syllabi - dr. harisingh gour university, sagar university

A

SYLLABI

For

B.A./B.Sc.

I, II, III, IV, V, VI Semester

COURSE

IN

MATHEMATICS

Session 2019-2020

DR. H. S. GOUR VISHWAVIDYALAYA

(A CENTRAL UNIVERSITY)

SAGAR (M.P.)

Introduction

Mathematics Profession

Mathematics is the backbone of all sciences. Along with many other branches

mathematics concerns with algebra, vector, calculus, differential, partial and integral

equations, which are frequently used in all physical sciences and discrete

mathematics in computer sciences, etc. Students possessing degrees Ph. D.,

M.Sc./M.A. in Mathematics and B.Sc./B.A. with Mathematics have a very large

numbers of job opportunities in the fields of Banking, Teaching, Software

Engineering, Actuaries, Defence and as Operation Research Analyst, Computer

System Analyst, etc. They can also get administrative jobs through UPSC and PSC.

The course is so designed that the student can also take employment worldwide.

1. Name of the Programme : B.Sc./B.A.

2. Duration of Programme : The duration of programme is six semesters spread

over a period of not less than 90 working days for each semester.

3. Structure of Programme: The course (elective and core) of study for

B.Sc./B.A., includes the subject. number of hrs. per week devoted to each

subject and credits for its theory practicals as per scheme attached.

4. Medium of instruction : English.

5. Each courses of B.Sc./B.A. is mark as a core/compulsory/elective course etc.

6. Credit allotted: Credit allotted followed through out of scheme of programme

(Total credit: 44)

(i) Core Course : 24

(ii) Elective Course : 20

7. Scheme of Examination:

(a) Mid Semester Examination : 20

(b) Internal Assessment : 20

(c) End Semester Examination : 60

Total : 100

DOCTOR HARISINGH GOUR VISHWAVIDYALAYA, SAGAR

(A Central University)

Department of Mathematics and Statistics

Academic Session 2019-2020 onwards

B.Sc.\ B.A. with Mathematics

Semester Code Course Title Credit Contact

Hours

1 MTS CC 111 Differential Calculus 6 90

2 MTS CC 211 Differential Equations 6 90

3 MTS CC 311 Real Analysis 6 90

MTS SE 311 Logic and Sets 2 30

MTS SE 312 Analytical Geometry 2 30

MTS SE 313 Integral Calculus 2 30

4 MTS CC 411 Algebra 6 90

MTS SE 411 Vector Calculus 2 30

MTS SE 412 Theory of Equations 2 30

MTS SE 413 Number Theory 2 30

5 MTS SE 511 Probability and Statistics 2 30

MTS SE 512 Mathematical Finance 2 30

MTS SE 513 Mathematical Modeling 2 30

MTS EC 511 Matrices 6 90

MTS EC 512 Mechanics 6 90

MTS EC 513 Linear Algebra 6 90

6 MTS SE 611 Boolean Algebra 2 30

MTS SE 612 Transportation and Game Theory 2 30

MTS SE 613 Graph Theory 2 30

MTS EC 611 Numerical Methods 6 90

MTS EC 612 Complex Analysis 6 90

MTS EC 613 Linear Programming 6 90

NOTE :-

1. Papers of CODE CC are compulsory in Semesters 1 to 4.

2. Choose any one paper of CODE SE in semesters 3 to 6.

3. Choose any one paper of CODE EC in semesters 5 and 6.




B.Sc.\ B.A.–I Semester

MTS CC 111 Differential Calculus L T P C 5 1 0 6

Max. Marks : 100

Mid Sem-20

Internal assessment-20

End Sem-60

Objectives:

(1) To understand continuity and discontinuity of function

(2) To find successive differentiations of product of two functions

(3) To calculate limit of indeterminate form

(4) To find the maximum and minimum value of function of one variable

(5) To trace the curve.

Unit I: (18 hours) Limit and Continuity of function (Using ε-δ definition), types of discontinuities, Differentiability

of function, Rolle’s theorem, Mean value theorem.

Unit II: (18 hours) Successive differentiation, Leibnitz’s theorem, Taylor’s theorem with Lagrange’s and Cauchy’s

forms of remainder, Maclaurin’s series of sin 𝑥, cos 𝑥, log(1 + 𝑥), (1 + 𝑥)𝑚 , 𝑒𝑥 etc.

Unit III: (18 hours) Indeterminate forms, Partial differentiation, Euler’s theorem on homogeneous functions,

Tangents and Normals.

