First Semester

*A student can choose any 1 of 4 elective courses offered by the departments of physics, chemistry, philosophy, geology to fulfill a total of 20 credit units. The elective course choosing by the students in this semester is Phys 1001(Physics).

Foundation Courses

မ-၁၀၀၁        မြန်မာစာ

Eng 1001         English

Core Courses

Math 1101       Algebra and Analytic Geometry

Math 1102       Trigonometry and Differential Calculus

Elective Courses

Phys  1001      Physics

Chem 1001      Chemistry

Phil 1005         Mathematical Logic I

Geo 1001        General Geology I

Math 1101      Algebra and Analytic Geometry

Course Description

This course deals with the following concepts.

- logic and sets; polynomial functions; mathematical induction; permutations, combinations and binomial theorem; mathematical induction; lines; circles; conic sections; translation of coordinate axes; rotation of coordinate axes

Learning Outcomes

At the end of the semester, the students should be able to:

• apply the basic concepts of logic in the set theory,
• understand the fundamental theorem of algebra,
• prove the mathematical statements or formulae which are true for all positive integers by using the method of mathematical induction,
• calculate the number of permutations and combinations with or without restrictions on combinatorial problems,
• find the equation of a line and of a circle from given information,
• determine the tangent and normal lines of the circle,
• understand the basic concept of parabola, ellipse and hyperbola; and
• find the equations of parabola, ellipse and hyperbola.

Math 1102      Trigonometry and Differential Calculus

Course Description

This course deals with the following concepts.

-  inverse trigonometric functions; limits; continuity; exponential, logarithmic, hyperbolic functions and their inverses; differentiation of inverse trigonometric functions; differentiation of exponential, logarithmic, hyperbolic functions and their inverses; L’Hospital’s rule; Taylor series

Learning Outcomes

At the end of the semester, the students should be able to:

• explain the inverse trigonometric functions,
• sketch the graphs of six inverse trigonometric functions,
• solve the inverse trigonometric functions and prove the inverse trigonometric identities,
• recognize and apply the properties of the exponential, logarithmic, hyperbolic functions and their inverse functions,
• calculate the derivatives of inverse trigonometric functions, exponential functions, logarithmic functions, hyperbolic functions and their inverse functions,
• calculate the limits of trigonometric functions and the limits at infinity of rational functions,
• understand the L’Hospital’s rule and calculate the values of the limits of functions by using L’Hospital’s rule,
• apply Taylor and Maclaurin series for finding some series of functions.

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Second Semester

*A student can choose any 1 of 4 elective courses offered by the departments of physics, chemistry, philosophy, geology to fulfill a total of 20 credit units. The elective course choosing by the students in this semester is Phys 1002(Physics).

Foundation Courses

မ-၁၀၀၂             မြန်မာစာ

Eng 1002         English

Core Courses

Math 1103       Algebra and Analytical Solid Geometry

Math 1104       Differential and Integral Calculus

Elective Courses

Phys  1002      Physics

Chem 1002      Chemistry

Phil 1006         Mathematical Logic II

Geo 1003        General Geology II

Math 1103      Algebra and Analytical Solid Geometry

Course Description

This course deals with the following concepts.

- determinants; matrices; complex numbers; polar coordinates; graphing in polar coordinates; areas and lengths in polar coordinates; three dimensional Cartesian coordinate system; lines; planes; quadric surfaces

Learning Outcomes

At the end of the semester, the students would be able to:

• determine whether or not a given matrix is invertible,
• solve the system of linear equations by using Gaussian elimination method, Gauss-Jordan reduction method and Cramer’s rule,
• explain the relationship between polar and rectangular coordinates,
• apply the methods of finding the n^th roots of complex numbers and the solutions of simple polynomial equations,
• find the Cartesian coordinates in three dimensions,
• find the equations of the lines and planes,
• analyze the main features of cylinders, ellipsoids, paraboloids and hyperboloids,
• identify and sketch the graphs of quadric surfaces.

Math 1104  Differential and Integral Calculus

Course Description

This course deals with the following concepts.

-  extreme values of functions; the mean value theorem; monotonic functions and the first derivatives test; methods of integration; improper integrals; applications of integration; partial differentiation; ordinary differential equation of first order

Learning Outcomes

At the end of the semester, the students should be able to:

• explain the extreme values of functions and the Mean Value Theorem,
• explain Monotonic functions and the first derivative test,
• understand the basic concept of definite integrals,
• know standard indefinite integrals and basic rules of indefinite integration,
• evaluate the integrals of functions by using the methods of integration,
• find the volumes of solids, lengths of the plane curves and areas of surfaces of revolution by using the methods of integration,
• calculate the improper integrals,
• calculate the partial derivatives by using the Chain rule,

·  identify the different types of first order differential equations and solve them.

First Semester

*A student can choose any 2 of 4 elective courses offered by the departments of mathematics, physics to fulfill a total of 21 credit units. The elective courses choosing by the students in this semester are Math 2104 and Math 2105.

Foundation Courses

Eng      2001  English 

Core Courses

Math    2101 Complex Variables I     

Math    2102 Calculus of Several Variables   

Math    2103 Vector Algebra and Statics       

Elective Courses

Phys    2003  Physics                                                                                  

Math    2104  Discrete Mathematics I                                                         

Math    2105  Theory of Sets I                                                                    

Math    2106  Spherical Trigonometry and Its Applications

Math  2101 Complex Variables I

Course Description

This course deals with the following concepts.

-  analytic functions; elementary functions; integrals; residues and poles

Learning Outcomes

At the end of the semester, the students would be able to:

• explain complex numbers algebraically and geometrically,
• define and analyze limits and continuity for complex functions as well as consequences of continuity,

• apply the concept and consequences of analyticity and the Cauchy-Riemann equations and of

   results on harmonic and entire functions including the fundamental theorem of algebra,

• evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem in its various versions, and the Cauchy integral formula,

• explain the functions as Taylor, power and Laurent series,
• classify singularities and poles, compute residues at these poles and evaluate complex integrals using the residue theorem.

Math  2102 Calculus of Several Variables

Course Description

This course deals with the following concepts.

- functions of two or more variables; partial derivatives; directional derivatives; chain rule for partial derivatives; total differential; maxima and minima; exact differentials; derivatives of integrals; double integrals in Cartesian coordinates and polar coordinates; triple integrals in Cartesian coordinates, cylindrical coordinates and spherical coordinates; applications of multiple integrals

Learning Outcomes

At the end of the semester, the students would be able to:

• discuss functions of two or more variables, partial derivatives and directional derivatives,
• apply chain rule for finding partial derivatives and total differential,
• find maximum and minimum values of a function,
• find the derivatives of integrals in Cartesian coordinates and polar coordinates, and
• apply the properties of multiple integrals.

Math  2103  Vector Algebra and Statics

Course Description

This course deals with the following concepts.

- dot products; cross products; triple scalar products; triple vector products; statics of a particle; tension of a string; friction; moment and couples; centre of gravity; statics of a rigid body in a plane; jointed rods; virtual works; stability

Learning Outcomes

At the end of the semester, the students would be able to:

• use vector algebra in the analysis of forces, moments and couple,
• identify special equilibrium situations of two forces and three forces in a plane,
• calculate the friction in equilibrium of a particle on a rough horizontal plane and rough inclined plane acted by external forces,

• solve the centre of gravity of a body, centroids of the surface and the centre of gravity of compound body,
• calculate the stable or unstable positions of equilibrium of the system by using the potential energy of any system.

Math  2104  Discrete Mathematics I

Course Description

This course deals with the following concepts.

