MA6251 MATHEMATICS II
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
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
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
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
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
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
OUTCOMES:
The subject helps the students to develop the fundamentals and basic concepts in vector calculus,
ODE, Laplace transform and complex functions. Students will be able to solve problems related to
engineering applications by using these techniques.
TEXT BOOKS:
1. Bali N. P and Manish Goyal, “A Text book of Engineering Mathematics”, Eighth Edition, Laxmi
Publications Pvt Ltd.,2011.
2. Grewal. B.S, “Higher Engineering Mathematics”, 41
st
Edition, Khanna Publications, Delhi,
2011.
REFERENCES:
1. Dass, H.K., and Er. Rajnish Verma,” Higher Engineering Mathematics”,
S. Chand Private Ltd., 2011.
2. Glyn James, “Advanced Modern Engineering Mathematics”, 3rd Edition, Pearson Education,
2012.
3. Peter V. O’Neil,” Advanced Engineering Mathematics”, 7th Edition, Cengage learning, (2012).
4. Ramana B.V, “Higher Engineering Mathematics”, Tata McGraw Hill Publishing Company, New
Delhi, 2008.
5. Sivarama Krishna Das P. and Rukmangadachari E., “Engineering Mathematics” Volume II,
Second Edition, PEARSON Publishing 2011.
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