Unit IV: (18 hours) Curvature, Asymptotes, Singular points of curves, Maxima and Minima.

Unit V: (18 hours) Tracing of curves, parametric representation of curves and tracing of parametric curves, Polar

coordinates and tracing of curves in polar coordinates.

Learning Outcomes: After completion the course student be able to understand application of Rolle’s

Theorem, Mean Value Theorem, Lagrange’s Theorem in industry. By plotting (trace) curves, they

understand the characteristics of different curves in Cartesian and polar coordinates.

Essential Readings:

1. H. Anton, I. Birens and S. Davis, Calculus, John Wiley and Sons, Inc., 2002.

2. G.B. Thomas and R.L. Finney, Calculus, Pearson Education, 2007.

Suggested Readings and links:

1. Video lecture on Introduction to Limits & Continuity

Link-https://www.youtube.com/watch?v=vEZsuZd9s88

2. Taylor's theorem with Problems

Link-https://www.youtube.com/watch?v=5uT1jbgwFqs

E Books Link :National Digital Library

https://www.youtube.com/watch?v=vEZsuZd9s88

https://www.youtube.com/watch?v=5uT1jbgwFqs




B.Sc.\ B.A.–II Semester

MTS CC 211 Differential Equations L T P C 5 1 0 6

Max. Marks : 100

Mid Sem-20 Internal assessment-20

End Sem-60 Objectives:

(1) To construct exact differential equation of first order.

(2) To understand the concept of linearly dependent and linearly independent by obtaining Wronskian.

(3) To solve the differential equation by method of variation of parameter.

(4) To construct the partial differential equations.

(5) To classify the second order partial differential equation and solving by charpit’s method .

Unit – I: (18 hours)

First order exact differential equations, Integrating factors, rules to find an integrating factor, First order

higher degree equations solvable for x, y and p, Methods for solving higher-order differential equations.

Unit–II: (18 hours)

Basic theory of linear differential equations, Wronskian and its properties, Solving a differential equation

by reducing its order, Linear homogenous equations with constant coefficients, Linear non-homogenous

equations.

Unit – III: (18 hours)

The method of variation of parameters, The Cauchy-Euler equation, Simultaneous differential equations,

Total differential equations.

Unit – IV: (18 hours)

Order and degree of partial differential equations, Concept of Linear and non-linear partial differential

equations, Formation of first order partial differential equations, Linear partial differential equation of

first order, Lagrange’s method.

Unit – V: (18 hours)

Charpit’s method, Classification of second order partial differential equations into elliptic, parabolic and

hyperbolic through illustrations only.

Learning Outcomes: After completion the course, students will able to construct PDE and

understand the application of it. Also they may identify linearly dependent solution and linearly

independent solution. They may capable to find linear and non linear solution.

Essential Readings:

1. Shepley L. Ross, Differential Equations, 3rd Ed., John Wiley and Sons, 1984.

2. Sneddon, Elements of Partial Differential Equations, McGraw-Hill, International Edition, 1967.

Suggested Readings and links 1. Video lecture on ordinary differential equation of first order-Exact differential equation

Link https://www.youtube.com/watch?v=suvzwN2Df7k

2. Video lecture on Partial Differential Equation - Charpit Method for Non Linear PDE Link https://www.youtube.com/watch?v=2_hfp8JPP30

E Books Link :National Digital Library

https://www.youtube.com/watch?v=suvzwN2Df7k

https://www.youtube.com/watch?v=2_hfp8JPP30




B.Sc.\ B.A.–III Semester

MTS CC 311 Real Analysis L T P C

5 1 0 6

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To understand the concept of real line and its properties.

(2) To learn about sequences and their boundedness.

(3) To learn the monotone sequence and their convergence.

(4) To find out the convergence of series using various test.

(5) To discuss about sequence and series of functions and uniform convergence.


Finite and infinite sets, examples of countable and uncountable sets. Real line, bounded sets,

supremum and infimum, completeness property of R, Archimedean property of R, intervals.

Unit – II: (18 hours)

Concept of cluster points and statement of Bolzano-Weierstrass theorem. Real Sequence, Bounded

sequence, Cauchy convergence criterion for sequences, Cauchy’s theorem on limits, order

preservation and squeeze theorem.


Monotone sequences and their convergence (monotone convergence theorem without proof), Infinite

series, Cauchy convergence criterion for series, positive term series, geometric series.


Comparison test, convergence of p-series, Root test, Ratio test, alternating series, Leibnitz’s test

(Tests of Convergence without proof). Definition and examples of absolute and conditional

convergence.