-  basic principle; permutations and combinations; algorithms for generating permutations and combinations; generalized permutations and combinations; binomial coefficients and combinatorial identities; the pigeonhole principle; solving recurrence relations, applications to the analysis of algorithms

Learning Outcomes

At the end of the semester, the students would be able to:

• recognize and apply the basic principle of permutations and combinations,
• apply the addition and multiplication principles for counting problems,
• apply combinatorial ideas to practical problems,
• apply Pigeonhole Principle to practical problems,
• solve recurrence relations and apply to the analysis of algorithms.

Math  2105  Theory of Sets I

Course Description

This course deals with the following concepts and theories.

-  cardinal numbers; partially and totally ordered sets

Learning Outcomes

At the end of the semester, the students would be able to:

• discuss the development of the axiomatic view of set theory,
• understand what the cardinal numbers is, and discuss the cardinal numbers of equivalent sets,
• discuss and prove Cantor’s Theorem,
• discuss the status of the Continuum hypothesis,
• characterize the elements such as first of last element as well as maximal or minimal elements in the ordered sets.

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Second Semester

A student can choose any 2 of 4 elective courses offered by the departments of mathematics, physics to fulfill a total of 21 credit units. The elective courses choosing by the students in this semester are Math 2110 and Math 2111.

Foundation Courses

Eng      2002    English                                                                 

Core Courses

Math    2107    Linear Algebra I                                                   

Math    2108    Ordinary Differential Equations                          

Math    2109    Vector Calculus and Dynamics                           

Elective Courses

Phys    2004    Physics                                                                 

Math    2110    Discrete Mathematics II                                      

Math    2111    Theory of Sets II                                                  

Math    2112    Astronomy                                                           

Math  2107    Linear Algebra I

Course Description

This course deals with the following concepts.

- vector spaces; subspaces; bases and dimensions; sums and direct sums; linear mapping; kernel  and image of linear map and their dimension; compositions of linear mapping

Learning Outcomes

At the end of the semester, the students would be able to:

• explain the properties of a vector space,
• explain why a set equipped with the necessary operations is or is not a vector space,
• explain why a subset of a given vector space V is or is not a subspace of V,
• recognize and use basic properties of subspaces and vector spaces,
• explain what is meant by the span of a set of vectors,
• explain what is meant by the dimension of a vector space,
• determine the existence of a basis of a vector space,
• determine a basis and the dimension of a finite-dimensional space,
• find a matrix representation for the linear transformation,
• describe geometrically significant linear transformations of the plane to itself,
• interpret a matrix product as a composition of linear transformations,
• discuss the kernel and image of a linear transformation in terms of nullity and rank of the matrix.

Math  2108    Ordinary Differential Equations

Course Description

This course deals with the following concepts.

-  general and particular solution; slope fields and solution curve; separable; exact; linear; homogeneous; reducible second order equations; Bernoulli equation

Learning Outcomes

At the end of the semester, the students would be able to:

• find general and particular solutions of ordinary differential equations,
• solve separable equations, exact equations, Bernoulli equations by using various methods,
• know how to solve the reducible second-order differential equations.

Math  2109    Vector Calculus and Dynamics

Course Description

This course deals with the following concepts.

- gradient; divergence; curl; line integrals; Green’s theorem; divergence theorem; Stoke’s theorem kinematics of a particle; relative velocity; mass; momentum and force; Newton’s law of motion; work; power and energy; simple harmonic motions; kinematics of a particle in two dimensions; kinetic of a particle in two dimensions

Learning Outcomes

At the end of the semester, the students would be able to: 

• calculate directional derivatives and gradients,
• apply gradient to solve the problem involving normal vectors to level surfaces,
• calculate the velocity, acceleration, mass, force, linear momentum, work done, power and energy of a body,
• solve the problems of simple harmonic motion,
• use the motion of a particle in two dimensions under the action of a central force.

Math  2110    Discrete Mathematics II

Course Description

This course deals with the following concepts.

-  introduction to graph theory; paths and cycles; Hamiltonian cycles and the traveling salesperson problem; a shortest path algorithm; representations of graphs; isomorphism of graphs; planar graphs; terminology and characterizations of trees; spanning trees; minimal spanning trees; binary trees

Learning Outcomes

At the end of the semester, the students would be able to: 

• discuss some important concepts in Graph Theory,
• find a shortest path by using shortest path algorithm,
• discuss on subclasses of trees and some applications of trees, such as spanning trees, game trees,
• characterize the various types of trees and apply the methods for traversing trees.

Math  2111    Theory of Sets II

Course Description

This course deals with the following concepts and theories.

-  well-ordered sets; ordinal numbers; axiom of choice; Zorn’s lemma; well-ordering theorem

Learning Outcomes

At the end of the semester, the students would be able to:

• explain the basic concepts of well-ordered sets and ordinals numbers,
• discuss the comparison of well-ordered sets,
• discuss the comparison of ordinal numbers,
• understand the relation between the ordinal and cardinal numbers,
• identify the axiomatic of theorems of set theory; for example the Zermelo’s Postulate, the Axiom of Choice.

First Semester

*A student can choose any 1 of 2 elective courses offered by the departments of mathematics to fulfill a total of 22 credit units. The elective course choosing by the students (3 yr students/1^st yr hons.  students) in this semester is Math 3105/ Math3205.

Foundation Courses

Eng      3001    English                                                                 

Core Courses

Math  3101         Analysis I       

Math  3102        Linear Algebra II       

Math  3103         Differential Equations

Math  3104         Differential Geometry

Elective Courses

Math  3105         Tensor Analysis                      

Math  3106         Number Theory I                   

Math  3101   Analysis I

Course Description

This course deals with the following concepts and theories.

-  elements of set theory; numerical sequences and series

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the concepts of finite, countable and uncountable sets,
• explain the Metric spaces together with Euclidean spaces as examples,
• characterize the behavior of compact sets, perfect sets and connected sets,
• explain the definition of a convergence and the relation between sequences and series,
• find the limits of convergent sequences and apply the tests for convergence of series.

Math  3102    Linear Algebra II

Course Description

This course deals with the following concepts and theories.

-  linear maps and matrices; determinants

Learning Outcomes

At the end of the semester, the students would be able to:

• find the matrix representing a linear transformation,
• understand the relationship between a linear transformation and its matrix representation,
• find the solution set of a system of linear equations by using Gauss elimination and Cramer’s rule,
• know about the determinants and their properties,
• determine the sign of a permutation,
• investigate the relation between permutations and determinant of a Square matrix,
• determine the determinant of a linear map to be a matrix representing this linear map.

Math   3103     Differential Equations

Course Description

This course deals with the following concepts and theories.

- elementary of series; series solution near ordinary points; regular singular points; method  of Frobenius; Bessel's equation

Learning Outcomes

At the end of the semester, the students would be able to:

• know the analytic function, ordinary point, regular singular point and irregular singular point,
• find the power series solutions for a differential equation near an ordinary point and near the regular singular point using the method of Frobenius,
• find the power series solutions of Legendre’s equation, Bessel’s equation and Frobenius series solution of a differential equation,
• understand the Gamma function and Bessel function,
• explain the Laplace transform of some elementary functions and inverse Laplace transform functions using either differentiation of transform or integration of transform,
• calculate the Laplace transform of convolution of two functions.

Math  3104   Differential Geometry

Course Description

This course deals with the following concepts and theories.

-  concept of a curve; curvature and torsion; the theory of curves; concept of a surface

Learning Outcomes

At the end of the semester, the students would be able to:

• analyse the equivalence of two curves applying some theorems,
• explain parametrization surfaces and tangent spaces of surfaces,
• explain differential maps between surfaces and find derivatives of such maps,
• integrate differential forms on surfaces,
• understand the concept of manifolds and investigate their properties.

Math  3105   Tensor Analysis

Course Description

This course deals with the following concepts and theories.

