MA 101 Mathematics-I L T P C

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MA 101 Mathematics
-
I




L T P C


First Semester ( All Branch )


3 1 0 8



Differential Calculus :

Successive differenti
ation, Leibnitz’s theorem & its application.

Indeterminate f
orms,
L. Hospital’s Rule
s
.

Rolle’s Theorem, Lagrange’s Mean value theorem, Taylor’s &
Maclaurin’s theorems with Lagrange’s form of remainder for a function of one variable.
Curvature, radius

& centre of curvature for Cartesian and polar curves.. Partial
differentiation, change of variables, Euler’s theorem & Jacobian.


Integral Calculus
:

Reduction Formulae.
Asymptotes for Cartesian and polar curves.

Curve tracing. Area &
length of plane cur
ves. Volume & surface area of solids of revolution (for Cartesian and
polar curves).


Differential Equation

:

Solution of o
rdinary differential equations of first order & of first degree
:

Homogeneous
equation, Exact

differential equation,
Integrating fa
ctors, Leibnitz’s linear equation,
Bernoulli’s equation.

Differential equation of first order
but

of higher degree, Clairaut’s equation. Differential
equations of second & higher order with constant coefficients.

Homogeneous Linear
equation
s
.


Reference B
ooks:


1. Differential Calculus

Das & Mukherjee U.N. Dhur & Sons Pvt. Ltd.

2. Integral Calculus

Das & Mukherjee U.N. Dhur & Sons Pvt. Ltd.

3. Elementary Engineering Mathematics

B.S. Grewal Khanna Publisher

4. Engineering Mathematics
-
II


Santi Narayan S. Chand & Co.



MA 102 Mathematics
-
II




L T P C



Second Semester ( All Branch )


3 1 0 8




Matrices :



R
ank of a matrix , Elementary transformations,

Consistency and

solutions of a system of
li
near equations by matrix methods .Eigen values & eigen vectors
.

Caley
-
Hamilton’s
theorem & its applications.


S
olid geometry :


S
traight lines
.
Shortest distance betwee
n skew lines . Sphere , cone,

cylinder

and
conicoid

.

Infinite & Fourier Series:

Con
vergence of infinite series & simple tests of convergence . Fourier series in any
interval .Half range sine & cosine series .

Complex Analysis
:


Function of a complex variable, Analytic functions , Cauchy
-
Reimann equations, Complex
line integral , Cauchy’
s theorem , Cauchy’s Integral formula. Singularities and residues,
Cauchy’s Residue theorem and its application to evaluate real integrals.

Differential calculus :


Taylor’s & Maclaurin’s theorems with Lagrange’s form of remainder for a function of two
variables, Expansions of functions of two variables . Errors & approximations. Extreme
values of functions of two & more variables

.

.


Reference Books:

1. Matrices Frank Ayres Mc Graw Hill
s

2. Solid Geometry


Santi Narayan S. Chand & Co.

3. Laplace Transforms M.R.Spiegel Mc Graw Hills

4
. Higher Engineering Mathematics B.S. Grewal Khanna Publisher

5
. Eng
ineering Mathematics


Bali & Iyengar Laxmi Publications Ltd.

6..Advanced Mathematical Analysis


Malik & Arrora S. Chand & Co.



MA 201 Mathematics
-
III




L T

P C


Third Semester ( All Branch )


3 1 0 8



Integral & Vector Calculus :

Double & triple integrals, Beta & Gamma function
s . Differentiation of vector functions of
scalar variables. Gradient of a scalar field , Divergence & Curl of a vector field and their
properties
, directional derivatives
. Line & surface integrals. Green’s theorem , Stokes’
theorem & Gauss’ theorem b
oth in vector & Cartesian forms ( statement only) with
simple applications.


Integral transforms :


Laplace
transform

:
Transform of elementary functions ,
I
nverse Laplace transforms
.
Solution of ordinary differential equation using Laplace transform.

Fourier transforms
: Definition, Fourier sine and cosine transforms, properties, relation
between Fourier and Laplace transforms.

Z
-
transform:
Definition, standard z
-
transforms, properties, initial and final value
theorems, convolution theorem. Inverse z
-
tr
ansform, application to difference equation.


Partial Differential Equation:

Formation of partial differential equations (PDE), Solution of PDE by direct integration.
Lagrange’s linear equation . Non
-
linear PDE of first order. Method of separation of
varia
bles. Heat, Wave & Laplace’s equations (Two dimensional Polar & Cartesian Co
-
ordinates).


Reference Books:


1.Vector Analysis Frank Ayres Mc Graw Hills

2.Advanced Mathematical Analysis



Malik & Arrora S. Chand & Co.

