Calculus : Partial derivatives, Euler’s theorem on homogeneous functions, Maclaurin’s and Taylor’s expansions of single and two variables, Maxima and minima of functions of several variables, Lagrangian method of multipliers; Multiple integrals and their use in obtaining surface areas and volumes of solids.
Infinite Series : Sequences and sub sequences and their convergence, Cauchy sequence, Infinite series and their convergence, Standard tests for convergence including p–test, Ratio test, Comparison test, Rabe’s test, Cauchy Integral test, Cauchy rot test, Gauss’s test, Absolute convergence, Alternating series and its convergence, Power series.
Vector Calculus : scalar and vector point functions, differentiation of vectors, gradient of a scalar field, line integral of a vector field, surface integral of a vector field, volume integral of a scalar field, gren’s theorem, stokes theorem, gauss divergence theorem (without profs) and their applications.
UNIT – I
Differential Equations : Exact differential Equation, Higher order linear Differential equations, ODE’s with constant coefficients. Laplace Transforms: Laplace transforms, Properties of Laplace transforms, Laplace transform of derivatives and differentiation theorem, Integration theorem, Laplace transform of Integrals, Inverse Laplace transform, Formulas for obtaining inverse Laplace transforms, Convolution theorem, The second shifting property.
Fourier Series and Fourier Transform : Fourier series expansion, Fourier series for even and odd functions, half range series, harmonic functions, Modulation theorem, Shifting properties, convolution theorems, sine and cosine transforms, Fourier transform of derivatives and integrals, inverse Fourier transform, applications to PDE’s & ODE’s .
Complex Analysis : De Moivre’s theorem with applications, Analytic functions, Cauchy– Rieman equations, Laplace equation, Cauchy’s integral theorem, Cauchy’s integral formula (without profs), Taylor series and Laurent series (without profs), Residues and their application in evaluating real improper integrals
Probability : Classical and axiomatic approach to the theory of probability, additive and multiplicative law of probability, conditional probability and bayes theorem. Random Variables: Random variable, probability mass function, probability density function, cumulative distribution function, function of random variable. Two and higher dimensional random variables, joint distribution, marginal and conditional distributions, Stochastic independence.
Expectation : Mathematical expectations and moments, moment generating function and its properties. Probability Distributions: Binomial, Poison, Uniform, Exponential, Gamma, Normal distribution, t–distribution, chi– square distribution, F–distribution.
Uniform Pseudo random number generation and random variable generation, generating random variate from standard statistical distribution (discrete and continuous distribution), Monte – Carlo integration.
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- Quality and experience of faculty members
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