About the subject
What Engineering Mathematics III is for
This is the most applied of the three mathematics courses. Fourier analysis breaks any repeating signal into sine waves, which is how audio is compressed, how a structure's vibration is studied and how a communication channel is described. The Z-transform does the same job for sampled, digital signals, and complex variable theory gives the tools that make both work.
The partial differential equation chapters model real systems directly: the wave equation for a vibrating string, the heat equation for conduction, and the continuity and Navier-Stokes equations for fluid flow. Civil students meet those equations again in fluid mechanics and hydraulics; Computer and BEI students meet the transforms again in signals and signal processing.
The objective of this course is to equip students with understanding and practical application of Fourier series, Fourier transform, function of complex variable, partial differential equations and obtaining mathematical models and Z- transform.
IOE's course objective for ENSH 201
- Taught to
- BCE, Semester 3 (Year II, Part I), BCT, Semester 3 (Year II, Part I) and BEI, Semester 3 (Year II, Part I)
- Weekly
- 3 lecture, 2 tutorial, 0 practical hours
- Marks
- Theory 40 internal + 60 final (3-hour exam); 100 in total
Where the marks are
ENSH 201 chapters, hours and final exam marks
IOE's evaluation scheme for the 60-mark final. IOE notes there may be minor deviation.
Scroll the table sideways for hours, marks and share.
Full syllabus
The complete ENSH 201 outline, with how to study each chapter
All 5 chapters and 34 topics as IOE lists them, each with ICE's advice on approaching it.
1Fourier Series and Fourier Transform12 hours
Fourier series and transforms together carry 18 of the 60 marks. Master odd and even functions first, because they halve the work in half-range series.
- 1.1 Review of periodic, odd and even functions
- 1.2 Fourier series of a function over an interval of length 2l and 2π; Euler’s formula, Dirichlet’s condition for uniform convergence of Fourier series, Fourier series of discontinuous functions
- 1.3 Half range Fourier sine and cosine series
- 1.4 Complex form of Fourier series; frequency and amplitude of a function
- 1.5 Fourier integral theorem, Fourier sine and cosine integrals, complex form of Fourier integral
- 1.6 Fourier transform, Fourier sine transform, Fourier cosine transform and their inversion formulas
- 1.7 Fourier transform of the derivative of a function
- 1.8 Relation between Fourier and Laplace transform
2Functions of Complex Variable12 hours
Also 18 marks. The Cauchy-Riemann equations, conformal mapping and residues are three separate skills; the residue theorem for evaluating real integrals is the one to practise most.
- 2.1 Intuitive idea of limit, continuity and differentiability of functions of complex variable
- 2.2 Analytic functions, the Cauchy Reimann equations both in Cartesian and polar form, construction of analytic functions
- 2.3 Harmonic functions, the orthogonal system
- 2.4 Application of analytic functions in flow problems
- 2.5 Transformation (Mapping), conformal mapping, translation, rotation and magnification; inversion, bilinear transformation
- 2.6 Complex integration, simply and multiply connected regions, Cauchy’s integral theorem and formula
- 2.7 Series of complex terms, power series, circle of convergence and radius of convergence, Taylor’s and Laurent’s series
- 2.8 Zeros, singularities, poles; residue at poles, Cauchy’s residue theorem and evaluation real and improper integrals
3Partial Differential Equations5 hours
Forming and solving first-order PDEs, including Lagrange's and Charpit's methods. A short chapter where following the method step by step is enough.
- 3.1 Definition and formation of partial differential equations
- 3.2 Partial differential equations solvable by direct integration
- 3.3 Linear partial differential equation of the first order, Lagrange’s linear equations and their solution
- 3.4 Nonlinear partial differential equation of first order; equations of the form 𝑓(𝑝, 𝑞) = 0, 𝑧= 𝑝𝑥+ 𝑞𝑦+ 𝑓(𝑝, 𝑞), 𝑓(𝑧, 𝑝, 𝑞) = 0, 𝑓ଵ(𝑥, 𝑝) = 𝑓ଶ (𝑦, 𝑞)
- 3.5 Charpit’s method of solving nonlinear partial differential equations of first order
4Modelling through Partial Differential Equation10 hours
The wave, heat and Laplace equations by separation of variables. The derivation is the same pattern each time: separate, solve the two ordinary equations, apply boundary conditions, then combine with a Fourier series.
