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MATHEMATICS -1
UNIT I
COMPLEX NUMBERS AND INFINITE SERIES: De Moivre’s theorem and roots of complex numbers. Euler’s theorem, Logarithmic Functions, Circular, Hyperbolic Functions and their Inverses. Convergence and Divergence of Infinite series, Comparison test d’Alembert’s ratio test. Higher ratio test, Cauchy’s root test. Alternating series, Lebnitz test, Absolute and conditioinal convergence.
UNIT II
CALCULUS OF ONE VARIABLE: Successive differentiation. Leibnitz theorem (without proof) McLaurin’s and Taylor’s expansion of functions, errors and approximation.
Asymptotes of Cartesian curves. Curveture of curves in Cartesian, parametric and polar coordinates, Tracing of curves in Cartesian, parametric and polar coordinates (like conics, astroid, hypocycloid, Folium of Descartes, Cycloid, Circle, Cardiode, Lemniscate of Bernoulli, equiangular spiral). Reduction Formulae for evaluating
Finding area under the curves, Length of the curves, volume and surface of solids of revolution.
UNIT III
LINEAR ALGEBRA – MATERICES: Rank of matrix, Linear transformations, Hermitian and skeew – Hermitian forms, Inverse of matrix by elementary operations. Consistency of linear simultaneous equations, Diagonalisation of a matrix, Eigen values and eigen vectors. Caley – Hamilton theorem (without proof).
UNIT IV
ORDINARY DIFFERENTIAL EQUATIONS: First order differential equations – exact and reducible to exact form. Linear differential equations of higher order with constant coefficients. Solution of simultaneous differential equations. Variation of parameters, Solution of homogeneous differential equations – Canchy and Legendre forms.
MATHEMATICS -2
UNIT - I
CALCULUS OF SEVERAL VARIABLES:
Partial differentiation, ordinary derivatives of first and second order in terms of partial derivaties, Euler’s theorem on homogeneous functions, change of variables, Taylor’s theorem of two variables and its application to approximate errors. Maxima and Minima of two variables, Langranges method of undermined multipliers and Jacobians.
UNIT - II
FUNCTIONS OF COMPLEX VARIABLES:
Derivatives of complex functions, Analytic functions, Cauchy-Riemann equations, Harmonic Conjugates, Conformal mapping, Standard mappings – linear, square, inverse and bilinear. Complex line integral, Cauchy’s integral theorem, Cauchy’s integral formula, Zeros and Singularities / Taylor series, Laurents series, Calculation of residues. Residue theorem, Evaluation and real integrals.
Unit - III
VECTOR CALCULUS:
Scalar and Vector point functions, Gradient, Divergence, Curl with geometrical physical interpretations, Directional: derivatives, Properties.
Line integrals and application to work done, Green’s Lemma, Surface integrals and Volume integrals, Stoke’s theorem and Gauss divergence theorem (both without proof).
UNIT - IV
LAPLACE TRANSFORMATION:
Existence condition, Laplace transform of standard functions, Properties, Inverse Laplace transform of functions using partial fractions, Convolution and coinvolution theorem. Solving linear differential equations using Laplace transform. Unit step function, Impulse function and Periodic function and their transforms.
TEXT BOOKS:
1. E. Kresyzig, “Advanced Engineering Mathematics”, John Wiley and Sons. (Latest edition).
2. R. K. Jain and S. R. K. Iyengar, “Advanced Engineering Mathematics”, Narosa, 2003 (2nd Ed.).
3. Dr. A. B. Mathur, V. P. Jaggi, “Advanced Engineering Mathematics”, Khanna Publishers.
REFERENCE BOOKS:
1. V. V. Mitin, M. P. Polis and D. A. Romanov, “Modern Advanced Mathematics for Engineers”, John Wiley and Sons, 2001.
2. R. Wylie, “Advanced Engineering Mathematics”, McGraw-Hill, 1995.
MATHEMATICS - 3:
UNIT – I
Laplace Transformation: Laplace Transformation, Inverse Laplace transformation Convolution Theorem, application to linear differential equations with constant coefficients, Unit step function, impulse functions / periodic functions. [No. of Hrs.: 11]
UNIT – II
Fourier Series: Fourier Series, Euler’s formulae, even and odd functions, having arbitrary periods, half range expansion, Harmonic Analysis.
Fourier Transforms: Fourier transform, Sine and Cosine transforms, Application to differential equations. [No. of Hrs.: 11]
UNIT – III
Special Functions: Beta and Gamma functions, Bessels functions of first kind, Recurrence relations, modified Bessel functions of first kind, Ber and Be functions, Legendre Polynomial, Rodrigue’s formula, orthogonal expansion of function. [No. of Hrs.: 11]
UNIT – IV
Partial Differential Equation: Formation of first and second order linear equations, Laplace, Wave and heat conduction equation, initial and boundary value problems. [No. of Hrs.: 11]
TEXT BOOKS:
1. E. Kresyig, “Advanced Engineering Mathematics”, 5th Edition, John Wiley & Sons, 1999.
REFERENCE BOOKS:
1. B.S. Grewal, “Elementary Engineering Mathematics”, 34th Ed., 1998.
2. H.K. Dass, “Advanced Engineering Mathematics”, S. Chand & Company, 9th Revised Edition, 2001.
3. Shanti Narayan, “Integral Calculus”, S. Chand & Company, 1999
4. Shanti Narayan, “Differential Caluculs”, S.Chand & Company, 1998
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