Unit –V: (18 hours) Sequences and series of functions, Point wise and uniform convergence, Mn-test, M-test, Statements

of the results about uniform convergence and integrability and differentiability of functions, Power

series and radius of convergence.

Learning Outcomes: After completion of this course the students will be able to understand

various types of sequences and series of numbers as well as the functions. They can also attempt the

questions related to the course in various competitive exams.

Essential Readings:

1. E. Fischer, Intermediate Real Analysis, Springer Verlag, 1983.

2. K.A. Ross, Elementary Analysis- The Theory of Calculus Series- Undergraduate Texts

inMathematics, Springer Verlag, 2003.

Suggested Readings and Links :

1. T. M. Apostol, Calculus (Vol. I), John Wiley and Sons (Asia) P. Ltd., 2002.

2. R.G. Bartle and D. R Sherbert, Introduction to Real Analysis, John Wiley and Sons (Asia) P.

Ltd., 2000.

E Books :National Digital Library





MTS SE 311 Logic and Sets L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To understand the concept of truth table.

(2) To learn about equivalence and quantifiers.

(3) To learn the set and its properties.

(4) To find out the operations on set and its identities.

(5) To discuss about relations and their types.


Introduction, propositions, truth table, negation, conjunction and disjunction. Implications,

biconditional propositions, converse, contra positive and inverse propositions and precedence of

logical operators.


Propositional equivalence: Logical equivalences. Predicates and quantifiers: Introduction, Quantifiers,

Binding variables and Negations.


Sets, subsets, Set operations, the laws of set theory and Venn diagrams. Examples of finite and

infinite sets. Finite sets and counting principle. Empty set, properties of empty set. Standard set

operations.


Classes of sets. Power set of a set. Difference and Symmetric difference of two sets. Set identities,

Generalized union and intersections.

Unit –V: (06 hours)

Relation: Product set, Composition of relations, Types of relations, Partitions, Equivalence Relations

with example of congruence modulo relation.

Learning Outcomes: After completion of this course the students will be able to understand the

basic concept of Boolean algebra and fundamental of sets and their types. The students will

also be able to attempt the questions of set theory in various competitive exams.

Essential Readings :

1. E. Kamke, Theory of Sets, Dover Publishers, 1950.

Suggested Readings and Links:

1. R.P. Grimaldi, Discrete Mathematics and Combinatorial Mathematics, Pearson

Education,1998.

2. P.R. Halmos, Naive Set Theory, Springer, 1974.

E Books Link: National Digital Library





MTS SE 312 Analytical Geometry L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To understand the concept of sketching for parabola, ellipse and hyperbola.

(2) To learn about reflection properties.

(3) To learn about classification.

(4) To study Spheres and Cylindrical surfaces.

(5) To discuss about illustration of graphing.


Techniques for sketching parabola, ellipse and hyperbola.


Reflection properties of parabola, ellipse and hyperbola.


Classification of quadratic equations representing lines, parabola, ellipse and hyperbola


Spheres, Cylindrical surfaces.

Unit –V (06 hours)

Illustrations of graphing standard quadric surfaces like cone, ellipsoid.

Learning Outcomes: After completion of this course the students will be able to understand the

basic concept of Analytic Geometry and they will be able to sketch the parabola, ellipse, hyperbola

for any equation.

Essential Readings:

1. S.L. Loney, The Elements of Coordinate Geometry, McMillan and Company, London.

2. R.J.T. Bill, Elementary Treatise on Coordinate Geometry of Three Dimensions, McMillan.

Suggested Reading and Links:

1. G.B. Thomas and R.L. Finney, Calculus, 9th Ed., Pearson Education, Delhi, 2005.

2. H. Anton, I. Bivens and S. Davis, Calculus, John Wiley and Sons (Asia) Pvt. Ltd., 2002.

India Ltd., 1994.

E Books Link: National Digital Library





MTS SE 313 Integral Calculus L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To understand the concept of integration by partial fraction.

(2) To learn about Reduction formulae.

(3) To learn about Reduction formulae for logarithmic functions and others.

(4) To find out the area, volume and surface.

(5) To discuss about double and triple integrals.

Unit (06 hours)

Integration by Partial fractions, integration of rational and irrational functions.

Unit - II: (06 hours)

Properties of definite integrals. Reduction formulae for integrals of rational, trigonometric,

exponential functions.

Unit - III: (06 hours)

Reduction formulae for logarithmic functions and others.

Unit - IV: (06 hours)

Areas and lengths of curves in the plane, volumes and surfaces of solids of revolution.