- 4.1 Second order partial differential equation and classification
- 4.2 One-dimensional wave equation
- 4.3 One-dimensional heat equation
- 4.4 Two-dimensional heat equation, Laplace equation in Cartesian form
- 4.5 Mass balance equation; equation of continuity in fluid dynamics, Navier- Stoke’s equation
- 4.6 Momentum balance equation; Euler’s equation of motion for inviscid fluid flow
5Z- transform and its Applications6 hours
The Z-transform mirrors the Laplace transform for sequences. Learn the standard pairs and the partial fraction method for the inverse, then apply them to difference equations.
- 5.1 Representation of a sequence and basic operations
- 5.2 Definition and existence of Z-transform, Z-transform of standard sequences
- 5.3 Properties of Z-transform; linearity, change of scale, shifting properties, initial and final value theorems
- 5.4 Differentiations of Z-transform
- 5.5 Inverse Z-transform; partial fraction and residue methods
- 5.6 Convolution of sequences, convolution of Z- transform
- 5.7 Difference equations, application of Z-transform to solve difference equations and to find the sum of series
Tutorials
Practising Engineering Mathematics III
Two tutorial hours a week and no laboratory. Because chapters 1 and 2 carry 36 of the 60 marks between them, they deserve the most practice, but the separation-of-variables method in chapter 4 depends on Fourier series, so study the chapters in order.
Before and after
How Engineering Mathematics III connects to other courses
Builds on
Leads to
BEI uses the transforms in Signals and Systems next semester, both BCT and BEI use them in Year IV signal processing, and Civil uses the flow equations in Fluid Mechanics and Hydraulics.
Reference books
Books IOE lists for ENSH 201
- Jeffery A. (2002). Advanced Engineering Mathematics (2nd edition). San Diego: Harcourt Academic Press.
- O’Neill, P.V. (2011). Advanced Engineering Mathematics (7th edition). India: Thompsons, USA/Baba Baghanath Printers.
- Kreyszig, A. (2020). Advanced engineering Mathematics (10th edition). USA: Wiley Publications.
- Sastry S.S. (2014). Engineering Mathematics vol I and II (4th edition). India: PHI Learning Pvt. Ltd.
- Wylie C., Barrett L. (1988). Advanced Engineering Mathematics (5th edition). McGraw Hill.
- Dutta, D. (2006). A text book of Engineering Mathematics Vol I and II (2nd edition). India: New Age International Publishers.
- Ogata, K. (2015). Discrete Time Control System (2nd edition). Pearson Publications.
- Sharma, Sanjay. (2017). Signals and Systems (9th edition). India: S.K.Kataria and Sons.
Quick answers
Engineering Mathematics III questions
Which chapters of Engineering Mathematics III carry the most marks?
Fourier Series and Fourier Transform and Functions of Complex Variable, 18 marks each out of 60 in IOE's scheme. Together they are 60% of the final paper.
Is the Z-transform part of the IOE Mathematics III syllabus?
Yes. Chapter 5 covers the Z-transform, its properties, the inverse by partial fractions and residues, and solving difference equations, for 8 of the 60 marks.
Why does Civil Engineering study Fourier and Z-transforms?
All three programmes take the same ENSH 201. For Civil, the partial differential equation chapters on heat, waves and fluid flow are the most directly used, in Fluid Mechanics and Hydraulics.
How many credits and marks is ENSH 201 Engineering Mathematics III?
3 credits and 100 marks: 40 internal and 60 in a 3-hour IOE final for theory. It is taught 3 lecture, 2 tutorial and 0 practical hours a week.
Source
Checked against IOE
The outline, references and marks are IOE's own, from the ENSH 201 syllabus PDF and IOE's curriculum structure. The study advice is ICE's. If IOE revises the course, its syllabus is what counts. IOE's BCE curriculum page.
