B.Tech Math Classes
5.4 hours (24 lessons)
About the course
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.
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.
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).
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.
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.
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.
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).
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 Lapla