PRE-REQUISITE: MAT 413**

Quantitative methods in management: branch and bound, maximal flow, minimal spanning tree, Hungarian assignment (HAM) or flood assignment (FAT) techniques. Graph theory and networks. Stock control. Queuing problems. PERT – project evaluation and review techniques.

**MAT 421** **NUMERICAL ANALYSIS II**

**CREDIT: 3** **SEMESTER: 1 ^{ST}**

**PRE-REQUISITE: MAT 321**

Numerical methods for the solution of ordinary and partial differential equations (including stability analysis for linear multi-step methods) Predictor-corrector algorithms and the Runge-Kutta methods. Finite difference approximation and applications to boundary value problems (BVP). Computation of eigenvalues and eigenvectors of symmetric matrices (emphasis on Rayleigh’s quotient). Monte-Carlo methods.

**MAT 423** **INTRODUCTION TO MATHEMATICAL MODELING**

**CREDIT 3** **SEMESTER** **1 ^{ST}**

**PRE-REQUISITE**

**MAT 202**

Methodology of model building; identification, formulation and solution of problems; casue-effect diagrams, Equation types: algebraic, differential (ordinary and partial) difference, integral and functional equations. Applications of mathematical modeling to the physical, biological, social and behavioural sciences. Epidemiology: dynamics of communicable infections (STIs – sexually transmitted infections, malaria, etc). Simulation.

**MAT 431:** **THEORY OF ORDINARY DIFFERENTIAL EQUATIONS**

**CREDIT: 3** **SEMESTER 1 ^{ST} PRE-REQUISTE: MAT 333**

Existence and uniqueness of solution; dependence of solution on initial and parameter. General theory of linear differential equations with constant coefficients. The two-point Sturm-Liouville boundary value problem; self adjointness. Sturm theory (Sturm comparison and Sonin-Polya theorems). Stability of solutions of nonlinear equations. Phase-plane analysis.

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**MAT 432:** **PARTIAL DIFFRENTIAL EQUATIONS**

**CRIDET: 3** **SEMESTER 2 ^{ND}**

**PRE-RE-REQUISITE: MAT 333**

Theory of first order partial differential equations. Partial differential equations in two independent variables with constant coefficients. The Cauchy-problem for the quasilinear first order PDE in two independent variables; existence and uniqueness of solutions. The Cauchy-problem for the linear second order PDE in two independent variables; existence and uniqueness of solutions. Normal forms. Boundary-and initial value problems for elliptic, hyperbolic and parabolic PDEs.

**MAT 434:** **SYSTEMS THEORY**

**CREDIT: 4** **SEMESTER: 2 ^{ND}**

**PRE-REQUISITE: MAT 431**

Lyapunov theorems. Solution of Lyapunov stability equation A^{T} B + BA = C. Controllability and absorbability. Theorems on the existence of solution to linear systems of differential equations with constant coefficients. Control theory.

**MAT 435:** **FIELED THEORY IN MATHERMATICAL PHYSICS**

**GREDIT: 3** **SEMESTER: 1 ^{ST}**

**PRE-REQUISITE: MAT 361**

Gradient, divergence and curl; further treatment and application of the differential definitions. The integral definition of gradient, divergence and curl. Line surface and volume integrals. Green’s, Gauss’s and Stoke’s theorems. Curvilinear coordinates. Simple notions of terrors; use of tensor notations.

**MAT 434: MATHERMATICAL METHODS 1**

**CREDIT: 3** **SEMESTER 2 ^{ND}**

**PRE-REQUESITE: MAT 331**

Orthogonal functions and orthonormal sets of functions. Gram-Schmidt orthonormalisation process. Eigenvalues, eigenvectors and eigenfunction expansion. Rayleigh’s quieted quadratic forms. Adjoint operators and adjoint manifolds. Green’s functions and application to the solution of differential problems. Laplace, Fourier and Hankel transforms. Introduction to variation calculus.

**MAT 436 MATHEMATICAL METHODS II**

**CREDIT : 3 SEMESTER 2 ^{ND} CO-REQUISITE: MAT 434**

Calculus of variation: Lagrange’s functional and associated density. Necessary condition for a weak relative extremum. Hamilton’s principles. Lagrange’s equation and geodesic problems. The Du-Bois-Raymond equation and corner conditions. Variable end-points and related theorems. Sufficient conditions for a minimum. Isoperimetric problems. Variational integral transforms. Laplace, Fourier and Hankel transforms. Complex variable methods; convolution theorems, applications to solutions of differential equations with initial/boundary conditions.

**MAT 441:** **ABSTRACT ALGEBR II (RING THEORY)**

**CREDIT: 3 SEMESTER 1 ^{ST}**

**PRE-REQUISTE: MAT 341**

Rings:- definition example including Z and Zn. Rings of polynomials matrices. Sobering and ideals. Quotient rings. Types of rings: principal Ideal Domains (PIDs), Unique Factorization Domains (UFD_{s}) Euclidean rings, integral domains, fields polynomial rings. Factorization; Euclidean algorithm for polynomials; GCD and LCM of polynomials. Irreducibility (including Eisenstein’s criterion).

**MAT 442:** **ABSTRACT ALGEBRA III (FIELD THEORY)**

**CREDIT: 3 SEMESTER 2 ^{ND}**

**PRE-REQUISITE: MAT 441**

Fields field extensions, degree of extension. Minimum polynomials. Algebraic and transcendental extensions. Constructibility (using compass and straight-edge). Splitters (splitting fields). Separability. Algebraic closure. Solvable (soluble) groups. Fundamental theorem of Galois theory. Solvability by radicals. Definition and examples of modules, submodules and quotient modules. Introduction to group representation theory.

