SYLLABUS OF UNDERGRADUATE CHEMISTRY …sc.uob.edu.ly/assets/.../d7903-discription-of-mathematics-1-.pdf · 13 1413 Axiomatic Set Theory 3 4 1207 14 1417 Fluid Mechanics 3 4 1201-1304

UNIVERSITY OF BENGHAZI

FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS

SYLLABUS OF UNDERGRADUATE MATHEMATICS COURSES

2016

Edited BY Dr. SOUAD MUGASSABI

1 | UNIVERSITY OF BENGHAZI - FACULTY OF SCIENCE - DEPARTMENT OF MATHEMATICS

Dr. Souad Mugassabi | BSc Mathematics Courses

Department of Mathematics Program

The requirement for B.Sc. degree in Mathematics are 118 credit hours. The distribution of total hours is shown below:

(A) Faculty requirement: 9 credit (B) Supporting courses : 25 credit (C) Compulsory courses : 66 credit (D) Elective courses : 18 credit

(A) Faculty requirement ( 9 credit )

(B) Supporting Courses ( 25 credit )

(C) Compulsory Courses (66 Credits)

No. Course Code Course Title Credits Lectures Prerequisites

1 1100 General Maths I 4 5 Nil

2 1101 General Maths II 4 5 1100

3 1110 Analytical Geometry 3 4 Nil

4 1200 Calculus I 3 4 1101

5 1201 Calculus II 3 4 1200

6 1202 Differential Equations I 3 4 1101

7 1203 Complex Variables I 3 4 1200

8 1204 Mechanics I 3 4 1101-4101


1 0010 Arabic 3 3 Nil

2 0101 English I 3 3 Nil

3 0102 English II 3 3 0101


1 1022 Elements of probability I 4 5 Nil

2 2109 General Statistics 4 5 Nil

3 4101 General physics I 3 3 Nil

4 4102 Practical physics I 1 3 Nil

5 4103 General physics II 3 3 4101

6 4104 Practical physics II 1 3 Nil

7 9103 Int. To Computer Prog. 3 4 Nil

8 9104 Int. To Computer Science 3 4 9103

9 9204 Numerical Methods I 3 4 1200 +9103



9 1205 Number Theory 3 4 1101

10 1207 Fundamentals of Math. 3 4 1101

11 1300 Linear Algebra I 3 4 1101

12 1301 Abstract Algebra I 3 4 1207-1300

13 1302 Differential Equations II 3 4 1200-1202

14 1304 Mechanics II 3 4 1204

15 1306 Real Analysis I 3 4 1200-1207

16 1309 Diff. & Transform Geo. 3 4 1201-1301

17 1310 Topology I 3 4 1306

18 1312 Foundations of Geometry 3 4 1101-1207-1300

19 1401 Abstract Algebra II 3 4 1301

20 1406 Real Analysis II 3 4 1306

21 1450 Project 4 4 Nil

( D) Elective Courses (69 Credits (Select 18 Credits)


1 1303 Complex Variables II 3 4 1203

2 1307 Mathematical Methods I 3 4 1203-1300

3 1311 Number Theory 3 4 1205

4 1316 Differential Geometry 3 4 1309 - 1310

5 1400 Linear Algebra II 3 4 1300

6 1402 Partial Diff. Equations 3 4 1201-1302

7 1404 Mechanics III 3 4 1304

8 1405 Integral Equations 3 4 1201-1302

9 1407 Mathematical Methods II 3 4 1203-1302

10 1408 Riemannian Geometry 3 4 1201-1207-1300

11 1409 Modern Analysis 3 4 1406

12 1410 Topology II 3 4 1310

13 1413 Axiomatic Set Theory 3 4 1207

14 1417 Fluid Mechanics 3 4 1201-1304

15 1418 Theory of Elasticity 3 4 1304

16 1419 Bio-Mathematics 3 4 1302-1309

17 1421 Field Theory 3 4 1401



(III) Minor Courses (12-14)

Minor Statistics Courses (14 Credits)

Minor Physics Courses (12 Credits)

Minor computer Courses (12 Credits)

18 1430 Combinational Analysis 3 4 1301

19 1431 Graph Theory 3 4 1301

20 1432 History of Mathematics 3 4 1406

21 1433 Mathematical Logic 3 4 1207

22 1440 Independent Study I 1 1 Nil

23 1441 Independent Study II 2 2 Nil

24 1442 Independent Study III 3 3 Nil


1 2106 Elements of probability II 4 5 2102

2 2107 Statistical Methods 4 5 2109

3 2207 Distribution Theory 3 4 2106

4 2307 Adv. Distribution Theory 3 4 2207


1 4202 Waves and oscillations 2 2 -41034101

2 4203 Therm. & kinetic Energy 2 2 -41034101

3 4204 Practical physics III 1 3 -41024101

4 4212 Electricity &magnetism 3 3 4103

5 4214 Alternating current theory 1 1 -42024103

6 4301 Modern Physics 3 3 4103-4201


1 9201 Design of digital computer 3 4 9103

2 9205 Pascal Language 3 4 9104

3 9302 Intro. To assembly language 3 4 9104+9201

4 9311 Data structures & appls. 3 4 9205



Description of B. Sc. Courses

(A) Faculty requirement ( 9 credit )

