GKV HARIDWAR B.TECH SYLLABUS

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B.SC III YEAR math Syllabus

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

B.SC II YEAR math Syllabus

BMG: 201 Max Marks: 50

Elementary Analysis, Differential Equations and Vector Calculus

Unit-I

Real –Valued functions, Equivalence, Countability, Real numbers, Least upper

bound,Sequence of real numbers, Series of real numbers.

Infinite series: Introduction, Alternating series with Leibnitz test, P-series test for

positive terms, Comparison test for positive terms, D’Alembert’s ratio test , Cauchy’s

root test , Raabe’s test , Logarithmic test,

Open sets, closed sets on R, Derivatives, Rolle ’s Theorem, The law of the mean.

Unit-II

Ordinary differential equation of the first order and first degree, Clairaut’s form of

differential equations, Orthogonal trajectories, Linear differential equations with constant

coefficient, Homogeneous linear differential equations.

Unit-III

Linear differential equation of second order with constant coefficients.

Unit-IV

Scalar and Vecor product of three vectors, Product of four vectors , Reciprocal vectors,

Vector differentiation, Directional derivatives, Gradient, Divergence and Curl.

Unit-V

Vector integration, Theorems of Gauss,Green,Stokes and problems based on these.

BOOKS SUGGESTED

1. R.R.Goldberg Methods of Real Analysis

2. Gorakh Prasad Integral Calculus

3. Shanti Narayan A text –Book of Vector Calculus

(S. Chand & Co. New Delhi)

4.G.F.Simmons Differential Equations

5. Murray R. Spiegel Vector Analysis, Schaum Pub. Company New york

6 S.L.Ross Ordinary Differential Equations

7 H.L.Royden Real Analysis

BMG: 202 Linear Algebra Max Marks: 50

Unit-I

Vector Space: Field, Vector space, Subspaces, Base and dimension, Coordinates,

Summary of rows equivalence, Computations concerning subspaces.

Unit-II

Linear Transformations: Linear transformations and their algebra. Isomorphism,

Representation of transformations by matrices.

Linear functionals, Double dual, Transpose of linear transformations.

Unit-III

Polynomials: Algebra of polynomials, Polynomial ideals, Determinant functions and

simple properties.

Unit-IV

Canonical Form: Characteristic values and Characteristic vectors, Annihilating

polynomials, Examples of invariant subspaces.

Diagonalization, Orthogonal diagonalization, Applications to differential equations.

Unit-V

Quadratic forms: Quadratic forms in two and n variables, Cross-product terms of the

quadratic form. Positive definite Quadratic form, Diagonalization of quadratic forms,

Application to conic sections.

Books Recommended:

1. H.Anton Elementary Linear Algebra , John Wiley & Sons.

2. Chatles W.Curtis(C.W.Curtis) Elementary Linear Algebra

BMG: 203 Mechanics Max Marks: 50

Unit-I

Virtual Works: Equilibrium of strings and chains (common catanary and catanary of

uniform strength),Stable and Unstable equilibrium, Moments and couples, Varignon’s

theorem of moments

Unit-II

Equiblirium of forces in three dimensions, central axis, Wrench and Screw, Pitch of the

wrench.

Unit-III

Kinematics, Motion in a straight line and a plane, Radial and transverse velocities and

accelerations.

Unit-IV

Angular velocity and acceleration, Tangential and normal velocities and acceleration,

Rectilinear motion with constant acceleration.

Unit-V

Simple harmonic motion, Hook’s law, Repulsion from a fixed point varying as the

distance from the point, Constrained motion on a smooth plane (Vertical circle and

cycloid) , Projection.