-  curvilinear coordinates; tensor analysis

Learning Outcomes

At the end of the semester, the students would be able to:

• find the tangent and normal of coordinate surfaces,
• express the divergence, gradient, curl of a vector or scalar field in terms of orthogonal curvilinear coordinates,
• understand the concept of tensor variables and why to use the tensor analysis,
• find tensor forms of gradient, divergence and curl,
• derive base vectors, metric tensors and strain tensors in an arbitrary coordinate system.

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Second Semester

*A student can choose any 1 of 2 elective courses offered by the departments of mathematics to fulfill a total of 22 credit units. The elective course choosing by the students (3 yr students/1^st yr hons.  students) in this semester is Math 3111/Math 3211.

Foundation Courses

Eng      3002    English                                                                 

Core Courses

Math    3107       Analysis II     

Math    3108       Linear Algebra III      

Math    3109       Mechanics      

Math    3110       Probability and Statistics       

Elective Courses

Math    3111       Complex Variables II                         

Math    3112       Number Theory II                  

Math  3107   Analysis II

Course Description

This course deals with the following concepts and theories.

-  continuity; differentiation

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the concepts of limits of functions, continuous functions and limits of continuity,
• explain the continuity on a compact set and on a connected set,
• discuss the discontinuity of a function,
• explain the definition of derivative of a real function and the continuity of derivatives,
• determine the derivatives of higher order.

Math  3108   Linear Algebra III

Course Description

This course deals with the following concepts and theories.

-  scalar products and orthogonality; matrices and bilinear maps

Learning Outcomes

At the end of the semester, the students would be able to:

• explain the concepts of scalar products, positive definite products and hermitian products,
• find the length of a vector in some vector spaces and the angle between two vectors,
• know the concepts of functional and dual spaces,
• investigate the rank of a matrix and apply this to determine the dimension of the solutions space of the homogeneous system,
• understand about the bilinear map defined on the vector spaces over the field K,
• know the concepts of bilinear form, quadratic form and hermitian form, and consider the matrix representing the bilinear form,
• explain the operators, such as symmetric operator, adjoint operator, hermitian operator and unitary operators.

Math  3109   Mechanics

Course Description

This course deals with the following concepts and theories.

- impulsive forces; central force motion; kinematics of plane rigid bodies; kinetics of plane rigid bodies; impact; dynamics of a particle in three dimensions; dynamics of system of particles; moment of inertia; polar coordinates; orbits

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the impulse and impulsive force,
• find the velocities of two body after direct impact or oblique impact of two body using the principle of linear momentum or Newton’s experimental law,
• calculate the moment of inertia of a rigid body about any axis using the parallel axes theorem and perpendicular axes theorem,
• solve the equations of two-dimensional motion of a rigid body,
• know the motion of a particle under a central force,
• understand the principles of energy and momentum applied to central orbits.

Math   3110    Probability and Statistics

Course Description

This course deals with the following concepts and theories.

-  introduction to probability theory; random variables; mean; median; mode; standard deviation; correlation; regression

Learning Outcomes

At the end of the semester, the students would be able to:

• recognize the role of probability theory,
• know the concepts of sample space, events and compute the probability and conditional probability of events,
• understand and apply the concepts of discrete and continuous random variables, the discrete and continuous probability distribution and the jointed probability distribution,
• apply theorems concerning the distributions of functions of random variables and the moment generating functions,
• evaluate several types of averages or measures of central tendency and the root mean square,
• compute the range, mean deviation, interquartile range, and the standard deviation,
• calculate the standard error of estimate,
• illustrate the basic concepts of regression lines and correlation theory.

Math  3111   Complex Variables II

Course Description

This course deals with the following concepts and theories.

-  conformal mapping; application of conformal mapping

Learning Outcomes

At the end of the semester, the students would be able to:

• recognize the mappings which are conformal and those which are not,
• describe conformal mappings between various plane regions,
• understand the mapping properties of elementary functions and some special transcendental functions,
• understand relations between conformal mappings and quadratic differentials and how to change geometric structures under conformal mappings, solve the boundary value problem involving laplace equation and boundary conditions.

First Semester

*A student can choose any 1 of 2 elective courses offered by the departments of mathematics to fulfill a total of 22 credit units. The elective course choosing by the students (4 yr students/2^nd yr hons. students) in this semester is Math 4105/ Math 4205.

Foundation Courses

Eng      4001    English                                                                 

Core Courses

Math    4101   Analysis III                                                  

Math    4102   Numerical Analysis I                                   

Math    4103   Linear Programming

Math    4104   Partial Differential Equations                                                                   

Elective Courses

Math    4105  Stochastic Process I                                                     

Math    4106  Fundamentals of Algorithms and Computer Programming                                

Math  4101  Analysis III

Course Description

This course deals with the following concepts and theories.

-  methods of mathematical research; the Riemann-Stieltjes integral

Learning Outcomes

At the end of the semester, the learners would be able to:

• know the definition and existence of the integrals,
• characterize the behavior of the Riemann-Stieltjes integral,
• understand how to use the theorems for differentiation and integration,
• find the integral of vector-valued functions.

Math  4102  Numerical Analysis I

Course Description

This course deals with the following concepts and theories.

-  numerical methods in general; numerical methods in linear algebra

Learning Outcomes

At the end of the semester, the students would be able to:

• explain the errors in computation, round-off rules and zero of a function,
• solve the equations by using numerical methods,
• solve the system of equations by using numerical methods,
• understand about interpolating and how to use the methods of interpolation and errors,
• compute the numerical differentiation and integration.

Math  4103    Linear Programming

Course Description

This course deals with the following concepts and theories.

- basic properties of linear programs; the simplex method; duality; dual simplex method and

  primal dual algorithms

Learning Outcomes

At the end of the semester, the students would be able to:

• explain the basic concept of optimization and the procedure for mathematical modeling of a linear programming,
• construct linear programming models for various type of problems,
• solve a problem modeled with linear program,
• distinguish the feasible solution, optimal solution and basic feasible solution,
• solve the linear programming models by using simplex Algorithm and its different types,
• associate the primal and dual models,
• construct the dual model of a given linear programming model.

Math  4104    Partial Differential Equations

Course Description

This course deals with the following concepts and theories.

- classification; canonical form of hyperbolic/ parabolic/elliptic; heat equation in one dimension;

   wave equation in one dimension

Learning Outcomes

At the end of the semester, the students would be able to:

• understand on the formulation of first and second order partial differential equations(PDEs) for three basic types of hyperbolic, parabolic and elliptic equations,
• know general classification of partial differential equations,
• apply analytical methods, and physically interpret the solutions,
• know concepts of partial differential equations and how to solve linear Partial Differential with different methods (such asusing Fourier series and Fourier Integral solving Homogeneous Heat, Wave , Laplace’s Equations, Characteristics Method),
• find the solutions of PDEs are determined by conditions at the boundary of the spatial domain and initial conditions at time zero,
• apply the technique of separation of variables to solve PDEs and analyze the behavior of solutions in terms of eigen function expansions,
• know how to transform PDE into canonical form and solve linear partial differential equations of both first and second order.

Math  4105    Stochastic Process I

Course Description

This course deals with the following concepts and theories.

-  stochastic processes in discrete case and continuous case; computing expectations by conditioning; computing variances by conditioning; computing probabilities by conditioning transition probabilities; Random walk model; a Gambling model

Learning Outcomes

At the end of  the semester, the students would be able to:

• sample any type of continuous or discrete time stochastic process on a computer,
• identify appropriate stochastic process model(s) for a given research or applied problem,
• apply the theory to model real phenomena and answer some questions in applied sciences,
• calculate the expectation by conditioning,
• calculate the distribution of a Markov chain at a given time,
• classify the states of a Markov chain and
• determine the stationary distributions of a Markov chain.

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Second Semester

*A student can choose any 1 of 2 elective courses offered by the departments of mathematics to fulfill a total of 22 credit units. The elective course choosing by the students (4 yr students/2^nd yr hons.  students)  in this semester is Math 4111/ Math 4211.

