**BMG: 301 Real Analysis and Mathematical Statistics Max Marks: 50**

Unit I

Concepts of metric space: Continuous functions, Open sets and closed sets in

metric spaces, connected sets, Bounded and Totally bounded sets, Complete metric

spaces, Contraction mapping, Compact metric spaces, Uniform continuity.

Unit II

Existence and properties of the Riemann integral, Fundamental theorem of

calculus, Improper integral.

Sequence and series of function: Point-wise and uniform convergence of sequence of

functions, Convergence and uniform convergence of series of function, Integration and

differentiation of series of functions.

Unit III

The metric space C[a,b] .The Weierstrass approximation theorem ,Picard’s

existence of differential equation, Fourier series ,Formulation of convergence problems,

Convergence of Fourier series .

Unit IV

Curve fitting-Method of least square, Introduction of Moments, Applications of ttest

, Z-test and F-test.

Unit V

Interpolation :( Newton’s and Lagrange’s formula), Correlation and regression,

Measures of correlation, The least square regression lines, Coefficient of correlation,

Rank correlation.

**Books Recommended:**

**1. R.R Goldberg Method of Real Analysis.(Relevant parts)**

**2. E.T.Coppson Metric spaces, Cambridge Univ .Press.**

**3 Ray & Sharma Mathematical Statistics.**

**4 Murray & Spiegel Statistics**

**5. C.E.Weatherban A text Book of Statistics**

**BMG: 302 Differential Geometry Max Marks: 50**

Unit-I

Curves With Torsion: Tangent, Principal normal-Curvature, Binormal -Torsion, Serret-Frenet

formulae, Locus of centre of curvature and examples. Spherical curvature, Locus of centre of

spherical curvature, Theorem: Curve determined by its intrinsic equation, Helices, Spherical

indicatrix of tangent, Involutes, Evolutes. Bertrand curves.

Unit-II

Envelopes, Developable Surfaces: Surfaces, Tangent plane –Normal, One –Parameter Family of

Surfaces: Envelope, Characteristics, Edge of regression, Developable surfaces, Developables

associated with a curve: Osculating development, Polar development, Rectifying development.

Two –parameter Family of Surface: Envelope, Characteristics points, and its examples.

Unit-III

Curvilinear Coordinates on a Surface Fundamental Magnitudes: Curvilinear Coordinates,

First order magnitude , Directions on a surface , The normal , Second order Magnitude

,Derivatives of n ,Curvature of normal section ,Meunier’s theorem and examples.

Unit-IV

Curves on a Surface: Lines of Curvature: Principal direction and curvatures, First and second

curvature, Euler’s theorem, Dupin’s indicatrix, The Surface z = f(x,y), Surface of revolution and

examples. Conjugate directions, Conjugate systems.

Asymptotic lines, Curvature and torsion, Isometric Parameters, Null Lines or Minimal curves and

examples.

Unit-V

The Equations of Gauss and of Codazzi: Gauss’s formula for r11, r12, r22 ,Gauss’s

characteristic equations ,Mainardi–Codazzi relations, Alternative expressions ,Bonnets theorem,

Derivation of an angle and examples.

Geodesic: Geodesic property, Equations of geodesics, Surface of revolution, Torsion of a

geodesic ,Curves in relation to Geodesics : Bonnet’s theorem ,Joachimsthal’s theorems ,Vector

curvature , Geodesic curvature and its other formulae ,Examples.

**Books Recommended:**

**1.C.E.Weatherburn Differential Geometry**

**2.Bansi Lal Differential Geometry, Atma Ram & Sons, Delhi**

**3.Andrew Presely Elementry Differential Geometry, Springer**

**BMG: 303 Linear Programming and Probability Max Marks: 50**

Unit-I

Convex sets and their properties, Introduction to linear programming problems,

Mathematical formulation, graphical method, Simplex method.

Unit-II

Concept of duality in linear programming, Framing of dual problems, Dual simplex

method, Sensitivity analysis.

Unit-III

Revised simplex method, Transportation problem and assignment problem.

Unit-IV

Definition of Probability, Addition and multiplication theorems, Conditional Probability,

Independent and dependent events, Mutually exclusive events, Mathematical expectation,

Introduction to axiomatic approach.

Unit-V

The Binomial distribution, Some properties of Binomial distribution , The Poisson

distribution, Some properties of Poisson distribution, Relation between binomial and

Poisson distributions and problems. The normal distribution, Some properties of normal

distribution, Relation between normal and binomial distributions.

**Books Recommended:**

**1. P.K.Gupta & Man Mohan Linear Programming**

**2. Ray & Sharma Mathematical Statistics**