Topics to be covered from July to September

Topics to be covered from October to December

CC-5:

Theory of Real Functions

Unit-1:

Limit & Continuity of functions

AM

• Limits of functions (Î-d approach), sequential criterion for limits. Algebra of limits for functions, effect of limit on inequality involving functions, one sided limits. Infinite limits and limits at infinity. Important limits like , , (a > 0) as x ® 0.

• Continuity of a function on an interval and at an isolated point. Sequential criteria for continuity. Concept of oscillation of a function at a point. A function is continuous at x if and only if its oscillation at x is zero. Familiarity with the figures of some well known functions: , |x|, sin x, cos x, tan x,

log x, . Algebra of continuous functions as a consequence of algebra of limits. Continuity of composite functions. Examples of continuous functions. Continuity of a function at a point does not necessarily imply the continuity in some neighbourhood of that point.

• Bounded functions. Neighbourhood properties of continuous functions regarding boundedness and maintenance

of same sign. Continuous function on [a, b] is bounded and attains its bounds. Intermediate value theorem.

• Discontinuity of functions, type of discontinuity. Step functions. Piecewise continuity. Monotone functions.

Monotone functions can have only jump discontinuity. Monotone functions can have atmost countably many points of discontinuity. Monotone bijective function from an interval to an interval is continuous and its inverse is also continuous.

• Uniform continuity. Functions continuous on a closed and bounded interval is uniformly continuous. A

necessary and sufficient condition under which a continuous function on a bounded open interval I will be

uniformly continuous on I. A sufficient condition under which a continuous function on an unbounded open interval I will be uniformly continuous on I (statement only). Lipschitz condition and uniform continuity.

Unit-2: Differentiability of functions

AM

&

SGD

• Differentiability of a function at a point and in an interval, algebra of differentiable functions. Meaning of sign of derivative. Chain rule.

• Darboux theorem, Rolle’s theorem, Mean value theorems of Lagrange and Cauchy--as an application of Rolle’s theorem. Taylor’s theorem on closed and bounded interval with Lagrange’s and Cauchy’s form of remainder deduced from Lagrange’s and Cauchy’s mean value theorem respectively. Expansion of , log(1 + x), , sin x, cos x with their range of validity (assuming relevant theorems). Application of Taylor’s theorem to inequalities.

• Statement of L’ Hospital’s rule and its consequences. Point of local extremum (maximum, minimum) of a function in an interval. Sufficient condition for the existence of a local maximum/minimum of a function at a point (statement only). Determination of local extremum using first order derivative. Application of

the principle of maximum/minimum in geometrical problems.

CC-6:

Ring Theory & Linear Algebra-I

Unit-1: Ring Theory

MDG

Definition and examples of rings, properties of rings, subrings, necessary and sufficient condition for a

nonempty subset of a ring to be a subring, integral domains and fields, subfield, necessary and sufficient condition for a nonempty subset of a field to be a subfield, characteristic of a ring. Ideal, ideal generated by a subset of a ring, factor rings, operations on ideals, prime and maximal ideals.

Ring homomorphisms, properties of ring homomorphisms. First isomorphism theorem, second isomorphism theorem, third isomorphism theorem, Correspondence theorem, congruence on rings, one-one correspondence between the set of ideals and the set of all congruences on a ring.

Unit-2:

Linear Algebra

SGD

Vector spaces, subspaces, algebra of subspaces, quotient spaces, linear combination of vectors, linear span,

linear independence, basis and dimension, dimension of subspaces. Subspaces of , dimension of subspaces of . Geometric significance of subspace.

Linear transformations, null space, range, rank and nullity of a linear transformation, matrix representation

of a linear transformation, change of coordinate matrix. Algebra of linear transformations. Isomorphisms.

Isomorphism theorems, invertibility and isomorphisms. Eigen values, eigen vectors and characteristic equation of a matrix. Cayley-Hamilton theorem and its use in finding the inverse of a matrix

CC-7:

Ordinary Differential Equation

& Multivariate Calculus-I

Unit-1:

Ordinary differential equation

MI

&

SG

• First order differential equations : Exact differential equations and integrating factors, special integrating factors and transformations, linear equations and Bernoulli equations, the existence and uniqueness theorem of Picard (Statement only).

• Linear equations and equations reducible to linear form. First order higher degree equations solvable for x, y and p. Clairaut’s equations and singular solution.

• Basic Theory of linear systems in normal form, homogeneous linear systems with constant coefficients: Two Equations in two unknown functions.

