COURSE STRUCTURE SYLLABI · Summation of series, Expansion of functions, Maclaurin’s theorem and Taylor’s theorem, Functions of two or more variables and introduction of partial

COURSE STRUCTURE

&

SYLLABI for three-year undergraduate programme

B.A./B.Sc. (Honours) (Mathematics)

(Main & Subsidiary) Approved by the Board of Studies held on 01.08.2019

Under

Choice Based Credit System (CBCS)

DEPARTMENT OF MATHEMATICS

ALIGARH MUSLIM UNIVERSITY

ALIGARH-202002

STRUCTURE OF MODEL CURRICULUM

For B.A./B.Sc.(Hons) Mathematics(Main & Subsidiary)Part: I & II

FIRST SEMESTER Course No. Course Title Credits Periods/Week Marks Assigned

(Sessional+Exams) MMB-151 Calculus 4 4 30+70=100

MMB-152 Geometry I 2 3 30+70=100

2 More Subjects 6+6=12

Total Credits

24 Foundation Courses

(Comp. English, Comp. Urdu,

Theology/INM)

2+2+2=6

SECOND SEMESTER Course No. Course Title Credits Periods/Week Marks Assigned

(Sessional+Exams) MMB-251 Numerical

Analysis

4 4 30+70=100

MMB-252 Geometry II 2 3 30+70=100


Total Credits


(Comp. English, Comp. Urdu

&Theology/INM)

2+2+2=6

THIRD SEMESTER

Course No. Course Title Credits Periods/Week Marks Assigned

(Sessional+Exams) MMB-351 Algebra I 4 4 30+70=100

MMB-352 Ordinary

Differential

Equations

2 3 30+70=100


Total Credits


(Comp. English & EVS)

2+4=6

FOURTH SEMESTER Course No. Course Title Credits Periods/Week Marks Assigned

(Sessional+Exams) MMB-451 Advanced

Calculus

4 4 30+70=100

MMB-452 Partial

Differential

Equations

2 3 30+70=100


Total Credits

20 Open Elective 2

STRUCTURE OF MODEL CURRICULUM

For B.A./B.Sc.(Hons) Mathematics(Main)Part: III

FIFTH SEMESTER

Credits for each course: 4

Total Credits: 24

Periods/Week for each course: 4

Maximum Marks assigned for each course: 100 (Sessional: 30 & Exams: 70)

S. No. Course No. Course Title

1. MMB-551 Real Analysis I

2. MMB-552 Group Theory

3. MMB-553 Set Theory and Number Theory

4. MMB-554 Geometry of Curves and Surfaces

5. MMB-555 Mechanics

6.

Elective (opt any ONE)

MMB-556 Tensor Analysis

MMB-559 Mathematical Methods

SIXTH SEMESTER

Credits for each course: 4

Total Credits: 24

Periods/Week for each course: 4

Maximum Marks assigned for each course: 100 (Sessional: 30 & Exams: 70)

S. No. Course No. Course Title

1. MMB-651 Real Analysis II

2. MMB-652 Ring Theory

3. MMB-653 Metric Spaces

4. MMB-654 Complex Analysis

5. & 6.

Elective (opt any TWO)

MMB-655 Programming in C and Matlab

MMB-656 Optimization

MMB-657 Discrete Mathematics

DEPARTMENT OF MATHEMATICS, ALIGARH MUSLIM UNIVERSITY, ALIGARH

Syllabus of B.A./B.Sc. I Semester approved in BOS: 01-08-2019

Course Title Calculus

Course Number MMB-151

Credits 4

Course Category Compulsory

Prerequisite Courses None

Contact Hours 4 Lectures/week

Type of Course Theory

Course Assessment Sessional Tests 30%

Semester Examination 70%

Course Objectives Calculus is the branch of mathematics that deals with rates of change and motion. It grew out of a desire

to understand various physical phenomena, such as the orbits of planets, and the effects of gravity. The

immediate success of calculus in formulating physical laws and predicting their consequences led to

development of a new division in mathematics called analysis, of which calculus remains a large part.

Today, calculus is the essential language of science and engineering, providing the means by which

physical laws are expressed in mathematical terms.

Calculus divides naturally into two parts, differential calculus and integral calculus. Differential

calculus is concerned with finding the instantaneous rate at which one quantity changes with respect to

another, called the derivative of the first quantity with respect to the second. Integral calculus deals with

the inverse of the derivative, namely, finding a function when its rate of change is known.

Course Outcomes The usefulness of calculus is indicated by its widespread application. For example, it is used in the

design of navigation systems, particle accelerators, and synchrotron light sources. It is used to predict

rocket trajectories, and the orbits of communications satellites. Calculus is the mathematical tool used to

test theories about the origins of the universe, the development of tornadoes and hurricanes,

and salt fingering in the oceans. It has even found extensive application in business, where it is used,

among other things, to optimize production.

Thus, as a result, students will learn basic theory of calculus and its applications in real life.

Contents of Syllabus No. of

Lectures

UNIT I: Differentiability and Expansion of Functions

Indeterminate forms, Hyperbolic functions, Inverse hyperbolic functions and their derivatives, Successive

differentiation: Derivatives of higher order, 𝑛𝑡ℎderivative of well-known functions, Leibnitz’s theorem,

Summation of series, Expansion of functions, Maclaurin’s theorem and Taylor’s theorem, Functions of

two or more variables and introduction of partial derivatives.

12

UNIT II: Homogeneous functions, Asymptotes and Curvature

Homogeneous functions, Euler’s theorem on homogeneous functions (with proof), Asymptotes of the

algebraic curves, parallel asymptotes, Asymptotes parallel to x-axis and y-axis, Curvature: Polar

coordinates; Formula for angle between radius vector and tangent (without proof); formula for 𝑑𝑠

𝑑𝑥

𝑑𝑠

𝑑𝑦

𝑑𝑠

𝑑𝑡 and

𝑑𝑠

𝑑𝜃; Angle of contingence; Definition of curvature, radius of curvature; Intrinsic formula for radius of

curvature (without proof); Cartesian, parametric and Polar formulae for radius of curvature (with proof);

Chord of curvature through the origin; Pedal formula for the radius of curvature (without proof); Circle

and coordinates of centre of curvature.

12

UNIT III: Tracing of Plane Curves

Normal, Cartesian subtangent and subnormal, Intercepts, Length of the tangent and normal, Lengths of the

polar subtangent and subnormal, Lengths of the perpendicular from the pole on tangent, Double points

and their types, convexity and concavity of a curve, Point of inflexion, Rough sketches of certain polar

curves, Tracing of Cartesian curves, Equation of the tangent(s) at the origin.

12

UNIT IV: Beta and Gamma functions, Rectification and Quadrature

Beta and Gamma functions and their applications to evaluate integration, Integration of the

type∫ 𝑆𝑖𝑛𝑚𝑥. 𝐶𝑜𝑠𝑛𝑥 𝑑𝑥𝜋

20

, Properties of Gamma functions, Length of curves for Cartesian, parametric and

polar equations, Intrinsic equation for Cartesian, parametric and polar equations, Volume and Surfaces of

solids of revolution for Cartesian, parametric and polar curves.

12

Total No. of Lectures 48

Text Books*/

Reference

Books

1. * Gorakh Prasad: Differential Calculus, 18th Ed. 2010, Pothishala Pvt Ltd, Allahabad.

2. * Gorakh Prasad: Integral Calculus, 17th Ed. 2015, Pothishala Pvt Ltd, Allahabad.

3. N. Piskunov: Differential and Integral Calculus.


Syllabus of B.A./B.Sc. I Semester approved in BOS: 01-08-2019

Course Title Two-dimensional Coordinate Geometry


Credits 2







Course Objectives The primary objective of this course is to introduce the basic tools of plane

geometry and geometric properties of different conic sections which are helpful

in understanding their applications to the real-world problems.

Course Outcomes This course will enable the students to:

• basic knowledge about pair of straight lines

• elementary properties of conic sections in in the cartesian and polar

coordinate systems.

• trace parabola, ellipse, hyperbola in a plane using its mathematical

properties.

Contents of Syllabus No. of Lectures

UNIT I: Pair of Straight Lines and Coordinate Transformation

Change of coordinate axes, Removal of xy-term, Invariants, Pair of straight line through

origin, Angle between two lines, Necessary and sufficient condition that general equation of

second degree represents a pair of straight lines, Lines joining the origin to the intersection

of a curve and a line.

10

UNIT II: General Equation of Second Degree

General equation of a conic section, Intersection of a straight line and a conic, Equation of

tangent and normal, Condition of tangency, Pair of tangents, Chord of contact of tangents,

Pole and polar, Conjugate points, conjugate lines and condition of conjugacy, Equation of

chord in terms of its middle point, Centre, Diameters and conjugate diameters.

