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Course Code             : AMAT 12253

Title                          : Numerical Methods I

Pre-requisites            : AMAT 11223

 Learning Outcomes:

At the end of the course, the student should be able to;

1. explain the issues of accuracy in floating-point representation
2. find roots of non-linear equations using appropriate numerical methods
3. identify the error in a given data set
4. apply appropriate interpolation and curve fitting techniques for a given application
5. solve a system of equations using direct methods
6. describe the limitations, advantages, and disadvantages of numerical methods.

 Course Contents: 

Introduction: Floating point number system, error in numerical computation, strategies for minimizing round-off
errors, Ill conditioning, condition number, the notion of an algorithm.
Solution of equations with one variable: Numerical solution of nonlinear equations using Bisection method, False
Position method, Fixed-Point iteration method, Newton-Raphson method, Secant method and modified secant
method, error analysis for iterative methods.
Difference Operators: Forward, Backward, Central and Averaging operators, symbolic relations of difference
operators, difference table and error propagation, difference equations, factorial polynomials.
Interpolation: Collocation polynomial and its properties, Newton’s Forward and Backward difference formulae,
Gauss’s Central Difference Formula, interpolation with unevenly spaced points: Lagrange’s and Newton’s
interpolation, Spline Interpolation: Linear, Quadratic and Cubic Spline Interpolation.
Least-square curve fitting techniques: linear functions: normal equations, coefficient of determination, non-
linear functions: exponential model, power model, Saturation Growth Rate model.
Solution of System of Linear Equations (Direct Methods): Matrix inversion, Naïve Gauss elimination, Gaussian
elimination with partial pivoting, Ill conditioning Matrices, Operation counts, Matrix Decomposition Techniques:
LU and QR Factorizations.

of Teaching and Learning: A combination of lectures and tutorial discussions

 Assessment: Based on tutorials, tests and end of course examination

 Recommended Reading      :

1. Burden, R.L., Faires, J.D, Burden, A.M. (10th Ed., 2015). Numerical Analysis, Cengage Learning.
2. Sastry, S.S. (5th Ed., 2012). Introductory Methods of Numerical Analysis, Prentice-Hall India.
3. Kreyszig, E. (10th Ed., 2010). Advanced Engineering Mathematics, John Wiley.
4. Sauer, T. (2012). Numerical Analysis, Pearson.
5. Epperson, J.F. (2013). An Introduction to Numerical Methods and Analysis, Wiley.
6. Faul, A.C. (2016). A Concise Introduction to Numerical Analysis, Chapman and Hall/CRC.

 

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Course Code    :  AMAT 12242

Title                   :  Vector Methods in Geometry

Pre-requisites   :  AMAT 11223

 Learning Outcomes  :

At the end of the course the student will be able to 

1. state the standard vector forms of lines and planes
2. calculate the tangent, normal, binormal vectors of curves and the characteristics of curves such as
curvature, torsion and radius of curvature
3. explain osculating, normal ad rectifying planes of curves
4. apply Frenet-Serret formulae
5. calculate unit vectors and scalar factors of curvilinear coordinates
6. describe the form of differential equations in curvilinear coordinates
7. calculate tangent plane and normal vectors of surfaces.

 Course Content  :

Lines and planes: Vector form of lines and planes, Parameterized curves. Condition for planarity of two lines,
bisecting the angle, between two given planes
Curves and Surfaces: Tangent, Normal and Binormal vectors, Curvature, torsion and radius of curvature, Frenet-
Serret formulae, Osculating Plane, Normal Plane, Rectifying Plane
Curvilinear Coordinates: Scalar Factors and unit vectors for curvilinear coordinates, Differential operators in
curvilinear coordinates, Length element, surface elements and volume elements in curvilinear coordinates, Scalar
Factors and unit vectors for spherical and cylindrical coordinate systems.
Theory of Surfaces: Concept of a surface, Implicit Equation of a Surface, Parametric Equation, Parametric Curves,
Curves on a Surface, Tangent Plane, Normal Vector, Surface of Revolution.

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment              :  Based on tutorials, tests and end of course examination.

 Recommended Reading      :

  1. Shanthi Narayan, Vector Algebra, S. Chands and Company, 2005
  2. L, Brand. Vector Analysis. Courier Corporation, 2012
  3. R. Shorter. Problems and Worked Solutions in Vector Analysis. Courier Corporation, 2014
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Course Code    :  AMAT 11223

Title                   :  Vector analysis

Pre-requisites   :  A/L Combined Mathematics

 Learning Outcomes  :

At the end of the course the student will be able to

1. demonstrate knowledge of coplanarity of three vectors, orthogonal vectors, vector product.
2. calculate unit vectors, scalar and vector products, triple scalar & vector products, arc lengths, unit
tangent and normal vectors.
3. define Gradient, Divergence, Curl Operators.
4. calculate derivatives of vector fields and normal and directional derivatives of surfaces.
5. identify conservative, solenoidal, rotational and irrotational vector fields.
6. apply Fundamental Theorem of Line Integrals to compute work.
7. establish the Divergence and Stoke’s theorems.

