Week 1: Differentiability, Lagrange’s mean value theorem, Cauchy’s mean value theorem, Taylor’s and Maclaurin’s theorem. Functions of several variables: Limit, continuity, partial derivatives and their geometrical interpretation, total differential and differentiability, Derivatives of composite and implicit functions, implicit function theorem, derivatives of higher order and their commutativity, Euler’s theorem on homogeneous functions, Taylor’s expansion of functions, maxima and minima, constrained maxima/minima problems using Lagrange’s method of multipliers
Week 2: Convergence of improper integral, test of convergence, Gamma and Beta functions, their properties, differentiation under the integral sign, Leibnitz rule of differentiation Double and triple integral, change of order of integration, change of variables, Jacobian transformation, Fubini theorem, surface, area and volume integrals, integral dependent on parameters applications, Surface and Volume of revolution. Calculation of center of gravity and center of mass.
Week 3: Differential Equations – first order, solution of first order ODEs, Integrating factor, exact forms, second order ODEs, auxiliary solutions
Week 4: Numerical analysis: Iterative method for solution of system of linear equations, Jacobi and Gauss-Seidal method, solution of transcendental equations: Bisection, Fixed point iteration, Newton-Raphson method.
Week 5: Finite differences, interpolation, error in interpolation polynomials, Newton’s forward and backward interpolation formulae, Lagrange’s interpolation, Numerical integration: Trapezoidal and Simpson’s 1/3rd and 3/8th rule.
Week 6: Vector spaces, basis and dimension, Linear transformation, linear dependence and independence of vectors, Gauss elimination method for system of linear equations for homogeneous and nonhomogeneous equations
Week 7: Rank of a matrix, its properties, solution of system of equations using rank concepts, Row and Column reduced matrices, Echelon Matrix, properties,
Week 8: Hermitian, Skew Hermitian and Unitary matrices, eigenvalues, eigenvectors, its properties, Similarity of matrices, Diagonalization of matrices,
Week 9: Scalar and vector fields, level surface, limit, continuity and differentiability of vector functions, Curve and arc length, unit vectors, directional derivatives,
Week 10: Divergence, Gradient and Curl, Some application to Mechanics, tangent, normal, binormal, Serret-Frenet Formulae, Application to mechanics
Week 11: Line integral, parametric representations, surface integral, volume integral, Gauss divergence theorem, Stokes theorem, Green’s theorem.
Week 12: Limit, continuity, differentiability and analyticity of functions, Cauchy-Riemann equations, line integrals in complex plane Cauchy’s integral formula, derivatives of analytic functions, Cauchy’s integral theorem, Taylor’s series, Laurent series, zeros and singularities, residue theorem, evaluation of real integrals