10

UNIT III: More on Conic Sections

Revisit of the concepts of circle, parabola, ellipse, hyperbola and their parametric equations,

tangents and normals, Equation of a circle when end points of a diameter are given, Length

of tangent to a circle, Common tangents to two circles, Sum (difference) of the focal

distances of a point on ellipse (hyperbola), Auxiliary circle and director circle of an ellipse,

Rectangular hyperbola, Asymptotes of a hyperbola, Conjugate hyperbola.

10

UNIT IV: Tracing of Conic Sections and Polar Equations

Nature of conic section, Tracing of parabola, ellipse and hyperbola, Polar equations of

straight line and circle, Polar equation of a conic referred to a focus as pole, Equations of

directrices, chord and tangent of a conic (in polar form).

10


Text

Books*/

Reference

Books

1. Ram Ballabh: A Text Book of Coordinate Geometry, Prakashan Kendra, Lucknow.

2. S. L. Loney: The Elements of Coordinate Geometry, AITBS Publishers.

3. E. H. Askwith: A Course of Pure Geometry, Merchant Books.

4. C. Smith: An Elementary Treatise on Conic Sections, MacMillon &Co.Ltd


Syllabus of B.A./B.Sc. II Semester approved in BOS: 01-08-2019

Course Title Numerical Analysis


Credits 4







Course Objectives Numerical Analysis is the study of algorithms using numerical

approximations for the problems of mathematical analysis.

This course is designed to achieve the following objectives:

1. The objectives of studying this module are to make the students familiarize

with the ways of solving complicated mathematical problems numerically.

2. Describing and understanding of the several errors and approximation in

numerical methods.

3. Obtaining numerical solutions to problems of mathematics. e.g. finding roots

of equations, numerical differentiation and integeration, solution of ordinary

differential equations.

4. The studying of Curve Fitting and Interpolation.

Course Outcomes To explore complex systems, physicists, engineers, financiers and

mathematicians require computational methods since mathematical models are

only rarely solvable algebraically. Numerical methods, based upon sound

computational mathematics, are the basic algorithms underpinning computer

predictions in modern systems science. Such methods include techniques for

simple optimisation, interpolation from the known to the unknown, linear

algebra underlying systems of equations, ordinary differential equations to

simulate systems, and stochastic simulation under random influences.


UNIT I: Numerical Solution of algebraic and transcendental equations

Absolute, relative and percentage errors, General error formula, Solution of algebraic and

transcendental equations by iteration methods namely: Bisection method, Regula falsi

method, Iterative method and Newton-Raphson method, Solution of system of linear

equations using direct methods such as matrix inversion, Gauss elimination and LU

decomposition including some iteration methods namely: Jacobi and Gauss-Siedel method.

12

UNIT II: Interpolation

Symbols of Δ,, E, E-1, D, and and their relations, Newton-Gregory interpolation

formulae, Forward difference, Backward difference, Gauss’s Forward difference, Gauss

Backward difference, Stirling’s formulae, Bessel’s formulae and Lagrange’s interpolation

formula, Divided Differences and their properties, Newton’s general interpolation formula,

Inverse interpolation formula.

12

UNIT III: Numerical Differentiation and Integration

Numerical differentiation of tabular functions including error estimations, Numerical

integration using Gauss quadrature formulae, Trapezoidal, Simpson’s 1/3- and 3/8-Rule and

Weddle’s Rule, Least squares curve fitting procedures and Least squares polynomial

approximation.

12

UNIT IV: Numerical Solution of Ordinary Differential Equations

Euler’s and modified Euler’s methods, Picard’s method, Taylor series method, Runge-Kutta

methods of 2nd and 4th order, Milne-Simpson method, Adams-Bashforth-Moulton method,

Solution of boundary value problems of ordinary differential equations using Finite

Difference method.

12


Text Books*/

Reference

Books

1. S. S. Sastry:Introductory Methods of Numerical Analysis, Prentice Hall of India,

New Delhi, 5th Ed, 2012.

2. M. K. Jain, S. R. K. Iyenger and R. K. Jain: Numerical Methods for Scientific and

Engineering Computation, New Age International (P) Ltd, 1999.


Syllabus of B.A./B.Sc. II Semester approved in BOS: 01-08-2019

Course Title Geometry II

Course Number MMB

Credits 2


Prerequisite Courses Two-dimensional Coordinate Geometry

Contact Course 3 Lectures/week

Type of Hours Theory



Course Objectives The main aim of this course to introduce the basic tools of space

geometry.

Course Outcomes This course will enable the students to:

• understand about lines in 3D, projections and planes.

• basic knowledge about different types of conicoids such as:

spheres, cone, cylinder, ellipsoid, hyperboloid and paraboloid.


UNIT I: Straight Line, Plane and Sphere

Direction cosines of a line, Projection of a segment, Angles between two lines, Distance of

a point from a line, Equation of a plane in various forms, Length of perpendicular from a

point to a plane, Equation of straight line (in symmetric and asymmetric forms), General

equation of sphere, Sphere on the join of two points as diameter, Tangent plane to a sphere.

10

UNIT II: Cylinder and Cone

Cylinder and its Equation, Right circular cylinder and its equation, Cone and its equation,

Cone with vertex at origin, Condition for general equation of second degree to represent a

cone, Tangent plane to a cone and condition of tangency, Reciprocal cone, Cone with three

mutually perpendicular generators, Number of mutually perpendicular generators.

10

UNIT III: Central Conicoids

Standard equation of central conicoids, Tangent plane, Condition of tangency of a plane,

Section with a given centre, Locus of the mid-points of a system of parallel chords, Polar

plane, Polar lines, Enveloping cone.

10

UNIT IV: Central Conicoids (cont.) and Paraboloids

Classification of central conicoids, Normal to an ellipsoid, Conjugate diametral plane and

diameters of ellipsoid, Paraboloids: Equation, Classification and Properties; Conicoids:

General equation and Examples.

10


Text

Books*/

Reference

Books

1. *Ram Ballabh: A Textbook of Coordinate Geometry, Prakashan Kendra,

Lucknow, 13th Revised Ed.

2. R. J. T. Bell: An Elementary Treatise on Coordinate Geometry, MacMillon &

Co Ltd, 1960.

3. Charles Smith: An Elementary Treatise on Solid Geometry, MacMillon & Co

Ltd, 1931.


Syllabus of B.A./B.Sc. III Semester approved in BOS: 01-08-2019

Course Title Algebra-I


Credits 4







Course Objectives This course aims to introduce students to the following concepts and cognitive skills.

In this course the students understand real vector spaces and subspaces and apply their

properties. Understand linear independence and dependence. Find the basis and dimension of a

vector space, and understand the change of basis. Find a basis for the row space, column space

and null space of a matrix and find the rank and nullity of a matrix. Compute linear

transformations, kernel and range, and inverse linear transformations, and find matrices of

general linear transformations. Find the dimension of spaces such as those associated with

matrices and linear transformations. Solve systems of linear equations using various methods.

Perform matrix algebra, invertibility, and the transpose and understand vector algebra in Rn.

Determine the relationship between coefficient matrix invertibility and solutions to a system of

linear equations and the inverse matrices. Verify an eigenvalue and an eigenvector of a given

matrix. Find the characteristic equation, and the eigenvalues and corresponding eigenvectors of

a given matrix use them in applications. Students also expected to gain an appreciation for the

applications of linear algebra to areas such as computer science, engineering, biology and

economics.

Course Outcomes On successful completion of this course, student should be able to:

• explain the concepts of vector space and subspace

• define vector operations for vectors in Rn.

• define the notion of vector spaces and subspaces

• explain the concept of Vector space and subspace

• analyze whether a set S of vectors in a vector space V is a spanning set of V.

• analyze whether a finite set of vectors in a vector space V is linearly independent.

• explain the concepts of base and dimension of vector space.

• explain the concept of the dimension of a vector space and express vector spaces in

different dimensions.

• explain bases concept of a vector space and properties of vectors on the bases.

• express row and column space of a matrix.

• explain some functions defined between vector spaces.

• express required conditions for a transformation to be a linear transformation.

• find kernel and image spaces of a linear transformation.

• express some of the algebra operations between linear transformations.

• explain matrix representation of a linear transformation.

• find the matrix representing a linear transformation.

• find the image set when a transformation matrix is given.

• Explain the system of linear equations is consistent or inconsistent and find the general

solution to a consistent system.

• explain the eigenvalues and eigenvectors of a linear transformation.

• explain concepts of eigenvalues and eigenvectors of a matrix.

• find characteristic polynomial, eigenvalues, and eigenvectors of a transformation

matrix.