Unit - V: 06 hours)

Double and Triple integrals.

Learning Outcomes: After completion of this course the students will be able to integration any

complex type of function and can find the area, volume and surface of revolution using

integrals.

Essential Readings:

1. H. Anton, I. Bivens and S. Davis, Calculus, John Wiley and Sons (Asia) P. Ltd., 2002.

Suggested Readings:


E Books: National Digital Library



Department of Mathematics and Statistics B.Sc.\ B.A.–IV Semester

MTS CC 411 Algebra L T P C

5 1 0 6

Max. Marks : 100

Mid Sem-20 Internal assessment-20

End Sem-60 Objectives: (1) To inculcate the basic features of basic Abstract algebra. (2) To teach various examples of groups. (3) To acquainted with Permutation group and group of symmetry. (4) To introduce finite group theory and applications. (5) To teach ring theory.

Unit – I: (18 hours) Definition and examples of groups, examples of abelian and non-abelian groups, the group Zn of integers under

addition modulo n and the group U(n) of units under multiplication modulo n.Cyclic groups from number

systems, complex roots of unity.

Unit – II: (18 hours) Circle group, the general linear group GLn (n,R), groups of symmetries of (i) an isosceles triangle, (ii) an

equilateral triangle, (iii) a rectangle, and (iv) a square, the permutation group Sym (n), Group of quaternions.

Unit – III: (18 hours) Subgroups, cyclic subgroups, the concept of a subgroup generated by a subset and the commutator subgroup

of group, examples of subgroups including the center of a group.

Unit – IV: (18 hours) Cosets, Index of subgroup, Lagrange’s theorem, order of an element, Normal subgroups: their definition,

examples, and characterizations, Quotient groups.

Unit – V: (18 hours) Definition and examples of rings, examples of commutative and non-commutative rings: rings from number systems, Zn the ring of integers modulo n, ring of real quaternions, rings of matrices, polynomial rings, and rings of continuous functions, Subrings and ideals, Integral domains and fields, examples of fields: Zp, Q, R,

and C. Field of rational functions.

Learning Outcomes: After completion of this course the students will be able to understand the basic features of basic Abstract algebra, various examples of groups, Permutation group and group of symmetry etc. Further, they will learn finite group theory and its applications to number theory and a basic ideas of ring theory.

Essential Readings: 1. Joseph A Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa, 1999.

2. George E Andrews, Number Theory, Hindustan Publishing Corporation, 1984.

Suggested Readings and Link: 1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002. 2. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011.

E Books Link : National Digital Library




B.Sc.\ B.A.–IV Semester

MTS SE 411 Vector Calculus L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To understand the concept of differentiation of vector function.

(2) To learn about partial differentiation of vector function.

(3) To learn about derivative of sum and dot product.

(4) To find out the derivative of cross product of two vectors.

(5) To discuss about gradient, divergence and curl.


Differentiation of a vector function.


Partial differentiation of a vector function.


Derivative of sum and dot product of two vectors and their properties.


Derivative of cross product of two vectors and their properties.


Gradient, divergence and curl.

Learning Outcomes: Upon completion of this course, students should be able to manipulate

vectors to perform geometrical calculations in three dimensions. They will also able to

calculate and interpret derivatives in up to three dimensions.


1. P.C. Matthew’s, Vector Calculus, Springer Verlag London Limited, 1998.

Suggested Readings :


2. H. Anton, I. Bivens and S. Davis, Calculus, John Wiley and Sons (Asia) P. Ltd. 2002.






MTS SE 412 Theory of Equations L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To understand the concept of polynomials.

(2) To learn about equations and their solutions.

(3) To learn about relationship between roots and coefficient of equation.

(4) To find out the solution of reciprocal and binomial equations.

(5) To discuss about solutions of cubic and biquadratic equations.


General properties of polynomials, Graphical representation of a polynomials, maximum and

minimum values of a polynomials.


General properties of equations, Descarte’s rule of signs positive and negative rule.


Relation between the roots and the coefficients of equations. Symmetric functions, Applications of

symmetric function of the roots.


Transformation of equations. Solutions of reciprocal and binomial equations.


Algebraic solutions of the cubic and biquadratic. Properties of the derived functions.

Learning Outcomes: After completion of this course the students will be able to find the solutions

of higher degree equations. They will also learn different techniques to solve various types of

equations.


1. C. C. MacDuffee, Theory of Equations, John Wiley & Sons Inc., 1954.

Suggested Readings :

1. W.S. Burnside and A.W. Panton, The Theory of Equations, Dublin University Press, 1954.






MTS SE 413 Number Theory L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To understand the concept of Lame’s and fundamental theorem of arithmetic.

