Teaching Scheme (in Hours)
Subject Credit : 5
Examination Scheme (in marks)
ESE Viva (V)
Syllabus Content Download
Unit-1: Indeterminate Forms and L'Haspital's Rule, Improper Integrals, Applications of definite integral
Indeterminate Forms and L’Hôspital’s Rule. Improper Integrals, Convergence and divergence of the integrals, Beta and Gamma functions and their properties. Applications of definite integral, Volume using cross-sections, Length of plane curves, Areas of Surfaces of Revolution
Unit-2: Convergence and divergence of sequences
Convergence and divergence of sequences, The Sandwich Theorem for Sequences. The Continuous Function Theorem for Sequences. Bounded Monotonic Sequences, Convergence and divergence of an infinite series, geometric series, telescoping series, nth term test for divergent series. Combining series, Harmonic Series, Integral test, The p - series, The Comparison test. The Limit Comparison test. Ratio test. Raabe's Test. Root test, Alternating series test, Absolute and Conditional convergence, Power series. Radius of convergence of a power series. Taylor and Maclaurin series.
Unit-3: Fourier Series
Fourier Series of 2PI periodic functions. Dirichlet's conditions for representation by a Fourier series. Orthogonality of the trigonometric system. Fourier Series of a function of period 2L. Fourier Series of even and odd functions, Half range expansions.
Unit-4: Functions of several variables
Functions of several variables, Limits and continuity, Test for non existence of a limit. Partial differentiation. Mixed derivative theorem. differentiability, Chain rule, Implicit differentiation, Gradient, Directional derivative, tangent plane and normal line, total differentiation, Local extreme values, Method of Lagrange Multipliers.
Multiple integral, Double integral over Rectangles and general regions, double integrals as volumes, Change of order of integration, double integration in polar coordinates, Area by double integration, Triple integrals in rectangular, cylindrical and spherical coordinates, Jacobian, multiple integral by substitution.
Unit-6: Elementary row operations in Matrix
Elementary row operations in Matrix, Row echelon and Reduced row echelon forms, Rank by echelon forms, Inverse by Gauss-Jordan method, Solution of system of linear equations by Gauss elimination and GaussJordan methods. Eigen values and eigen vectors, Cayley-Hamilton theorem, Diagonalization of a matrix.
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