Contents of Syllabus

No. of Lectures

UNIT I: Vector Spaces

Binary operations, Definition of Field with examples, Definition of Vector space with examples,

Subspaces, Span of a set, Sum of subspaces, Linear dependence, and independence, Basis and

Dimensions of a vector space, Coordinates of a vector relative to the ordered basis, Dimension

Theorem.

12

UNIT II: Linear Transformations

Linear transformation and its properties, Range and kernel of a linear transformation, Rank and nullity

of a linear transformation, Rank-nullity Theorem, Inverse of linear transformation.

12

UNIT III: Composition and Matrix Representation of Linear Maps

Vector space L(U,V) and its dimension, Composition of linear transformations, Matrix associated with

a linear transformation, Linear transformation associated with a matrix, Rank and nullity of a matrix.

12

UNIT IV: Elementary operations and Eigen-values

Elementary row operations and row-reduced echelon form, Inverse of a matrix through elementary

row operation, Solution of a system of linear equations, Eigen-values, Eigen-vectors.

12


Text Book*/

References books

1. *V. Krishnamurty, V. P. Mainra and J. L. Arora: An introduction to Linear Algebra, East

West Press, New Delhi, 2002.

2. S Lang, Introduction to Linear Algebra (2nd edition), Springer, 2005

3. S. Kumaresan: Linear Algebra- A Geometric Approach, Prentice Hall of India, 1999.

4. S. Lipschutz and M. Lipson: Linear Algebra, Schuam’s Outline Series


Syllabus of B.A./B.Sc. III Semester approved in BOS: 01-08-2019 Course Title Ordinary Differential Equations


Credits 2


Prerequisite Courses Calculus





Course Objectives The main objectives of this course are to introduce the students to the

exciting world of Differential Equations, Mathematical Modeling and

their applications.

Course Outcomes On successful completion of this course, student should be able to

solve first order nonlinear differential equation, second order boundary

value problems and linear differential equations of higher order using

various techniques.


UNIT I: First Order Differential Equations

Bernoulli equations, Exact differential equations, Integrating factors, Change of variables,

Orthogonal trajectory of a given family of curves, Equations of the first order and higher

degree, Equations solvable for p, y and x, Clairaut’s equation, Lagrange’s equation,

Singular solutions.

10

UNIT II: Higher Order Differential Equations

Homogeneous and non-homogeneous linear differential equations of order n with constant

coefficients, Complementary functions and particular integrals, Variation of parameters,

Linear differential equations of second order with variables coefficients, Reduction of order,

Cauchy-Euler and Legendre linear differential equations, Series solution of differential

equations: Frobenius method.

10

UNIT III: Differential equations in Three Variables

Total differential equations, Condition for integrability, Different methods of solving

Pdx+Qdy+Rdz=0; Simultaneous total differential equations, Equations of the form

dx/P=dy/Q=dz/R, Methods of grouping and multipliers; Solution of a system of linear

differential equations with constant coefficients, Solution of a triangular system of linear

differential equations, Degenerate system of linear differential equations.

10

UNIT IV: Laplace Transform Method

Laplace transform, Linearity of Laplace transform, First shifting property, Inverse Laplace

transform, Laplace transform of derivative and integrals, Unit step function and its Laplace

transform, Second shifting property, Unit impulsive function and its Laplace transform,

Convolution and periodic function theorems, Solution of linear differential equations as well

as system of linear differential equations with constant coefficients using Laplace transform

methods.

10


Text-Book*/

Reference

Books

1. *Zafar Ahsan: Differential Equations and their Applications, Prentice Hall of India,

New Delhi, 3rd Ed, August 2016.

2. George F. Simmons:Differential Equations with Applications and Historical Notes,

Tata McGraw Hill Comp Ltd, New Delhi, 1974.

3. Dennis G. Zill: A first course in differential equations, Cengage Learning.

4. W.E. Boyee and R.C. DiPrima: Elementary Differential Equations and Boundary

Value Problems, John Wiley and Sons (1977).

https://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=George+F.+Simmons&text=George+F.+Simmons&sort=relevancerank&search-alias=books


Syllabus of B.A./B.Sc. IV Semester approved in BOS: 01-08-2019 Course Title Advanced Calculus


Credits 4







Course Objectives To focus on general concepts of limit, continuity and differentiability.

Idea of directional derivative and its physical interpretation. Introduce

the idea of multiple integral, line and surface integrals and connection

among all integrals (Green’s and Stoke’s theorems).


• Express the physical problems containing more variables.

• Understand the idea of directional derivative and find

extremum of physical problems

• Find area and volume of nonrectangular regions and ready to

solve problems arise in mathematical physics.


UNIT I: Limit, Continuity and Differentiability

Functions of several variables, Contour curves, level curves and level surfaces, Limits and

continuity for functions of two variables, Partial derivatives, Partial derivatives and

continuity, Differentiability, Chain rule for functions of two and three variables.

12

UNIT II: Applications of Partial Derivatives

Directional derivatives, Gradient vectors, Tangent planes, Linearization and differentials,

Extreme values and saddle points, Local and absolute maxima / minima, Method of

Lagrange multipliers (with one constraint only), Taylor’s formula for function of two

variables, Partial derivatives with constrained variables.

12

UNIT III: Multiple Integrals

Double integrals over rectangles, Double integrals as volumes, Iterated integrals, Double

integrals over general regions, Fubini’s Theorem, Area by double integration, Double

integrals in Polar form, Triple integrals in Rectangular, Cylindrical and Spherical co-

ordinates, Applications of triple integrals.

12

UNIT IV: Line and Surface Integrals

Line integrals of scalar fields and vector fields, Applications of line integrals: Work,

Circulation and Flux; Green’s Theorem in the plane, Evaluation of line integral using

Green’s Theorem, Surfaces and Area, Surface integrals, Stoke’s Theorem.

12


Text Book*/

Reference

Books

1. *G. B. Thomas Jr., J. Hass, C. Heil and M. D. Weir: Calculus, Pearson Education

Services Pvt Ltd, 12th Ed, 2009.

2. D. V. Widder: Advanced Calculus, Prentice Hall of India Pvt. Ltd., New Delhi, 2nd

Ed, 2012.

3. N. Piskunov: Differential and Integral Calculus, Vol. I and II, CBS Publishers and

Distributors, New Delhi, 1996.

DEPARTMENT OF MATHEMATICS, ALIGARH MUSLIM UNIVERSITY

Syllabus of B.A./B.Sc. IV approved in BOS: 01-08-2019 Course Title Partial Differential Equations


Credits 2


Prerequisite Courses Advanced Calculus, Ordinary Differential Equations





Course Objectives The main objectives of this course are to teach students to form and

solve partial differential equations and use them in solving some

physical problems.


• Formulate and classify partial differential equations.

• Solve linear and nonlinear partial differential equations using

various methods and apply these methods in solving some

physical problems.

• Cauchy problems for first and second order PDE and their

solutions by using method of characteristic.

• Solve of BVP by using Fourier series and Laplace transform.


UNIT I: First order PDE

Definition of a partial differential equation, Formation of partial differential equations,

Classification of first order partial differential equations and their solutions, Equations

easily integrable, Solution of quasilinear partial differential equations of first order by

Lagrange’s method, Nonlinear partial differential equation of first order and their different

forms, Charpit’s method, Jacobi’s Method.

10

UNIT II: Characteristics and Canonical Forms

Integral surfaces of first order partial differential equations through a given curve, Cauchy’s

problem for first order partial differential equations, Surfaces orthogonal to a given system

of surfaces, Cauchy’s method of characteristics, Compatible system of first order partial

differential equations, Classification of second order linear partial differential equations,

Canonical forms for hyperbolic, parabolic and elliptic equations.

10

UNIT III: Higher order PDE

Homogeneous linear partial differential equations of higher order with constant coefficients,

Different cases for complimentary functions and particular integrals, Non-homogeneous

partial differential equations of higher order with constant coefficients, Equations reducible

to linear partial differential equations with constant coefficients, Solution of quasilinear

partial differential equation of second order by Monge’s Method, Method of separation of

variables.

10

UNIT IV: Boundary Value Problems

Fourier series, Dirichlet’s conditions, Even and odd functions and their Fourier series,

Change of interval, One dimensional wave equation, one dimensional heat equation,

Laplace equation and their Fourier series solutions, Solutions of second order BVP using

Laplace transform method.

10


Text-Books*/

Reference

Books

1. *Zafar Ahsan: Differential Equations and their Applications, Prentice Hall of India,

New Delhi, 3rd Ed, August 2016.

2. *K. S. Rao: Introduction to Partial Differential Equations, Prentice Hall of India,

New Delhi, 3rd Ed, 2009.