Foundation Courses

Eng      4002  English                                                                   

Core Courses

Math    4107    Analysis IV                                                  

Math    4108   General Topology I                                      

Math    4109   Abstract Algebra I                                       

Math    4110   Hydromechanics                                          

Elective Courses

Math    4111  Stochastic Process II                                                    

Math    4112  Integer Programming                                                   

Math  4107   Analysis IV

Course Description

This course deals with the following concepts.

-  sequences and series of functions

Learning Outcomes

At the end of  the semester, the students would be able to:

• know how to manipulate the notation of uniform convergence,
• differentiate the differences between pointwise convergence and uniform convergence,
• explain the relations between uniform convergence and continuity, integrability, differentiablity,
• understand how to use the Stone-Weierstrass theorem.

Math  4108  General Topology I

Course Description

This course deals with the following concepts and theories.

-  topology of the line and plane; topological spaces; bases and subbases

Learning Outcomes

At the end of the semester, the students would be able to:

• know properties of points and set in line,
• find the sequence of integers especially Cauchy sequences,
• discuss the continuous properties on R and R²,
• understand some properties of points and sets in topological space and discuss the neighborhood system of points in topological space,
• explain base and subbase for a topology on topological space.

Math  4109  Abstract Algebra I

Course Description

This course deals with the following concepts and theories.

- definitions and examples of groups; some simple remarks; subgroups; Lagrange’s theorem; homomorphisms and normal subgroups; factor groups; the homomorphism theorems

Learning Outcomes

At the end of the semester, the students would be able to:

• know about the concepts on groups, subgroups and related properties of them,
• explain the equivalence relation into group and its properties, the cosets of a group and know how to use Lagrange’s theorem,
• construct and manipulate group homomorphism and isomorphism,
• know about the cyclic groups and their related properties with homomorphism,
• explain about the normal subgroups of a group and consider the relation with homomorphism,
• create factor groups using normal subgroups and homomorphism theorem and interpret element of factor groups accurately.

Math  4110  Hydromechanics

Course Description

This course deals with the following concepts and theories.

- density and specific gravity; theorems on fluid-pressure under gravity; pressure of heavy fluids;  thrusts on curved surfaces and floating bodies;  stability of floating bodies;  equation of continuity; equation of motion;  some three dimensional flows

Learning Outcomes

At the end of the semester, the students would be able to:

• Solve hydrostatic problems,
• calculate the pressure distribution for incompressible fluids,
• calculate the hydrostatic pressure and force on plane and curved surfaces,
• demonstrate a foundation in the fundamentals of fluid mechanics and relevant analytical, numerical and experimental approaches,
• understand the dynamics of fluid flows and the governing non-dimensional parameters,
• apply concepts of mass, momentum and energy conservation to flows,
• derive fluid equations of mass, momentum and energy conservation, Euler’s and Bernoulli’s equations for inviscid fluid equations, irrotational Bernoulli Equation,
• find the stream functions for incompressible flows and exact solutions,
• make into model of different flows from a combination of uniform flows, sources, sinks and doublets,
• use the continuity equation to determine whether an inviscid flow is incompressible.

Math  4111   Stochastic Process II

Course Description

This course deals with the following concepts.

-  Markov chains; the exponential distribution;  the Poisson process

Learning Outcomes

At the end of the semester, the students would be able to:

• describe the gambler’s ruin problem in terms of a discrete time Markov chain,
• calculate the probability of ruin and the expected duration of a game in the gambler’s ruin problem,
• compute the probability and percentile and notice the Lack of Memory property for exponential, understand the structure of the Poisson Processes, and the special properties such as the interarrival time follows Exponential distribution.

First Semester

*A student can choose any 1 of 2 elective courses offered by the departments of mathematics to fulfill a total of 22 credit units. The elective course choosing by the students (3 yr students/1^st yr hons.  students) in this semester is Math 3105/ Math3205.

Foundation Courses

Eng      3001    English                                                                 

Core Courses

Math  3201         Analysis I       

Math  3202        Linear Algebra II       

Math  3203         Differential Equations

Math  3204        Differential Geometry

Elective Courses

Math  3205         Tensor Analysis                      

Math  3206         Number Theory I                   

Math  3201   Analysis I

Course Description

This course deals with the following concepts and theories.

-  elements of set theory; numerical sequences and series

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the concepts of finite, countable and uncountable sets,
• explain the Metric spaces together with Euclidean spaces as examples,
• characterize the behavior of compact sets, perfect sets and connected sets,
• explain the definition of a convergence and the relation between sequences and series,
• find the limits of convergent sequences and apply the tests for convergence of series.

Math  3202    Linear Algebra II

Course Description

This course deals with the following concepts and theories.

-  linear maps and matrices; determinants

Learning Outcomes

At the end of the semester, the students would be able to:

• find the matrix representing a linear transformation,
• understand the relationship between a linear transformation and its matrix representation,
• find the solution set of a system of linear equations by using Gauss elimination and Cramer’s rule,
• know about the determinants and their properties,
• determine the sign of a permutation,
• investigate the relation between permutations and determinant of a Square matrix,
• determine the determinant of a linear map to be a matrix representing this linear map.

Math   3203     Differential Equations

Course Description

This course deals with the following concepts and theories.

- elementary of series; series solution near ordinary points; regular singular points; method  of Frobenius; Bessel's equation

Learning Outcomes

At the end of the semester, the students would be able to:

• know the analytic function, ordinary point, regular singular point and irregular singular point,
• find the power series solutions for a differential equation near an ordinary point and near the regular singular point using the method of Frobenius,
• find the power series solutions of Legendre’s equation, Bessel’s equation and Frobenius series solution of a differential equation,
• understand the Gamma function and Bessel function,
• explain the Laplace transform of some elementary functions and inverse Laplace transform functions using either differentiation of transform or integration of transform,
• calculate the Laplace transform of convolution of two functions.

Math  3204   Differential Geometry

Course Description

This course deals with the following concepts and theories.

-  concept of a curve; curvature and torsion; the theory of curves; concept of a surface

Learning Outcomes

At the end of the semester, the students would be able to:

• analyse the equivalence of two curves applying some theorems,
• explain parametrization surfaces and tangent spaces of surfaces,
• explain differential maps between surfaces and find derivatives of such maps,
• integrate differential forms on surfaces,
• understand the concept of manifolds and investigate their properties.

Math  3205   Tensor Analysis

Course Description

This course deals with the following concepts and theories.

-  curvilinear coordinates; tensor analysis

Learning Outcomes

At the end of the semester, the students would be able to:

• find the tangent and normal of coordinate surfaces,
• express the divergence, gradient, curl of a vector or scalar field in terms of orthogonal curvilinear coordinates,
• understand the concept of tensor variables and why to use the tensor analysis,
• find tensor forms of gradient, divergence and curl,
• derive base vectors, metric tensors and strain tensors in an arbitrary coordinate system.

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Second Semester

*A student can choose any 1 of 2 elective courses offered by the departments of mathematics to fulfill a total of 22 credit units. The elective course choosing by the students (3 yr students/1^st yr hons.  students) in this semester is Math 3111/Math 3211.

Foundation Courses

Eng      3002    English                                                                 

Core Courses

Math    3207       Analysis II     

Math    3208       Linear Algebra III      

Math    3209       Mechanics      

Math    3210       Probability and Statistics       

Elective Courses

Math    3211       Complex Variables II                         

Math    3212       Number Theory II                  

Math  3207   Analysis II

Course Description

This course deals with the following concepts and theories.

-  continuity; differentiation

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the concepts of limits of functions, continuous functions and limits of continuity,
• explain the continuity on a compact set and on a connected set,
• discuss the discontinuity of a function,
• explain the definition of derivative of a real function and the continuity of derivatives,
• determine the derivatives of higher order.