Books Recommended:

1.R.S.Verma Statics

2.S.L.Loney Statics

3.M.Ray Dynamics

B.SC I YEAR math Syllabus

SYLLABUS

B. Sc. Mathematics

BMG: 101 ABSTRACT ALGEBRA Max Marks: 50

Unit-I
Sets and Logic (No question should be asked on this part). The well-ordering principle.
The division algorithm. The fundamental theorem of arithmetic, Congruence modulo.
Equivalance relations and Equivalance classes.
Unit-II
Groups: Definition, Examples and Properties, Permutation and Permutation group,
Subgroups and their properties.
Unit-III
Cosets and Coset decomposition, Lagrange’s theorem and its corollaries, Farmat’s
theorem, Cyclic group.
Unit-IV
Normal subgroup, Centre of a group, Quotient group, Homomorphism and Isomorphism,
Fundamental theorem of homomorphism, Cayley’s theorem.
Unit-V
Ring, Examples and simple properties, Different types of rings, Subring and Ideals,
Divisibility in an integral domain, Polynomial ring, Field and simple properties.


Books Recommended:
1. Thomas A.Whitelaw Introduction to Abstract (Blakie & Son Ltd.)
2. R.S. Mishra and N.N.Bhattacharya Fundamental structure in Modern Algebra
3. Bhattacharya ,Jain & Nagpal Abstract Algebra (Cambridge Uni. Press)

BMG: 102 CALCULUS Max Marks: 50

Unit-I
Successive differentiation: Expansion of functions, Maclaurin’s and Taylor’s theorems.
Unit-II
Maxima and minima up to two independent variables, Indeterminate form , Jacobian of
three functions, Partial differentiation.
Unit-III
Asymptotes, Curvature, Envelop, Double point and curve tracing (Polar and Cartesian).
Unit-IV
Standard reduction formula, Integration, as the limit of sum, Simple definite integrals.
Unit-V
Rectification, Quadrature, Volumes and Surfaces solids of revolution, Beta and Gamma
functions with their properties.


Books Recommended:
1. Gorakh Prasad Differential Calculus
2. Gorakh Prasad Integral Calculus

BMG: 103 3-D Coordinate Geometry & Trigonometry Max Marks: 50

Unit-I
System of coordinates, Direction Cosine, Angle between two lines, Projections, Distance of a
point from a line.
The plane: General form,Normal form, Intercept form, Reduction of the general form to normal
form , Equation of plane through three points, Angle between two planes, Parallel planes,
Perpendicular distance of a point from the planes, Pair of the planes ,Area of a triangle and
volume of a tetrahedron.
Unit-II
The straight line: Equation of a line in general form, Symmetric form, Two point form,
Reduction of the general equation to the symmetrical form, Straight line and the planes,
Conditions of parallelism and perpendicularity of a line and a plane ,Plane through a given line,
Perpendicular distance formula for the line, Projection of a line on a given plane containing them,
Equation of a straight line intersecting two given lines, Perpendicular distance formula for the line
and coordinates of the foot of the perpendicular, Shortest distance between two lines.
Unit-III
Sphere: General equation of a sphere, Plane section of a sphere, Intersection of two spheres,
Sphere through a given circle, Intersection of a straight line and a sphere, Equation of a tangent
plane to sphere, Condition of tangency. Plane of contact, Polar plane of a given plane, Angle of
intersection of two spheres, Length of tangent, Radical plane, Coaxial system of spheres.
Unit-IV
Cone:Equation of a cone whose vertex is at origin, Equation of a cone with a given vertex and a
given conic as base, Condition that general equation of second degree represent a cone, Equation
of a tangent plane, Condition of tangency of a plane and a cone, Reciprocal cone, Right circular
cone.
Unit-V
Expansions of sine and cosine of multiple angles in series, Expansion of sine, cosine and tan in
power of angles, Expansion of power of sines and cosines in multiples angles, Exponential series
for complex numbers, Circular and inverse circular functions for complex numbers, Hyperbolic
functions, Inverse hyperbolic functions, Logarithm of complex numbers, Summation of
trigonometric series.
Books Recommended:
1. Shanti Narayan Coordinate Geometry of 3-Dimension.
2. S.LL.Loney Trigonometry part III
3. Schaum’s series Coordinate Geometry