3. T. Amaranath: An Elementary Course in Partial Differential Equations , Narosa

Publishing House, New Delhi (2nd Edition, reprint July 2014).


Syllabus of B.A./B.Sc. IV Semester approved in BOS: 01-08-2019 Course Title Elementary Mathematics


Credits 2

Course Category Open Elective






Course Objectives The primary objective of this course is to provide essential knowledge

of basic tools of set operations, number systems, differential calculus,

integral calculus, matrices and determinants for those students which

have not mathematical background at +2 level.

Course Outcomes This course will enable the students to provide understanding in

different topics of elementary algebra and beginning calculus.


UNIT I: Sets and Numbers

Sets and elements, Empty and Non-empty sets, Subset of a set, Union and intersection of

two sets, Venn diagram, Universal set, Revisit to number system (natural numbers, whole

numbers, integers, rational numbers and real numbers), Standard notations to represent the

sets of different class of numbers, Principle of mathematical induction and its simple

applications, Complex numbers, Real and imaginary parts of a complex numbers, Algebra

of complex numbers, Conjugate and modulus of a complex number, Cube roots of unity.

10

UNIT II: Functions

Definition and examples of a function, Domain, Codomain and Range of a function, Image

and preimage of an element, One-one and Many-one functions, Onto and into functions,

Composite functions, Inverse of a function, Pictorial representation of a function, Real

valued function of a real variable, Some elementary real functions such as: linear, quadratic,

power, polynomial, rational, absolute value, signum and greatest integer functions with their

graphs, domains and ranges; Properties of trigonometric, logarithmic and exponential

functions, Even and odd functions, Periodic functions, Sum, difference, product and

quotients of real functions, Limit of real functions, Left-hand and right-hand limits,

Continuous functions.

10

UNIT III: Calculus

Derivative of a function and its geometrical explanation and properties, Differentiation from

the first principle, Derivative of function of function, Differentiation by substitution,

Differentiation of implicit functions, Logarithmic differentiation,Parametric differentiation,

Higher order derivatives, Idea of integration of a function, Indefinite integration, Some basic

formulas for differentiation and integration of functions, Integration by substitution,

Integration of the functions of certain forms, Integration by partial fractions, Integration by

parts, Definite integration and its properties.

10

UNIT IV: Matrices and Determinants

Introduction to matrices, Order of a Matrix, Different types of matrices, Properties of

matrices (transpose, symmetry and skew symmetry), Algebra of Matrices, Product of two

matrices, Concept of elementary row and column operations, Determinant of a square

matrix (up to matrices of order 3 x 3), Minors and cofactors, Properties of the determinants,

Applications of determinants in finding the area of a triangle and solving a system of linear

equations, Inverse of a real matrix and its uniqueness.

10


Text

Books*/

Reference

Books

1. *Z. Ahsan and N. Ahsan: Mathematical Methods, Real World Education Publishers,

2016.

2. *R. Steege and K. Bailey: Intermediate Algebra, Schaum’s Outline Series, 2nd Ed.


Syllabus of B.A./B.Sc. V approved in BOS: 01-08-2019

Course Title Real Analysis


Credits 4







Course Objectives The course will develop a deep and rigorous understanding of real line ℝ and of

defining terms to prove the results about convergence and divergence of sequences

and series of real numbers.


• Understand many properties of the real line, especially axiomatic theory

and point set topology on ℝ.

• Recognize bounded, convergent, divergent, Cauchy and monotonic

sequences and to calculate their limit superior, limit inferior, and the limit

of a bounded sequence.

• Apply the ratio, root, alternating series and limit comparison tests for

convergence and absolute convergence of an infinite series of real numbers.


UNIT I: Fundamental Properties of Real Numbers

Interval and its different kinds, Bounded and unbounded sets, Supremum and infimum,Field

axioms, Order axioms and Completeness axioms on ℝ, Mathematical induction, Well

ordering principle, Archimedean property of real numbers, Denseness property of rational

numbers in ℝ, Dedekind theory of real numbers, Absolute value of real numbers,Properties

of modulus, Inequalities of Cauchy-Schwartz, Minkowski and Holder, Finite and Infinite

sets, Countable and uncountable sets.

12

UNIT II: Sequence of Real Numbers

Concept of sequence, Limit points of a sequence, Bolzano Weierstrass theorem for

sequence, Limit inferior and superior, Convergent, divergent and oscillate sequences,

Bounded and unbounded sequences, Cauchy’s general principle of convergence, Algebra of

sequences, Subsequences, Monotonic increasing and decreasing sequences, Cantor’s theory

of real numbers.

12

UNIT III: Series of Real Numbers

Introduction to series of real numbers, Sequence of partial sums and convergence of infinite

series, Necessary condition for the convergence of an infinite series, Positive term series,

Comparison tests (first type and limit form), Cauchy root test, D’Alembert’s ratio test with

their applications, Alternating series, Leibnitz test, Absolute and conditional convergence,

Series of arbitrary terms, Abel’s and Dirichlet’s tests, Rearrangement of series.

12

UNIT IV: Elements of Point Set Topology on ℝ

Neighbourhood of a point, Interior points, Open sets, Limits points and derived set,

Bolzano-Weierstrass Theorem, Adherent point and Closure of a set, Closed sets and their

sequential characterization, Compact sets and their sequential characterization, Heine-Borel

theorem, Connected sets, Dense sets, Perfect sets.

12


Text Books*/

References

Books

1. *R. G. Bartle and D. R. Sherbert: Introduction to Real Analysis, John Wiley and Sons,

Singapore, 3rd Ed, 2003.

2. * S. C. Malik and S. Arora: Mathematical Analysis, New Academic Science Ltd, 5th

Ed, 2017.

3. W. Rudin: Principles of Mathematical Analysis, Third Edition, McGraw Hill, New

York, 3rd Ed, 1976.

4. A. Kumar and S. Kumaresan: A Basic Course in Real Analysis, CRC Press, 2014.


Syllabus of B.A./B.Sc. V approved in BOS: 01-08-2019

Course Title Group Theory


Credits 4








The definition of the basic concepts of abstract algebra. The application of the concept of

binary operations and groups. The proof of the basic properties of groups and subgroups.

Exploring different types of subgroups and cyclic groups. The definition of cosets and the

Theorem of Lagrange. Analysis of the concept of permutation groups. The definition of

group homomorphism and factor groups. Computing direct products of groups. Analysis of

the finitely generated abelian groups. We are improving the student’s abstract and logical

thinking capabilities, applying students' mathematical ability to handle proofs.

Course Outcomes On successful completion of this course, the student should be able to:

• Prove the basic structural properties of groups and subgroups.

• Test the group axioms in different problems.

• Verify the basic properties of subgroups and cosets.

• Prove or disprove the validity of some group-theoretic statements, including isomorphic

groups, normal subgroups, and simple groups.

• Prove Lagrange’s theorem and some of its corollaries.

• Apply Lagrange’s theorem to some exercises.

• Explore the groups of permutations and the alternating groups.

• Compute the direct products of groups.

• Analyze finitely generated abelian groups.

• Examine the subgroup and normal subgroup structure of a group.

• Decide whether two groups are isomorphic.

• Decide whether a given subgroup of a group is normal.

• Determine the automorphism group of any cyclic group.

• Prove Cayley’s theorem and its generalization.

• Prove isomorphism’s theorems.


UNIT I: Basic concepts

Binary relation, Function, Binary Operation; Groups, its examples and basic properties, Order of an

element in a group, Subgroups, its examples and some basic properties, Centre of a group,

Normaliser of a set, Product of two subgroups, Cyclic groups, Generators, its examples and related

results.

12

UNIT II: Lagrange’s Theorem, Homomorphisms and Normal Subgroups

Cosets, Lagrange’s theorem and its related results, Index of subgroup of a group, Euler’s theorem,

Fermat’s theorem, Isomorphism and homomorphism of groups with examples and related results,

Inner automorphism; Normal subgroups and simple Groups, their examples and related results.

12

UNIT III: Quotient Groups, Isomorphism Theorems and Direct Product

Commutator subgroup and some basic properties, Quotient groups with examples, First, second and

third isomorphism theorems and their related results, Internal and External direct product of groups

and their related results, Characterization of a group as a direct product of its two subgroups.

12

UNIT IV: Permutation Groups

Permutations, even and odd permutations, Order of a permutation, Transposition, Cycle and its

length, Disjoint cycles and their examples, Permutation groups, Alternating groups and their related

results, Signature of a permutation, Cayley’s theorem, Cauchy’s theorem for finite abelian groups.

12


Text Book*/

Reference Book

1. *N. S. Gopalakrishnan: University Algebra.

