## DEPARTMENT OF MATHEMATICS AT UNIVERSITY OF GHANA

COURSE DESCRIPTIONS AND PREREQUISITES

Prerequisite for 100 LEVEL: SHS (or equivalent) grade B in elective mathematics.

LEVEL 100

MATH 101: General Mathematics I (Non-Mathematics students)

Indices and Logarithms. Equations and inequalities. Functions and graphs. Arrangements and

selections. Binomial theorem. Limits, differentiation and integration.

MATH 121: Algebra and Trigonometry

This course is a precalculus course aiming to develop the students ability to think logically, use

sound mathematical reasoning and understand the geometry in algebra. It examines: logic and

concept of mathematical proof; sequences and series; elementary set theory; the algebra of surds,

indices and logarithms; the concept of a function, identifying domain and range and injective and

surjective functions; trigonometric functions, their inverses, their graphs, circular measure and

trigonometric identities.

MATH 123: Vectors and Geometry

Vectors may be used very neatly to prove several theorems of geometry. This course is about

applying vector operations and the method of mathematical proof (of MATH 121) to geometric

problems. The areas of study include: vector operations with geometric examples; components of

a vector and the scalar product of vectors. Coordinate geometry in the plane including normal

vector to a line, angle between intersecting lines, reflection in a line, angle bisectors and the

equation of a circle, the tangent and the normal at a point.

MATH 122: Calculus I

Calculus is the mathematics of change and motion. This course develops the mathematics of

change and the course MATH 124 will explicitly develop the mathematics of motion. In this

course we address: limits and continuity of real valued functions of a single real variable; the

derivative as a limit, algebraic rules of differentiation, implicit differentiation and the derivative

of a function in parametric form; integration and the solution to first order differential equations.

MATH 124: Mechanics

In this course we deal with the calculus involved with motion. Kinematics deals with the study of

motion without reference to the cause of the motion. So we first study kinematics and then we add

the ingredient of the cause of the motion, force. So, we study statics, where the sum of the forces

on the body is zero and we study dynamics, where the sum of the forces on the body is no longer

zero.

LEVEL 200

Prequisite for 200 LEVEL MATH courses- Passes in MATH 121 and MATH 122.

MATH 221 Algebra

Polar coordinates; conic sections. Complex numbers: algebra, Argand diagram, roots of unity.

Algebra of matrices and determinants, linear transformations. Transformations of the complex

plane. Coordinate geometry in 3 dimensions. Vector product and triple products. Geometry of the

sphere.

MATH 222 Vector Mechanics

1-dimensional kinematics. Forces acting on a particle. 1-dimensional dynamics. Newton’s laws of

motion; motion under constant acceleration, resisted motion, simple harmonic motion. 3-

dimensional kinematics. Relative motion. 2-dimensional motion under constant acceleration.

Work, energy and power. Impulse and linear momentum.

MATH 223 Calculus II

Second derivative of a function of a single variable. Applications of first and second derivatives.

Hyperbolic and inverse hyperbolic functions. Methods of integration. Applications of the definite

integral. Ordinary differential equations, first order and second order (with constant coefficients).

Higher derivatives, Taylor (Maclaurin) series expansion of elementary functions.

MATH 224 Introductory Abstract Algebra

Equivalences, partial order. Construction of R from Z. Elementary number theory. Axiomatically

defined systems; groups, rings and fields. Morphisms of algebraic structures. Vector spaces.

Homomorphism of vector spaces.

MATH 226 Introductory Programming for Computational Mathematics

Variables, functions, arrays and matrices, classes, introduction to Graphical User Interfaces

(GUI’s). Introduction to symbolic computing. Visualization in mathematics.

LEVEL 300

MATH 351 Linear Algebra-prerequisite MATH 221 or MATH 224

Spanning sets. Subspaces, solution spaces. Bases. Linear maps and their matrices. Inverse maps.

Range space, rank and kernel. Eigenvalues and eigenvectors. Diagonalization of a linear operator.

Change of basis. Diagonalizing matrices. Diagonalization theorem. Bases of eigenvectors.

Symmetric maps, matrices and quadratic forms.

MATH 354 Abstract Algebra I-prerequisite MATH 224

Subgroups, cyclic groups.The Stabilizer-Orbit theorem.Lagrange’s theorem. Classifying groups.

Structural properties of a group. Cayley’s theorem. Generating sets. Direct products. Finite abelian

groups. Cosets and the proof of Lagrange’s theorem. Proof of the Stabilizer-Orbit theorem.

