Course Content:

Linear Algebra: Vector space and its basis; Matrices as coordinate-dependent linear transformation; null and range spaces; Solution of linear algebraic equations: Gauss elimination and Gauss-Jordon methods, LU decomposition; Determinants; Eigenvalues and eigenvectors of matrices and their properties; Similarity transformation; Jordon canonical form and orthogonal diagonalization,symmetric matrices. Vector Calculus: Curves and surfaces; Gradient, divergence and curl, Line, surface and volume integrals; Gauss (divergence), Stokes and Green's theorems. Ordinary Differential Equations (ODEs): Techniques of the separation of variable and the integrating factor for 1st order ODEs; Solutions of linear, 2nd order ODEs with constant coefficients and Euler-Cauchy ODEs; System of 1st order ODEs, Homogeneous, linear, 2nd order ODEs with variable coefficients: power series and Frobenius methods; Sturm-Louville problem; Laplace transform method for non-homogeneous, linear, 2nd order ODEs. Partial Differential Equations (PDEs): Fourier series, integrals, and transforms; String vibration equation; heat equation; Laplace's equation.

Lecture wise Breakup (based on 50 min per lecture)

I. Linear Algebra (15 Lectures)

• Vector spaces: definition, linear independence of vectors, basis, dimension, inner product and inner product space, orthogonality, least square approximations, Gram-Schmidt procedure, projections, subspaces. [2 lectures]

• Matrices: matrix operations, inverse, null and range spaces. [1 lecture]

• Linear algebraic equations: existence and uniqueness of solution, elementary row/ column operations, Gauss elimination and Gauss Jordon methods, Echelon form, pivoting, LU factorization, complete solution to Ax=b. [4 lectures]

• Determinants: cofactors, inverses, Cramer's rule, volumes. [1 lecture]

• Eigenvalues and eigenvectors: Eigenvalue spectrum and linear independence of eigenvectors, similarity transformation and Jordon canonical form, eigenvalues/ eigenvectors of symmetric matrices: positive definite matrices. [3 lectures]

• The singular value decomposition (SVD): bases and matrices in SVD, principal component analysis, image processing. [2 lectures]

• Applications of linear algebra: matrices in engineering, data sciences, Markov matrices, linear programming. [2 lectures]

II. Vector Calculus (03 Lectures)

• Curves: parametric representation, tangent vector, arc length, curvature, principal normal vector, osculating plane, bi-normal vector; Surfaces: parametric representation, tangent vector and tangent plane. [2 lectures]

• Scalar fields: gradient, directional derivative, potential; Vector fields: divergence, curl, solenoidal and irrotational vector fields; Line integral and path independence; Surface and volume integrals; Gauss (divergence), Stokes and Green's theorems (without proof). [1 lecture]

III. Ordinary Differential Equations (ODEs) (14 Lectures)

• Initial value problem (IVP) of a 1st order ODE: existence, uniqueness and continuity with initial conditions. [1 lecture]

• Methods of solving 1st order ODEs: separation of variable technique, change of variable to make ODE separable; exact ODEs, integrating factor to make ODE exact, linear 1st order ODEs. [2 lectures]

• Homogeneous, linear, 2nd order ODEs: existence and uniqueness, 2 fundamental (linearly independent) solutions and Wronskian, superposition for obtaining general solution, fundamental solutions of ODEs with constant coefficients, method of reduction of order to find 2nd linearly independent solution, fundamental solutions of Euler-Cauchy ODEs. [2 lectures]

• Non-homogeneous, linear, 2nd order ODEs: existence and uniqueness, methods of undermined coefficients and variation of parameters, introduction to higher order ODEs. [2 lectures]

• System of 1st order ODEs: existence and uniqueness of IVP, solution of the homogeneous system with constant coefficients, generalized eigenvector to find other fundamental solutions, method of variation of parameters. [2 lectures]

• Homogeneous, linear, 2nd order ODEs with variable coefficients: power series method, solution of Legendre equation; Frobenius method, solution of Bessel equation; Sturm-Louville problem with regular, periodic and singular (homogeneous) boundary conditions and use of its eigenfunctions as an orthogonal basis for the representation of functions. [3 lectures]

• Laplace transform method for IV problem involving non-homogeneous, linear, 2nd order ODEs, properties of transform, inverse transform using tables, discontinuous right-hand sides involving unit step, impulse and Dirac-delta functions, t-shifting theorem. [2 lectures]

IV. Partial Differential Equations (PDEs) (8 Lectures)

• Fourier series, integrals and transforms. [2 lectures]

• Vibrating string; Separation of variables solution, solution by Fourier series, d'Alembert's solution, wave equation. [2 lectures]

• Heat equation: Solution by Fourier series, integrals and transforms. [2 lectures]

• Laplace's equation: Solution using series and potential methods. [2 lectures]

References:

1. Advanced Engineering Mathematics by E. Kreyszig, John Wiley and Sons.
2. Introduction to Linear Algebra by G. Strang, Wellesley-Cambridge press.