 Course Content:

Vector Algebra: Introduction to vectors, Condition for coplanarity of three vectors and collinearity of two
vectors, Orthogonal triads of unit vectors, Scalar and vector products, Triple scalar and vector products, Solution
of vector equations.


Vector Analysis: Scalar and vector fields, Differentiation of vector functions, sketch curves defined by position
vectors, arc length, unit tangent and unit normal vectors of curves, Gradient, Divergence and Curl operators and
identities involving them, Surfaces and normal, Directional Derivative, Conservative, solenoidal, rotational &
irrotational vector Fields, Sketching 3D objects,
Line Integrals, Fundamental Theorem of Line Integrals, Area Integrals, Surface Integrals, Volume Integrals,
Divergence and Stokes’ theorems.

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment              :  Based on tutorials, tests and end of course examination.

 Recommended Reading      :

  1. Spiegel, M. & S. Lipschutz. Vector Analysis, (2e) McGraw-Hill Education, 2009.
  2. Chatterjee, D.  Vector Analysis, PHI Learning private limited, India, 2009.
  3. Davis, H.F. & Snider, A.D. An Introduction to Vector Analysis, C. Brown, New York, 1992.
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Course Code    : AMAT 11232

Title                   :  Mechanics I

Pre-requisites   :  A/L Combined Mathematics

 

Learning Outcomes  : 

Upon successful completion of this course, the student should be able to

1. consolidate the understanding of fundamental concepts in mechanics such as force, energy, momentum etc
2. describe and apply concepts of inertial frames and transformations between inertial frames
3. define properties of particle motion
4. apply Newton’s law for motion of particles under conservative forces
5. describe and apply Kepler’s law
6. expand and exercise the Newton’s laws in solving problems related to the motion of a particle in inertial
frames, rotating frames and relative to rotating earth.

 Course Content:

Newtonian Kinematics: Inertial frames, Transformations between inertial frames (Lorentz and Galilean
transformation), Relative motion of particles, Relative motion of frames of reference.


Motion of a Particle: Mass, Momentum, Torque and angular momentum, Velocity and acceleration in polar,
cylindrical and spherical coordinates, Equation of motion in vectorial form, One dimensional motion, Integrals of
motion, Work, kinetic energy and potential energy, Impulse, Motion under a conservative forces, Motion under a
central force, Kepler’s laws, Rotating frames of reference, Motion relative to rotating earth.


System of Particles: Centre of mass, External and internal forces, Integrals of motion, Momentum, Angular
momentum, Work, kinetic energy & potential energy, Conservative systems, Constants of motion.

 Method of Teaching and Learning           : A combination of lectures and tutorial discussions.

 Assessment:            Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Rao, A.V. (2006). Dynamics of Particles and Rigid Bodies: A Systematic approach, Cambridge University
Press.
2. Chorlton, F. (2nd Ed., 2019). Textbook of Dynamics, D. Van Nostrand.
3. Chirgwin, B.H. & Plumpton, C. (2013). Advanced Theoretical Mechanics: A Course of Mathematics for
Engineers and Scientists, Volume 6. Elsevier.
4. Desloge, E.A. (1982). Classical Mechanics, John Wiley, New York.
5. Greiner, W. (2nd Ed., 2010). Classical Mechanics: Systems of Particles and Hamiltonian Dynamics,
Springer.
6. Goldstein, H., Poole, C. P. & Safko, J. (3rd Ed., 2011). Classical Mechanics, Pearson.
7. Strauch D. (2009). Classical Mechanics, An Introduction, Springer.
8. Rao, K.S. (2003). Classical Mechanics, Universities Press.

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Course Code     : PMAT 42843

Title                   : Group Theory

Pre-requisites    : PMAT 21553

 Learning Outcomes:

At the end of the course the student should be able to demonstrate knowledge of the structure of Groups and to apply the knowledge in solving problems in different areas in Algebra.

 Course Contents:

Groups, Subgroups, Normal subgroups, Quotient groups, Permutation groups, Derived group, Homomorphisms, Automorphisms, Isomorphism theorems, Sylow's theorems, Internal direct product, Structure theory of finite Abelian Groups, Groups of small order.

 Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

 Assessment: Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading: 

  1. Khanna, V.K. & Bhambri, S.K., (2016) A Course in Abstract Algebra, Vikas Publishing House.
  2. Frakeigh, J.B., (2003) A first course in Abstract Algebra, Pearson Education India.
  3. Baumslag, B. & Chandler, B., (1968) Group theory, McGraw-Hill, New York.
  4. Narayan, S. & Pal, S., (1992) A Text Book of Modern Abstract Algebra, S.Chands, India. (1992).
  5. Rotman, J.J., (4th edition, 2014) An Introduction to the Theory of Groups, Springer-Verlag.
  6. Linda Gilbert, (8th edition, 2014) Elements of Modern Algebra (8e) Cengage Learning.
  1. PMAT 42833: Measure Theory
  2. PMAT 43976: Research/Study Project
  3. PMAT 44962: Research Methodology
  4. PMAT 41823: Topology

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