MAIR11 Matrices and calculus 

Course plan: click here

Class Timings:  

BE  (CIvil Engineering A section): Monday 8:30-9:20; Wednesday 9:20-10:10; Thursday  11:20-12:10

BE (Production Engineering B Section): Monday 10:30-11:20; Wednesday 8:30-9:20;  Thursday 9:20-10:10

Assessments: 

There will be four assessments of 100 marks in total; 50 for class tests and 50 for end semester final examination. Each unit will carry 20 marks.   It includes 2 written class tests of 20 marks each of one hour duration and one assignment test of 10 marks. Final examination will be of 50 marks of 3 hours duration. One compensation test (with syllabus of both CT1 and CT2) will be conducted for those who miss one or both CTs. If one miss both CTs, then he or she miss 20 marks automatically as only one compensation test of 20 marks will be conducted. 

The section  B.12.0 of BE Regulation says the following about "Final Assessment": Every theory/laboratory course should have a final assessment on the entire syllabus with at least 30% weightage conducted for duration of three hours. A student must score a minimum of 20% in the final assessment (for all courses) to complete the course.

Attendance: There will be 40 lectures. Anyone who miss 11 or more classes will be prevented from writing final examination and shall be assigned V grade. Four days of OD is accepted. 

Course Notes:  I will teach using PDF that can be found at the bottom of this page. 

Objectives of the course is to

• 1. study the properties of  eigen value and eigen vectors and determine canonical form of given quadratic form.

• 2. discuss the convergence of infinite series by applying various tests.

• 3. analyze and discuss the extrema of the functions of several variables.

• 4. evaluate the multiple integrals and apply in solving problems.

• 5. study differential operator for scalar/vector functions and verify Green’s, Stoke’s and divergence theorems.

Outcome. On completion of the course, student will be able to

1.  diagonalize a given matrix and use it to compute higher powers of the given matrix and to analyze quadratic forms

2.  test the convergence of infinite series by applying various tests.

3.  compute maxima/minima  for functions of  two and three variables   

4.  evaluate multiple integrals

5.  verify  Green’s, Gauss divergence and Stoke’s theorems  

Syllabus
Eigenvalues and eigenvectors;  Diagonalization  of matrices; Cayley-Hamilton Theorem; Quadratic forms.

Sequence and series: Convergence of sequence. Infinite series-Tests for convergence-Integral test, comparison test, Ratio test, Root test, Raabe’s test, Logarithmic test, and Leibnitz’s test; Power series.

Functions of two variables: Limit, continuity and partial derivatives; Total derivative, Jacobian, Taylor series,  Maxima, minima  and saddle points; Method of Lagrange multipliers.

Double and triple integrals, change of variables, multiple integral in cylindrical and spherical coordinates.

Gradient, divergence and curl; Line and surface integrals; Green's theorem, Stokes theorem and   Gauss divergence theorem (without proofs).

References

1.  Dennis Zill, Warren. Wright, Michael R. Cullen, Advanced Engineering Mathematics, Jones & Bartlett Learning, 2011

2.  Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 2019.

3.  Jerrold E. Marsden, Anthony Tromba, Vector Calculus, W. H. Freeman, 2003

4.  Strauss M.J, G.L. Bradley and K.J. Smith, Multivariable calculus, Prentice Hall, 2002.

5.  Ward Cheney, David Kincaid, Linear Algebra: Theory and Applications, Jones & Bartlett Publishers, 2012.