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MA6251                                                 MATHEMATICS – II                                                   L  T  P  C
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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).


OUTCOMES:

TOTAL (L:45+T:15): 60 PERIODS

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.

st
  1. Grewal. B.S, “Higher Engineering Mathematics”, 41
2011.

REFERENCES:

Edition, Khanna Publications, Delhi,

  1. Dass,  H.K.,     and     Er.     Rajnish     Verma,”     Higher     Engineering     Mathematics”,
  2. S. Chand Private Ltd., 2011
  3. Glyn James, “Advanced Modern Engineering Mathematics”, 3rd Edition, Pearson Education,
2012.
  1. Peter V. O?Neil,” Advanced Engineering Mathematics”, 7th Edition, Cengage learning, 2012.
  2. Ramana B.V, “Higher Engineering Mathematics”, Tata McGraw Hill Publishing Company, New
Delhi, 2008.
  1. Sivarama Krishna Das P. and Rukmangadachari E., “Engineering Mathematics” Volume II,
Second Edition, PEARSON Publishing 2011.

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