0010 ARABIC LANGUAGE (3 Credits) Basic Grammar : definition and structure of sentence, definition of verb. Some Quranic verses. Poetry (some poems) . Prerequisite: (Nil)

0101 ENGLISH I (3 Credits) Special materials in English language introduced and practiced through reading , writing, and speaking. Reading sections taken from authentic scientific text books. Strategies of academic reading: skimming and scanning. Simple instructions and basic strategies of writing: description, definition and classifications. Prerequisite: (Nil)

0102 ENGLISH II (3 Credits) Grammar and vocabulary. Introducing grammar through listening, reading and writing. How grammar integrates with the skills of reading and writing. How grammar integrates with communicative functions: instruction, description and narration. Vocabulary: simple structure with everyday vocabulary. Structure of scientific English vocabulary. Prerequisite: (0101)

(B) Supporting courses : 25 credit

2102 Elements of Probability Distributions I (4 credits) Permutation and Combinations, Combinatorial problems, Sample Space, Events, Counting of sample points. Classical and axiomatic definitions of probability, Marginal and conditional probability. Baye’s theorem. Concept of random variable. Probability function. Expectation and variance of random variables. Moments, Moment Generating Function, Characteristic Function. Ideas of Bernoulli, Binomial and Poisson distributions. Prerequisite: (Nil)

2109 General Statistics (4 credits) Definitions, importance, scope and limitations of Statistics. Sources of data collection, Classification and Representation of Data. Frequency Distribution. Measures of Central Tendency, Dispersion and Skewness. Moments and Kurtosis. Ideas about Attributes and Association. Correlation ( Simple, Rank, Partial and Multiple ). Simple and Multiple Regression ( up to three variables ). Least squares method of estimation. Fitting of Polynomial curve up to 2nd degree and solution of Simultaneous equations. Index Numbers. Whole-Sale Price and Cost of Living Index Numbers. Constructions of Index Numbers. Sources of Errors in Constructions of Index Numbers, Different Tests of Index Numbers. Prerequisite: (Nil)

4101 General Physics I (3 credits) Mechanics: Standards and systems of units; vectors: representation, addition, subtraction, and multiplication; kinematics of linear motion: quantities of motion,



motion with constant acceleration, motion with variable acceleration; dynamics: force, inertia, linear momentum, Newton's law of motion, gravitation, simple harmonic motion, work, energy, conservation of energy, conservation of linear momentum; kinematics of rotation, circular motion, rotation of rigid bodies, moment of inertia, conservation of angular momentum. Properties of matter: Elasticity, modulli of elasticity, energy stored, Poisson's ratio, hydrostatic pressure, atmospheric pressure, surface tension with application hydrodynamics, Bernoulli's equation and application, viscosity, Poiseuille's law and Stokes's law. Heat: Temperature and temperature scale, thermal expansion, quantity of heat, heat exchange, heat transfer, heat and work, the first law of thermodynamics with applications. Prerequisite: (Nil)

4102 Practical Physics I (1 credits) Experiments are based on topics covered in 4101. Prerequisite: (Nil)

4103 General Physics II (3 credits) Electricity and Magnetism: Coulomb's law, electric field, Gauss's law with applications, electric potential, equipotentials, Capacitors and dielectrics, current electricity, simple DC-circuits, magnetic field of a current, magnetic force on a conductor, electromagnetic induction, magnetic properties of matter, simple a-c circuits. Optics: Reflection and refraction of light at plane and spherical surfaces, defects of images, optical instruments, photometry, spectroscopy velocity of light, introduction to wave theory, interference of light waves. Modern Physics: Birth of modern physics, quantization of energy with application to Bohr's atom and photoelectric effect, production and uses of X-ray, radioactivity, decay law, nuclear radiations, fission, fusion, electron motion in electric and magnetic fields. Prerequisite: (4101)

4104 Practical Physics II (1 credits) Experiments are based on topics covered in 4103. Prerequisite: (Nil)

9103 Introduction to Computer programming (3 credits) Understand the history of how computer technology unfolded, with particular emphasis on the “generations”. Understand how people and events affected the development of computers. Identify the basic components of computer system: input, processing, output and storage. Understand the difference between the difference types of software. Identify the components of the central processing unit and how they work together to form a system and interact with memory. Appreciate the need for the different applications included in the accessories of an Operating System. Know how the commands work in both GUI (Graphical User Interface) and CUI (Character User Interface) environment. Prerequisite: (Nil).

9104 Introduction to Computer Science (3 credits) This course is to provide an introduction to basic programming techniques including the following: Problem solving skills. Understand flowcharting tools. Use



the proper tool for proper operation. Learn the necessary properties of algorithms: input, output, definiteness, correctness, finiteness, effectiveness, and generality. Understand how to analyze the given problem scientifically and not by intuition. Understand how to write an algorithm to solve a given problem. Convert the algorithm into flowchart and ultimately to a given programming language. Prerequisite: (9103).