Math  3208    Linear Algebra III

Course Description

This course deals with the following concepts and theories.

-  scalar products and orthogonality; matrices and bilinear maps

Learning Outcomes

At the end of the semester, the students would be able to:

• explain the concepts of scalar products, positive definite products and hermitian products,
• find the length of a vector in some vector spaces and the angle between two vectors,
• know the concepts of functional and dual spaces,
• investigate the rank of a matrix and apply this to determine the dimension of the solutions space of the homogeneous system,
• understand about the bilinear map defined on the vector spaces over the field K,
• know the concepts of bilinear form, quadratic form and hermitian form, and consider the matrix representing the bilinear form,
• explain the operators, such as symmetric operator, adjoint operator, hermitian operator and unitary operators.

Math 3209   Mechanics

Course Description

This course deals with the following concepts and theories.

- impulsive forces; central force motion; kinematics of plane rigid bodies; kinetics of plane rigid bodies; impact; dynamics of a particle in three dimensions; dynamics of system of particles; moment of inertia; polar coordinates; orbits

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the impulse and impulsive force,
• find the velocities of two body after direct impact or oblique impact of two body using the principle of linear momentum or Newton’s experimental law,
• calculate the moment of inertia of a rigid body about any axis using the parallel axes theorem and perpendicular axes theorem,
• solve the equations of two-dimensional motion of a rigid body,
• know the motion of a particle under a central force,
• understand the principles of energy and momentum applied to central orbits.

Math 3211    Probability and Statistics

Course Description

This course deals with the following concepts and theories.

-  introduction to probability theory; random variables; mean; median; mode; standard deviation; correlation; regression

Learning Outcomes

At the end of the semester, the students would be able to:

• recognize the role of probability theory,
• know the concepts of sample space, events and compute the probability and conditional probability of events,
• understand and apply the concepts of discrete and continuous random variables, the discrete and continuous probability distribution and the jointed probability distribution,
• apply theorems concerning the distributions of functions of random variables and the moment generating functions,
• evaluate several types of averages or measures of central tendency and the root mean square,
• compute the range, mean deviation, interquartile range, and the standard deviation,
• calculate the standard error of estimate,
• illustrate the basic concepts of regression lines and correlation theory.

Math 3211   Complex Variables II

Course Description

This course deals with the following concepts and theories.

-  conformal mapping; application of conformal mapping

Learning Outcomes

At the end of the semester, the students would be able to:

• recognize the mappings which are conformal and those which are not,
• describe conformal mappings between various plane regions,
• understand the mapping properties of elementary functions and some special transcendental functions,
• understand relations between conformal mappings and quadratic differentials and how to change geometric structures under conformal mappings, solve the boundary value problem involving laplace equation and boundary conditions.

First Semester

*A student can choose any 1 of 2 elective courses offered by the departments of mathematics to fulfill a total of 22 credit units. The elective course choosing by the students (4 yr students/2^nd yr hons. students) in this semester is Math 4105/ Math 4205.

Foundation Courses

Eng      4001    English                                                                 

Core Courses

Math    4201   Analysis III                                                  

Math    4202   Numerical Analysis I                                   

Math    4203   Linear Programming

Math    4204   Partial Differential Equations                                                                   

Elective Courses

Math    4205  Stochastic Process I                                                     

Math    4206  Fundamentals of Algorithms and Computer Programming                                

Math  4201  Analysis III

Course Description

This course deals with the following concepts and theories.

-  methods of mathematical research; the Riemann-Stieltjes integral

Learning Outcomes

At the end of the semester, the learners would be able to:

• know the definition and existence of the integrals,
• characterize the behavior of the Riemann-Stieltjes integral,
• understand how to use the theorems for differentiation and integration,
• find the integral of vector-valued functions.

Math  4202   Numerical Analysis I

Course Description

This course deals with the following concepts and theories.

-  numerical methods in general; numerical methods in linear algebra

Learning Outcomes

At the end of the semester, the students would be able to:

• explain the errors in computation, round-off rules and zero of a function,
• solve the equations by using numerical methods,
• solve the system of equations by using numerical methods,
• understand about interpolating and how to use the methods of interpolation and errors,
• compute the numerical differentiation and integration.

Math  4203    Linear Programming

Course Description

This course deals with the following concepts and theories.

- basic properties of linear programs; the simplex method; duality; dual simplex method and

  primal dual algorithms

Learning Outcomes

At the end of the semester, the students would be able to:

• explain the basic concept of optimization and the procedure for mathematical modeling of a linear programming,
• construct linear programming models for various type of problems,
• solve a problem modeled with linear program,
• distinguish the feasible solution, optimal solution and basic feasible solution,
• solve the linear programming models by using simplex Algorithm and its different types,
• associate the primal and dual models,
• construct the dual model of a given linear programming model.

Math  4204    Partial Differential Equations

Course Description

This course deals with the following concepts and theories.

- classification; canonical form of hyperbolic/ parabolic/elliptic; heat equation in one dimension;

   wave equation in one dimension

Learning Outcomes

At the end of the semester, the students would be able to:

• understand on the formulation of first and second order partial differential equations(PDEs) for three basic types of hyperbolic, parabolic and elliptic equations,
• know general classification of partial differential equations,
• apply analytical methods, and physically interpret the solutions,
• know concepts of partial differential equations and how to solve linear Partial Differential with different methods (such asusing Fourier series and Fourier Integral solving Homogeneous Heat, Wave , Laplace’s Equations, Characteristics Method),
• find the solutions of PDEs are determined by conditions at the boundary of the spatial domain and initial conditions at time zero,
• apply the technique of separation of variables to solve PDEs and analyze the behavior of solutions in terms of eigen function expansions,
• know how to transform PDE into canonical form and solve linear partial differential equations of both first and second order.

Math  4205    Stochastic Process I

Course Description

This course deals with the following concepts and theories.

-  stochastic processes in discrete case and continuous case; computing expectations by conditioning; computing variances by conditioning; computing probabilities by conditioning transition probabilities; Random walk model; a Gambling model

Learning Outcomes

At the end of  the semester, the students would be able to:

• sample any type of continuous or discrete time stochastic process on a computer,
• identify appropriate stochastic process model(s) for a given research or applied problem,
• apply the theory to model real phenomena and answer some questions in applied sciences,
• calculate the expectation by conditioning,
• calculate the distribution of a Markov chain at a given time,
• classify the states of a Markov chain and
• determine the stationary distributions of a Markov chain.

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Second Semester

*A student can choose any 1 of 2 elective courses offered by the departments of mathematics to fulfill a total of 22 credit units. The elective course choosing by the students (4 yr students/2^nd yr hons.  students)  in this semester is Math 4111/ Math 4211.

Foundation Courses

Eng      4002  English                                                                   

Core Courses

Math    4207    Analysis IV                                                  

Math    4208   General Topology I                                      

Math    4209   Abstract Algebra I                                       

Math    4210   Hydromechanics                                          

Elective Courses

Math    4211  Stochastic Process II                                                    

Math    4212  Integer Programming                                                   

Math  4207   Analysis IV

Course Description

This course deals with the following concepts.

-  sequences and series of functions

Learning Outcomes

At the end of  the semester, the students would be able to:

• know how to manipulate the notation of uniform convergence,
• differentiate the differences between pointwise convergence and uniform convergence,
• explain the relations between uniform convergence and continuity, integrability, differentiablity,
• understand how to use the Stone-Weierstrass theorem.

Math  4108  General Topology I

Course Description

This course deals with the following concepts and theories.

-  topology of the line and plane; topological spaces; bases and subbases

Learning Outcomes

At the end of the semester, the students would be able to:

• know properties of points and set in line,
• find the sequence of integers especially Cauchy sequences,
• discuss the continuous properties on R and R²,
• understand some properties of points and sets in topological space and discuss the neighborhood system of points in topological space,
• explain base and subbase for a topology on topological space.