2. Joseph A. Gallian: Contemporary Abstract Algebra.


Syllabus of B.A./B.Sc. V approved in BOS: 01-08-2019 Course Title Set Theory and Number Theory


Credits 4








The first two units of this course will consist of the basics of mathematical set theory,

including well-orderings, cardinality, Characteristic functions and choice functions,

Equipotent sets, Cantor’s theorem, Schroeder Bernstein theorem proved and Maximal and

minimal elements.

Next, two units of this course will consist of Mathematical induction, divisibility, prime

numbers, congruences, factorization, arithmetic functions, quadratic reciprocity, primitive

roots, Diophantine equations, Chinese remainder Theorem, Algebraic congruence mod p,

Lagrange’s theorem, Wilson theorem, and their applications.

Course Outcomes On successful completion of this course, students should be able to:

• be able to formalize mathematical statements in ZF set theory

Understand cardinal and ordinal arithmetic

• be able to apply variants of the axiom of choice

• be able to carry out proofs and constructions by transfinite induction and recursion

• be familiar with various paradoxes in naive set theory and understand the need for

formalization of set theory

• know independence results for the continuum hypothesis and the axiom of choice

• be able to present mathematical arguments to others

• knowledge of the basic definitions and theorems in number theory

• the ability to apply number theory algorithms and procedures to basic problems

• the ability to think and reason about abstract mathematics skills at writing

mathematical proofs

• analyze and solve problems involving the distribution of primes

• analyze and solve problems involving primitive roots

• solve systems of Diophantine equations using the Chinese Remainder Theorem &

the Euclidean algorithm

• understand the basics of modular arithmetic

• state and prove Lagrange’s theorem, Wilson theore& its generalization using Euler's

function.


UNIT I: Binary Relations and Functions

Relations and their representations, Inverse relation, Composition of relations and their properties,

Equivalence relation and partition, Cross Partition, Fundamental theorem of equivalence relation,

Functions their restrictions and extensions, Invertible functions, Characteristic functions and choice

functions, Equipotent sets.

12

UNIT II: Countability and Partial Ordering

Infinite sets, Denumerable sets, Countable sets, Continuum, Cardinals, Cardinal arithmetic,

Inequalities of cardinal numbers, Cantor’s theorem, Schroeder Bernstein theorem, Continuum

hypothesis, Partially ordered sets, Totally ordered sets, Similar sets and Well-ordered sets, First and

last elements, Maximal and minimal elements.

12

UNIT III: Divisibility theory and Prime Integers

Division algorithm and derived results, Least common multiple, Greatest common divisor, Euclid’s

algorithm, Prime numbers and related results, Fundamental theorem of arithmetic, Relatively prime

integers, Euler’s function.

12

UNIT IV: Theory of Congruences

Congruences, Euler’s Theorem, Fermat’s theorem, Order of an integer (mod m), Linear

congruences, Chinese remainder Theorem, Algebraic congruence mod p, Lagrange’s theorem,

Wilson theorem, Algebraic congruences with composite number.

12


Text Books*/

Reference

Books

1. *Seymour Lipschutz: Set Theory and Related Topics, Schuam’s Outline Series.

2. *J. Hunter: Number Theory.

3. P. R. Halmos: Naive Set Theory.

4. David M. Burton: Elementary Number Theory, 6th Ed.

5. G. B. Mathews: Theory of Number Part-I.


Syllabus of B.A./B.Sc. V Semester approved in BOS: 01-08-2019 Course Title Geometry of Curves and Surfaces


Credits 4


Prerequisite Courses Calculus, Coordinate Geometry





Course Objectives The primary objective of this course is to understand the notion of space

curve, surfaces, geodesics, Weingarten maps, parametrization of

surfaces, areas, volumes and Goddazi equation and Gauss theorem.

Course Outcomes After studying this course the student will be able to:

• understand the concepts of graphs, level sets as solutions of

smooth real valued functions, tangent space and normal.

• comfortably familiar with orientation, Gauss map and

geodesic.

• learn about linear self-adjoint Weingarten map and curvature

of a plane curve with applications in geometry and physics.

• deal with parametrization and be familiar with well-known

surfaces as equations in multiple variables, able to find area

and volumes.

• study surfaces with boundary and be able to solve various

problems and the Gauss theorem.


UNIT I: Space Curves

Space curves, Examples, Plane curves, Parameterization of curves (Generalized and natural

parameters), Change of parameter regular curves and singularities,Contact of curves,

Contact of a curve and a plane, Frenettrihedron, Osculating plane, Serret-Fernet formulae,

Involutes and Evolutes, Fundamental Theorem for space curves.

12

UNIT II: Surface in ℝ3

Surfaces in ℝ3, Implicit and explicit forms of the equation of a surface, Parametric curves

on surfaces, Tangent plane, First fundamental form, Angle between two curves on a surface,

Area of a surface, Invariance under co-ordinate transformation.

12

UNIT III: Extrinsic Geometry

Second fundamental form on a surface, Gauss map and Gaussian curvature, Gauss and

Weingrten formulae, Christoffle symbols, Some co-ordinate transformations, Goddazi

equation and Gauss theorem, Fundamental theorem of surface Theory.

12

UNIT IV: Curves on a Surface

Curvature of a curve on a surface, Geodesic curvature and normal curvature, Geodesics,

Principal directions and lines of curvature, Rodrigue formula, Asymptotes lines, Conjugate

directions.

12


Text

Books*/

Reference

Books

1. *A. Goetz: Differential Geometry, Springer Verlag.

2. *S.I. Husain: Lecture notes on Differential Geometry, Seminar Library, Department of

Mathematics, AMU, Aligarh.


Syllabus of B.A./B.Sc. V Semester approved in BOS: 01-08-2019

Course Title Mechanics


Credits 4







Course Objectives The aim of this course is to:

1) Develop an ability to grasp the concepts of vectorial force,

equilibrium and tension.

2) To make them keen in learning the concepts of gravity and then using

calculus to solve the related problems.

3) To help them get a hold of components of vector

quantities and understand Projectile Motion.

4) To develop an understanding of the fundamentals and principles of

motion of particles, pendulums and laws of planetary motion.

Course Outcomes On successful completion of this course, students should be able to:

• Understand the vector representation of forces and equilibrium.

• Analyse the problems involving tension in a string and

simultaneously solve them.

• Illustrate laws of motion, kinematics of motion and their

interrelationship.

• Explain the concepts of motion of particles and stability of orbits.

Get a hold of SHM of simple pendulum.


UNIT I: Method of Analytic Statics

Vector moment of a force, Varignon’s Theorem on moments, Resultant of a couple,

Resultant of coplanar forces, Equation of line of action of the resultant, Equilibrium of a

rigid body, Conditions of equilibrium of three force body, Cables, Suspension and

parabolic cables, Intrinsic and Cartesian equation of a catenary, Sag and span, maximum

tension in a cable.

12

UNIT II: Virtual Work and Centre of Gravity

Virtual work, Principle of virtual work, Determination of tension in a string and thrust in a

rod, Solutions of problems involving equilibrium by principle of virtual work, Centre of

gravity. Determination of Centre of gravity by integration, Centre of gravity of arcs, plane

areas, enclosed areas, solids of revolution and surfaces of revolution.

12

UNIT III: Kinematics and Kinetics

Tangential, normal, radial and transverse components of velocity and acceleration, Motion

of projectile without resistance, Projectile motion up and down an inclined plane, Tangent

problems, Motion in a resisting medium including projectiles, Upward and downward

motion in a resisting medium.

12

UNIT IV: Applications of Plane Dynamics

Motion of particles in central orbits, Stability of circular orbits, Kepler’s laws of planetary

motion, Plane impulsive motion, Direct and oblique impact, Loss of energy during impact,

Impact against a fixed plane, Simple harmonic motion, Motion of a simple pendulum.

12


Text Books*/

Reference Books

1. *J L. Synge and G. B. Griffith: Principle of Mechanics.

2. *M. A. Pathan: Statics. Jhonson and Beer: Vector Mechanics for Engineers.

3. Zafar Ahsan: Lectures Notes on Mechanics.


Syllabus of B.A./B.Sc. V Semester approved in BOS: 01-08-2019 Course Title Tensor Analysis


Credits 4

Course Category Optional

Prerequisite Courses Calculus, Linear Algebra





Course Objectives The course aim at understanding the various relations which remain valid on change

of coordinate system. It also emphasizes the utilitarian aspect and intended to help

the learners of relativity, differential geometry, engineering mathematics etc.


• The relations between other papers of Mathematics;

• To study and to learn the cause-effect related to these;

• Visualization of the transformation of the mathematical quantities from

one space to other and their expressions.

• The applications in observing and relating real situations/structures.