9204 Numerical Methods I (3 credits) Techniques for finding numerical approximations to solutions to mathematical problems of various types. Considerations of algorithm efficiency and error control in numerical procedures. The use of packaged computer algorithms in the solution of numerical problems. To set up and solve linear systems of equations manually as well as perform Gaussian elimination on small systems. To write programs using any software tool that evaluates and plot analytical functions. To explore topics related to nonlinear equations in one variable. Use interpolation techniques to solve different mathematical problems. Prerequisite: (9103 + 1200).

(C) Compulsory Courses and Elective Courses (66 Credits)

1100 General Mathematics I (4 credits) Cartesian coordinates, distance between two points, division of line segment, slope, angle between two lines, parallel and perpendicular lines, equation of line, distance from point to line, standard equation of circle, parabola, ellipse and hyperbola. Sets and subsets, basic set operations (union, intersection, and difference), real line, order, intervals. Inequalities and absolute values. Cartesian product of two sets, relations and functions, domain and range of functions, graph of functions, composition of function, one-to-one and onto functions, inverse functions. Limits, one-sided limits and continuity, derivative: differentiation of algebraic function, chain rule, parametric equations and higher order derivatives, differentiation of trigonometric, inverse of trigonometric, exponential and logarithmic functions application of derivative: tangent and normal, differentials, related rates, maxima and minima, curve tracing. Prerequisite: (Nil)

1101 General Mathematics II (4 credits) Hyperbolic functions, Rolles and mean value theorems with applications, generalized (Cauchy) mean value theorem, L'Hospital's rule and indeterminate form, extended mean value theorem and Taylor's expansion formula, approximation and errors. Standard Taylor series of xexp , xsin , xx 1/1,cos ,

mean value theorem and Newton's method for approximating solutions of equations, indefinite integrals, definite integrals and fundamental theorem of calculus, mean value theorem for integral, differentiation under integral sign. Various techniques of integrations, improper integrals, numerical integration of indefinite integrals. Application of definite integrals, arc length, area, volume, area of surfaces of revolution. Prerequisite: (1100)



1110 Analytic Geometry (3 credits) Vectors in plane and space (sums, differences, norm, direction, inner product). Straight line, circle, conic sections, transformation, rotation and general second degree equations of plane curves. Surfaces (sphere, cylinder, ellipsoid, paraboloid), transformation of coordinates in space, cylindrical and spherical coordinates. Equations of straight line and plane. Prerequisite: (Nil)

1200 Calculus I (3 credits) Sequences and infinite series, tests of convergence, power series, Taylor series with remainder, polar coordinates curve tracing, conic sections, angle between radius vector and tangent line, length of curve, area of region in polar coordinates, Cartesian and polar forms of curvature, functions of several variables: limits, continuity, partial derivatives, tangent plane, normal line, directional derivatives, gradient, chain rule, total differential, maxima and minima, methods of Lagrange multipliers, higher order derivatives. Prerequisite: (1101)

1201 Calculus II (3 credits) Integral calculus of functions of several variables, multiple integral, double and triple integrals in Cartesian, polar, cylindrical and spherical coordinates. Applications of double and triple integrals for calculating mass, area, volume, surface area, center of mass, moment of inertial, line integral, dependence on path, vector differential operators (grade, dive, curl) and relative formulas. Theorems of Green, Gauss, and stokes. Fourier series, half range series. Prerequisite: (1200)

1202 Differential Equations I (3 credits) Ordinary differential equation, basic concepts, separable equations, homogeneous equations, exact equations, integrating factors, linear first order equations of second order, fundamental theory of solution, solution of equations with constant coefficients, homogenous equations. Particular solution of non-homogenous equations, methods of undetermined coefficients, equations with constant coefficients of higher order, solution of equations with variables coefficients, method of variation of parameters, application of second order equations. Prerequisite: (1101)

1203 Complex variables I (3 credits) Complex numbers, function of complex variables, limits and continuity, sequences and series of complex numbers, analytic functions, elementary functions and mappings by them, conformal mappings, complex line integral, Cauchy-Goursat, theorem and Cauchy integral formula (without Proofs). Prerequisite: (1200)

1204 Mechanics I (3 credits) The axioms of mechanics, static of partial, vectors, composition and resolution of concurrent forces in plane and forces in space, equilibrium in plane and in space,



center of mass and center of gravity of some common bodies, theorems of papas-guldinus, center of mass, composite bodies. Rigid body, equivalent systems of forces, moment of force, moment of couple, reduction of system of forces friction, coefficient of friction, angle of friction. Differentiation of vectors, velocity and acceleration, rectilinear motion of particle, simple harmonic motion. Curvilinear motion of particle, motion of projectiles, circular motion, momentum of energy, work power, kinetic and potential energy, principle of impulse and momentum. Prerequisite: (1101+1110)

1205 Numbers Theory (3 credits) Divisibility, prime and composite numbers, fundament theorem of arithmetic, greatest common divisor and least common multiple. Congruencies and their properties, complete residue systems and reduced residue systems. Euler -

function, computation of n . Theorems of Euler, Fermat and Wilson. Solution of

linear congruence, Chinese remainder theorem, quadratic residues. Basic properties of polynomials, algebraic equations, roots and doefficients, symmetric polynomials and Newton's formula. Rational roots, irrational roots and complex roots of certain equations by radicals of cubic, quadratic and reciprocal equations. Case of equations of degree >5 (statement only). Separation of roots and sturm's theorem. Approximation of real roots (Newton's method). Prerequisite: (1101)