3. Advanced Differential Equations M.D.Rai Singhaniya S. Chand & Co.

4. Complex Analysis.

M.R.Spiegel. Schuam’s out line Series

5. Hi
gher Engineering Mathematics B.S. Grewal Khanna Publisher

6. Engineering Mathematics




Bali & Iyengar

Laxmi Publications

Ltd.



MA 202 Probability Theory & Stochastic Processes

L T
P C


Third Semester ( ET & CS Branch )


3 1
0 8



Probability :

Introduction, joint probability, conditional probability

, total probability, Baye’s theorem ,
multiple events, independent events.


Random Variable:

Introduction, discrete and continuous random variables, distribution function , mass / density
function , Binomial, Poisson , Uniform, Exponential, Gaussian an
d Gamma random variables,
conditional distribution and density function, function of a random variable .

Bivariate distributions, joint distribution and density, marginal distribution and density
functions, conditional distribution and density, statistical

independence, distribution and
density of a sum of random variables.

Operation on one Random variable:

Expected value of a random variable , conditional expected value , moments about the origin
, central moments , moment generating function, variance ,

skewness and
Kurtosis,covariance, correlation and regression, monotonic and non
-
monotonic
transformation of a random variable (both discrete and continuous).

Stochastic

Processes:


Definition of a stochastic process , classification of states, Random w
alk, Markov chains,
poisson process
, Wiener
process

,stationary and independence, distribution and density
functions, statistical independence,

Kolmogorov equations,

first order stationary processes,
second order and wide sense stationary, time average
s and ergodicity , correlation functions,
covariance function.

Spectral characteristics of random processes:

Power density spectrum and its properties, bandwidth of the power density spectrum,
relationship between power spectrum and autocorrelation functio
n, cross power spectral
density and its properties.

Noise:

White Noise , shot noise, thermal noise, noise equivalent bandwidth.


Reference Books:

1. An Introduction to Probability Theory and its Applications (Vol. 1 & II)
-
W Feller (John
Wiley & Sons).



2. Probability , Random


Variables & Stochastic processes Papoulis McGraw Hill

3. Probability & Stochastic processes C.W.Helstrom McMillan, New York


for en
gineers

4. Probability & Random processes A.Leon
-
Garcia Addison Wisley


for electrical engineers

5. The Theory of Stochastic Processes


D R Cox and H D Miller ( Chapman & Hall
Ltd.)

6. An Introduction to Probability Theory and its Applications (Vol. 1 & II)
-
W Feller (John
Wiley & Sons).







MA 203 Discrete Mathematics L T P
C



Fourth Semester ( CS Branch )


3 1
0 8




Boolean algebra
:
-

Binary relation , equivalence relation

, Partial order
relations, PO
-
set, Totally orde
red set, Maximal and Minimal elements. Well
ordered set. Lattice, bounded lattices, sublattice, distributive lattice, modular
lattice, irreducible elements, complemented lattice.

Boolean Algebra, Boolean functions & expression , minimization of Boolean
f
unctions & expressions.(Algebraic method and Karnaugh map method)

Logic gates :
-

Introduction, Design of digital circuits and application of
Boolean algebra in switching circuits.


Graph theory:
-
.
Introduction,
Basic definition, incidence and degree, ad
jacency,
paths and cycles, matrix representation of graphs( directed and non
-
directed).
Digraphs. Trees.

Mathematical Logic:

Statement Calculus
-

sentential connectives, Truth tables, Logical
equivalence, Deduction theorem.

Predicate Calculus
-

Symbolizing

everyday language., validity and
consequence.

Modern Algebra:

Algebraic structures, Semi group, Monoid, Group, Cyclic group, Subgroup,
Normal subgroup, Quotient group, Homomorphism of groups.

Ring, Integral domain, Field. Vector space , Linear dependence

&
independence . Basis & Dimension.

Recurrence relations & Generating functions.


Reference Books:

1. Set Theory and Logic R.R Stoll. S. Chand.&
Co.

2. Discrete Mathematical Structures G. S. Rao

New age
International

3.Discrete Mathematics and Structures S. Balgupta


Laxmi
Publications

4. Modern Algebra


Herstein New age
International

5. Graph theory

Harary Narosa
Publishing House








































MA 204 Mathematics
-
IV






L T
P C



Fourth Semester ( CE & ME Branch )


2 1
0 6



Statistics :

Measures of central tendency, dispersion, moments, skewness & kurtosis.


Probability density function, distribution f
unction, Binomial, Poisson & Normal
distributions.

Curve fitting
-

Method of Least squares, fitting of straight line & parabola.

Correlation & Regression
-

determination of correlation & regression
coefficients & determination of lines of regression.


