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30th July 2014, 09:24 AM
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Join Date: Apr 2013
Re: GSAT M.Sc Applies Mathematics Syllabus

Detailed information regarding the Syllabus of GSAT M.Sc Applies Mathematics is given below :

Real number system, sequences and series :

Field axioms, Dedikind’s axiom, Bolzano weistrass theorem, countability of sets, Sequences and their limits, subsequences, convergence and divergence of sequences, limit of a sequence, Cauchy sequences, Cauchy general principle of convergence, Definition of infinite series, necessary condition for convergence, Cauchy general principle of convergence, comparision test, nth root test, ratio test, integral test, Alternating series, Leibnitz test, Absolute convergence and conditional convergence.

Limits, Continuity, Differentiation And Integration

Real valued Functions, Limit of a function, Algebra of limits, continuity of a function at a point, Uniform continuity. The derivative, The mean value theorems, Taylor’s Theorem. Riemann integral , Riemann integrable functions, Fundamental theorem on integral calculus.

Vector Calculus :

Vector differentiation. Ordinary derivatives of vectors, Space curves, Continuity, Differentiability, Gradient, Divergence, Curl operators, Formulae involving these operators. Vector integration, Theorems of Gauss and Stokes, Green’s theorem in plane and applications of these theorems.


Section – B ( 20 bits : 20 Marks )

Rings : Definition and basic properties, Fields, Integral domains, divisors of zero and Cancellation laws, Integral domains, The characteristic of a ring, some non – commutative rings, Matrices over a field, Homomorphism of Rings – Definition and elementary properties, Maximal and Prime ideals, Prime fields.

Vector Spaces : Vector spaces, Vector subspaces, Linear span, linear sum of two subspaces, Linear independence and dependence of vectors, Basis of vector space, Finite dimensional vector spaces, Linear transformations, linear operators, Range and null space of linear transformation, Rank and nullity of linear transformations.

Inner Product Spaces : Inner product spaces, Euclidean and unitary spaces, Norm or length of a vector, Schwartz inequality, Orthogonality, Orthonormal set, complete orthonormal set, Gram – Schmidt orthogonalisation process.

Section – C : ( 25 bits : 25 Marks )

Differential Equations : Linear differential equations; Exact differential equations; Simultaneous differential equations; Orthogonal trajectories. Equations solvable for p, x, y, Solution of homogeneous linear differential equations of order n with constant coefficients. Method of variation of parameters.

Solid Geometry : Equations of plane, Equations of a line, Angle between a line and a plane, The shortest distance between two lines. Definition and equation of the sphere, Intersection of a sphere and a line. Power of a point; Tangent plane. Plane of contact. Polar plane, Pole of a plane, Conjugate points, Conjugate planes; Radical plane. Coaxial system of spheres.

Matrices : Determinants, properties of determinants, Elementary Matrix operations and Elementary Matrices, The rank of a matrix and matrix inverses, system of linear equations, eigen values and eigen vectors, diagonalisation, Cayley – Hamilton theorem.

Groups : Groups, Subgroups and cyclic subgroups. Permutations, Isomorphism – Definition and elementary properties, Cayley’s theorem, Groups of cosets, Normal subgroups – Factor groups, The fundamental theorem of homomorphisms.


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