(2) To learn some theorems on prime number and prime counting function.

(3) To learn about concept of residues and divisors.

(4) To understand the Dirichlet product.

(5) To discuss about Euler’s Phi function and greatest integer function.


Division algorithm, Lame’s theorem, linear Diophantine equation, fundamental theorem of

Arithmetic.


Prime counting function, statement of prime number theorem, Goldbach conjecture, binary and

decimal representation of integers, linear congruence.


Complete set of residues. Number theoretic functions, sum and number of divisors.


Totally multiplicative functions. Definition and properties of the Dirichlet product.


The Möbius inversion formula, the greatest integer function, Euler’s phi-function.

Learning Outcomes: After completion of this course the students will be able to solve the problem

related to prime numbers and prime functions. They can also find the number of divisors.

Essential Readings:

1. Richard E. Klima, Neil Sigmon, Ernest Stitzinger, Applications of Abstract Algebra with

Maple, CRC Press, Boca Raton, 2000.

2. Neville Robinns, Beginning Number Theory, 2nd Ed., Narosa Publishing House Pvt. Limited,

Delhi, 2007.

Suggested Readings:

1. David M. Burton, Elementary Number Theory 6th Ed., Tata McGraw-Hill Edition, Indian

reprint, 2007.





B.Sc.\ B.A.–V Semester

MTS SE 511 Probability and Statistics L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To understand the basics of Probability theory. (2) To be able to comprehend MGF and CF of a distribution. (3) To learn some basic discrete and continuous distributions. (4) To study marginal and conditional distributions. (5) To learn about conditional expectations.

Unit–I: (06 hours)

Sample space, probability axioms, real random variables (discrete and continuous), cumulative

distribution function, probability mass/density functions.


Mathematical expectation, moments, moment generating function, characteristic function.

Unit–III: (06 hours)

Discrete distributions: uniform, binomial, Poisson, continuous distributions: uniform, normal,

exponential.

Unit–IV: (06 hours)

Joint cumulative distribution function and its properties, joint probability density functions,

marginal and conditional distributions.

Unit–V (06 hours) Expectation of function of two random variables, conditional expectations, independent random

variables.

Learning Outcomes:

After completion of this course the students will be able to understand the basics of probability

theory which is necessary to study any higher course in Statistical analysis.

Essential Readings: 1. Irwin Miller and Marylees Miller, John E. Freund: Mathematical Statistics with Application,

7th Ed., Pearson Education, Asia, 2006.

2. Sheldon Ross: Introduction to Probability Model, 9th Ed., Academic Press, Indian Reprint,

2007.

Suggested Readings and Link :

1. Robert V. Hogg, Joseph W. McKean and Allen T. Craig: Introduction to Mathematical

Statistics, Pearson Education, Asia, 2007.






MTS SE 512 Mathematical Finance L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60 Objectives: (1) To study interest rate & time value of money. (2) To explain the NPV and IRR in contaxt of portfolio optimization. (3) To understand portfolio returns with the concept of immunization. (4) To study mean variance analysis. (5) To explore Markowitz model.


Basic principles: Comparison, arbitrage and risk aversion, Interest (simple and compound, discrete

and continuous), time value of money.


Inflation, net present value, internal rate of return (calculation by bisection and Newton-Raphson

methods), Comparison of NPV and IRR. Bonds, bond prices and yields.


Floating-rate bonds, immunization, Asset return, short selling, portfolio return.


Brief introduction to expectation, variance, covariance and correlation, Random returns, portfolio

mean return and variance.

Unit–V: (06 hours)

Diversification, portfolio diagram, feasible set, Markowitz model (review of Lagrange multipliers for

1 and 2 constraints).

Learning Outcomes: After studying the syllabus the student will be able to understand basics of

portfolio management as an investment science.

Essential Readings: 1. John C. Hull, Options, Futures and Other Derivatives, 6th Ed., Prentice-Hall India, Indian

reprint, 2006.

2. Sheldon Ross, An Elementary Introduction to Mathematical Finance, 2nd Ed., Cambridge

University Press, USA, 2003

Suggested Readings: 1. David G. Luenberger, Investment Science, Oxford University Press, Delhi, 1998.






MTS SE 513 Mathematical Modeling L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60 Objectives: (1) To study vibrating strings. (2) To understand resonance. (3) To study simultaneous differential equations. (4) To explain traffic flow & study vibrating membrane. (5) To study potential & conservation law.

