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MA6251
MATHEMATICS-II
S.NO
CONTENTS
PAGE NO
UNIT-1 VECTOR CALCULUS
1.1
Gradient-Directional Derivative
1
Problems
1.2.
Divergence And Curl –Irrotational And Solenoidal Vector
Fields Divergence
4
Problems
1.3
Vector Integration
7
Problems
1.4
Green’s Theorem In A Plane;(Excluding proof)
9
Problems
1.5
Gauss Divergence Theorem:(Excluding proof)
10
Problems
1.6
Stoke’s Theorem(Excluding proof)
12
Problems
UNIT –II ORDINARY DIFFERENTIAL EQUATIONS
2.1
Higher Order Linear Differential Equations With
14
Constant Coefficients
Problems
2.2
Method of Variation of Parameters
22
Problems
2.3
Differential Equations for the Variable Coefficients
24
Problems
2.4
Simultaneous First Order Linear Equations with Constant Coefficients
Problems
26

MA6251
3.1
MATHEMATICS-II
UNIT –III LAPLACE TRANSFORMATION
Laplace transformation-Conditions and existence
28
Problems
3.2
Transforms of Elementary functions-Basic Properties
29
Problems
3.3
(a) Transforms of derivatives
31
(b)Derivatives and integrals of Transforms
(c)Integrals of Transforms
Problems
3.4
Transforms of the unit step functions and impulse function
34
Problems
3.5
Transforms of periodic functions
35
Problems
3.6
Inverse Laplace Transform
36
Problems
3.7
Convolution theorem
42
Problems
3.8
Initial and final value theorems
45
3.9
Solution of linear ODE of Second Order with constant coefficients
47
Problems
UNIT-IV
ANALYTIC FUNCTIONS
4.1
Introduction –Function of A Complex Variable
49
4.2
Analytic Functions(C-R Equations)
49
Problems
4.3
Harmonic and Orthogonal Properties Of Analytic Functions
Problems
51

MA6251
4.4
MATHEMATICS-II
Construction of Analytic Functions
56
Problems
4.5
Conformal Mapping
58
Problems
4.6
Bilinear Transformation
61
Problems
UNIT V- COMPLEX INTEGRATION
5.1
Prerequisite
62
5.2
Introduction
62
5.3
Cauchy’s Theorem
62
Problems
5.4
Taylor’s and Laurent’s Series Expansion.
64
Problems
5.5
Singularities
67
Problems
5.6
Residues
69
Problems
5.7
Evaluation of real definite Integrals as contour integrals
72
Problems
5.8
Applications
79
APPENDICES
A
Question Bank
B
University Questions

MA6251 MATHEMATICS – II REGULATION 2013 SYLLABUS
MA6251 MATHEMATICS – II L T P C
3104
OBJECTIVES:
• To make the student acquire sound knowledge of techniques in solving ordinary differential
equations that model engineering problems.
• To acquaint the student with the concepts of vector calculus, needed for problems in all
engineering disciplines.
• To develop an understanding of the standard techniques of complex variable theory so as to
enable the student to apply them with confidence, in application areas such as heat conduction,
elasticity, fluid dynamics and flow the of electric current.
• To make the student appreciate the purpose of using transforms to create a new domain in
which it is easier to handle the problem that is being investigated.
UNIT I VECTOR CALCULUS 9+3
Gradient, divergence and curl – Directional derivative – Irrotational and solenoidal vector fields
– Vector integration – Green’s theorem in a plane, Gauss divergence theorem and Stokes’
theorem (excluding proofs) – Simple applications involving cubes and rectangular
parallelopipeds.
UNIT II ORDINARY DIFFERENTIAL EQUATIONS 9+3
Higher order linear differential equations with constant coefficients – Method of variation of
parameters – Cauchy’s and Legendre’s linear equations – Simultaneous first order linear
equations with constant coefficients.
UNIT III LAPLACE TRANSFORM 9+3
Laplace transform – Sufficient condition for existence – Transform of elementary functions –
Basic properties – Transforms of derivatives and integrals of functions - Derivatives and
integrals of transforms - Transforms of unit step function and impulse functions – Transform of
periodic functions. Inverse Laplace transform -Statement of Convolution theorem – Initial and
final value theorems – Solution of linear ODE of second order with constant coefficients using
Laplace transformation techniques.
UNIT IV ANALYTIC FUNCTIONS 9+3
Functions of a complex variable – Analytic functions: Necessary conditions – Cauchy-Riemann
equations and sufficient conditions (excluding proofs) – Harmonic and orthogonal properties of
analytic function – Harmonic conjugate – Construction of analytic functions – Conformal
mapping: w = z+k, kz, 1/z, z2, ez and bilinear transformation.
UNIT V COMPLEX INTEGRATION 9+3
Complex integration – Statement and applications of Cauchy’s integral theorem and Cauchy’s
integral formula – Taylor’s and Laurent’s series expansions – Singular points – Residues –
Cauchy’s residue theorem – Evaluation of real definite integrals as contour integrals around unit
circle and semi-circle (excluding poles on the real axis).
TOTAL: 60 PERIODS

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