BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI- Hyderabad CampusINSTRUCTION DIVISION, SECOND SEMESTER 2010-2011

Course Handout(Part II)

Dated: 07/01/2011In addition to Part I (General Handout for all courses appended to the time table) this portion gives further specific details regarding the course.

Course Number : MATH C192Course Title : MATHEMATICS IIInstructor-in-charge : T KURMAYYA

Instructors: D. Tripathi, B Mishra, K Venkata Ratnam, MS Radhakrishnan, PK Sahoo, , TSL Radhika.

1. Scope and Objective of the Course:The course is meant as an introduction to Linear Algebra and Theory of Complex Variable Functions and their applications. Students are encouraged to study MATLAB’s capabilities for solving linear algebra problems given in the Text Book.

2. Course Description :Vector spaces, basis and dimension of vector spaces. Linear transformations, range and kernel. Elementary row operations. System of linear equations. Eigenvalues and eigenvectors. Complex functions and their analyticity. Elementary complex functions. Complex integration. Taylor and Laurent series expansions. Calculus of residues and its applications.

3. Text Books:(a) Introductory Linear Algebra with Applications by B. Kolman

and D.R. Hill, 8th edition, 2005, Pearson Education, Inc.(b) Complex Variables and applications by R.V. Churchill and J.W.

Brown, 8th edition, 2008, McGraw-Hill.

4. Reference Books:(a)Linear Algebra and its Applications by D.C. Lay, 3rd edition,

Pearson Education, Inc.(b)Complex Variables with Applications by A.D. Wunsch, 3rd edition,

Pearson Education, Inc.

5. Course Plan:Lec. No.

Learning Objectives Topics to be covered Ref. Sec. No. of Text Book

A. Linear Algebra1,2. Solving system of linear

equations. Solutions of linear systems of equations, The inverse of a matrix.

1.6, 1.7.

3 - 11. Introduction to abstract Vector spaces, 6.1- 6.7.

vector spaces, finite and infinite dimensional vector spaces and related concepts.

subspaces, linear independence, basis and dimension. Rank and inverse of a matrix and applications. Coordinates and change of basis.

12 - 14.

Introduction to linear transformations, examples of linear transformations.

Definition and examples, kernel and range of linear transformation.

1.5,10.1,10.2.

15,16. Understanding the link between linear transformations and matrices.

The matrix of a linear transformation.

10.3.

17,18. Acquiring more knowledge of linear transformations.

Composite and invertible linear transformations.

Appendix B2

19,20. Computing eigenvalues and eigenvectors.

Eigenvalues and eigen vectors.

8.1.

B. COMPLEX VARIABLES 21,22. Revising the knowledge

of complex numbers.Review 1-11

23 Evaluation of limit of functions of complex variables at a point. Testing continuity of such functions.

Functions of a complex variable. Limit and continuity

12,15-18

24-26 Introduction to analytic functions. Finding out singular point of a function.

Derivative, CR-equations, analytic functions.

19-24,26

27,28 Study of elementary functions. These functions occur frequently all through the complex variable theory.

Exponential, trigonometric and hyperbolic functions.

29-35

29,30 Understanding Multiple Valued Function, branch cut branch point

Logarithmic functions, complex exponents, inverse functions.

29-35

31,32 Integrating along a curve in complex plane.

Contour integrals, anti-derivatives .

37-43

33,34,35

Learning techniques to find integrals over particular contours of different functions.

Cauchy-Goursat Theor

44-46, 48-51

2

em,

Cauchy Integral Formula, Morera’s Theorem. No proof

36 To study application of complex variable theory to algebra.

Liouville’s Theorem, Fundamental Theorem of Algebra.

53,54

37 Series expansion of a function analytic in an annular domain. To study different types of singular points.

Laurent series. No proof

60-62

38,39 Calculating residues at isolated singular points.

Residues, Residue Theorem.

68-73

40,41 To study application of complex integration to improper real integral.

Improper real integrals. 78-81

6. Evaluation Scheme:ECNo.

Evaluation Component

Duration

Weightage %

Date Time & Venue

Nature of Component

1. Test I 50 min.

25 29/01 8.00 – 8.50 AM CB

2. Test II 50 min.

25 26/02 8.00 – 8.50 AM OB

3. Quiz 30 min

10 26/03 8.00 – 8.50 AM CB

5. Compre. Exam.

3 hrs. 40 13/05 FN CB

7. Make-up : Make-up will be given only in genuine cases.

8. Chamber consultation hour: To be announced in the class.

9. Notices: All notices regarding MATH C192 will be put on LTC Notice Board.

Instructor-In-Charge MATH C192

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