Objective

To round out the students' preparation for more sophisticated applications with an introduction to linear algebra, Fourier series, Laplace Transforms, integral transformation theorems and linear programming.

Syllabus

1. Determinants and Matrices
   1. Determinant and its properties
   2. SOlution of system of linear equations
   3. Algebra of matrices
   4. Complex matarices
   5. Rank of matrices
   6. System of linear equations
   7. Vector spaces
   8. Linear transformations
   9. Eigen value and Eigen vectors
   10. The Cayley-Hamilton theorem and its uses
   11. Diagonalization of matrices and its applications
2. ** Line, Surface and Volume Integrals**
   1. Line integrals
   2. Evalutation of line integrals
   3. Line integrals independent of path
   4. Surfaces and surface integrals
   5. Green's theorem in the plane and its applications
   6. Stoke's theorem (without proof) and its applications
   7. Volume integral; Diverfence theorem of Gauss (without proof) and its applications
3. Laplace Transform
   1. Definitions and properties of Laplace Transform
   2. Derivations of basic formulae of Laplace Transform
   3. Inverse Laplace Transform: Definition and standard formulae of inverse Laplace Transform
   4. Theorems on Laplace transform and its inverse
   5. Convolution and related problems
   6. Applicaitons of Laplace Transform to ordinary differential equations
4. Fourier Series
   1. Fourier Series
   2. Periodic functions
   3. Odd and even functions
   4. Fourier series for arbitary range
   5. Half ranfe Fourier series
5. Linear Programming
   1. System of Linear Inequalities in two variables
   2. Linear Programming in two dimensions: A Geometrical Approach
   3. A Geometric introduction to the Simplex method
   4. The Simplex method: Maximization with Problem constraints of the form "<="
   5. The Dual: Maximization with Problem constraints of the form ">="
   6. Maximization and Minimization with mixed constraints. The two-phase method (An alternative to Big M Method)