UNIT I: Relevant Concepts of Linear Algebra

Dummy indices, Free indices, Summation convention, Kroneckersymbol, Permutation symbols,

Differentiation of a determinant, Linear equations, Cramer’s Rule, Functional determinants,

Functional matrices, Dual spaces and bilinear forms, Dual basis, Quadratic forms, Real quadratic

forms, Signature, index and nature of quadratic forms, Pairs of quadratic forms, Quadratic

differential forms.

12

UNIT II: Algebra of Tensors

Transformations of coordinates, Contravariant vectors, Scalar invariants, Covariant vectors, Scalar

product of two vectors, Tensors of the second order, Tensors of any order, Symmetric and skew

symmetric tensors, Addition and multiplication of tensors, Contraction, Composition of tensors,

Quotient law, Reciprocal symmetric tensors of the second order.

12

UNIT III: Riemannian Space

Riemannian space, Fundamental tensors, Metric tensor, Raising and lowering of indices,

Magnitude of a vector, Associate covariant and contravariant vectors, Inclination of two vectors,

Orthogonal vectors, Relative and absolute tensors, Tensor density.

12

UNIT IV: Covariant Derivative and Curvature Tensor

Christoffel symbols, Transformation law and their properties, Equation of geodesics (without

proof), Covariant differentiation of contravariant and covariant vectors, Covariant differentiation of

tensors, Divergence of a vector, Curl of a vector, Riemann curvature tensor, Properties of curvature

tensor, Ricci tensor, Scalar curvature, Einstein tensor.

12


Text

Books*/

Reference

Books

1. *C. E. Weatherburn: Riemannian geometry and The Tensor Calculus, CUP, 1938.

2. *Zafar Ahsan: Tensors-Mathematics of Differential Geometry and relativity, PHI, New Delhi,

2015.

3. I. S. Sokolnikoff: Tensor Analysis-Theory and Applications, Chapman and Hall, 1951.

4. U. C. De, A. A. Shaikh and J. Sengupta: Tensor Calculus, Narosa Publication, New Delhi, 2nd

Ed, 2008.

5. R. S. Mishra: A Course in Tensors with Riemannian Geometry, PothishalaPvt Ltd, Allahabad,

4th Ed. Reprint, 2013.


Syllabus of B.A./B.Sc. V Semester approved in BOS: 01-08-2019 Course Title Mathematical Methods


Credits 4


Prerequisite Courses Differential Equations





Course Objectives The main objectives of this course are to introduce the methods and concepts for solving linear

integral equations, to study Laplace and Fourier transforms with their applications to ODE, PDE

as well as integral equations and to provide an understanding the problems through calculus of

variations.

Course Outcomes On successful completion of this course, student should be able to: determine the solutions to

Volterra as well as Fredholm integral equations by method of resolvent kernel, method of

successive approximations, method of integral transforms, understand with eigen values and

eigen functions of homogeneous Fredholm integral equations, calculate the Laplace transform,

Fourier transform and their inverse transforms of common functions and understand the

formulation of variational problems, the variation of a functional and its properties, extremum of

functional, necessary condition for an extremum.


UNIT I: Linear Integral Equations

Definition, examples and classification of integral equations, Relation between differential

and integral equations, Solution of Volterra as well as Fredholm integral equations of second

kinds by the method of successive substitutions and successive approximations, Iterated and

resolvent kernels, Reduction of Volterra integral equations of first into second kind, Solution

of Volterra integral equations of first kind.

12

UNIT II: More on Fredholm Equations

Solution of Fredholm integral equations with separable kernels, Eigenvalues and eigen

functions of Homogeneous Fredholm integral equations, Solution of integral equations with

symmetric kernels, Fundamental properties of Eigenvalues and Eigen functions for

symmetric equations.

10

UNIT III: Integral Transforms

Revisit to Laplace transform, Solution of PDEs and integral equation by Laplace transform

method, Revisit to Fourier series, Complex form of Fourier series, Fourier integrals, Fourier

sine and cosine integrals, Fourier transform and inverse Fourier transform, Fourier transform

of elementary functions, Properties of Fourier transform, Fourier sine and cosine transform,

Solution of ODEs, PDEs and integral equations by Fourier transform method.

14

UNIT IV: Calculus of Variations

Functional and its variation and extremal, Variational principle, Euler’s equation for

functionals containing first order derivatives and one independent variable, Functionals

depending on higher order derivatives, Functionals depending on several independent

variables, Parametric form, Isoperimetric problem, Functionals depending on partial

derivatives and Ostrogradsky’s equation, Invariance of Euler’s equation under coordinates

transformation, Simple applications in physical problems.

10


Text-

Books*/

Reference

Books

1. *R. P. Kanwal: Linear Integral Equations, Birkhäuser, Inc, Boston, 2nd Ed, 1997. (For Unit I & II).

2. *Pinkus Allan and Samy Zafrany: Fourier Series and Integral Transforms, Cambridge University

Press, 1997. (For Unit III).

3. *M. Gelfand and S. V. Fomin: Calculus of Variations, Dover Books, 2000. (For Unit IV).

4. R. K. Jain and S. R. K. Iyenger: Advanced Engineering Mathematics, Narosa Publishing House,

2009.

5. M. D. Raisinghania: Integral equations and Boundary Value Problems, S. Chand and Co Ltd, New

Delhi, Reprint, 2017.

6. Zafar Ahsan: Differential Equations and their Applications, Prentice Hall of India, New Delhi, 3rd

Ed, 2016.


Syllabus of B.A./B.Sc. VI Semester approved in BOS: 01-08-2019

Course Title Real Analysis-II


Credits 4







Course Objectives It is a basic course on the study of real valued functions that would develop an analytical ability

to have a more matured perspective of the key concepts of calculus, namely: limits, continuity,

differentiability, integrability and their applications.

Course Outcomes On successful completion of this course, student should be able to have a rigorous understanding

of the concept of limit of a function, the geometrical properties of continuous functions on

closed intervals, compact sets and connected sets, the applications of mean value theorem and

Taylor’s theorem, some of the families and properties of Riemann integrable functions, and the

applications of the fundamental theorems of integration, the valid situations for the inter-

changeability of differentiability and integrability with infinite sum, and approximation of

transcendental functions in terms of power series and to derive a Fourier series of a given

periodic function by evaluating Fourier coefficients.


UNIT I: Limits, Continuity, Boundedness and Monotonicity

Limit of a function, Infinite limits and limits at infinity, Continuous functions, Algebra of

limits and continuous functions, Sequential criterion for limits and continuity,

Characterizations of continuous functions via open sets and closed sets, Types of

discontinuities, Properties of continuous functions on closed intervals, compact sets and

connected sets; Uniform continuous functions, Bounded functions and Monotonic

functions.

12

UNIT II: Differentiation and Bounded Variation

Derivative of a function, Relation between differentiability and continuity, Relation

between differentiability and monotonicity, Darboux’s theorem, Rolle’s theorem, Mean

value theorems of differential calculus, Taylor’s theorem, Maculaurin’s theorem, Functions

of bounded variation and their properties, Variation function, Jordon theorem.

12

UNIT III: Some Special Functions

Power series, Radius and interval of convergence, Cauchy’s Hadamard theorem, Term-wise

differentiation and integration of power series, Uniform convergence, Abel’s theorem,

Taylor’s theorem; Exponential, Logarithmic, Generalized power, Trigonometric, Inverse

trigonometric functions and their properties; Fourier series and Fourier coefficients,

Periodic functions, Bessel’s inequality, Dirichlet’s criteria of convergence of Fourier series,

Fourier series for even and odd functions, Half-range series, Fourier series on arbitrary

intervals.

12

UNIT IV: Riemann Integration

Definition and existence of Riemann integral, Inequalities for Riemann integrals,

Refinement of partitions, Darboux’s theorem, Conditions of integrability, Integrability of

the sum, difference, quotient, product, modulus and square of integrable functions,

Riemann integral as a limit of sums, Classes of Riemann integrable functions, Primitive of

a function, Fundamental theorem of calculus, Mean value theorems of integral calculus,

Integration by parts, Change of variables.

12


Text Books*/

Reference

Books

1. *R. G. Bartle and D. R. Sherbert: Introduction to Real Analysis, John Wiley and Sons,

Singapore, 3rd Ed, 2003.

2. *S. C. Malik and S. Arora: Mathematical Analysis, New Academic Science Ltd, 5th Ed, 2017.

3. D. Somasundaram and B. Choudhary: A First Course in Mathematical Analysis, Narosa, 1999.

4. K. A. Ross: Elementary Analysis: The Theory of Calculus, Under graduate Texts in

Mathematics, Springer (SIE), Indian reprint, 2004.



