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 and Cholesky method, Gauss-Seidel/ Jacobi iterative methods; Condition number; Minimum norm and least square error solutions; Eigenvalues and eigenvectors of matrices and their properties; Similarity transformation; Jordon canonical form and orthogonal diagonalization; Mises power method for finding eigenvalues/eigenvectors of symmetric matrices. Tensor Algebra and Index Notation. Vector and Tensor Calculus: Curves and surfaces; Gradient, divergence and curl, Line, surface and volume integrals; Gauss (divergence), Stokes and Green’s theorems. Topics in Numerical Methods: Solution of a non-linear algebraic equation and system of equations; Interpolation methods, Regression; Numerical Integration. 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; Numerical methods for solving 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: discontinuous right-hand sides

Lecturewise Breakup (based on 50 min per lecture)

I. Introduction (1 Lecture)

• Introduction to the course. [1 Lecture]

II. Linear Algebra (12 Lectures)

• Vector spaces: definition, linear independence of vectors, basis, inner product and inner product space, orthogonality, Gram-Schmidt procedure, subspaces. [2 Lectures]

• Matrices: coordinate-dependent linear transformations, null and range spaces. [1 Lectures]

• Linear algebraic equations: existence and uniqueness of solution, elementary row/column operations, Gauss elimination and Gauss Jordon methods, Echelon form, pivoting, LU decomposition and Cholesky method, Gauss-Seidel and Jacobi iterative methods, condition number, minimum norm and least square error solutions. [4.5 Lectures]

• Eigenvalues and eigenvectors of matrices: properties like multiplicity, eigenspace, spectrum and linear independence of eigenvectors, similarity transformation and Jordon canonical form, eigenvalues/eigenvectors of symmetric matrices: orthogonal diagonalization. [3 Lectures]

• Iterative methods to find eigenvalues/eigenvectors of symmetric matrices: forward iteration and Mises power method, inverse iteration. [1.5 Lectures]

III. Tensor Algebra (4 Lectures)

• Index Notation and Summation Convention. [1 Lectures]

• Tensor algebra: tensor as a linear vector transformation, dyadic representation, transformation of components, product of tensors, transpose, decomposition into symmetric and antisymmetric parts, invariants, decomposition into isotropic and deviatoric parts, inner product and norm, inverse, orthogonal tensors, eigenvalues and eigenvectors, square-root, positive definite symmetric tensor, polar decomposition, tensors of higher order. [3 Lectures]

IV. Vector and Tensor Calculus (4 Lectures)

• Review of multi-variable calculus. [0.5Lecture]

• Curves: parametric representation, tangent vector, arc length, curvature, principal normal vector, osculating plane, bi-normal vector; Surfaces: parametric representation, tangent vector and tangent plane. [1.5Lecture]

• 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). [1.5Lecture]

• Tensor calculus: tensor derivative of a scalar field, gradient of a vector field, divergence of a tensor field. [0.5Lecture]

V. Topics in Numerical Methods (5 Lectures)

• Solution of non-linear algebraic equations, Newton-Raphson method for a system of non-linear algebraic equations. [1.5Lectures]

• Lagrange and Hermite interpolation methods. [1Lectures]

• Regression: linear least-square method. [1Lectures]

• Numerical integration: trapezoidal and Simpson’s rules, Gauss quadrature. [1.5Lectures]

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

• Initial value problem (IVP) of a 1st order ODE: existence, uniqueness and continuity with initial conditions. [0.5 Lectures]

• 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. [1.5 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. [1.5 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 [1.5 Lectures]

• Numerical methods for solving IVP of ODEs: Euler and Runge-Kutta methods, stability of numerical methods. [1.5 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. [3.5 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]

  References:

  1. Advanced Engineering Mathematics by E. Kreyszig, John Wiley and Sons, International 8th Revised Edition, 1999.

  2. Applied Mathematical Methods by B. Dasgupta, Pearson Education, 2006.