1207 Fundamentals of Mathematics (3 credits) Mathematical logic: statement calculus and predicate calculus, rules of inference, logical equivalence and logical implication, proof and methods of proof, direct and indirect proof set theory and subsets, set operations, ordered pairs and Cartesian product. Relations, equivalence relations, equivalence classes and partition, ordering relations, partial, total and well-order, supremum and infimum elements, functions: one-to-one and onto properties, composition and inverse functions, image and inverse image of sets, binary operations, types of binary operations, finite and infinite sets, countable and uncountable sets, integers, divisibility and congruence relations, residue and reduced residue classes, rational and irrational numbers, decimal representations of real numbers. Prerequisite: (1101)

1300 Linear Algebra (3 credits) Matrix algebra: elementary row operations, rows-reduced echelon form, rank of matrix, inverse matrix. Determinants: properties and computations. Classical adjoint and inverse matrix. System of linear equations, homogeneous and non-homogeneous case, crammer's rule. Vector space, linear independence, basis and dimension, linear transformations and matrices. Change of basis for vectors and linear transformations: similar matrices, eigenvalues and eigenvectors, Cayley-IIamilton theorem. Diagonalization of matrix, inner product spaces, orthogonal bases, Gram-Schmidt theorem. Bilinear forms, quadratics forms, Hermitian forms and normal forms. Prerequisite: (1101)



1301 Abstract Algebra (3 credits) Groups, Isomorphic Groups, Cyclic Groups, Subgroups and Lasrange’s Theorem, Quotient Groups, Group Homomorphism and First Isomorphism Theorem. Permutation Groups and Cayley’s Theorem. Dihedral and Quaternion Groups. Rings and Subrings, Characteristics, Matrix Rings and Polynomial Rings. Integral Domains, Division Rings and Fields. Ring Homomorphis. Field of Quotients. Prerequisite: (1207 + 1300 + 1302)

1302 Differential Equations II (3 credits) Series solution of second order equations, Taylor series Forbenius methods. Bessel and Legendre equations, methods of Laplace transform, system of equations, existence and uniqueness theorems for first and second order equations (statement and illustrations only). Finite difference equations, partial differential equations of first order. Prerequisite: (1202)

1303 Complex Variables II (3 credits) Cauchy-Goursat theorem, Cauchy integral formula, derivatives of analytic function, Morera's theorem, Liouville's theorem, fundamental theorem of algebra, maximum modulus theorem, isolated singularities and their classification, Laurent series expansion, calculation of residues and applications, evaluation of certain real improper integrals and definite integrals of trigonometric function, argument theorem, Rouche's theorem. Mittag's-Leffler's expansion theorem. Prerequisite: (1203) 1304 Mechanics II (3 credits)

Attraction and potential at point due to some common bodies, gravitational potential energy, moments of inertia of some common bodies, parallel and perpendicular-axes theorems, moments of inter of composite bodies, product of inertia, principal axes and principal moments of inertia, ellipsoid of inertia. Forces in beam and distributed loads, parabolic cable, methods of virtual work, potential energy and equilibrium, stability of equatilibrium, D'Alembert's principle. Angular momentum of particle, equation of motion of particle terms of radical and transverse components. Motion under central force, Kepler's laws of planetary motion, impulsive motion, impact, oblique central impact, moving coordinate system, velocity and acceleration in moving system. Prerequisite: (1204)

1306 Real Analysis I (3 credits) Axiomatic description of real number system as complete order field, Archimedean property, dense properties of rational and irrationals, limit point and Bolzano-Weierstrass theorem (statement only). Sequences and series of numbers, convergence, theorems and tests, limit, continuity and convergence, uniform continuity, intermediate value theorem. Derivative and its properties, mean value theorems, implicit function theorem (without proof), Riemann-Stielitjes integrals and their properties. Prerequisite: (1201)



1307 Mathematical Methods I (3 credits) Orthogonal curvilinear coordinates and expression of gradient, divergence, Curl and Laplacian in cylindrical and spherical coordinates, Laplace transform Dirac delta function, Fourier series and Fourier Integral. Prerequisite: (1201 + 1203 + 1300)

1309 Differential and Transformational Geometry (3 credits) Vector function of real variables (limits, continuity, and derivative), curves in (equivalent representations, are-length and natural representations), unit tangent vector, curvature, principal normal vector, binomial moving Trihedron, torsion, Frenet equations, the fundamental existence and uniqueness theorem

for curves in nR . Transformations and geometric transformations, groups of invariance, Euclidean transformations (isometries), rotations and Glide-reflections, basic properties of isometries, analytic and matrix representations, Euclidean group and its basic subgroups, Euclidean invariant, similarities (homotheties), properties and analytic representations, similarity group, Affine and projective transformations, their groups and invariant, topological transformations, Klein's concept of geometry, inverse geometry, analytic representations of invariant, elementary facts of plane geometry proved be transformations. Prerequisite: (1301)

