Hindi - Himachal Pradesh

a matrix. Applications of matrices to solve a system of linear homogeneous /nonhomogeneousequations.Vector space, linear dependence and independence, Subspaces, Bases, dimensions. Finitedimensional vector spaces.Linear transformations, the algebra of linear transformations, isomorphism, representationof transformations by Matrices, linear functionals. The double dual and the transpose of alinear transformation.Inner product spaces. Cauchy-Schwarz inequality. Orthogonal vectors. Orthogonalcomplements. Orthonormal sets and orthonormal bases. Bessel’s inequality for finitedimensional spaces. Gram-Schmidt orthogonalization process. Linear functionals andadjoints.Calculus:Real numbers, limits, continuity, differerentiability, mean-value theorems. Taylor'stheorem with remainders. Indeterminate forms, maxima and minima, asymptotesCurvature, Concavity, Convexity, Points of inflexion and tracing of curves.Functions of two variables: continuity, differentiability, partial derivatives, Euler’stheorem for homogeneous functions, Jacobian, maxima and minima. Lagrange's method ofmultipliers. Riemann's definition of definite integrals. Indefinite integrals, infinite andimproper integrals, beta and gamma functions. Double and triple integrals. Areas, surfaceand volumes.Analytic Geometry:Cartesian and polar coordinates in two and three dimensions, second degree equations intwo and three dimensions, reduction to canonical forms, straight lines, shortest distancebetween two skew lines. Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloidof one and two sheets and their properties.SECTION -BOrdinary Differential Equations:Formulation of differential equations, order and degree, equations of first order and firstdegree, integrating factor, equations of first order but not of first degree, Clairaut'sequation, singular solution.Higher order linear equations with constant coefficients, complementary function andparticular integral, general solution, Euler-Cauchy equation.Second order linear equations with variable coefficients, determination of completesolution when one solution is known, method of variation of parameters.Solution by Power series method and its basis, solution of Bessel and Legendre’sequations, properties of Bessel and Legendre functions.Vector Analysis:Scalar and vector fields, triple products, differentiation of vector function of a scalarvariable, gradient, divergence and curl in Cartesian, cylindrical and spherical coordinatesand their physical interpretations. Higher order derivatives, vector identities and vectorequations.Applications to Geometry: curves in space, curvature and torsion. Serret-Frenet'sformulae, Gauss’ and Stokes' theorems, Green's identities.Statics:Analytical conditions of equilibrium of coplanar forces, virtual work.