Math  4209  Abstract Algebra I

Course Description

This course deals with the following concepts and theories.

- definitions and examples of groups; some simple remarks; subgroups; Lagrange’s theorem; homomorphisms and normal subgroups; factor groups; the homomorphism theorems

Learning Outcomes

At the end of the semester, the students would be able to:

• know about the concepts on groups, subgroups and related properties of them,
• explain the equivalence relation into group and its properties, the cosets of a group and know how to use Lagrange’s theorem,
• construct and manipulate group homomorphism and isomorphism,
• know about the cyclic groups and their related properties with homomorphism,
• explain about the normal subgroups of a group and consider the relation with homomorphism,
• create factor groups using normal subgroups and homomorphism theorem and interpret element of factor groups accurately.

Math  4210  Hydromechanics

Course Description

This course deals with the following concepts and theories.

- density and specific gravity; theorems on fluid-pressure under gravity; pressure of heavy fluids;  thrusts on curved surfaces and floating bodies;  stability of floating bodies;  equation of continuity; equation of motion;  some three dimensional flows

Learning Outcomes

At the end of the semester, the students would be able to:

• Solve hydrostatic problems,
• calculate the pressure distribution for incompressible fluids,
• calculate the hydrostatic pressure and force on plane and curved surfaces,
• demonstrate a foundation in the fundamentals of fluid mechanics and relevant analytical, numerical and experimental approaches,
• understand the dynamics of fluid flows and the governing non-dimensional parameters,
• apply concepts of mass, momentum and energy conservation to flows,
• derive fluid equations of mass, momentum and energy conservation, Euler’s and Bernoulli’s equations for inviscid fluid equations, irrotational Bernoulli Equation,
• find the streamfunctions for incompressible flows and exact solutions,
• make into model of different flows from a combination of uniform flows, sources, sinks and doublets,
• use the continuity equation to determine whether an inviscid flow is incompressible.

Math  4211   Stochastic Process II

Course Description

This course deals with the following concepts.

-  Markov chains; the exponential distribution;  the Poisson process

Learning Outcomes

At the end of the semester, the students would be able to:

• describe the gambler’s ruin problem in terms of a discrete time Markov chain,
• calculate the probability of ruin and the expected duration of a game in the gambler’s ruin problem,
• compute the probability and percentile and notice the Lack of Memory property for exponential, understand the structure of the Poisson Processes, and the special properties such as the interarrival time follows Exponential distribution.

First Semester

Math  5201  Analysis V

Course Description

This course deals with the following concepts.

- power series; the exponential and logarithmic functions; the trigonometric functions; fourier               series

Learning Outcomes

At the end of the semester, the students would be able to:

• relate concepts of uniform continuity, differentiation, integration and uniform convergence,
• apply the definition of convergence to sequence, series and functions,
• illustrate the convergence properties of power series and fourier series,
• define and analyze limits and continuity for functions as well as consequences of continuity,
• explain Perseval’s theorem, Gamma function and Stirling’s formula.

Math  5202  General Topology II

Course Description

This course deals with the following concepts.

- continuity and topological equivalence; metric and normed spaces

Learning Outcomes

At the end of the semester, the students would be able to:

• know sets and operations, Euclidean space and functions,
• explain the properties of metric and normed spaces,
• understand the definitions and theorems related to topology,
• characterize the behavior of inner product space,
• apply theoretical concepts in topology space and other related spaces.

Math  5203  Abstract Algebra II

Course Description

This course deals with the following concepts.

-   Cauchy’s theorem; direct products; finite abelian groups; conjugacy and Sylow’s theorem; preliminaries on symmetric groups; cycle decompositions; odd and even permutations

Learning Outcomes

At the end of the semester, the students would be able to:

• know Cauchy’s theorem, direct products, finite abelian groups, conjugacy and Sylow’s theorem,
• explain the preliminaries on symmetric groups, cycle decompositions, odd and even permutations.

Math  5204  Hydrodynamics I

Course Description

This course deals with the following concepts.

- axisymmetric flow; Stoke's stream function; some two-dimensional flows; general motion of cylinder

Learning Outcomes

At the end of the semester, the students would be able to:

• calculate the Stokes stream functions corresponding to a uniform stream, simple source, doublet, uniform line source and doublet in uniform stream,
• study some irrotational flows and steady flows in two-dimensional motion of incompressible inviscid fluid,
• know the uniform flow past a stationary circular cylinder and moving cylinder with uniform velocity in a fluid at rest,
• understand the concepts of the images of source, sink, doublet in two-dimensional motion and know how to apply the circle theorem,
• calculate the action of the moving fluid pressure on the fixed cylinder by Blasius theorem,
• find some flows in new plane applying the conformal transformation.

Math  5205  Numerical Analysis II

Course Description

This course deals with the following concepts.

- numerical methods in linear algebra; numerical methods for differential equations

Learning Outcomes

At the end of the semester, the students would be able to:

• know how to use the numerical methods in linear algebra and the least square method,
• find the eigenvalues of the matrix using the power method and QR-Factorization method,
• apply numerical methods for differential equations,
• derive and implement the numerical methods such as Euler’s method, Multistep method, Runge-Kutta method and Taylor method,
• find the numerical solution of differential equation using the above method,
• derive and implement the finite difference method for solving three types of partial differential equations namely Elliptic equations, hyperbolic equations and parabolic equations,
• derive the Liebmann’s procedure and find the interior points using Liebmann’s procedure.

Math  5206  Qualitative Theory of Ordinary Differential Equations I

Course Description

This course deals with the following concepts.

-  systems of differential equations; linear systems with an introduction to phase space analysis

Learning Outcomes

At the end of the semester, the students would be able to:

• know the coupled mass-spring systems,
• use vector-matrix notation for systems,
• discuss existence, uniqueness and continuity and Grownwall inequality,
• determine the existence and uniqueness for linear systems,
• explain linear homogeneous and non-homogeneous systems,
• construct the autonomous system, phase space two-dimensional systems.

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Second Semester

Math  5207  Analysis VI

Course Description

This course deals with the following concepts and theories.

-  continuous transformations of metric spaces; euclidean spaces; continuous functions of several real variables; partial derivatives; linear transformations and determinants; the inverse function theorem; the implicit function theorem; functional dependence

Learning Outcomes

At the end of the semester, the students would be able to:

• know about the properties of linear transformation for sets of vectors in Euclidean n-space Rⁿ,
• recognize the concepts of the term span, linear independence, basis and dimension and apply these concepts to various vector spaces,
• determine the continuity, differentiability and integrability of functions defined on subsets of the n-space Rⁿ,
• illustrate the effect of uniform convergence on the limit function with respect to continuity, differentiability and integrability.

Math  5208   General Topology III

Course Description

This course deals with the following concepts and theories.

-  separation axioms; compactness; concept of product topology and examples

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the definitions and existence of separation axioms and compact set,
• explain the compactness and concepts of product topology and examples,
• discuss some properties of compact set in topological space,
• apply theoretical concepts of separation axioms in topological space.

Math  5209   Abstract Algebra III

Course Description

This course deals with the following concepts and theories.

- definitions and examples of rings; some simple results on rings; ideals, homomorphism; and quotient rings; maximal ideals

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the definitions and examples corresponding to rings and solve some simple problems on rings by using these definitions,
• explain the concepts of ideals, homomorphism, quotient rings and maximal ideals.

Math  5210   Hydrodynamics II

Course Description

This course deals with the following concepts.