1310 Topology I (3 credits) Metric space, open ball and open set, topological space, closed set, neighborhood, Hausdroff space, interior, limit, closure and boundary points of set, Bolzano-Weierstrass theorem (statement and examples), continuity, homeomorphism and topological properties, subspace, finite product space, quotient space, brief account on connectedness, path-connectedness and compactness. Prerequisite: (1200 + 1207)

1311 Number Theory (3 credits) Revision of divisibility congruencies, linear congruencies and Chinese remainder theorem, primitive, roots and indices, quadratic residues, Legendre symbol and quadratic reciprocity law, polynomial congruencies, number theoretic functions, basic Diophantine equations, infinite continued fractions and rational approximations to real numbers, Pell's equation, distribution of primes. Prerequisite: (1205)

1312 Foundations of Geometry (3 credits) Euclid's original system and its effects, axiomatic method, proof and diagrams, Hilbert's axiomatic system of Euclidean geometry, brief construction of elementary geometry, advanced Euclidean geometry of triangle and circle (Menalau's and Cave's theorems, nine point circle). A brief historical survey of axiom of parallelism. Hyperbolic geometry up to proving that angle sum of triangle . Elliptic geometry up to proving that angle sum of triangle . Models and consistency of non-Euclidean geometry, measurements in non-Euclidean geometry, projective plane, principle of duality, Desargue's theorem



and Papua's theorem (statements and diagrams only). Quadrangle axiom, separation axiom, axiom of continuity, homogeneous coordinates, projective lien and cross ratio, subgeometries (Affine, Euclidean, hyperbolic and elliptic) of projective geometry. Prerequisite: (1300)

1316 Differential Geometry (3 credits)

Elementary topology in nR (open closed, connected, compact sets, continuous functions, homeomorphisms), vector functions of a vector variable (continuity,

limits, derivative, chain rule, inverse function theorem). Surfaces in nR . Definition, tangent plane, normal line topological properties, oriented surfaces. First and second fundamental forms (properties and application), osculating paraboloid, classification of points on surface, normal curvature, principal curvature, Gauss-Weigarten equations, the fundamental theorem of surfaces. Transformations between surfaces, isometrics, invariant of isometrics, Geodesic curvature, geodesics coordinates, Euler characteristic, Gauss-Bonnet theorem. Prerequisite: (1309+1310)

1400 Linear Algebra II (3 credits) Vector space, subspace and quotient spaces, sum and direct sum of vector space, bases and dimensions, inner product spaces and Gram-Schmidt theorem, orthogonal complement and orthogonal projection, linear forms, bilinear forms and quadratic forms, symmetric, Hermitian and normal forms, linear operators, adjoint operators, self-adjoint and skew-adjoint operators, types of linear operators and their respective matrix representations. Unitary, orthogonal, normal and positive definite operators and their matrix representations, generalized eigenvalues of matrix, Jordon canonical forms and triangulation of matrices, infinite series of matrix calculus. Prerequisite: (1300)

1401 Abstract Algebra II (3 credits) External and Internal Products, Fundamental Theorem of Finite Abelian Groups. Conjugate Classes, Cauchy Theorem and P-groups. Sylow’s Theorems and Its Applications. Solvable Groups. Ideals and Quotient Rings, First Isomorphism Theorem; Principal, Prime and Maximal Ideals. Field Extensions. Finite Fields. Prerequisite: (1301)

1402 Partial Differential Equations (3 credits) Linear and Quasi-linear first order equations, Cauchy problem, solution by methods of Lagrange and Jacobi, linear and quasi-linear second order equations, reduction to canonical forms and solution in special cases. Wave equation: solution of Cauchy problem using D'Alembert formula and by separation of variables. Potential equation: solution of Drichlet problem by Poisson's integral formula and by separation of variables on square region. Heat equation: boundary and initial-value problem and solution by method of eighenfunciton. Prerequisite: (1302)



1404 Mechanics III (3 credits) System of particles, momentum of system of particles, motion of center of mass, angular momentum of system of particle, total external torque acting on system of particles, kinetic energy of system of particles, work potential energy holonomic and non-holonomic constraints, plane motion of rigid body, Euler's theorem, Chasles theorem, the compound pendulum, general plane motion of rigid body, D'Alembert principle, space motion of a rigid bodies, Euler's equation, force free motion, the invariable line and plane, Poinsot's construction, Euler angles, angular velocity and kinetic energy, motion of spinning top, gyroscopes. Lagrange's equation, Hamiltonian theory. Prerequisite: (1304)

1405 Integral Equations (3 credits) Physical and mechanical problems leading to integral equations, linear integral equations: classification and solution, differential equations and integral equations, Fredholm integral equations (1st and 2nd kind), Volterra integral equations (1st and 2nd kind), Characteristic behavior of integral equations, application to ordinary differential equations. Prerequisite: (1201 + 1302)