Course Title Ring Theory


Credits 4







Course Objectives This course aims to introduce students to the following concepts and cognitive skills:

This is a second course in modern algebra which deals with ring theory . Some basics concept

of ring theory like rings, subrings, ideals, ring homomorphisms and their properties. The

application of the concepts of ring theory to important examples of rings. The definition of

Ideals, Nil ideals, quotient rings, prime and maximal ideals. The homomorphisms of rings

and various theorems of ring homomorphisms, the embedding of rings and ring of

endomorphisms of an abelian group. . We will then discuss classes of rings that have some

additional nice properties (e.g. Euclidean domains, principal ideal domains and unique

factorization domains). The fundamental theorems of algebraic structures are explained.

Also, explore the concepts of Polynomial rings, UFD, ED, PID.


• able to understand the standard computations of ring theory.

• to learn the elementary theorems and proof techniques of ring theory.

• to apply the theorems, proof techniques and standard computations of ring theory to

solve problems.

• Demonstrate knowledge of polynomial rings and asscoateed properties.

• Derive and apply Gauss lemma, Eistentein criterion for irreduciube of rationals.

• Factrorization and ideal theory in the polynomial rings; the structure of a primirive

polynomials.

• Utilize the Polynomial rings, UFD, ED, PID to solve different related problems


UNIT I: Basics of Rings, Special Kinds and Ideals

Rings, Zero divisors, Integral domains, Division rings, Fields, Subrings and Ideals, Congruence

modulo a subring relation in a ring, Simple ring, Algebra of ideals, Ideal generated by a subset,

Nilpotent ideals, Nil ideals, Quotient rings, Prime and Maximal ideals.

12

UNIT II: Homomorphisms and Embedding of Rings

Homomorphism in rings, Natural homomorphism, Kernel of a homomorphism, Fundamental

theorem of homomorphism, First and second isomorphism theorems, Field of quotients,

Embedding of rings, Ring of endomorphisms of an abelian group.

12

UNIT III: Factorization in Integral Domains

Prime and irreducible elements, H.C.F. and L.C.M. of two elements of a ring, Principal ideals

domains, Euclidean domains, Unique factorisation domains, Different relations between Principal

ideal domains, Euclidean domains and Unique factorization domains.

12

UNIT IV: Rings of Polynomials

Polynomials rings, Algebraic and transcendental elements over a ring, Factorization in polynomial

ring R[x], Division algorithm in R[x], where R is a commutative rings with identity, Properties of

polynomial ring R[x] if R is a field or a U.F.D., Gauss lemma, Gauss Theorem (statement only),

Eisenstein irreducibility criteria and its applications, Division algorithm for polynomial ring F[x],

where F is a field, Reducibility test for polynomials of degree 2 and 3 in F[x].

12


Text

Book*/

References

Books

1.* Surjeet Singh and QuaziZameeruddin: Modern Algebra.

2. J. B. Fraleigh: A first Course in Abstract Algebra.

3. Joseph A Gallian: Contemporary Abstract Algebra.



Course Title Metric Spaces


Credits 4


Prerequisite Courses A Course of Real Analysis





Course Objectives To give the idea of distance between two elements in a set and to extend the

concepts, namely, open sets, closed sets, convergence of sequences, compact

sets, continuity of functions etc, from real line to a metric space. The course

focuses on basic notions of metric spaces and their properties.

Course Outcomes Knowledge: A will student know:

• basic notions of metric spaces,

• methods and techniques of proving basic theorems on metric spaces

and continuous mappings

• equivalent methods for introducing a metric in a set

Skills: A student can:

• check if a given function is a metric,

• check if a given function is continuous,

• check if a given set is open, closed, dense, compact

Final course output - social competences

A student knows the importance of metric spaces in mathematics and its

applications in different areas.


UNIT I: Basic Concepts

Definition and examples of metric spaces, Bounded and unbounded metric spaces, Distance

between sets, Diameter of a set, Open and closed balls, Interior points and interior of a set,

Open set, Neighbourhood of a point, Limit point of a set, Closure of a set, Closed set,

Boundary points and boundary of a set, Exterior points and exterior of a set, Subspace of a

metric space.

12

UNIT II: Completeness and Separability

Sequences and subsequences in a metric space, Convergent and Cauchy sequences,

Complete metric spaces, Relation between completeness and closedness, Cantor

Intersection Theorem, Completion Theorem, Dense sets, Separable spaces, Nowhere dense

sets, Categories and Baire Category Theorem.

12

UNIT III: Compactness and Connectedness

Cover of a metric space, Compact metric spaces, Compact sets and their criterion,

Properties of compact sets, Relation between compactness, completeness and closedness,

Finite Intersection property, Bolzano-Weierstrass property,Sequential compactness, Totally

bounded spaces; Separated sets, Connected and disconnected metric spaces, Properties of

connected sets.

12

UNIT IV: Continuity

Continuous functions between two metric spaces, Characterizations of Continuous

functions, Continuous functions on compact spaces and connected space, Uniform

continuous functions, Homeomorphism and Isometry.

12


Text

Book*/

Reference

Books

1. *Q. H. Ansari: Metric Spaces Including Fixed Point Theory and Set-valued Maps,

Narosa Publishing House, New Delhi. 2010.

2. E. T. Copson: Metric spaces, Cambridge University Press, 1968.

3. M. O. Searcoid: Metric spaces, Springer, 2007.

4. S. Kumaresan: Topology of Metric Spaces, Narosa Publishing House, 2nd Ed, 2011.



Course Title Complex Analysis


Credits 4







Course Objectives This course aims to provide knowledge about the analytical aspects of

complex functions in complex variables with visualization through relevant practicals.

Particular emphasis has been laid on Cauchy’s theorems, series expansions and calculation of

residues.


• Understand the significance of differentiability of complex functions leading to the

understanding of Cauchy-Riemann equations.

• Evaluate the contour integrals and understand the role of Cauchy-Goursat theorem

and the Cauchy integral formula.

• Expand some simple functions as their Taylor and Laurent series, classify the nature

of singularities, find residues and apply Cauchy Residue theorem to evaluate

integrals.


UNIT I: Basic Concepts

Revision of complex number system, Triangle inequality and its applications, Polar and

Exponential forms of complex numbers: De-Moivre’s formulae and Euler’s formulae, Products and

quotients in exponential form, Roots of complex numbers, Point sets and Regions in complex

plane, Extended complex plane, Spherical representation of complex numbers: Stereographic

projection.

12

UNIT II: Functions of a complex variable

Limits, Continuity and differentiability of functions of a complex variable, Cauchy-Riemann (CR)

equations, Sufficient conditions for differentiability, Polar form of CR equations, Analytic

functions, Harmonic functions, Harmonic conjugate, Polar form of Laplace equation, Exponential,

Logarithmic, Trigonometric and Hyperbolic functions of complex variables, Complex exponents,

Inverse trigonometric and inverse hyperbolic functions.

12

UNIT III: Complex Integration

Definite integral of a complex valued function of a real variable, Contour integrals, Cauchy-

Goursat theorem (without proof), Consequences of Cauchy-Goursat theorem, Cauchy’s integral

formula, Cauchy’s integral formula for higher order derivatives, Morera’s theorem, Cauchy’s

inequality, Liouville’s theorem, Fundamental theorem of algebra, Gauss’ mean value theorem.

12

UNIT IV: Complex Sequence and Series

Sequence and series of complex numbers and their convergences, Sequence and series of complex

functions and their convergences, Power series and its convergence (Absolute/Uniform), Taylor’s

series, Laurent’s series (without proof), Singular points and its classifications, Zeros and poles of

order m, Residues, Calculation of residues, Residue theorem.

12


Text

Books*/

Reference

Books

1. *R. V. Churchill and J. W. Brown: Complex Variables and Applications, New York

*McGraw Hill, 9th Ed, 2013.

2. H. S. Kasana: Complex Variables -Theory and Applications, Prentice Hall of India, New

Delhi, 2nd Ed, 2008.

3. Murray R. Spiegel: Theory and Problems of Complex Variables, Schaum’s Outline Series.



Course Title Programming in C and Matlab


Credits 4







Course Objectives 1. To familiarize student the concept of programming in C and exploring software like

MATLAB etc.

2. To enable the student on how to approach for solving problems using simulation tools.

3. To prepare the students to use MATLAB in their project works.

4. To provide a foundation in use of this softwares for real time applications.

Course Outcomes On Successful completion of this course, students should be able to:

• Ability to express programming & simulation.

• Ability to find importance of this software for Lab Experimentation.

• Articulate importance of software’s in research by simulation work.

• Ability to write basic mathematical problems in MATLAB.


