MA101 : Mathematics - I (4-0-0)

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MA101 :
Mathematics
-

I

(4
-
0
-
0)


4 credits




UNITI


1
5

hrs


Differential

Calculus:

Determination of n
th

derivative of standard functions. Leibnitz’s theorem and
Problems.

Polar curves. Pedal equations of polar curves. Radius

of curvature


Cartesian, parametric, polar and
pedal forms.


Partial Derivatives, Euler’s Theorem. To
tal differentiation. Differentiation of Composite and implicit
functions. Jacobians and their properties.

Errors and approximations.




UNIT II



1
1

Hrs


Integral Calculus:

Reduction formulae for the integration of sin
n
x, cos
n
x, tan
n
x, cot
n
x, sec
n
x, cosec
n
x

a
nd sin
m
x cos
n
x. Substitutional problems.

Tracing of standard curves in Cartesian, Parametric and Polar forms.

Applicatio
ns to find area and length of given curves.




UNIT

III




10 hrs


Analytical Solid Geometry:

Direction cosines and direction ratio
s. Planes and Straight lines,

shortest distance between two skew lines. Right Circular Cone and Right Circular Cylinder.






UNIT

IV


0
8
hrs


Differential Equations of First order:

Solution for Homogeneous, exact and linear differential
equations.

Differential equations reducible to above types.

Orthogonal


traject
ories of Cartesian and Polar curves.




UNITV


08
hrs


Differential Equations of Second & Higher Order:

Linear differential eq
uations of second and higher
order with constant coefficients. Method of Variation of Parameters. Cauchy’s and Legendre’s linear
differential equations.







Text Book
:
“Higher Engineering Mathematics”, B. S. Grewal,
, Khanna Publications,
40
th

Edition,
2005


(Ch. No’s 2,3 & 4)

Reference Books
:

1. “
Advanced Engineering Mathematics”, K. A. Stroud, 5
th

Edition.


2. “
A
dvanced
Engineering Mathematics”,

Erwin Kreyszig
, 8
th

Edition







1



MA151

:
Mathematics
-

II

(4
-
0
-
0)



4 credits






UNIT I


1
3

hrs




Differential Calculus
:

Rolle’s theorem

Geomertical Interpretat
ion
.


Lagrange’s and Cauchy’s mean value theorems. Taylor’s


Theorem for a function of a single variable
and Maclaurin’s series expansions.

Indeterminate Forms
-

L’Hospital’s rule
.

Taylor’s theorem for functions of two variables,


Maxima and Minima for f
unction
s

of two variables.

Lagrange’s method of undetermined multipliers

for extreme values (with one subsidiary condition).



UNIT II

1
3

hrs


Laplace Transforms

:

Definition
-

Transforms of elementary functions. Periodic function. Inverse
Laplace Transforms. Convolution theorem. Laplace transforms of derivatives and integrals of function
s
.
Solutions of linear differential equations. Unit

step function and unit impulse function.

Simple examples
on LRC circuits.




UNIT III

08

hrs


Integral Calculus

:
Multiple Inte
grals
-

Evaluation by change of order of integration


and change of
variables.Applications
t
o area and volume.





UNIT IV


06 hr
s













Special functions
:
Beta and gamma
f
unctions.

Differentiation under

the integral sign





UNIT V

1
2
hrs


Ve
ctor Calculus
:
Introduction to vector calculus. Directional derivatives. Gradient of a scalar field,
Divergence and Curl of a vector field.

Integral Theorems: Green’s, Stoke’s

and Ga
uss’s theorems
with
examples
.





Text Book :


“Higher Engineering Mathe
matics” B. S. Grewal,
Khanna Publication
s,

40
th

Edition,

2005


(Ch. No’s 2, 5 & 7)

Reference Books

:

1. “Advanced Engineering Mathematics” K. A. Stroud, , 5
th

Edition.


2. “A
dvanced
Engineering M
athematics


Erwin Kreyszig
, 8
th

Edition.






