MATH 353 Analysis I-prerequisite MATH 223

Normed vector spaces. Limits and continuity of maps between normed vector spaces. The

algebra of continuous functions. Bounded sets of real numbers. Limit of a sequence.

Subsequences. Series with positive terms.Convergence tests. Absolute convergence. Alternating

series. Cauchy sequences and complete spaces.

MATH 356 Analysis II-prerequisite MATH 223

Sequences of functions. Pointwise and uniform convergence. Power series. The contraction

mapping theorem and application. Real analysis. Definition of integral and condition for

integrability. Proof of the fundamental theorem of calculus and major basic results involved in its

proof.

MATH 361 Classical Mechanics -prerequisite MATH 222

1-dimensional dynamics: damped and forced oscillations. Motion in a plane: projectiles, circular

motion, use of polar coordinates and intrinsic coordinates. Two-body problems, variable mass.

Motion under a central, non-inertial frame. Dynamics of a system of particles.

MATH 366 Electromagnetic Theory I

Scalar and vector fields, grad, div and curl operators. Orthogonal curvilinear coordinates.

Electrostatics: charge, Coulomb’s law, the electric field and electrostatic potential, Gauss’s law,

Laplace’s and Poisson’s equations. Conductors in the electrostatic field. Potential theory.

MATH 362 Analytical Mechanics- prerequisite MATH 222

Rigid body motion, rotation about a fixed axis. General motion in a plane, rigid bodies in contact,

impulse. General motion of a rigid body. Euler-Lagrange equations of motion.

MATH 364 Introductory concepts in Financial Mathematics-prerequisite MATH

223/STAT 221/224

Probability functions, random variables and their distributions, functions of random variables;

basic theorems for functions of independent random variables, characteristic function of a random

variable; central limit theorem, random walks and martingales; Markov chain, Markov process,

queuing theory.

MATH 350 Differential Equations I-prerequisite MATH 223

Differential forms of 2 and 3 variables. Exactness and integrability conditions. Existence and

uniqueness of solution. Second order differential equations with variable coefficients. Reduction

of order, variation of parameters. Series solution. Ordinary and regular singular points. Orthogonal

sets of functions. Partial differential equations.

MATH 355 Calculus of Several Variables-prerequisite MATH 223

Functions of several variables, partial derivative. Directional derivative, gradient. Local extema,

constrained extrema. Lagrange multipliers. The gradient, divergence and curl operators. Line,

surface and volume integrals. Green’s theorem, divergence theorem, Stokes’ theorem.

MATH 359 Discrete Mathematics-prerequisite MATH 224

Boolean algebra. Combinatorics languages and grammars. Recurrence relations, generating

functions and applications. Problems of definition by induction: no closed form, infinite loops and

the halting problem. Algorithms: correctness, complexity, efficiency. Graph theory: planarity,

Euler circuits, shortest-path algorithm. Network flows. Modelling computation: languages and

grammars, models, finite state machines, Turing machines.

MATH 357 Computational Mathematics I

Error analysis. Rootfinding; 1 and 2 point methods. Linear systems of equations, matrix algebra,

pivoting. Analysis of algorithms. Iterative methods. Interpolation, polynomial approximation,

divided differences. Initial value problems, single and multistep methods. Numerical integration.

MATH 358 Computational Mathematics II

Multi-dimensional root-finding. Optimization. Non-linear systems of equations. Eigenvalues.

Numerical methods for ordinary differential equations and for partial differential equations.

MATH 368 Introductory Number Theory

This course covers results of elementary number theory. Topics include: divisibility and

factorization; congruences; arithmetic functions; quadratic residues; the primitive root theorem;

continued fractions and topics from computational number theory.

MATH 441 Advanced Calculus

Linear and affine maps between normed vector spaces. Limits, continuity, tangency of maps and

the derivative as a linear map. Component-wise differentiation, partial derivatives, the Jacobian as

the matrix of the linear map. Generalized mean value theorem. Inverse map theorem. Implicit

function theorem.

MATH 442 Integration theory and Measure

Generalisation of the Riemann (R) integral (eg Kurzweil-Henstock (KH) integral). Lebesgue (L)

integral. Relationship between the KH-integrable, L-integrable and R-integrable functions.

Convergence theorems. Measurability. Measure.

