TEW Workshop for Mathematics Teachers of Engineering Colleges (2016)

1. Complex Analysis by Sushmita Venugopalan:

The topics covered in the complex analysis course include:

1. holomorphicity of complex functions,
2. power series and their discs of convergence
3. Cauchy’s integral formula and
4. classification of isolated singularities of holomorphic functions.

The course consisted of six lectures and two tutorial sessions, In the tutorial sessions, teachers were given a problem sheet, and they came up to the board to solve the problems in the sheet.A geometric and visual approach was adopted throughout the course, with the hope of aug-menting the teachers’ understanding of the topics. The course started o↵ with a detailed discussion of the concept of derivative for multi-variable functions. This was then used to explain complex linearity of derivative. Thus the link between the Cauchy-Riemann equations and its geometric con-sequence of conformality was explained. After this, the study of the transformations of the complex plane defined by various holomorphic functions, like linear functions, powers of z, followed by the exponential and logarithm functions were discussed. An entire lecture was spent in conveying a geometric intuition of the exponentialfunction, explaining why the log function cannot be defined on all of the complex plane, and why the log function can be defined on a spiral ramp that is a covering space of the complex plane minus the origin. The concept of analyticity was explained as being a source of rigidity in complex functions. An example of a smooth real function that is not analytic to explain the contrast, was then discussed. The tutorial problem sheets contained many examples of computations of power series and their radius of convergences. These examples were used to illustrate the fact that the disc of convergence was the largest possible disc that was contained in the domain on which the function can be defined holomorphically. Examples of the log function, and non-integer powers of z were discussed from this point of view. A visual proof of the Cauchy integral formula was presented. Various examples on the application of the Cauchy integral formula were discussed in the tutorial and the lecture. Using the geometric discussions of the log function, the integral of 1/z on the unit circle was evaluated. The final part of the course dealt with the types of isolated singularities using various examples.

2. Laplace transforms by Indrava Roy:
In this course standard material concerning Laplace transforms were covered comprising of the following topics:

• Background and definition of Laplace Transforms with definition
• Computing basic examples involving elementary functions for illustration
• Integral transforms
• Properties of Laplace transforms
• Computations involving special functions
• Limiting behaviour- Initial and Final value theorems
• Inverse Laplace transform and the Bromwich integral; use of residue theorem
• Applications to solve linear ordinary and partial di↵erential equations with constant coefficients

Some non-standard material was also covered, which helped bring out technical subtleties and richness of the theoretical aspects as well as applications in real-world problems of Laplace transforms. These include:

• Convergence issues for improper Riemann integrals
• Criteria for exchanging limits and integrals; Dominated convergence theorem for improper integrals
• Applications of Laplace transform to physical systems; examples of damped oscillations and bending of beams were illustrated
• Applications to solve integral and integro-di↵erential equations; example of Abel’s solution for the tautochrone problem and PID controllers in control system engineering were mentioned.
• Applications to solve recurrence relations, e.g. Fibonacci sequence.

3. Vector Calculus by Parameswaran Sankaran
Dot and cross product of vectors. Work done by a force–rotation and angular velocity. Moment of a force about a point acting on a body. Gradient of a vector field. Geometric and physical significance. Scalar potential. Constrained extrema–Lagrange multipliers. Line integral. Work done by a force field along a path. Divergence of a vector field. Surface integral. Flux of a vector field across a surface. Curl of a vector field. Green’s theorem, Stokes’ theorem, divergence theorem.

How to reach

The Chennai neé Madras, and to IMSc, the Institute of Mathematical Sciences. Here is some information on getting to the Institute. Address and phone numbers are

THE INSTITUTE OF MATHEMATICAL SCIENCES
C I T CAMPUS, TARAMANI, CHENNAI 600 113
Telephone Numbers

Here are some maps to give you a rough idea of the layout of Madras and the area around the Institute.

• A simplified schematic map of Madras, showing the major roads and Hotels and marking the area in which the CIT Campus is located.
• A postscript version of this figure if you want to download it to print at your end.
• A detailed Madras map.
• A gzipped postscript version of this figure
• A schenatic map of the location of IMSc within CIT Campus and its environs
• Detailed map of the location of IMSc within CIT Campus and other landmarks in the Greater Adyar area.

The Institute of Mathematical Sciences is located in a campus-like area variously known as the ``CIT Campus'' or the ``CPT Campus''. Many people in Madras may not have heard of the Institute, but almost everyone would have heard of the M.G.R. Film City and Tidel Park which are further down in the same campus area. IMSc has two complexes. One houses the main Institute building, the library building and the new building. The other has the students' hostel and Guest House. These two complexes are across the road from each other.