UNIT I: MATLAB Windows and Mathematical Operations with Arrays

Introduction to MATLAB, Standard MATLAB windows (Command Window, Figure

Window, Editor Window, help window), The semicolon (;), The clc command, Using

MATLAB as calculator, Display formats, Elementary math built in functions, The zeroes,

ones and eye commands, The transpose operators, Using a colon, Adding elements to

existing variables, Deleting elements, Creating arrays (one dimensional & two

dimensional), Built in functions for handling arrays, Array multiplication, Inverse of a

matrix, Solving three linear equations (array division), Element by element operations, Built

in function for analysing arrays, Generation of random numbers, Creating and saving a

script files, output commands.

12

UNIT II: User-Defined Functions and Function Files

User-defined functions and function files, Creating a Function File, Structure of a function

file, Function definition, Input and output arguments, Function body, Local and global

variables, Saving a function file, Using a user-defined function, Examples of simple user-

defined Functions, Comparison between script files and function files, Anonymous and

inline functions, Anonymous functions, Inline functions, Using function handles for passing

a function into a function, Using a function name for passing a function into a function,

Subfunctions nested functions.

12

UNIT III: Programming in MATLAB and Plots with Special Graphics

Programming in MATLAB, Relational and logical operators, Conditional statements: The

If-End structure, The If-Else-End structure, The If-Elseif-Else-End structure, The switch-

Case statement, Loops: For-End Loops, While-End Loops, Nested Loops And Nested

Conditional statements, The break and continue commands,Two dimension and three

dimensional plots, Line plots, Mesh and surface plots, Plots with special graphics.

12

UNIT IV: Symbolic Math and Applications in Numerical Analysis

Solving an equation with one variable, Finding a minimum or a maximum of a function,

Numerical integration, Ordinary differential equations, Interpolation etc, Symbolic Math:

symbolic objects and symbolic expressions, Creating symbolic objects, Creating symbolic

expressions, Changing the form of an existing symbolic expression, Integration, Solving an

ordinary differential equation, Plotting symbolic expressions, Numerical calculations with

symbolic expressions, Examples Of MATLAB applications etc. .

12


Text Books*/

Reference Books

1. *Amos Gilat: MATLAB-An Introduction and its Applications, Wiley India Edition.

2. *E. Balagurusamy: Programming in ANSI C, McGraw Hill Education, 8th Ed.

https://www.amazon.in/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=Balagurusamy&search-alias=stripbooks


Syllabus of B.A./B.Sc. VI Semester approved in BOS: 01-08-2019 Course Title OPTIMIZATION


Credits 4







Course Objectives This course begins with applications and scope of O.R. Formulation of linear

programming problem and then different methods to solve them will be discussed.

Duality in LPP will be introduced. Introduction to NLPP and some solving methods

will be covered. At the end KKT Conditions and convex optimization

techniqueswill be discussed.


Students will understand the concept of LPP and NLPP and will be able to solve some

reallife problems using optimization techniques. This course will make them capable

to attend different competitions problems mainly asked in GATE and NET.


UNIT I: Introduction to LPP and Graphical Method

Definitions and scope of O.R.(see Ch-1 in [1]), Linear programming problem (Sec:2.2 in

[1]), Formulation of linear programming problem (Sec:2.3, 2.4 in [1]), Graphical solution of

L.P.P. (Sec:3.2 in [1]), Some exceptional cases (Sec:3.3 in [1]), General L.P.P. and some

definitions (Sec:3.4 in [1]), Canonical and standard form of L.P.P. (Sec:3.5 in [1]),

Hyperplanes, Convex sets and their properties (Sec:0.13 in [1]).

12

UNIT II: Simplex Method and Duality

Some definitions (Sec:4.1 in [1]), Fundamental theorem of linear programming (Theo 4.3 in

[1]), Simplex method (Sec:4.3 in [1]), Two-phase method, Big M method (Sec:4.4 in [1]),

Duality in L.P.P., General Primal-Dual pair (Sec:5.2, 5.4 in [1]), Weak duality theorem,

Strong duality theorem (Sec:5.5 in[1]), Dual simplex method (Sec:5.9 in [1]).

12

UNIT III: Introduction to NLPP and Some Solving Methods

Convex functions and their properties (Sec: 7.2 in [2]), General NLPP (Sec: 27.3 in [1]),

Formulation of NLPP (Sec: 27.2 in [1]), Methods for solving NLPP: Graphical method

(Sec: 28.2 in [1]), Method of Lagrange's multipliers (Sec: 27.4 in [1]), The Steepest Descent

method (unconstrained opt. prob.) (Sec: 9.4 in [2]), Newton's method (unconstrained opt.

prob.) (Sec: 9.5 in [2]).

12

UNIT IV: KKT Conditions and Convex Optimization

KKT necessary/sufficient optimality conditions, Solution of NLPP using KKT conditions

(Sec: 27.5 in [1], Sec: 8.5 in [2]), Quadratic programming (Sec: 28.4 in [1]), Wolfe's method

for quadratic programming (Sec: 28.5 in [1], Sec: 7.7 in [2]), Convex programming

problems (Sec: 7.4 in [2]).

12


Text

Books*/

Reference

Books

1. *Kanti Swarup, P.K. Gupta, Man Mohan, Operations Research, Sultan Chand & Sons,

2009.

2. *S. Chandra, Jayadeva, Aparna Mehra, Numerical Optimization with Applications,

Narosa.

3. Hamdy A. Taha, Operations Research, An Introduction, 9th Edition, Pearson.

4. M.S. Bazarra, H.D. Sheral and C.M. Shetty, Nonlinear Programming theory and

Algorithms.


Syllabus of B.A./B.Sc. VI approved in BOS: 01-08-2019

Course Title Discrete Mathematics


Credits 4







Course Objectives This course aims to introduce students to the following concepts and cognitive

skills: The definition of different types and families of graphs. The application of a graph as a

mathematical model for many real life situations. The definition of isomorphism of

graphs. Exploring different types of subgraphs of a graph. The definition of basic

concepts related to graphs and graph operations. Representation of a graph using

adjacency and incidence matrices. Computing the number of spanning trees of a graph

using the matrix tree theorem and the deletion contraction method. Prim’s and

Kruskal’s Algorithms to construct Minimum Spanning Trees. Introduce coding and

discuss "What is coding?" and "Why do we use it?". Discuss simple error models,

Hamming distance, and coding gain. Introduce the student to applications in

Communication, Weight of code word, and Distance between the code word. Course Outcomes On successful completion of this course, student should be able to:

• Prove the hand shaking lemma and its corollaries.

• Apply the handshaking lemma to different problems.

• Test whether two given graphs are isomorphic.

• Calculate the order and size of line graph and product

• graphs using data of given ones.

• Compute the independence, covering and dominating numbers of a graph.

• Utilize suitable algorithms to find the complement, line, powers, closure and dual

of some given graph.

• Prove some criteria for Eulerian and Hamiltonian graphs.

• Represent graphs using adjacency and incidence matrices.

• The student has knowledge of properties of and algorithms for coding and

decoding of linear block codes

• The student is able to apply various algorithms and techniques for coding and

decoding. Contents of Syllabus No. of Lectures

UNIT I: Introduction to Graphs

Definition of a graph, Finite and infinite graphs, Incidence of vertices and edges, Types of

graphs, Subgraphs, Representing graphs and graph isomorphism, Matrix representation of

graphs, Incidence and adjacency matrices of graphs, Degree sequences.

12

UNIT II: More on Graph Theory

Walks, Trails, Paths, Connected graphs, Distance, Cut-vertices, Cut-edges, Block, Euler’s

path and circuit, Hamiltonian path and circuit, Eulerian and Hamiltonian graphs, Planar

graphs.

12

UNIT III: Trees

Introduction to Trees and characterizations, Applications and properties of Trees, Rooted

and binary trees, Spanning Trees, Weighted graphs, Prim’s Algorithm to construct

Minimum Spanning Trees, Kruskal’s Algorithm to to construct Minimum Spanning Trees,

Dijktrals Algorithm to find the shortest Path.

12

UNIT IV: Coding Theory

Introduction to coding theory, Error Correction and decoding, Communication channel,

Coding problem, Block codes, Hamming distance, Nearest neighbour/ minimum distance

decoding, Group codes, Weight of code word, Distance between the code word.

12


Text

Books*/

Reference

Books

1. *W. B. West: Introduction to Graph Theory, Prearson Education, Singapore (Unit I-III).

2. G. Charteand and P. Zhang: Introduction to Graph Theory, Tata McGraw Hill, 2007.

3. *S. Ling and C. P. Xing: Coding Theory-A First Course, Cambridge University Press,

Cambridge, 2004 (Unit IV).

COURSE STRUCTURE SYLLABI · Summation of series, Expansion of functions, Maclaurin’s theorem and Taylor’s theorem, Functions of two or more variables and introduction of partial

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