MAT 210 Programming

Syllabus for MAT 210 – Programming

This is a compulsory course for BS Mathematics students in their 3^rd semester.

From XKCD. Thanks to Aman Agarwal for the reference!

Credits (Lec:Tut:Lab)= 1:0:1 (One lecture hour and three lab hours weekly)

Prerequisites: None

Overview: This course provides an introduction to formal programming languages via the medium of Python 3.0. The programming activities will be centered around mathematical models involving differential equations, algebraic systems, iterative processes, linear transformations, random processes etc. The course begins with Python language constructs and moves to an in-depth exploration of the SCIPY and NUMPY packages that hold the key to the desired mathematical simulations.

Detailed Syllabus:

1. Basics of the PYTHON programming language:

• Input and output statements, formatting output, copy and assignment, arithmetic operations, string operations, lists and tuples, control statements
• User defined functions, call by reference, variable number of arguments
• One dimensional arrays, two dimensional arrays, random number generation
• Classes, static data, private data, inheritance, scope of variables, nested functions

2. The NUMPY and SCIPY packages:

• Numpy numerical types, data type objects, character codes, dtype constructors.
• Mathematical libraries, plotting 2D and 3D functions, ODE integrators, charts and histograms, image processing functions.
• File I/O, loading data from CSV files
• Using SCIPY/NUMPY to solve models involving difference equations, differential equations, finding limit at a point, approximation using Taylor series, interpolation, definite integrals.

Main References:

1. John Zelle, Python Programming: An Introduction to Computer Science. Franklin, Beedle & Associates Inc., Second Edition, 2010.
2. Ivan Idris, Numpy 1.5 Beginner’s Guide. Packt Publishing, 2011.
3. Hans Petter Langtangen, A Primer on Scientific Programming on Python. Springer, Second Edition, 2011.

Other References:

1. Hans Petter Langtangen, Python Scripting for Computational Science. Springer, 2010.
2. David M. Beazley, Python Essential Reference, 3^rd Edition. Pearson, 2009.
   
   

MAT 202 - Mathematical Methods

Syllabus for MAT 202 – Mathematical Methods

This is a compulsory course for B.Tech students in their 3rd semester.
 
Credits (Lec:Tut:Lab)= 3:0:0 (3 lectures weekly)

Prerequisites: Class XII mathematics

Brief Description: The first part is an introduction to multivariable calculus, finishing with the various versions of Stokes' theorem. The second part deals with series of numbers and functions (such as power series and Fourier series) and their applications to solving differential equations. The concepts and techniques covered here are used extensively in the social and natural sciences as well as in engineering to study systems with many dimensions.

Detailed Syllabus:

1. Computer Algebra System (CAS): Equations, solving linear system, function definition, function evaluation, two and three dimensional plots, differentiation, integration, matrices, matrix algebra, simplification of expressions

2. Differential calculus in several variables: Space curves and arc length, functions of several variables, level curves and surfaces, limits and continuity, partial derivatives, tangent planes, chain rule, directional derivatives, gradient, Lagrange multipliers, extreme values and saddle points, 2^nd derivative test

3. Double and triple integrals: Double integrals over rectangles, double integrals over general regions, double integrals in polar coordinates, center of mass, triple integrals, triple integrals in cylindrical coordinates, triple integrals in spherical coordinates, change of variables

4. Vector Integration: Vector fields, line integrals, fundamental theorem, independence of path, Green's theorem, divergence, curl, parametric surfaces, area of a parametric surface, surface integrals, Stokes' theorem, Gauss' divergence theorem.

5. Series and Applications: Limits of sequences, algebra of limits, series, divergence test, comparison and limit comparison tests, integral test, alternating series test, absolute convergence, root & ratio tests, power series, Taylor polynomials and series, power series method for solving ODEs, Legendre's equation, Bessel's equation, orthogonal functions and Sturm-Liouville problem, periodic functions and trigonometric series, Fourier series, half-range expansions, Fourier integral, heat equation

Main References:

• Essential Calculus – Early Transcendentals, by James Stewart. Cengage, India Edition. (Chapters 8 to 13)
• Advanced Engineering Mathematics, Erwin Kreyszig, 9^th edition, Wiley India, 2011.

Supplementary References:

• Advanced Engineering Mathematics, Dennis Zill and Warren Wright, 4^th ed., Jones & Bartlett, 2011.
• Calculus and Analytic Geometry by G B Thomas and R L Finney, 9^th edition, Pearson.
• Basic Multivariable Calculus by J E Marsden, A J Tromba and A Weinstein, 1^st edition, Springer (India), 2011.

Journals for Students

The SNU library now offers online access to many maths journals from the campus. The main repositories are JSTOR and SpringerLink. Some are particularly good for students:

American Mathematical Monthly "Publishes articles, as well as notes and other features, about mathematics and the profession. Its readers span a broad spectrum of mathematical interests, and include professional mathematicians as well as students of mathematics at all collegiate levels."