• Linear differential equations of second order, Wronskian: its properties and applications, Euler equation, method of undetermined coefficients, method of variation of parameters.

• System of linear differential equations, types of linear systems, differential operators, an operator method for linear systems with constant coefficients.

• Planar linear autonomous systems : Equilibrium (critical) points, Interpretation of the phase plane and

phase portraits.

• Power series solution of a differential equation about an ordinary point, solution about a regular singular point (up to second order).

Unit-2:

Multivariate Calculus-I

MDG &

MI

• Concept of neighbourhood of a point in (n > 1), interior point, limit point, open set and closed set in (n > 1).

• Functions from (n > 1) to (m³ 1), limit and continuity of functions of two or more variables. Partial derivatives, total derivative and differentiability, sufficient condition for differentiability.

Chain rule for one and two independent parameters, directional derivatives, the gradient, maximal and normal property of the gradient, tangent planes. Extrema of functions of two variables, method of Lagrange multipliers, constrained optimization problems.

SEC-A

C Programming language

SG

• An overview of theoretical computers, history of computers, overview of architecture of computer, compiler,

assembler, machine language, high level language, object oriented language, programming language and importance of C programming.

• Constants, Variables and Data type of C-Program: Character set. Constants and variables data types, expression, assignment statements, declaration.

• Operation and Expressions : Arithmetic operators, relational operators, logical operators.

• Decision Making and Branching: decision making with if statement, if-else statement, Nesting if statement, switch statement, break and continue statement.

• Control Statements: While statement, do-while statement, for statement.

• Arrays: One-dimension, two-dimension and multidimensional arrays, declaration of arrays, initialization of one and multi-dimensional arrays.

• User-defined Functions: Definition of functions, Scope of variables, return values and their types, function

declaration, function call by value, Nesting of functions, passing of arrays to functions, Recurrence of function.

• Introduction to Library functions: stdio.h, math.h, string.h stdlib.h, time.h etc.

GE-3

Unit-1: Integral Calculus

SGD

• Evaluation of definite integrals.

• Integration as the limit of a sum (with equally spaced as well as unequal intervals).

• Reduction formulae of ,

, and associated problems (m and n are non-negative integers).

• Definition of Improper Integrals : Statements of (i) μ-test (ii) Comparison test (Limit from excluded)--Simple problems only. Use of Beta and Gamma functions (convergence and important relations being assumed).

• Working knowledge of double integral.

• Applications: Rectification, Quadrature, volume and surface areas of solids formed by revolution of plane curve and areas problems only.

Unit-2:

Numerical Methods

AM

&

MDG

• Approximate numbers, Significant figures, Rounding off numbers. Error: Absolute, Relative and percentage.

• Operators - D, Ñ and E (Definitions and some relations among them).

• Interpolation: The problem of interpolation Equispaced arguments Difference Tables, Deduction of Newton’s

Forward Interpolation Formula, remainder term (expression only). Newton’s Backward interpolation Formula (Statement only) with remainder term. Unequally- spaced arguments Lagrange’s Interpolation Formula (Statement only). Numerical problems on Interpolation with both equally and unequally spaced

arguments.

• Numerical Integration : Trapezoidal and Simpson’s one-third formula (statement only). Problems on Numerical

Integration.

• Solution of Numerical Equation : To find a real root of an algebraic or transcendental equation. Location

of root (tabular method), Bisection method, Newton-Raphson method with geometrical significance,

Numerical Problems. (Note : Emphasis should be given on problems)

Unit-3:

Linear Programming

MDG

&

MI

• Motivation of Linear Programming problem. Statement of L.P.P. Formulation of L.P.P. Slack and Surplus variables. L.P.P. is matrix form. Convex set, Hyperplane, Extreme points, convex Polyhedron, Basic solutions and Basic Feasible Solutions (B.F.S.). Degenerate and Non-degenerate B.F.S.

• The set of all feasible solutions of an L.P.P. is a convex set. The objective function of an L.P.P. assumes its optimal value at an extreme print of the convex set of feasible solutions, A.B.F.S. to an L.P.P. corresponds to an extreme point of the convex set of feasible solutions.

• Fundamental Theorem of L.P.P. (Statement only) Reduction of a feasible solution to a B.F.S. Standard form of an L.P.P. Solution by graphical method (for two variables), by simplex method and method of penalty. Concept of Duality. Duality Theory. The dual of the dual is the primal. Relation between the objective values of dual and the primal problems. Dual problems with at most one unrestricted variable, one constraint of equality. Transportation and Assignment problem and their optimal solutions