1406 Real Analysis II (3 credits) Sequences and series of functions, Points-wise and uniform convergence, uniform convergence and properties of continuity, derivative and integration, power series, uniform approximation and Weierstrass approximation theorem, measurable sets and functions, Lebesgue integral and its properties, convergence

theorems, relation of Lebesguc and Riemann integrals, pL -apace and Minkowski inequality. Prerequisite: (1306)

1407 Mathematical Methods II (3 credits) Legendre polynomials, Bessel functions, spherical harmonics, Hermite polynomials, Lagurre polynomials, Chebyshev polynomials, hypergeometric functions, group representation. Prerequisite: (1201+1203)

1408 Riemannian Geometry (3 credits) Atlases, charts, manifolds, morphisms, tangent vectors, tangent spaces, tangent bundle, tensor bundle partition of unity, vector fields and differential equations, differential operators, brackets, lie derivative, differential forms, exterior derivative, closed and exact forms, Poincare lemmas, Riemannian metric, integration of differential forms, oriented manifolds, stokes theorem, canonical Riemannian volume form, divergence theorem. Prerequisite: (1201+1207+1300)

1409 Modern Analysis (3 credits) Continuous functions on metric space, Stone-Weierstrass theorem, fixed point theorem, normed linear spaces, Holder and Minkowski's inequalities, liner transformations, completeness, inner product space, orthogonal sets, Gram-Schmidt orthogonalizations process, Hilbert space. Prerequisite: (Nil)



1410 Topology II (3 credits) Separation axioms, compactness and compactifiation, connectedness and path-connectedness, topological group, groups, group action and orpit space. Homotopy, function spaces, manifold and surfaces. Prerequisite: (1301)

1413 Axiomatic Set Theory (3 credits) Bases and informal set theory and its paradoxes, axiomatization of set theory, ZF axiomatic set theory, algebra of set, functions and relations, Peano's systems and abstract natural number. Number systems, cardinal numbers, ordinal numbers. VNB axiomatic set theory, status of axiom of choice and continuum hypothesis, consistency of set theory, statement of Godel incompleteness theorems. Prerequisite: (1207) 1417 Fluid Mechanics I (3 credits) Fundamentals of fluid mechanics, fluid statics, basic hydrostatic equation, hydrostatic forces on plane and curved summerged surfaces, Lagrangian and Eulerian systems, particle derivative, particle acceleration in cartesian, cylindrical systems. Stream, path and streak lines, Bernonlli's theorem. Hydroyynamic pressure, Pitot and Venturi tube, equation of continuity and equation of motion, rotation, circulation, Stlke's theorem function, complex potential, sources, sinks, and doublets, circle theorem and Blasius theorem. Prerequisite: (1403+1418)

1418 Theory of Elasticity I (3 credits) Basic difinations, stress theory, equilibrium equation, principal surfaces and stresses, strain theory, compatibility strain conditions, principal strains, strain potential energy, relations between stresses and strains, (Hook's law). Basic problems of theory of elasticity, Lame's and Belt, Ramy equations, plane stress and strain fields, torsion of rods, applications. Prerequisite: (Nil)

1419 Biomathematics (3 credits) Mathematical modeling, dynamical systems as differential equations, typical forms of solution of dynamic differential equations, growth equations, Lotka-Volterra prey-predator equations, competition for fixed resources, application of expotential functions in biology, growth of insect population, half-life of radioactive substance, concentration-time curve in blood of dengs injected intramuscularly, subcontanevesly of interperitoneally. Common growth functions, exponential curve, monomolecular curve, logistic curve, carbon dixide time carry in lung during breath-holding. Measurement of cardiac output by Dye-Dilution method, cybermetics and entropy. Relevance of thermodynamic principles to living systems, forms of arterial pulse. Oxygen dept: heart function test, growth of isolated, non-isolated, two confecting populations, differential equations of epidemics. Bacteriology: birth and death process, diffusion through memberance. Prerequisite: (1302+1309)



1421 Field Theory (3 credits) Some basic properties of fields, sub-field and prime fild, polynomial ring xF ,

ideal structure in xF , UFD, PID, field extension-simple, finite and algebraic,

algebraic closure of a field, basic isomorphism theorem of algebraic field theory,

automorphisms and fixed field, finite fields, galosi field of order nPGF , normal

extenstion of field galois group, statement (no proof) of fundamental theorem of galois theory with some illustrations. Prerequisite: (1401)

1430 Combinatorial Mathematics (3 credits) Arrangements, selections and distributions of objects, inclusion-exclusion principle, generating functions, Poy's counting theorem and its applications, recurrence relations. Prerequisite: (1301)

1431 Introduction to Graph Theory (3 credits) Basic concepts of graphs and digraphs, trees, Euclerian property, Hamiltonian property, connectivity, planarity, coloring graphs, counting graphs, and applications. Prerequisite: (Nil)

1432 History of Mathematics (3 credits) Mathematics in civilizations of Babylon and Ancient Egypt, contributions of Indians and Chinese, Greek mathematics, development of mathematics in Arabic and Islamic civilizations, transition of mathematics to Europe, European renaissance, invention of analytic geometry and calculus in 17th and 18th. Centuries, mathematics in 19th. Century: non-Euclidean geometry, theory, abstract algebra. Asprcts of 20th. Century mathematics. Throughout the course main scholars should be mentioned. Prerequisite: (1406)

1433 Mathematical Logic (3 credits) Informal statement calculus, informal predicate calculus, formal statement calculus, formal predicate calculus, first order systems with equality. Mathematical system: theory of groups, arithmetic and formal set theory, Godel on completeness theorems Prerequisite: (1207).