College Mathematics Journal "Emphasizes the first two years of the college curriculum. The journal contains a wealth of material for teachers and students."

Mathematical Gazette "The original journal of the Mathematical Association and it is now over a century old. Its readership is a mixture of school teachers, college and university lecturers, educationalists and others with an interest in mathematics" 

Mathematical Intelligencer "Not only does The Mathematical Intelligencer inform a broad audience of mathematicians and the wider intellectual community, it also entertains. Throughout, the journal, humor, puzzles, poetry, fiction, and art can be found. The journal also features information on emergent mathematical communities around the world, new interdisciplinary trends, and relations between mathematics and other areas of culture."

Once on campus, you should be able to access all of these!

CCC 801 - Art of Numbers

Syllabus for CCC 801 – Art of Numbers

Credits (Lec:Tut:Lab) = 1.5:0:0 (3 lectures weekly over a half-semester)

Overview: This course deals with two aspects of numbers. In the first part of the course we will take up some unexplored patterns that exist in nature, study them and understand some of their applications. The second part looks at numbers as carriers of information about our lives. Here we learn how to analyze and present data in ways that help us make sense of our lives. We'll use the spreadsheet program in Open Office to analyze the data in depth.

Detailed Syllabus:

Part A: Fun with Numbers

1. Moessner’s Magic
2. Permutation, Combinations
3. Pascal Triangle, Binomial Theorem
4. Fibonacci Sequence
5. Some applications

Part B: Handling Data

6. Interacting with real time data
7. Descriptive Statistics like mean, median, mode, range, standard deviation, percentiles, quartiles
8. Introduction to a Spreadsheet program (Open Office or Excel)
9. Charts – Bar Charts, Histograms, Line Charts, Pie Charts
10. Simulations
11. Case Studies

Assessment:

Assignments	20%
Presentations	40%
Term Paper	40%

References:

1. The Book of Numbers by John Horton Conway, Richard K. Guy. 2^nd edition, Copernicus.
2. The Heart of Mathematics: An Invitation to Effective Thinking by Edward B. Burger, Michael Starbird. 3^rd edition, Wiley.
3. The Visual Display Of Quantitative Information by Edward Tufte. 2^nd edition, Graphics Press.
4. Excel 2007 for Starters: The Missing Manual by Matthew MacDonald. Shroff/O'Reilly.
5. Analyzing Business Data with Excel by Gerald Knight. Shroff/O'Reilly.

CCC 101 - Mathematics in India

Syllabus for CCC 101 – Mathematics in India

Credits (Lec:Tut:Lab) = 1.5:0:0 (3 lectures weekly over a half-semester)

Prerequisites: None

Overview: Mathematics had a rich history in ancient and medieval India. Indian mathematicians made original contributions to algebra, number theory and geometry; while the Kerala school made fundamental discoveries related to differential calculus and infinite series two centuries before their full development by Newton and Leibniz. This course will provide an overview of the story of mathematics in India, and also incorporate the social context and the connections with other civilizations.

Detailed Syllabus: Issues of dating, translation and interpretation; prehistory; the ancient civilizations of Egypt, Iraq, China and America; Indus Valley Civilization; Mathematics in the Vedas and Puranas; Pythagoras theorem; Applications to grammar, logic, astronomy and technology; Medieval mathematicians and schools of mathematics; Universities; Invention of Zero; Trigonometry; Rates of change; π; Connections with Greece, China and the Arabs; The Kerala school.

Assessment:

Assignments	20%
Class Performance	10%
Term Paper	40%
Presentation	30%

References:

1. Mathematics in India by Kim Plofker, Princeton University Press.
2. Studies in the History of Indian Mathematics by C S Seshadri (ed.), Hindustan Book Agency.
3. Contributions to the History of Indian Mathematics by Gerard G Emch et al (ed.), Hindustan Book Agency.
4. History of Mathematics by Carl B Boyer and Uta C Merzbach, Wiley.

Sliding Ladder

The locus of the midpoint of a sliding ladder placed against a wall is a circle, whose centre is at the point of intersection of the wall and the floor.

Proof:

The end points of the ladder are (x, 0) and (0, y).

If the length of the ladder is 1 unit, then using Theorem of Pythagoras,

x^2 + y^2 = 1

Let the midpoint of the ladder is (x', y') or (x/2, y/2) and

r is the distance between the midpoint and the centre (where the wall and floor meets).

Using distance formula:

(x/2 - 0)^2 + (y/2 - 0)^2 = r^2

(x^2 + y^2)/4 = r^2

r^2 = 1/4

Therefore, (x' - 0)^2 + (y' - 0)^2 = 1/4

x'^2 + y'^2 = 1/4

which is the equation of a circle.

"Happy Abstract Algebra Classes"

This article by John Fraleigh on how he revamped the evaluation processes for his Abstract Algebra classes is quite interesting. His book on Abstract Algebra first made me happy about the subject, so I am certainly biased towards accepting his claims...