- vorticity; vortex line; vortex tube and vortex filament; rectilinear vortices; two vortex filaments; vortex pair; vortex doublet; motion of any vortex; image of a vortex filament in a plane; vortex inside an infinite circular cylinder; vortex outside a circular cylinder; image of a vortex outside/inside a circular cylinder; vortex rows; Karman vortex street; Rankine’s combined vortex;

- Mathematical representation of a wave motion; standing or stationary waves; types of liquid waves; surface waves; the energy of progressive waves; the energy of stationary waves; waves at the interface of two liquids; waves at the interface of two liquids with upper surface free; group velocity

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the vortex line, vortex tube, rectilinear vortices, vortex row and their properties,
• calculate the velocity and angular velocity at any point in the fluid when two vortex filaments are same sense and opposite sense,
• study the motion of any vortex in an infinite liquid and image of a vortex filament in a plane and in a quadrant,
• know the infinite number of parallel vortices of the same strength in one row and two infinite rows of parallel rectilinear vortices,
• express the mathematical representation of wave motion of a liquid acted upon by gravity,
• analyze the surface waves, the energy of progressive waves and stationary waves,
• find the velocity of propagation of waves at the interface of two liquids,
• explain the properties of groups of waves.

Math  5211   Graph Theory

Course Description

This course deals with the following concepts.

- graphs and subgraphs; trees

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the basic concepts of graphs, directed graphs, and weighted graph,
• find the degree of a vertex,
• know how to use handshaking lemma in practice,
• understand and apply the fundamental concept of graph theory which contains  Eulerian trails, Hamiltonian cycles, bipartite graphs, planar graphs, and Euler characteristics and classical theorems in graph theory,
• determine whether graphs are Hamiltonian and/or Eulerian,
• understand various types of trees and methods for traversing tree,
• identify induced subgraphs and cut edges and bonds, cut vertices, Cayley's Formula.

Math  5212   Qualitative Theory of Ordinary Differential Equations II

Course Description

This course deals with the following concepts.

-  existence theory; stability of linear and almost linear systems

Learning Outcomes

At the end of the semester, the students would be able to:

• know the existence theory for systems of first-order equations,
• determine the uniqueness of solutions and continuous of solutions,

·           explain about the definition of stability, almost linear system and conditional stability.

First Semester

Math  5201  Analysis V

Course Description

This course deals with the following concepts.

- power series; the exponential and logarithmic functions; the trigonometric functions; fourier               series

Learning Outcomes

At the end of the semester, the students would be able to:

• relate concepts of uniform continuity, differentiation, integration and uniform convergence,
• apply the definition of convergence to sequence, series and functions,
• illustrate the convergence properties of power series and fourier series,
• define and analyze limits and continuity for functions as well as consequences of continuity,
• explain Perseval’s theorem, Gamma function and Stirling’s formula.

Math  5202  General Topology II

Course Description

This course deals with the following concepts.

- continuity and topological equivalence; metric and normed spaces

Learning Outcomes

At the end of the semester, the students would be able to:

• know sets and operations, Euclidean space and functions,
• explain the properties of metric and normed spaces,
• understand the definitions and theorems related to topology,
• characterize the behavior of inner product space,
• apply theoretical concepts in topology space and other related spaces.

Math  5203  Abstract Algebra II

Course Description

This course deals with the following concepts.

-   Cauchy’s theorem; direct products; finite abelian groups; conjugacy and Sylow’s theorem; preliminaries on symmetric groups; cycle decompositions; odd and even permutations

Learning Outcomes

At the end of the semester, the students would be able to:

• know Cauchy’s theorem, direct products, finite abelian groups, conjugacy and Sylow’s theorem,
• explain the preliminaries on symmetric groups, cycle decompositions, odd and even permutations.

Math  5204  Hydrodynamics I

Course Description

This course deals with the following concepts.

- axisymmetric flow; Stoke's stream function; some two-dimensional flows; general motion of cylinder

Learning Outcomes

At the end of the semester, the students would be able to:

• calculate the Stokes stream functions corresponding to a uniform stream, simple source, doublet, uniform line source and doublet in uniform stream,
• study some irrotational flows and steady flows in two-dimensional motion of incompressible inviscid fluid,
• know the uniform flow past a stationary circular cylinder and moving cylinder with uniform velocity in a fluid at rest,
• understand the concepts of the images of source, sink, doublet in two-dimensional motion and know how to apply the circle theorem,
• calculate the action of the moving fluid pressure on the fixed cylinder by Blasius theorem,
• find some flows in new plane applying the conformal transformation.

Math  5205  Numerical Analysis II

Course Description

This course deals with the following concepts.

- numerical methods in linear algebra; numerical methods for differential equations

Learning Outcomes

At the end of the semester, the students would be able to:

• know how to use the numerical methods in linear algebra and the least square method,
• find the eigenvalues of the matrix using the power method and QR-Factorization method,
• apply numerical methods for differential equations,
• derive and implement the numerical methods such as Euler’s method, Multistep method, Runge-Kutta method and Taylor method,
• find the numerical solution of differential equation using the above method,
• derive and implement the finite difference method for solving three types of partial differential equations namely Elliptic equations, hyperbolic equations and parabolic equations,
• derive the Liebmann’s procedure and find the interior points using Liebmann’s procedure.

Math  5206  Qualitative Theory of Ordinary Differential Equations I

Course Description

This course deals with the following concepts.

-  systems of differential equations; linear systems with an introduction to phase space analysis

Learning Outcomes

At the end of the semester, the students would be able to:

• know the coupled mass-spring systems,
• use vector-matrix notation for systems,
• discuss existence, uniqueness and continuity and Grownwall inequality,
• determine the existence and uniqueness for linear systems,
• explain linear homogeneous and non-homogeneous systems,
• construct the autonomous system, phase space two-dimensional systems.

-----------------------------------------------------------------------------

Second Semester

Math  5207  Analysis VI

Course Description

This course deals with the following concepts and theories.

-  continuous transformations of metric spaces; euclidean spaces; continuous functions of several real variables; partial derivatives; linear transformations and determinants; the inverse function theorem; the implicit function theorem; functional dependence

Learning Outcomes

At the end of the semester, the students would be able to:

• know about the properties of linear transformation for sets of vectors in Euclidean n-space Rⁿ,
• recognize the concepts of the term span, linear independence, basis and dimension and apply these concepts to various vector spaces,
• determine the continuity, differentiability and integrability of functions defined on subsets of the n-space Rⁿ,
• illustrate the effect of uniform convergence on the limit function with respect to continuity, differentiability and integrability.

Math  5208   General Topology III

Course Description

This course deals with the following concepts and theories.

-  separation axioms; compactness; concept of product topology and examples

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the definitions and existence of separation axioms and compact set,
• explain the compactness and concepts of product topology and examples,
• discuss some properties of compact set in topological space,
• apply theoretical concepts of separation axioms in topological space.

Math  5209   Abstract Algebra III

Course Description

This course deals with the following concepts and theories.

- definitions and examples of rings; some simple results on rings; ideals, homomorphism; and quotient rings; maximal ideals

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the definitions and examples corresponding to rings and solve some simple problems on rings by using these definitions,
• explain the concepts of ideals, homomorphism, quotient rings and maximal ideals.

Math  5210   Hydrodynamics II

Course Description

This course deals with the following concepts.

- vorticity; vortex line; vortex tube and vortex filament; rectilinear vortices; two vortex filaments; vortex pair; vortex doublet; motion of any vortex; image of a vortex filament in a plane; vortex inside an infinite circular cylinder; vortex outside a circular cylinder; image of a vortex outside/inside a circular cylinder; vortex rows; Karman vortex street; Rankine’s combined vortex;

- Mathematical representation of a wave motion; standing or stationary waves; types of liquid waves; surface waves; the energy of progressive waves; the energy of stationary waves; waves at the interface of two liquids; waves at the interface of two liquids with upper surface free; group velocity

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the vortex line, vortex tube, rectilinear vortices, vortex row and their properties,
• calculate the velocity and angular velocity at any point in the fluid when two vortex filaments are same sense and opposite sense,
• study the motion of any vortex in an infinite liquid and image of a vortex filament in a plane and in a quadrant,
• know the infinite number of parallel vortices of the same strength in one row and two infinite rows of parallel rectilinear vortices,
• express the mathematical representation of wave motion of a liquid acted upon by gravity,
• analyze the surface waves, the energy of progressive waves and stationary waves,
• find the velocity of propagation of waves at the interface of two liquids,
• explain the properties of groups of waves.