Nume
rical Analysis:

Finite differences, Interpolation & extrapolation. Newton’s forward & backward
formulae, Lagrange’s formula & Newton’s divided difference formula for
unequal intervals.(statements & applications of the formulae only)

Numerical differentiat
ion & integration, Trapezoidal rule, Simpson’s 1/3
rd

&
3/8
th

rules.

Numerical solution of transcendental & algebraic equations
-

Method of
Iteration & Newton
-
Raphson method.


Solution of system of linear equations :


Gaussian elimination method, Gauss Seid
al method, LU decomposition &
Cholesky decomposition.& their application in solving system of linear
equations, matrix inversion by Gauss
-
Jordan method .



Reference Books :

1.
Numerical Mathematical Analysis James B Scarborough Oxford &
IBH P
ublishing

2. Numerical Analysis B.S. Grewal Khanna
Publishers

3. Finite Differences H.C. Sexena S. Chand
& Co.


& Numerical Analysis

4. Probability
& Statistics M.R. Spiegel Mc Graw
Hill

5. Engineering Mathematics


Bali & Iyengar Laxmi
Publications Ltd.


MA 205 Probability Theory & Statistical Methods L

T P C


Fourth Semester ( EE Branch )


3 1 0 8



Various Statistical Measures
:

Measures of central tendency & dispersi
on ,Co
-
efficient of variation, moments,
skewness & kurtosis.

Probability Theory:

Introduction to probability. Additive & multiplicative Laws of probability, conditional
probability, independent events. Baye’s theorem.

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u瑩tn func瑩tnI B楮om楡氬 mo楳獯n C 乯牭a氠
d楳瑲ibu瑩tns⸠

Curve fitting:

Method of Least squares , fitting of straight line & parabola.

Correlation & Regression
:


Determination of correlation & regression coefficients & determination of lines of
regressi
on. Multiple correlation.

Mathematical Expectation:

Discrete Random variables. Function of random variables. Expectation of random
variables. Jointly distributed random variables.

Descriptive & inferential statistics
:

Estimation

-

concept of sampling,
sampling distributions & standard errors, confidence
interval, estimation of mean & variance, maximum likely
-
hood method of estimation for
parameters of distribution,


Testing of hypothesis
:


Z
-
test , t
-
test , F
-
test &

2

-

test .

Reference Books:

1.

Probability & Statistics M.R. Spiegel Mc Graw Hill

2.
Mathematical Statistics


Kapoor & Sexena S.Chand.



MA 301 Numerical Methods &

L T
P C


Computations


Fifth Semester ( ET & CS Branch )


2 1
0 6






Numerical Analysis:


Finite differences, Interpolation & extrapolation. Newton’s forward & backward
formulae, Lagrange’s formula & Newton’s divided difference formula for
unequal intervals.(statements & applications of the formulae only), evaluation
of functions , minimization & maximization of functions .


Numerical differentiation & integration, Newton’s general quadrature formula,
Trapezoidal rule, Simpson’s 1/3
rd

& 3/8
th

rules.


Numerical solution of transcendental & algebraic equations
:
-

Method

of
Iteration & Newton
-
Raphson method.


Numerical Solution of a system of linear equations

:


Gaussian elimination method with pivoting strategies , Gauss
-
Jordan method
& Gauss
-
Seidel method. LU decomposition & Cholesky decomposition.&
their applicatio
n in solving system of linear equations. Matrix inversion by
Gauss
-
Jordan method .


Numerical solution of ordinary differential equations with initial value:



Taylor’s series method , Eulers & modified Eulers method , Runge
-
Kutta
method of 4
th

order.



Re
ference Books :


1. Numerical Mathematical Analysis

James B Scarborough Oxford &
IBH Publishing


2. Numerical Analysis


B.S. Grewal Khanna
Publishers


3. Finite Differences H.C. Sexena

S.
Chand & Co.


& Numerical Analysis.


4. Engineering Mathematics


Bali & Iyengar Laxmi
Publications Ltd.



MA 441 Modern Algebra L T P
C


Eighth Semester (Elective

III, Open ) 3 0 0
6



Posets & Lattices :

Partial order relations, Po
-
set, Lattices & Boolean algebra.

Groups :

Groups ,
Subgroups , Normal subgroups , Permutation group . Lagrange’s Theorem .
Cyclic groups Quotient group , Homomorphism of groups , First three isomorphism
theorems . Inner Automorphism . Normalizer / Centralizer of an element , Centre of a
group . Conjugacy
relation , Class equation , Sylow’s


Theorems. Subnormal &
Normal series , Solvable group , Commutators . Nilpotent groups .Free groups
.