2

MA
201: LINEAR ALGEBRA









52 Hrs




Unit 1

M
atrices

and Gaussian Elimination









12

Hrs

Introduction

to 3
-
D geometr
y, geometry of system of linear equations
,
Gaussian Elimination
,

Matrix notation and multiplication, Triangular factorization, Inverses and transposes.




Unit

2

Vector spaces

& Linear Equations









14 Hrs

Vector spaces
, subspaces,
linear independe
nce, b
asis and dimension,

fundamental

subspaces, linear transformations.



Unit

3

Orthogonality







12

Hrs

Orthogonal subspaces
,

inner products, projections and l
east squares approximations.

Gram
-

Schmidt Orthogonalization.



Unit

4

Eigen Values and Eigen Vectors


06

Hrs

Eigen values and eigen

vectors, Cayley
-
Hamilton theorem, Power method of finding the

dominant eigenvalue and the corresponding eigen vector, Diagonalization.




Unit 5

Fourier Series





08 Hrs

Periodi
c and orthogonal functions. General orthonormal expansions. Even and odd functions.

Dirichlet’s conditions. Fourier Series expansions, harmonics. Half
-
range Series.







Text Book

:

G. Strang, Linear Algebra and its applications,
4
th


edition,
Thomson

Brooks/Cole
,



2
nd

Indian reprint
200
7.(
ch.nos.1,2,3,5,6)

Ref
erence Book
:


B S Grewal, Higher

Engineering Mathematics
39
th

Edition , Khanna Publishers,2005 .


(ch. nos. 2, 10)









MA251:
METHODS OF APPLIED MATHEMATICS



52 Hrs



Unit 1

Complex Variables
:










11 Hrs

Limits, Continuity and differentiability of functions of complex variables. Analytic functions.

Cauchy

Riemann equations in Cartesia
n and Polar forms
-

consequences..

Conformal transformations: z
2

,
e
Z
, trigonometric functions, z + a
2

/ z (z ≠ 0) and

Bilinear transformations.


Unit

2

Complex Integration










09 Hrs



Line integral, Cauchy’s theorem
and corollaries, Cauchy’s integral formula.

Taylor’s and Laurent’s series expansions of functions.

Isolated singularities and Cauchy’s Residue theorem
.


Unit

3


Numerical Methods







17 Hrs

Roots of transcendental equations using Bisection, Regula
-

Falsi and Newton
-
Raphson Methods.

Finite Differences, Newton


Gregory forward and backward interpolation formulae.

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

and Newton’s divided difference interpolation formula.

Numerical Integration using Trapezoidal Rule, Simpson’s one third rule, Simpson’s three

eighth rule and Weddle’s rule.

Numerical solutions of first order Ordinary Different
ial Equations
-

Taylor’s series method,

Modified Euler’s , Runge
-

Kutta 4
th

order methods.


Unit 4


Partial Differential Equations (P.D.E) Part 1:






08 Hrs

Formation of Partial differential equations by elimination of arbitrary constan
ts

and arbitrary functions. Solution by direct integration.

Solutions of equations of the type: Pp + Qq = R and Charpit’s method.
s


Unit

5

Partial Differential Equations(P.D.E)Part :2

07 Hrs

Solution of PDE’s by the method of separation of variables, one dimensional Heat and Wave equations.
Solution of two dimensional and three dimensional Laplace’s equations in Cartesian and Polar systems.

Solution of Homogeneous and Non
-
Homogeneo
us Linear PDEs.




Text Book :

B S Grewal, Higher

Engineering Mathematics 39
th

Edition, Khanna Publishers.


(ch. nos. 17, 18, 19, 20,27,28,30)

Reference Book
s
:

1
. KA Stroud, Advanced Engineering Mathematics 4
th

Edition, MacMillan Publi
cations,2003.

2
. Murray R Spiegel, Complex Variables, Schaum’s Outline Series, Tata McGraw


Hill Publications, 1981