Unit–I: (06 hours) Applications of differential equations: the vibrations of a mass on a spring, mixture problem.


Free damped motion, forced motion, resonance phenomena.


Electric circuit problem, mechanics of simultaneous differential equations.


Applications to Traffic Flow. Vibrating string, vibrating membrane.


Conduction of heat in solids, gravitational potential, conservation laws.

Learning Outcomes: After completion of this course the students will be able to use differential

equations and apply it in real life situation.

Essential Readings: 1. I. Sneddon: Elements of Partial Differential Equations, McGraw-Hill, International Edition,

1967.

Suggested Readings: 1. Shepley L. Ross: Differential Equations, 3rd Ed., John Wiley and Sons, 1984.






MTS EC 511 Matrices L T P C

5 1 0 6

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To learn basics of the R1, R2 , R3 vector space over R. (2) To study matrices and its properties. (3) To understand and calculate rank of the matrix.

(4) To reduce a Matrix into diagonal form.

(5) To apply matrix theory in various disciplines.


R1, R2 , R3 as vector spaces over R. Standard basis for each of them. Concept of Linear Independence

and examples of different bases. Subspaces of R2, R3. Translation, Dilation, Rotation, Reflection in a

point, line and plane.


Matrix form of basic geometric transformations. Interpretation of eigen values and eigen vectors for

such transformations and eigen spaces as invariant subspaces. Types of matrices.


Rank of a matrix. Invariance of rank under elementary transformations. Reduction to normal form,

Solutions of linear homogeneous and non-homogeneous equations with number of equations and

unknowns upto four.


Matrices in diagonal form. Reduction to diagonal form upto matrices of order 3.


Computation of matrix inverses using elementary row operations. Solutions of a system of linear

equations using matrices. Illustrative examples of above concepts from Geometry, Physics,

Chemistry, Combinatorics and Statistics.

Learning Outcomes: After studying the syllabus the student will be able to understand basics of

vector space and properties of matrix and application of these in other fields of study.

Essential Readings: 1. S. H. Friedberg, A. L. Insel and L. E. Spence: Linear Algebra, Prentice Hall of India Pvt. Ltd.,

New Delhi, 2004.

Suggested Readings: 1. A.I. Kostrikin: Introduction to Algebra, Springer Verlag, 1984.

2. Richard Bronson: Theory and Problems of Matrix Operations, Tata McGraw Hill, 198






MTS EC 512 Mechanics L T P C

5 1 0 6

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To learn coplanar forces acting on a rigid Body.

(2) To study equilibrium under forces. (3) To explain rank of the matrix.

(4) To understand Velocity and acceleration.

(5) To study various motions.

Unit–I (18 hours)

Conditions of equilibrium of a particle and of coplanar forces acting on a rigid Body, Laws of

Friction.

Unit–II (18 hours)

Problems of equilibrium under forces including friction, Centre of gravity.

Unit–III (18 hours)

Work and potential energy. Velocity and acceleration of a particle along a curve: radial and

transverse components (plane curve).

Unit–IV (18 hours)

Velocity and acceleration of a particle along a curve: tangential and normal components (space

curve), Newton’s Laws of motion.


Simple harmonic motion, Simple Pendulum, Projectile Motion.

Learning Outcomes: After studying the syllabus the student will be able to understand laws of

motion and various forces action in physical sciences.

Essential Readings: 1. A.P. Roberts: Statics and Dynamics with Background in Mathematics, Cambridge University

Press, 2003.

Suggested Readings:

1. A.S. Ramsay, Statics: CBS Publishers and Distributors (Indian Reprint), 1998.






MTS EC 513 Linear Algebra L T P C

5 1 0 6

Max. Marks : 100

Mid Sem-20


End Sem-60 Objectives: (1) To study some spaces. (2) To understand dimension and basis of a subspace. (3) To explain matrix representation of a linear transformation.

(4) To understand dual space & characteristics polynomial.

(5) To study isomorphism & change of coordinate matrix.

Unit–I (18 hours)

Vector spaces, subspaces, algebra of subspaces, quotient spaces.

Unit–II (18 hours) Linear combination of vectors, linear span, linear independence, basis and dimension, dimension of

subspaces.

Unit–III (18 hours) Linear transformations, null space, range, rank and nullity of a linear transformation, matrix

representation of a linear transformation, algebra of linear transformations.


Dual Space, Dual basis, Double Dual, Eigen values and Eigen vectors, Characteristic Polynomial.