459

MATH 443 Differential Geometry

Arclength, curvature and torsion of a curve. Geometry of surfaces. Curvature, first and second

fundamental form, Christofel symbols. Geodesics. Parallel vector fields. Surfaces of constant

Gaussian curvature. Introduction to manifolds, tangent spaces and tangent bundles. Vector fields

and Lie brackets. Parallel vector fields on manifolds and Riemannian manifolds.

MATH 444 Calculus on Manifolds

Manifold, submanifold, differentiability of maps between manifolds, the tangent space, the tangent

bundle and the tangent functor. Vector bundle. The exterior algebra, the notion of a differentiable

form on a manifold, singular n-chains and integration of a form over a chain. Partition of unity.

Application to Stokes’ theorem.

MATH 440 Abstract Algebra II

Finite groups. Sylow theorems and simple groups. Composition series and Jordan-Holder theorem.

Direct and semi-direct products. Abelian groups, torsion, torsion-free and mixed abelian groups.

Finitely generated group and subgroups. P-groups, nilpotent groups and solvable groups.

Introduction to module theory.

MATH 446 Module Theory

Modules, submodules, homomorphism of modules. Quotient modules, free (finitely generated)

modules. Exact sequences of modules. Direct sum and product of modules. Chain conditions,

Noetherian and Artinian modules. Projective and injective modules. Tensor product, categories

and functors. Hom and duality of modules.

MATH 447 Complex Analysis

Elementary topology of the complex plane. Complex functions and mappings. The derivative.

Differentiability and analyticity. Harmonic functions. Integrals. Maximum modulus, Cauchy-

Gorsat, Cauchy theorems. Applications. Taylor and Laurent series, zeros and poles of a complex

function. Residue theorem and consequences. Conformal mapping, analytic continuation.

MATH 438 Topology

Topological spaces. Basis for a topological space. Separation and countability properties. Limit

points, closure and interior. Connectedness, compactness, subspace topology. Homeomorphism,

continuity, metrizability. General product spaces and the general Tychonoff theorem.

MATH 443 Differential Geometry

Arclength, curvature and torsion of a curve. Geometry of surfaces. Curvature, first and second

fundamental form, Christofel symbols. Geodesics. Parallel vector fields. Surfaces of constant

curvature. Introduction to manifolds, tangent space, tangent bundle. Vector fields and Lie

brackets. Parallel vector fields on manifolds. Riemannian manifold.

MATH 451 Introduction to Quantum Mechanics

(Principle of least action, Hamilton’s equation, Poisson brackets. Liouville’s equation.) Canonical

transformations. Symmetry and conservation laws. Postulates of quantum mechanics, the wave

formalism. Dynamical variables. The Schrodinger equation in one-dimension; free particles in a

box, single step and square well potentials. Orbital angular momentum. The 3-dimensional

Schrodinger equation; motion in a central force field, the 3-d square well potential, the hydrogenic

atom. Heisenberg’s equation of motion, harmonic oscillator and angular momentum.

460

MATH 448 Special Relativity

Galilean relativity, postulates of special relativity; Lorentz transformations. Lorentz-Fitzgerald

contraction, time dilation. 4-vectors, relativistic mechanics, kinematics and force, conservation

laws; decay of particles; collision problems, covariant formulation of electrodynamics.

MATH 445 Introductory Functional Analysis

Finite dimensional normed vector spaces. Equivalent norms. Banach spaces.Infinite-dimensional

normed vector spaces–Hamel and Schauder bases; separability. Compact linear operators on a

Banach space. Complementary subspaces and the open-mapping theorem. Closed Graph theorem.

Hilbert spaces. Special subspaces of l¥ and 1 l and the dual space. The completion of a normed

vector space. Reflexive Banach spaces

MATH 450 Differential Equations II

Classification of second order partial differential equations. Legendre’s equation/polynomial.

Gamma function and Bessel equation. Laplace transforms/equations. Fourier series. Boundary

value problems. Application to heat conduction, vibrating strings and the 1-dimensional wave

equation.

MATH 451 Introduction to Algebraic Field Theory-prerequisite MATH 354

Algebraic numbers. Extending fields. Towers of fields. Irreducible polynomials. Constructible

numbers and fields. Transcendence of π and є. Residue rings and fields.

MATH 452 Introduction to Lie Groups and Lie Algebras

Vector fields and groups of linear transformations. The exponential map. Linear groups and their

Lie algebras. Connectedness. Closed subgroups. The classical groups. Manifolds, homogeneous

spaces and Lie groups. Integration and representation.