About the subject
What Engineering Mathematics II is for
Where Mathematics I dealt with one variable, this course moves to fields that vary in space: a temperature across a plate, a force field around a charge, water flowing through a pipe. Vector calculus, the largest chapter, gives the language for all of them, and Green's, Gauss's and Stokes's theorems connect what happens inside a region to what happens on its boundary.
The Laplace transform and matrices are the other two pillars. The Laplace transform turns a differential equation into algebra, which is how circuit and control problems are solved later. Matrices and eigenvalues sit underneath structural analysis, computer graphics and machine learning.
After completion of the course students will be able to apply knowledge of partial differentiation, multiple integrals, vector calculus, optimization, matrices and infinite series in their corresponding study area.
IOE's course objective for ENSH 151
- Taught to
- BCE, Semester 2 (Year I, Part II), BCT, Semester 2 (Year I, Part II) and BEI, Semester 2 (Year I, Part II)
- 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 151 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 151 outline, with how to study each chapter
All 6 chapters and 28 topics as IOE lists them, each with ICE's advice on approaching it.
1Calculus of Two and More Variables6 hours
Partial derivatives and Lagrange multipliers. Treat optimisation problems as a recipe: set up the function and constraint, then solve the system.
- 1.1 Partial differentiation
- Partial derivatives of first and higher order
- Homogeneous function: Euler’s theorem for two and three variables
- Total derivatives and differentials, differentiation of composite and implicit functions
- Jacobians and their properties
- 1.2 Extreme values of two and three variables. Lagrange’s multiplier
- 1.3 Application in optimization of function of two variables in one constraint
2Multiple Integrals7 hours
Changing the order of integration is the skill examiners test most in double integrals. Always sketch the region first, then choose Cartesian, polar, cylindrical or spherical coordinates to match its shape.
- 2.1 Double integrals in Cartesian and Polar form, change of order of integration
- 2.2 Triple integrals in Cartesian, cylindrical and spherical coordinates
- 2.3 Area, volume, moment of inertia, mass and centroid by double and triple integrals
3Vector Calculus12 hours
The centre of the course at 12 hours and 18 of the 60 final marks. Learn gradient, divergence and curl as physical ideas, not only formulas, and practise converting between line, surface and volume integrals with the three theorems.
- 3.1 Review of scalar and vector products, scalar and vector triple product, scalar and vector product of four vectors
- 3.2 Vector differentiation and integration, their geometrical meaning, velocity and acceleration
- 3.3 Vector differential operators: Gradient, directional derivatives, divergence and curl
- 3.4 Line integrals, independent of path, conservative and irrotational vector fields, scalar potential
- 3.5 Introduction to Green’s theorem and its application
- 3.6 Surface integrals, calculation of flux
- 3.7 Volume integrals, Gauss divergence theorem (Without proof) and its application in evaluation of surface integrals
- 3.8 Introduction to Stoke’s theorem and its application
4Laplace Transform7 hours
Build a table of standard transforms and properties and use it constantly. The final section, solving differential equations, is where the chapter pays off.
- 4.1 Definition of Laplace transform, condition for existence, Laplace transforms of some elementary functions, properties of Laplace transform, shifting and change of scale properties
- 4.2 Inverse Laplace transform, uniqueness of inverse Laplace transform, properties of inverse Laplace transform
- 4.3 Laplace transform of derivatives and integral, multiplication and division by tn the convolution theorem
- 4.4 Laplace transform of Heaviside’s unit function, Dirac-delta function and periodic functions
- 4.5 Application of Laplace transform to ordinary differential equations
5Matrices8 hours
Rank, linear dependence and eigenvalues. Work several full diagonalisation examples by hand; the steps are mechanical once practised.
- 5.1 Review of algebra of real and complex matrices
- 5.2 Rank of matrices and its application in system of linear equations
- 5.3 Vector space, linear dependence and independence
- 5.4 Eigen values: Cayley Hamilton theorem and its applications
- 5.5 Eigen vectors, diagonalization of matrices
- 5.6 Reduction of quadratic forms into canonical forms (Three variables only)
6Solution of Differential Equation in Series and Special Functions5 hours
Power series solutions and the Bessel and Legendre functions. The shortest chapter; focus on the method and the main properties.
- 6.1 Power series method
- 6.2 Bessel’s functions: Introduction, properties and application
- 6.3 Legendre’s function: Introduction, properties and application
Tutorials
Practising Engineering Mathematics II
As in Mathematics I, the work happens in tutorials, with two tutorial hours a week and no laboratory. Because vector calculus carries the most marks, give it the most practice time, and keep a single sheet of Laplace transform pairs that you build as the chapter goes.
Before and after
How Engineering Mathematics II connects to other courses
Builds on
Leads to
Mathematics III follows in Semester 3. Electromagnetics relies on vector calculus, and circuit and control courses use the Laplace transform.
Reference books
Books IOE lists for ENSH 151
- Kreyszig, E. (2011). Advanced engineering mathematics. John Wiley & Sons.
- Jeffrey, A. (2002). Advanced engineering mathematics. Academic Press.
- O’Neil, P.V. (2011). Advanced engineering mathematics. Cengage Learning.
- Sastry, S.S. (2008). Engineering mathematics (Vols. I–II). PHI Learning.
- Wylie, C.R, Barrett, L.C. (1995). Advanced engineering mathematics (Latest Edition). McGraw-Hill.
- Dutta, D. (2006). Textbook of engineering mathematics (Vols. I–II). New Age International.
Quick answers
Engineering Mathematics II questions
Which chapter carries the most marks in Engineering Mathematics II?
Vector Calculus, with 18 of the 60 final marks and 12 of the 45 lecture hours in IOE's evaluation scheme. Matrices come next with 12 marks.
Is the Laplace transform in Mathematics II or III?
In Mathematics II (ENSH 151), chapter 4. Mathematics III then covers the Fourier and Z-transforms and relates the Fourier transform back to Laplace.
Do all three programmes take ENSH 151?
Yes, Civil, Computer and BEI all take it in Semester 2 with the same outline.
How many credits and marks is ENSH 151 Engineering Mathematics II?
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 151 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.