Rings :

Ring , Integral domain , Field . Ideals & Quotient rings , Homomorphism of Rings ,
Maximal Ideal , Minimal
Ideal , Prime Ideal , Principal Ideal , Principal Ideal Ring /
Domain (PIR / PID) , Euclidean Domain , Polynomial Rings. Field of quotient of an
integral domain. Field extensions.


1.
Modern Algebra Surjit singh &

Zameeruddin Vikas Pub
lishing
House


2. Modern Algebra I.N. Herstein


New age
International


3. Modern Algebra Khanna & Bhamri

Vikas Publishing
House





********************
********************







MA 442


F
unctional Analysis L T P
C



Eighth Semester (Elective

III, Open) 3 0 0
6







Matric Space :

Definition and
Examples of metric space . Open Sphere, Open Set & Closed
Set. Convergence of sequences, Cauchy sequence, Complete Metric Spaces,
Sequentially Compact Metric Space, Continuous mappings.


Topological Space

:

Definition and examples, Trivial and non
-
trivial

topology, Cofinite topology,
Usual Topology with special reference to R. Continuity and homeomorphism.


Functional Analysis

:


Linear space, subspace, basis, dimension, normed linear space, Banach space,
continuous linear transformation, Conjugate space,
Inner product spaces, Hillbest
space, Orthogenality, orthonormal sets, Cauchy’s Schwartz’s inequality, Bessel’s in
equality.


Linear operators, Self adjoint operator, normal and unitary operators, Projections,
Spectrum of an operator. The spectral theorem.


******************************

Reference Books:


1. Introduction to Topology and


Modern Analysis

Simmon G.F.

Tata McGraw Hill

2. Functional Analysis

B.K. Lahiri

World Press Pvt. Ltd.

3. Gene
ral Topology

Lipschutz

Schaum Outline Series, McGraw
Hill Book Company.





MA 443 Mathematical Modeling L T P C


Eighth Semester (Elective

III, Open )

3 0
0 6



Mathematical modelling techniques, classification with simple illustration.

Mathematical modelling through ordinary differential equations.

Modellin
g through difference equations.

Modelling through partial differential equations.

Modelling through integral and differential
-

difference equations.

Modelling through calculus of variations and dynamic programming.

Modelling through mathematical programm
ing, maximum principle and
maximum entropy principle.


***********************************************

Reference Books:


1. Mathematical Modelling

J.N. Kapur

New age International

2. Advanced Engineering


Mathematics

E. Kreyszig

New age International

3. Higher Engineering Mathematics

B.S. Grewal

Khanna Publishers


4. Operations Research, Methods


and Practice

5. Numerical Methods for


Engineering Problems


C.K. Mustafi


N.K. Raju & K.U.
M
uthu


Wiley Eastern


Macmillan India
Limited




MA 444 Operation Research L T P
C


Eighth Semester (Elective

III, Open )

3 0 0
6




Introduction to Operation Research (O.R):

Meaning of O.R. Principles of Modelling. Features and Phases of O.R.


Linear Programming:

Introduction, Formulation of Linea
r Programming Problems (L.P.P), Graphical
solution procedure. Idea of Convex set & convex combination of two points,
Fundamental Theorem of L.P.P. (proof not required ). Solutions of L.P.P. Simplex
Method . Big
-
M methods.

Transportation Problems(T.P):

Int
roduction. Mathematical formulation. Definitions of Balanced, Unbalanced T. P.
Rules to find initial Basic feasible Solution (B.F.S) of a T.P.
-

North West Corner Rule,
Vogel’s approximation Method. Solution algorithm of T.P. Solution technique for
unbala
nced

T. P. Resolution of degeneracy. Examples.


Assignment Problems(A.P):

Introduction , Mathematical Formulation. Reduction theorem ( proof not required).
Definitions of Balanced and Unbalanced A.P. Hungarian Algorithm for solving A.P.
Solution technique

for unbalanced A.P. Examples.


Sequencing Problems:

Introduction. Definition. Solution of Sequencing problems. Processing n jobs through
2 machines, 2 jobs through m machines ( Graphical method), Processing n jobs
through m machines.

Integer Programmin
g Problems(I.P.P):

Introduction. Pure and mixed integer programming problems. Gomory’s Cutting
Plane technique for solving I.P.P. Examples.


Reference Book:


1.Operations Research


Kanti Swarup



Sultan Chand &
Sons

2.Operations Research


S.D. Sharma



Kh
anna
Publishers

3.Operations Research


J.K. Sharma



MacMillan India
Ltd

4.Operations Research


Hira and Gupta


Sultan Chand &
Sons

5.Operations Research


Hamady A Taha


Prentice Hall of
India

6. Linear Programming

P.M.Karak New Cent
ral Book
Agency