OBJECTIVES:

• To make the student acquire sound knowledge of techniques in solving ordinary 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 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 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).

MA6251 Mathematics II Previous Year Question Paper f 