Unit–V ( 18 hours) Isomorphisms, Isomorphism theorems, invertibility and Isomorphisms, change of coordinate matrix.

Learning Outcomes: After studying the syllabus the student will be able to understand vector

space and Dual space properties with Isomorphisms.

Essential Readings: 1. S. Lang: Introduction to Linear Algebra, 2nd Ed., Springer, 2005.

2. Gilbert Strang: Linear Algebra and its Applications, Thomson, 2007.

Suggested Readings: 1. Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence: Linear Algebra, 4th Ed., Prentice-

Hall of India Pvt. Ltd., New Delhi, 2004.

2. David C. Lay: Linear Algebra and its Applications, 3rd Ed., Pearson Education Asia, Indian

Reprint, 2007. E Books Link : National Digital Library




B.Sc.\ B.A.–VI Semester

MTS SE 611 Boolean Algebra L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60 Objectives: (1) To study basic properties of ordered sets & maximal and minimal elements. (2) To understand algebraic structures of lattices. (3) To learn properties of lattices.

(4) To understand Boolean algebras & Quinn-McCluskey method.

(5) To study applications of switching circuits.

Unit–I (06 hours)

Definition, examples and basic properties of ordered sets, maps between ordered sets, duality

principle, maximal and minimal elements.


Lattices as ordered sets, complete lattices, lattices as algebraic structures, sub lattices. Products and

homomorphisms.

Unit–III (06 hours)

Definition, examples and properties of modular and distributive lattices, Boolean algebras.


Boolean polynomials, minimal forms of Boolean polynomials, Quinn-McCluskey method.


Karnaugh diagrams, switching circuits and applications of switching circuits.

Learning Outcomes : After completion of this course the students will be able to understand the basics of Boolean algebras &

applications of it in lattices and switches.

Essential Readings: 1. Rudolf Lidl and Günter Pilz: Applied Abstract Algebra, 2nd Ed., Undergraduate Texts in

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

Suggested Readings: 1. B A. Davey and H. A. Priestley: Introduction to Lattices and Order, Cambridge University Press,

Cambridge, 1990.






MTS SE 612 Transportation and Game Theory L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To find initial basic feasible solution of transportation problems

(2) To find optimal solution of transportation problems

(3) To find solution of an assignment problems.

(4) To understand the concept of game theory.

(5) To find solution of game having no saddle point.

Unit – I: . (06 hours)

Transportation problem and its mathematical formulation, northwest-corner method, least cost

Method.


Vogel approximation method for determination of starting basic solution, Algorithm for solving

transportation problem.


Assignment problem and its mathematical formulation, Hungarian method for solving

assignment problem.


Game theory: formulation of two person zero sum games, solving two person zero sum games.


Games with mixed strategies, graphical solution procedure.

Learning Outcomes:

After completion the course, student may identify the application of operation research

technique in business as well as in real life. They may solve the problems based on transportation

problem assignment problem and game theory. Essential Readings:

1. Mokhtar S. Bazaraa, John J. Jarvis and Hanif D. Sherali, Linear Programming and Network

Flows, 2nd Ed., John Wiley and Sons, India, 2004.

2. F. S. Hillier and G. J. Lieberman, Introduction to Operations Research, 9th Ed., Tata McGraw

Hill, Singapore, 2009.

3.. Hamdy A. Taha, Operations Research, An Introduction, 8th Ed., Prentice‐Hall India, 2006.

Suggested Readings and links 1. Video lecture on Transportation Problems

Link. https://youtu.be/eC-H8_wQ6ig 2. Video lecture on Game theory #1||Pure & Mixed Strategy

Link. https://www.youtube.com/watch?v=fSuqTgnCVRg


https://youtu.be/eC-H8_wQ6ig

https://www.youtube.com/watch?v=fSuqTgnCVRg





MTS SE 613 Graph Theory L T P C

2 0 0 2

Max. Marks : 100

Mid Sem-20


End Sem-60 Objectives: (1) To study the various graphs.

(2) To study the Isomorphism of graphs, paths & circuits.

(3) To study adjacency matrix & weighted graph .

(4) To study Travelling salesman’s problem .

(5) To study Dijkstra’s & Warshall algorithm.


Definition, examples and basic properties of graphs, pseudo graphs, complete graphs.


Bipartite graphs, Isomorphism of graphs, paths and circuits.


Eulerian circuits, Hamiltonian cycles, the adjacency matrix, weighted graph.


Travelling salesman’s problem, shortest path.


Dijkstra’s algorithm, Floyd‐Warshall algorithm.