Minor Courses (12-14) Minor Statistics Courses (14 Credits)

2106 Elements of Probability Distributions II (4 Credits) All univariate discrete distributions: Bernoulli, binomial, Poisson, geometric, hyper geometric, negative binomial, multinomial, distributions of exponential, uniform. Law of large numbers. Central limit theorem. Chebychev inequality, normal distribution. Joint distribution. Marginal and conditional distributions. Conditional Expectation and variance. Prerequisite: (2102)

2107 Basic Statistical Methods (4 Credits) Population, sample and sampling distributions point estimation and interval estimation .properties of estimators, biased and unbiased estimators, consistent,



sufficient and efficient estimators. Test of hypothesis: null and alternative hypothesis, critical region, type I and type ll errors, level of significance, degree of freedom. Power of a test, test statistics, one and two tailed tests. Test of hypothesis concerning means, proportions. Equality of means and proportions, variances and their interval estimation .association of attributes, contingency tables and test of independence. And goodness of fit. Elements of analysis of variance: one-way and two-way classification. Prerequisite: (2109)

2207 Distribution theory (3 Credits) Review joint distribution. Marginal distribution and Conditional distribution, Functions of Random variables. Beta and Gamma distributions transformation of variables .Distribution of mean and variance for normal population. Distribution of student’s t , χ2 and F statistics and their properties. Large sample approximations. Prerequisite: (2106)

2307 Advanced Distribution Theory (3 Credits) Bivariate (normal, binomial and Poisson) distributions, multivariate normal distributions, order statistics; distributions of median, quartiles and other order statistics and their properties. Interval limits, tolerance limits, and their limiting distributions (asymptotic properties). Truncated and censored distributions. Ideas of compound distributions (binomial and Poisson). pearsonian types of curves. Prerequisite: (2207)

Minor Physics Courses (12 Credits)

4202 Waves and Oscillations (2 credits) Oscillations: Simple harmonic oscillations of mechanical and electrical oscillators, vector representation of SHM, superposition of SHM's by vector addition and complex exponentials, Lissajous figures, beats. Damped Oscillations: Damped oscillations in mechanical and electrical oscillators, heavy damping, critical damping, damped S.H. oscillations, logarithmic decrement, relaxation time, Q-values Forced Vibrations and Resonance: Undamped oscillator with a harmonic force, forced vibrations with damping, transient phenomena, power absorbed by a driven oscillator, resonance Coupled Oscillations: Spring coupled pendulums, normal coordinates and normal modes of vibrations, superposition of normal modes. Waves: Wave equation, mathematical representation of waves, types of waves, speeds of some mechanical waves, interference of waves, reflection and transmission of transverse waves at a boundary, standing waves, wave groups, and phase and group velocities. Sound waves: Audible, ultrasonic and infrasonic waves, speed of sound waves, vibrating systems and sources of sound, intensity of sound, the Doppler effect. . Prerequisite: ( 4101 + 4103)



4203 Thermodynamics & the Kinetic Thy of Gases (2 credits) Fundamental Concepts: Thermal Equilibrium, the zeroth Law, temperature scales, thermodynamic equilibrium and processes, equations of state, equation of state of an ideal gas, equation of state of a real gas, expansivity and compressibility. P.V.T. surface for an ideal gas The First Law of Thermodynamics: Work in a volume change, other forms of work, the first Law of Thermodynamics, the mechanical equivalents of heat, Heat capacity, Heats of transformation, enthalpy, energy equation of steady flow. Some consequences of the First Law: The energy equation, the Gay-Lussac-Joule experiments and the Joule-Thomson experiment, the Carnot cycle, the heat engine and the refrigerator. Entropy and the Second Law of Thermodynamics: The Second Law of Thermodynamics, Thermodynamic temperature, Entropy, the principle of increase of entropy, the Clausius and Kelvin-Planck statements of the Second Law. Thermodynamics Potentials: The Helmholtz function and Gibbs function, thermodynamic potentials, the Maxwell relations, the Clausius-Clapeyron equation, the third Law of thermodynamics Introductory Kinetic theory: Basic assumptions, molecular flux, equation of state of an ideal gas, the principle of equi-partition of energy, Classical theory of specific heat capacity, specific heat capacity of a solid. Prerequisite: ( 4101)