**MAT 457** **COMPLEX ANAYSIS I**

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**CREDIT:3** **SEMESTER: 1 ^{ST}**

**PRE-REQUISITE: MAT 252**

Function of a complex variable. Limits and continuity of functions of a complex variable. Analytic functions, bilinca transformations and conformal mappings. Contour integrals. Cauchy’s theorems and its main consequences. Convergence of sequence and series of a complex variable. Power series. Taylor series. Laurent series expansions; poles, singularities – isolated, removable and essential. Residues and residue calculus.

**MAT 451** **GENERAL TOPOLOGY**

**CREDITS: 3 SEMESTER 1 ^{ST}**

**PRE-REQUISITS: MAT 353**

Topological spaces, definition and examples, open and closed sets. Neighborhoods. Coarser and finer topologies. Basis and sub-basis. Separation axioms (Trenungsaxiomes). Compactness, local compactness, countable compactness, sequential compactness and the Lindelof property. Connectedness. Construction of new topological spaces from given ones. Sub-spaces and quotient spaces. Continuous functions. Product spaces and product topologies.

**MAT 452** **FUNCTIONAL ANALYSIS**

**CREDIT: 3 SEMESTER: 2 ^{ND} PRE-REQUESITE MAT 451**

A survey of the classical theory of metric spaces – including Baire’s category theorem, compactness, separability, isometries and completion. Elements of Banach and Hilbert spaces:- parallelogram law and polar identities in Hilbert spaces H; the natural embeddings of normed linear spaces into the second dual, and H onto H. Properties of operators including the open-mapping and closed graph theorems. The spaces C(X), the sequence (Banach) spaces l_{p} and L_{p} and c (convergent sequences).

**MAT 453** **LEBESGUE MEASURE AND INTEGRATION**

**CREDIT: 3 SEMESTER: 2 ^{ND} PRE-REQUISITE: MAT 351**

Lebesgue measure; measurable and non-measurable sets. Measurable functions. Lebesgue integral, Integration of non-negative functions. The general integral convergence theorems.

**MAT 454:** **COMPEX ANALYSIS II**

**CREDIT: 3** **SEMESTER 2 ^{ND} PRE-REQUISITE:**

**MAT 457**

Meromophic function zeros and poles. Argument principle Rouche’s theorem. Maximum modulus principle. Analytic continuation and elementary Riemann surfaces. Hurwits theorem and the inverse function theorem. Boundary-value problems. Poisson’s humulate.

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**MAT 456:** **MEASURE THEORY**

**CREDIT: 3 SEMESTER 2 ^{ND}**

**PRE-REQUISITE MAT. 453**

Abstract integration in Lp -spaces.

**MAT 461** **QUANTUM MECHANICS**

**CREDIT: 3** **SEMESTER 1 ^{ST} PRE-REQUSISTE: MAT 333**

Particle-wave duality. Quantum postulated. Schrödinger equations of motion. Potential steps and well in 1-dimension. Heisenberg formulation. Classical limit of quantum mechanics. Computer brackets. Linear harmonic oscillator. Angylar momentum. 3-dimensional square well potential. The hydrogen atom. Collision in 3-dimensions. Approximation methods for stationary problems.

**MAT 463** **GENERAL RELATIVITY:**

**CREDIT: 3 SEMESTER 1 ^{ST} PRE-REQIESIT**

**MAT 361**

Particles in a gravitational field. Curvilinear coordinates. Intervals. Covariant differentiation; Christ fell symbol and metric tensor: the constant gravitational field. The energy-momentum tensor. Newton’s laws. Motion in a centrally symmetric gravitational field. The energy-momentum pseudo-tensor. Gravitational waves. Gravitational fields at large distances from bodies. Isotropic space. Space-time metric in the closed and open isotropic models.

**MAT 462** **ELECTROMAGNETISM**

**CREDIT:** **SEMESTER 2 ^{ND}**

**PRE-REQISITE MAT 461**

Maxwell’s field equations. Electromagnetic waves and electromagnetic theory of lights. Place electromagnetic waves in non-conducting media, reflection and refraction at place-boundary. Waves and resonant cavities. Simple radiating systems. The Lorentz-Einstein transformation. Energy and momentum. Electromagnetic 4-vectors. Transformation of (E-H) fields. The Lorentz force.

**MAT 464** **ANALYTICAL DYNAMICS II**

**CREDIT**:** 3** **SEMESTER 2 ^{ND} PRE-REQUISITE MAT 361**

Lagrange’s equations for non-holonomic systems. Lagrange’s multipliers. Variational principles. Calculus of variation. Hamilton’s principle. Lagrange’s equation from Hamilton’s principles. Canonical transformations. Normal modes of vibration. Hamilton-Jacobi equations.

**MAT 466** **FLUID DYNAMICS II**

**CREDIT**:** 3** **SEMESTER 2 ^{ND} PRE-REQUISITE MAT 461**

Real and ideal fluids. Differentiation following motion of fluid particles. Equations of notion and continuity for incompressible in viscid fluids. Velocity potentials and Stoke stream function. Bernoulli’s equations with applications to flows along curved paths.

**MAT 465** **ELASTICITY**

**CREDIT**:** 3** **SEMESTER 1 ^{ST} PRE-REQUISITE MAT 363**

Stress and strain analysis, constitutive rotations, equilibrium and compatibility equations. Principles of minimum potential and complementary energy. Principles of virtual work. Variational formulation. Extension, bending torsion of beams. Elastic waves.