Vector Calculus: scalar and vector fields – level surfaces – directional derivatives, gradient, curl, divergence – Laplacian – line and surface integrals – theorems of Green, Gauss, and Stokes.
Sequences and Series of Functions: complex sequences – sequences of functions – uniform convergence of series – test for convergence – uniform convergence for series of functions.
Ordinary Differential Equations: first order ordinary differential equations – classification of differential equations – existence and uniqueness of solutions of initial value problem – higher order linear differential equations with constant coefficients – method of variation of parameters and method of undetermined coefficients – power series solutions – regular singular point – Frobenius method to solve variable coefficient differential equations.
Special Functions: Legendre polynomials, Bessel's function, gamma function and their properties – Sturm‐Liouville problems.
. Ross, S. L., Differential Equations, Blaisedell (1995).
2. Kreyszig, E., Advanced Engineering Mathematics, 9th ed., John Wiley (2005).
3. Stewart, J., Calculus: Early Transcendentals, 5th ed., Brooks/Cole (2007).
Greenberg, M.D., Advanced Engineering Mathematics, Pearson Education (2007).
Jain, R.K. and Iyengar, S.R.K., Advanced Engineering Mathematics, Narosa (2005).
Course Outcomes (COs):
CO1: Differentiate between pointwise and uniform convergence, check whether a given series is pointwise or uniformly convergent series, and apply the techniques to integral and differential calculus.
CO2: Analyze and solve ODEs, confirm existence and uniqueness of solutions of IVP. Find power series solution of linear homogeneous ODE with variable coefficients and Frobenius method for equations with regular singular point. Know the special functions like Legendre polynomial, Bessel function etc. and properties. Finding eigenvalues and eigenfunctions for Sturm-Liouville problems.
CO3: Verify continuity/differentiability of scalar/vector-valued function. Calculate line/surface integration of scalar/vector-valued functions. Apply fundamental theorems to understand the nature of vector fields and check if a given vector field is conservative.