Learning Outcomes : After completion of this course the students will be able to understand

graphs, paths, circuits which is very useful in daily life. Also find out shortest path by Travelling

salesman’s problem & Dijkstra’s algorithm it is used in making travelling plan road maps, etc.

Essential Readings: 1. Rudolf Lidl and Günter Pilz, Applied Abstract Algebra, 2nd Ed., Undergraduate Texts in

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

Suggested Readings: 1. Edgar G. Goodaire and Michael M. Parmenter, Discrete Mathematics with Graph Theory

2nd Ed., Pearson Education (Singapore) P. Ltd., Indian Reprint, 2003.






MTS EC 611 Numerical Methods L T P C

5 1 0 6

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To learn False position and Fixed point iteration method. (2) To study some iteration methods.

(3) To study interpolation. (4) To study differences.

(5) To study Simpson’s rule, Euler’s method.

Unit – I: (18 hours) Algorithms, Convergence, Bisection method, False position method, Fixed point iteration

method.


Newton’s method, Secant method, LU decomposition, Gauss-Jacobi, Gauss-Siedel and SOR

iterative methods.


Lagrange and Newton interpolation: linear and higher order, finite difference operators.


Numerical differentiation: forward difference, backward difference and central difference.


Integration: trapezoidal rule, Simpson’s rule, Euler’s method

Learning Outcomes:

After completion of this course the students will be able to understand basics of numerical

techniques so that these can be applied.

Essential Readings: 1. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering

Computation, 5th Ed., New age International Publisher, India, 2007.

Suggested Readings: 1. B. Bradie, A Friendly Introduction to Numerical Analysis, Pearson Education, India, 2007.






MTS EC 612 Complex Analysis L T P C

5 1 0 6

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To learn limit continuity in complex plane. (2) To study analyticity and C-R Equations . (3) To learn complex integral.

(4) To learn fundamental theorem of algebra.

(5) To study power series and its convergence.

Unit – I: (18 hours) Limits, Limits involving the point at infinity, continuity. Properties of complex numbers, regions

in the complex plane, functions of complex variable, mappings. Derivatives, differentiation

formulas.


Cauchy-Riemann equations, sufficient conditions for differentiability. Analytic functions,

examples of analytic functions, exponential function, Logarithmic function, trigonometric

function, derivatives of functions, definite integrals of functions.


Contours, Contour integrals and its examples, upper bounds for moduli of contour integrals.

Cauchy-Goursat theorem.


Cauchy integral formula, Liouville’s theorem and the fundamental theorem of algebra.

Convergence of sequences and series.


Taylor series and its examples. Laurent series and its examples, absolute and uniform

convergence of power series.

Learning Outcomes: After completion of this course the students will be able to understand analytic functions and their integral in complex plane.

Essential Readings: 1. Joseph Bak and Donald J. Newman, Complex analysis, 2nd Ed., Undergraduate Texts in

Mathematics, Springer-Verlag New York, Inc., New York, 1997.

Suggested Readings: 1. James Ward Brown and Ruel V. Churchill, Complex Variables and Applications, 8th Ed.,

McGraw – Hill International Edition, 2009.






MTS EC 613 Linear Programming L T P C

5 1 0 6

Max. Marks : 100

Mid Sem-20


End Sem-60

Objectives: (1) To learn Linear Programming Problems.

(2) To study simplex method.

(3) To discuss two-phase method, Big-M method.

(4) To learn dual problem.

(5) To study sensitivity analysis.


Linear Programming Problems, Graphical Approach for Solving some Linear Programs. Convex

Sets, Supporting and Separating Hyper planes.


Theory of simplex method, optimality and unboundedness, the simplex algorithm, simplex

method in tableau format.


Introduction to artificial variables, two-phase method, Big-M method and their comparison.


Duality, formulation of the dual problem, primal- dual relationships.


Economic interpretation of the dual, sensitivity analysis.

Learning Outcomes:

After completion of this course the students will be able to understand LPP, Big-M method and

sensitivity analysis.

Essential Readings: 1. F.S. Hillier and G.J. Lieberman, Introduction to Operations Research, 8th Ed., Tata McGraw

Hill, Singapore, 2004.

2. Hamdy A. Taha, Operations Research, An Introduction, 8th Ed., Prentice-Hall India, 2006.

Suggested Readings: 1. Mokhtar S. Bazaraa, John J. Jarvis and Hanif D. Sherali, Linear programming and Network

Flows, 2nd Ed., John Wiley and Sons, India, 2004.


syllabi - dr. harisingh gour university, sagar university

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