Math  5211   Graph Theory

Course Description

This course deals with the following concepts.

- graphs and subgraphs; trees

Learning Outcomes

At the end of the semester, the students would be able to:

• understand the basic concepts of graphs, directed graphs, and weighted graph,
• find the degree of a vertex,
• know how to use handshaking lemma in practice,
• understand and apply the fundamental concept of graph theory which contains  Eulerian trails, Hamiltonian cycles, bipartite graphs, planar graphs, and Euler characteristics and classical theorems in graph theory,
• determine whether graphs are Hamiltonian and/or Eulerian,
• understand various types of trees and methods for traversing tree,
• identify induced subgraphs and cut edges and bonds, cut vertices, Cayley's Formula.

Math  5212   Qualitative Theory of Ordinary Differential Equations II

Course Description

This course deals with the following concepts.

-  existence theory; stability of linear and almost linear systems

Learning Outcomes

At the end of the semester, the students would be able to:

• know the existence theory for systems of first-order equations,
• determine the uniqueness of solutions and continuous of solutions,

·           explain about the definition of stability, almost linear system and conditional stability.

First Semester

Math 611  Analysis I

Course Description

This course deals with the following concepts and theories.

- concept of measurability; simple functions; elementary properties of measures; integration of positive functions; Lebesgue’s monotone convergence theorem; integration of complex functions; Lebesgue’s dominated convergence theorem; Riesz representation theorem; regularity properties of Borel measures; Lebesgue measures; continulity properties of measurable functions; convex functions and inequalities; the L^p-spaces; approximation by continuous functions

Learning Outcomes

At the end of the semester, the students would be able to:

• explain the concepts of measurable sets, measurable function and measure,
• understand how to construct measure theory in spaces other than the real line,
• explain the definition of the derivative of a measurable function,
• define the concept of Lebesgue Measure and Lebesgue Integral,
• explain the difference between the Lebesgue Integral and the Riemann Integral,
• distinguish the difference between uniform convergence and pointwise convergence, and apply the Lebesgue,
• know the regular properties of Borel Measure,
• explain the definition of Lp spaces, the Riesz representation theorem, and the completeness of the Lp spaces.

Math 612  Abstract Algebra

Course Description

This course deals with the following concepts and theories.

-  polynomial rings; polynomials over the rational; field of quotients of an integral domain;

   examples of fields; a brief excursion into vector spaces; field extensions

Learning Outcomes

At the end of the semester, the students would be able to:

• assess properties implied by the definitions of rings,
• analyze and demonstrate examples of rings and quotient rings,
• construct the structure of polynomial rings and extend it to quotient ring, field and integral domain,
• construct the extension fields and then prove the relationship between two fields which are itself and its extension,
• produce rigorous proofs of propositions arising in the setting of Abstract Algebra.

Math 613   Qualitative Theory of Ordinary Differential Equations

Course Description

This course deals with the following concepts.

-  Lyapunov’s second method; applications of ODE

Learning Outcomes

At the end of the semester, the students would be able to:

• know Lyapunov’s theorem and proof,
• explain invariant set and stability,
• construct the structure of nonautonormous systems,
• understand the undamped oscillator and the pendulum,
• find self-oscillations periodic solutions of the linear equations.

Math 614   Discrete Mathematics

Course Description

This course deals with the following concepts.

-  network models; a maximal flow algorithm; the max flow; min cut theorem; matching; combinatorial circuits; properties of combinatorial circuits; Boolean algebras; Boolean functions and synthesis of circuits; applications

Learning Outcomes

At the end of the semester, the students would be able to:

• know how to construct network models and Petri nets, that make use of directed graphs,
• find a maximal flow produced by Maximal flow algorithm,
• compute the value of maximal flow in transport network,
• determine whether the cut is minimal or not by using max flow-min cut theorem,
• find a maximal flow in a network by the concept of matching,
• explain the relationship between flows and matchings,
• know how to use Petri nets when the computer program is considered.

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Second Semester

Math 621   Analysis II

Course Description

This course deals with the following concepts and properties.

-  Banach spaces; continuous linear transformation; functional;  dual space N ^*  of a normed space N; Hahn Banach theorem;  duals; natural imbedding of N in   N ^** , reflexive spaces; weak topology; weak^*  topology; open mapping theorem; closed graph theorem;  uniform boundedness  theorem; conjugate of an operator

- inner product space; Hilbert space; Schwarz inequality; orthogonal complement; orthonormal sets; Bessel's inequality; Parscal's equation; conjugate space H^* of a Hilbert space H. representation of functionals in H^*; adjoint of an operator; self-adjoint operator; normal and unitary operators-projectory

Learning Outcomes

At the end of the semester, the students would be able to:

• recognize the fundamental properties of normed spaces and of the transformations between them,
• express the finite dimensional normed spaces, examples of Banach spaces,
• understand Topological dual of classical spaces and weak topology and reflexive spaces,
• understand open mapping theorem, uniform boundedness theorem and closed graph theorem,
• understand and apply fundamental theorems from the theory of normed and Banach spaces, including the Hahn-Banach theorem, the open mapping theorem, the closed graph theorem, and the Stone-Weierstrass theorem,
• familiar with the theory of linear operators on a Hilbert space, including adjoint operators, self-adjoint and unitary operators with their spectra and Riesz-Fréchet theorem, variational problems, the Dirichlet principle, bases in Hilbert spaces, orthogonality,
• know the L₂-theory of Fourier series, the classical theory of Fourier series and other orthogonal expansions, Spectral theorem for self-adjoint compact operators.

Math 622  Linear Algebra

Course Description

This course deals with the following concepts and properties.

-  eigenvectors and eigenvalues; polynomials and matrices; triangulations of matrices and linear maps

Learning Outcomes

At the end of the semester, the students would be able to:

• identify the eigenvectors and eigenvalues on vector spaces,
• compute the eigenvectors, eigenvalues and eigenspaces with the characteristic polynomial and apply the basic diagonalization result,
• compute the scalar products and determine orthoganility on vector space, including Gram-Schmidt orthogonalization,
• apply the spectral theorem and orthogonal decomposition of scalar product spaces.

Math 623  Partial Differential Equations                     

Course Description

This course deals with the following concepts.

-  integral curves and surfaces of vector fields;  theory and applications of quasi-linear and linear equations of first order; series solutions; linear partial differential equations; equations of mathematical Physics

Learning Outcomes

At the end of the semester, the students would be able to:

• identify the integral curves and surface of vector fields,
• explain the theory and applications of Quasi-linear and linear equations of first order,
• find the series solutions by using Cauchy-Kovalevsky theorem,
• solve linear partial differential equations and find characteristic, classification and canonical forms,
• solve the equations of mathematical physics.

Math 624  Graph Theory

Course Description

This course deals with the following concepts.

-  connectivity; Euler tours; Hamilton cycles

Learning Outcomes

At the end of the semester, the students would be able to:

• know the concept of connection in graphs,
• understand connectivity and edge connectivity of graph,
• explain the relationship between connectivity and edge connectivity by using the connectivity number,
• solve the connector problem by Kruskal’s algorithm,
• identify the usefulness of Euler tour and Hamilton cycle in graph theory,
• know the conditions for a connected graph to be eulerian,
• explain the sufficient conditions for a graph to be Hamiltonian, describe a good algorithm for constructing the closure of a graph and finding a Hamilton cycle if the closure is complete.

First Semester

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Second Semester