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# B.TECH Mathematics

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## MATHEMATICS-I

Section-A

Infinite series: Convergence and divergence, comparison tests, D’ Alembert’s ratio test, integral test, Raabe’s test, logarithmic and Cauchy root tests, Gauss’s test, alternating series, absolute and conditional convergence.

Section-B

Matrices & Its Application : Rank of a matrix, elementary transformations, elementary matrices, inverse using elementary transformations, normal form of a matrix, linear dependence and independence of vectors, consistency of linear system of equations, linear and orthogonal transformations, eigenvalues and eigenvectors, properties of eigenvalues, Cayley- Hamilton theorem and its applications, diagonalization of matrices, similar matrices, quadratic forms.

Section-C

Differential and Integral Calculus; Successive differentiation, Leibnitz theorem, and applications, Taylor’s and Maclaurin’s series, curvature, asymptotes, curve tracing. Functions of two or more variables, limit, and continuity, partial derivatives, total differential and differentiability, derivatives of composite and implicit functions, jacobians, higher-order partial derivatives, homogeneous functions, Euler’s Theorem and applications. Taylor’s series for functions of two variables (without proof) maxima-minima of the function of two variables. Lagrange’s method of undetermined multipliers, differentiation under the integral sign (Leibnitz rule).

Section-D

Integral Calculus:  Beta and gamma functions and the relationship between them. Applications of single integration to find the volume of solids and surface area of solids of revolution. Double integral, change of order of integration, double integral in polar coordinates, applications of double integrals to find the area enclosed by plane curves, triple integrals, change of variables, the volume of solids, Dirichlet’s integral.

### MATHEMATICS-II

Section-A

Vector Calculus: Differentiation of vectors, scalar, and vector point functions. The gradient of a scalar field and directional derivative, divergence, and curl of a vector field and their physical interpretations. Integration of vectors, line integral, surface integral, volume integral, Green, Stoke’s and Gauss theorems (without proof) and their applications.

Section-B

Ordinary Differential Equations and Applications: Exact differential equations, equations reducible to exact differential equations. Applications of differential equations of first order & first degree to simple electric circuits, Newton’s law of cooling, heat flow and orthogonal trajectories, linear differential equations of second and higher-order. Complete solution, complementary function, and particular integral, method of variation of parameters to find particular integral, Cauchy’s and Legendre’s linear equations. Simultaneous linear equations with constant coefficients. Applications of linear differential equations to the simple pendulum, oscillatory electric circuits.

Section-C

Laplace Transforms and its Applications: Laplace transforms of elementary functions. Properties of Laplace transforms, existing conditions transforms of derivatives, transforms of integrals, multiplication in by t n, division by t. Evaluation of integrals by Laplace transforms. Laplace transform of a unit step function, unit impulse function and periodic function. Inverse transforms, convolution theorem, application to linear differential equations and simultaneous linear differential equations with constant coefficients and applications to integral equations.

Section-D

Partial Differential Equations and Its Applications: Formation of partial differential equations, Lagrange’ linear partial differential equation, first-order non-linear partial differential equation, Charpit’s method. Method of separation of variables and its applications to wave equation, one-dimensional heat equation and two-dimensional heat flow (steady-state solutions only).

### MATHEMATICS-III

Section-A

Fourier Series and Fourier Transform: Euler’s formulae, conditions for a Fourier expansion, change of interval, Fourier expansion of odd and even functions, Fourier expansion of square wave, a rectangular wave, saw-toothed wave, half and full rectified wave, half range sine, and cosine series. Fourier integrals, Fourier transforms, Shifting theorem (both on time and frequency axes), Fourier transforms of derivatives, Fourier transforms of integrals, Convolution theorem, Fourier transform of Dirac-delta function.

Section -B

Functions of Complex Variable: Definition, Exponential function, Trigonometric and Hyperbolic functions, Logarithmic functions. Limit and Continuity of a function, Differentiability, and Analyticity. Cauchy-Riemann equations, necessary and sufficient conditions for a function to be analytic, polar form of the Cauchy-Riemann equations. Harmonic functions, application to flow problems. Integration of complex functions. Cauchy-Integral theorem and formula.

Section -C

Power series, radius, and circle of convergence, Taylor’s Maclaurin’s and Laurent’s series. Zeroes and singularities of complex functions, Residues. Evaluation of real integrals using residues (around unit and semicircle only). Probability Distributions and Hypothesis Testing: Conditional probability, Bayes theorem, and its applications, the expected value of a random variable. Properties and application of Binomial, Poisson and Normal distributions.

Section -D

Testing of a hypothesis, tests of significance for large samples, Student’s t-distribution (applications only), and Chi-square test of goodness of fit. Linear Programming: Linear programming problems formulation, solving linear programming problems using the Graphical method (ii) Simplex method (iii) Dual simplex method.

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