Matrices, row and column reductions, echelon forms. Eigenvalues, eigenvectors and characteristic equation of a matrix. Cayley-Hamilton theorem and its applications, rank of a matrix. Applications of matrices to solve a system of linear homogeneous /nonhomogeneous equations.
Vector space, linear dependence and independence, Subspaces, Bases, dimensions. Finite dimensional vector spaces.
Linear transformations, the algebra of linear transformations, isomorphism, representation of transformations by Matrices, linear functionals. The double dual and the transpose of a linear transformation.
Inner product spaces. Cauchy-Schwarz inequality. Orthogonal vectors. Orthogonal
complements. Orthonormal sets and orthonormal bases. Bessel’s inequality for finite dimensional spaces. Gram-Schmidt orthogonalization process. Linear functionals and adjoints.
Real numbers, limits, continuity, differentiability, mean-value theorems. Taylor’s theorem with remainders. Indeterminate forms, maxima and minima, asymptotes Curvature, Concavity, Convexity, Points of inflexion and tracing of curves.
Functions of two variables: continuity, differentiability, partial derivatives, Euler’s theorem for homogeneous functions, Jacobian, maxima and minima. Lagrange’s method of multipliers. Riemann’s definition of definite integrals. Indefinite integrals, infinite and improper integrals, beta and gamma functions. Double and triple integrals. Areas, surface and volumes.
Cartesian and polar coordinates in two and three dimensions, second degree equations in two and three dimensions, reduction to canonical forms, straight lines, shortest distance between two skew lines. Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.
Ordinary Differential Equations:
Formulation of differential equations, order and degree, equations of first order and first degree, integrating factor, equations of first order but not of first degree, Clairaut’s equation, singular solution.
Higher order linear equations with constant coefficients, complementary function and particular integral, general solution, Euler-Cauchy equation.
Second order linear equations with variable coefficients, determination of complete solution when one solution is known, method of variation of parameters.
Solution by Power series method and its basis, solution of Bessel and Legendre’s equations, properties of Bessel and Legendre functions.
Scalar and vector fields, triple products, differentiation of vector function of a scalar variable, gradient, divergence and curl in Cartesian, cylindrical and spherical coordinates and their physical interpretations. Higher order derivatives, vector identities and vector equations.
Applications to Geometry: curves in space, curvature and torsion. Serret-Frenet’s formulae, Gauss’ and Stokes’ theorems, Green’s identities.
Analytical conditions of equilibrium of coplanar forces, virtual work.
Forces in three dimensions, Poinsot’s central axis, Wrenches, Null lines and planes, Stable and unstable equilibrium.
Simple harmonic motion, motion on rough curve, tangential & normal accelerations, motion in a resisting medium, motion when the mass varies, velocity along radial and transverse directions, central orbits.
Kepler’s laws of motion, motion of a particle in three dimensions, acceleration in terms of Polar and Cartesian co-ordinate systems.
(Note: Use of Scientific non-programmable calculators will be allowed in this paper for numerical analysis part.)
Mappings, elementary properties of integers. Definition of a Group and Subgroup their examples and properties. Normal subgroups, Quotient Groups. Homomorphism, Groupautomorphisms, Cayley’s theorem, permutation Groups.
The Riemann integral: Definition and existence of integral, refinement of partitions, Darboux’s theorem, condition of integrability. Integrability of the sum and difference of integrable functions. The fundamental theorem of calculus, first and second mean value theorems of calculus.
Improper integrals and their convergence, comparison tests, Abel’s and Dirichlet’s tests.
Sequences and series:
Definition of a sequence, theorems on limits of sequences, bounded and monotonic sequences and their convergence. Cauchy’s convergence criterion, algebra of sequences, main theorems, monotonic sequences, series of non-negative terms, comparison test, Cauchy’s Integral test, Ratio test, Raabe’s test, logarithmic test, Gauss’s test, alternating series, Leibnitz’s test. Absolute and conditional convergence.
Definition and examples of metric spaces. Limits in metric spaces. Functions continuous on metric spaces. Open sets. Closed sets. Connected sets. Complete metric spaces. Compact metric spaces. Continuous functions on compact metric spaces, uniform continuity.
Complex numbers, Geometric representation of Complex numbers. Analytic function, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula. Conformal mapping, Bilinear Transformation (Mobius transformation).
Partial Differential Equations:
First order partial differential equations: Partial differential equations of the first order in two independent variables, formulation of first order partial differential equation, solution of linear first order partial differential equations (Lagrange’s Method), integral surfaces passing through a given curve, surfaces orthogonal to a given system of surfaces, solution of non-linear partial differential equations of first order by Charpit’s method.
Second order partial differential equations: Origin and classification of second order partial differential equation, solution of linear partial differential equation with constant coefficients. Monge’s method to solve the non-linear partial differential equation Rr+Ss+Tt = V.
Introduction, basic theory of Laplace transforms, solution of initial value problem using Laplace transforms, shifting theorems, unit step function, Dirac-delta function.
Differentiation and integration of Laplace transforms. Convolution theorem.
Calculus of Variations:
Variation problems with fixed boundaries-Euler’s equation for functionals containing first order derivative and one independent variable. Extremals. Functionals dependent on higher order derivatives. Functionals dependent on more than one independent variable.
Variational problems in parametric form. Invariance of Euler’s equation under coordinates transformation.
Variational problems with moving boundaries-functionals dependent on one and two functions.
Sufficient conditions for an Extremum-Jacobi and Legendre conditions.
Numerical Analysis and computer programming:
Numerical Methods: Solution of algebraic and transcendental equations of one variable by Bisection, Secant, Regula Falsi, Newton-Raphson Method, Roots of Polynomials.
Linear Equations: Solution of system of linear equations by Gaussian elimination method, Gauss-Seidel iterative method.
Interpolation: Lagrange and Newton interpolation, divided differences, difference schemes, interpolation formulas using differences.
Numerical Differentiation: Solution of ordinary differential equations by Euler’s method, Runge-Kutta’s II and IV order method.
Numerical Integration: Simpson’s 1/3 rule, Simpson’s 3/8 rule, Trapezodial rule, Gaussian quadrature formula.
Programming in C: Algorithms and flow-charts for solving numerical problems.
Developing simple programs in C language for problems involving techniques covered in the numerical analysis.