4204 Practical Physics III (1 credits) Experiments are based on topics covered in 4101. Prer.: (4101+4102)

4212 Electricity and Magnetism (3 credits) Electrostatics: The electric charge, Coulomb's Law, the electric fields and potentials, conductors and insulators, Gauss's law and applications, the electric dipole, Poisson's and Laplace's equations and their solutions in one independent variable, solution to Laplace's equation in spherical coordinates- zonal harmonics and applications Electrostatic field in Dielectric Media: Polarization, fields outside and inside a dielectric, media, Gauss's Law in a dielectric, the electric displacement vector, electric susceptibility and dielectric constant Electrostatic Energy: Potential energy of a group of point charges, electrostatic energy of a charge distribution, energy density of an electrostatic field, Coefficients of capacitance and induction, capacitance Electric current : Nature of the current, current density, equation of continuity, conductivity, Ohm's law Magnetic Field of Steady Currents: The definition of magnetic induction, forces on current carrying conductors, the law of Biot and Savart and applications, Ampere's circuital Law and applications, the magnetic vector and scalar potentials. Faraday's law, motional e.m.f., self-inductance, mutual inductance, inductances in series and in parallel. Prerequisite: (4103)



4214 Alternating Current (1 credits) Slowly varying currents: Kirchhoff's law, elementary transient behavior, steady state behavior of a simple series circuits, series and parallel connection of impedances, RC and RLC circuits, power and power factors, resonance Theorems of circuit analysis: Mesh current network analysis, Node voltage network analysis, Thevein's and Norton's theorem superposition and reciprocity theorems. A.C motors and transformers. Prerequisite: (4103+4202)

4301 Modern Physics (3 credits) Relativity: The Michelson-Morley experiment, Fundamental Postulates of the special theory of relativity, time dilation and length contraction, clock synchronization and simultaneity, the Doppler effect, the Lorentz transformation, the Twin Paradox, relativistic momentum, relativistic energy, mass and Binding energy. The Origin of Quantum Theory: Quantization of electric charge, black body radiation, the photoelectric effect, X-rays and the Compton effect The Old Quantum Theory: Rutherford scattering, Thomson and Bohr models of the hydrogen atom, X-ray spectra. Prerequisite.: (4103 + 4201 + 4202)

Minor computer Courses (12 Credits)

9201 Design of digital computer (3 credits) Understand basics of Logic design techniques. Acquire sound knowledge in the key principles and practices used in the design and analysis of computer logic. Analyze related sequential circuits. Understand the different types of Flip-Flops. Understand the operation and characteristics of Synchronous and Asynchronous counters. Recognize and understand the operation of various types of registers. Distinguish among the various types of memories. Describe the difference between read/write operations. Describe the components of microprocessor. Prerequisite: (9103).

9205 Pascal Language (3 credits) Know the basic skills needed in programming. Be able to write, compile, debug and run a program in Pascal. Understand the uses of all data types in Pascal and will be able to declare data variables of all types and constants in a program. Understand the use of functions and write functions in Pascal. Be able to use pointers in their programs. Be able to use input/output statements in a program. Prerequisite: (9104).

9302 Introduction to assembly language (3 credits) Demonstrate through descriptions, discussions, and or illustrations an understanding of the purpose of an assembly language. Understand the relationship between Assembly language, Machine language and High level language. Discuss the issues involved in hardware and software implementation of an instruction in a digital computer instruction set. Demonstrate the ability to discuss, with appropriate illustrations, the concepts of subroutine calls and



interrupts at the assembly level. Be able to explain the concepts of assembly language directives, operators, macros, and program structure. Students will understand the differences among the main types of programming languages relative to their effect on a digital computer’s processing speed. Prerequisite: Prerequisite: (9104+9201). 9311 Data structures & applications (3 credits) Data structures is an essential area of study for computer scientists and for anyone who will ever undertake a serious programming course. Use and implement fundamental data structures including stacks, queues, priority queues, lists, trees and hash tables. Use specified programming tool to develop and implement computer-based solutions to problems. Develop software by using different data structures studied. Learn to implement search and sorting algorithms including the quick sort, the heap sort and hashing. Learn to use recursion to solve problems. Do a Big-Oh analysis for their implementations of basic data structures. Prerequisite: (9205).

Mathematics Courses for other departments (7 Credits)

1000 General Mathematics for Biology (4 credits) Sets and subsets, basic set operations (union, intersection, and difference), real line. Fractions. Exponentials. Equations, Solve first and second order equations. Functions, domain and range of functions, graph of functions, Limits, one-sided limits and continuity, derivative: differentiation of algebraic function, chain rule, parametric equations and higher order derivatives, differentiation of trigonometric, exponential and logarithmic functions application of derivative. Prerequisite: (Nil) 1206 Calculus and Differential Equations (3 credits) Polar coordinates, polar equations and graphs, intersection of graphs of polar coordinate, area in polar coordinate, the angle , parametric equations, arc

length, curvature, surface area of a solid of revolution. Functions of two or more variables, partial derivatives, limit and continuity. Multiple integral, the double integral, evaluation of double integral, iterated integrals, double integrals in polar coordinates, the triple integral. Equations of first order and first degree, separable equations, homogenous equations, linear equations exact differentials, application of first order equations, second order equations with constant coefficients, application of second order equations, other second order equations. Prerequisite: (1101)

Documents

SYLLABUS OF UNDERGRADUATE CHEMISTRY …sc.uob.edu.ly/assets/.../d7903-discription-of-mathematics-1-.pdf · 13 1413 Axiomatic Set Theory 3 4 1207 14 1417 Fluid Mechanics 3 4 1201-1304