**MA6251 – ENGINEERING MATHEMATICS – II SYLLABUS (REGULATION 2013) ANNA UNIVERSITY **

**UNIT I: VECTOR CALCULUS **

(MA6251) 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).

### Course Curriculum

Unit 1 | |||

U1 Introduction | FREE | 00:15:00 | |

Gradient, Directional derivative, normal derivative,unit normal vector | FREE | 00:00:00 | |

Angle b/w the surface , scalar potential | FREE | 00:30:00 | |

Divergence, solenoidal, curl & irrotational vector | FREE | 00:30:00 | |

Laplace operator | 00:00:00 | ||

Vector integration, line integration | 00:00:00 | ||

surface integral | 00:00:00 | ||

Volume integral | 00:00:00 | ||

Gauss divergence | 00:00:00 | ||

Stokes theorem | 00:00:00 | ||

Green’s theorem | 00:00:00 | ||

Unit 2 | |||

U2- Introduction | 00:00:00 | ||

Higher order differential equation with constant co efficients | 00:00:00 | ||

Type 1 | 00:00:00 | ||

Type 2 | 00:00:00 | ||

Type 3 | 00:00:00 | ||

Type 4 | 00:00:00 | ||

Type 5 | 00:00:00 | ||

Type 6 | 00:00:00 | ||

Method of variation of parameter | 00:00:00 | ||

Homogeneous equation of Euler type Cauchy’s type | 00:00:00 | ||

Homogeneous equation of legendre’s type | 00:00:00 | ||

Simultaneous first order linear differential equation with constant co efficient | 00:00:00 | ||

Unit 2 Conclusion | 00:00:00 | ||

Unit 3 | |||

Laplace transforms introduction | 00:30:00 | ||

Laplace transformation basic problems | 00:00:00 | ||

First shifting theorem | 00:30:00 | ||

Transforms of derivatives & integrals of functions | 00:00:00 | ||

Integrals of transform | 00:00:00 | ||

Laplace transform of integrals | 00:00:00 | ||

Transform of periodic function | 00:00:00 | ||

Inverse laplace transform | 00:00:00 | ||

Inverse laplace transform of derivatives of F(s) | 00:00:00 | ||

Partial fraction method | 00:00:00 | ||

Convolution theorem | 00:30:00 | ||

solution of linear ODE of second order with constant co-efficient | 00:00:00 | ||

Initial value theorem, final value theorem | 00:00:00 | ||

Unit 4 | |||

Analytic function introduction | 00:30:00 | ||

Analytic function problems | 00:30:00 | ||

Harmonic conjugate | 00:00:00 | ||

Harmonic conjugate problems | 00:00:00 | ||

Construction of analytic function | 00:30:00 | ||

Conformal mapping | 00:30:00 | ||

Bilinear transformation | 00:30:00 | ||

Bilinear transformation problem | 00:30:00 | ||

unit 5 | |||

Complex integration introduction | 00:30:00 | ||

Complex integration theorem | 00:30:00 | ||

Cauchy’s integral formula | 00:30:00 | ||

Cauchys integral formula for derivatives | 00:30:00 | ||

Taylor and laurents series | 00:00:00 | ||

Singularities | 00:00:00 | ||

Residues | 00:00:00 | ||

Cauchys residues theorem | 00:30:00 | ||

Contour integration Type 1 | 00:30:00 | ||

Contour integration type 2 | 00:30:00 | ||

Contour integration type 3 | 00:30:00 |

### Course Reviews

**3495 STUDENTS ENROLLED**

## gud

5easy to learn

## Very usefull

5Thanks for creating this system.