## MA2111 ENGINEERING MATHEMATICS

(Common to all branches of B.E. / B.Tech. Programmes)L:T:P:M=3:1:0:100

## Aim

The course is aimed at developing the basic Mathematical skills of Engineering students that are imperative for effective understanding of Engineering subjects. The topics introduced will serve as basic tools for specialized studies in many Engineering fields, significantly in fluid mechanics, field theory and Communication Engineering.## Objectives

On completion of the course the students are expected- to identify algebraic eigenvalue problems from practical areas and obtain the eigensolutions in certain cases.
- To understand solid geometry concepts
- to understand maxima and minima concept.
- to solve differential equations of certain types, including systems of differential equations that they might encounter in the same or higher semesters.
- to understand double and triple integration and enable them to handle integrals of higher orders.
- to know the basics of vector calculus comprising of gradient, divergence & curl and line, surface & volume integrals along with the classical theorems involving them.
- to understand analytic functions and their interesting properties.
- to know conformal mappings with a few standard examples that have direct application.
- to grasp the basics of complex integration and the concept of contour integration which is important for evaluation of certain integrals encountered in practice.
- to have a sound knowledge of Laplace transform and its properties.
- to solve certain linear differential equations using the Laplace transform technique which have applications in other subjects of the current and higher semesters.

## Unit - I Matrices, Solid Geometry and Differential Calculus (18+6)

Eigenvalue problem – Eigenvalues and Eigenvectors of a real matrix – Characteristic equation – Properties of Eigenvalues and Eigenvectors – Cayley-Hamilton theorem (excluding proof) - Similarity transformation (Concept only) – Orthogonal transformation of a symmetric matrix to diagonal form – Quadratic form – Orthogonal reduction to its canonical form.Sphere, Right circular cylinder and right circular cone.Maxima / Minima for functions of two variables – Method of Lagrangian multiplier – Jacobians.

## Unit – II Multiple Integrals and Vector Calculus (20+6)

Special functions-Beta, Gamma functions.Double integration – Cartesian and polar co-ordinates – Change of order of integration – Change of variables between Cartesian and polar co-ordinates – Triple integration – Area as a double integral-Volume as a triple integral.

Gradient, Divergence and Curl – Directional derivative – Irrotational and solenoidal vector fields – Vector integration – Green's theorem in a plane, Gauss divergence theorem and Stoke's theorem (excluding proof) – Simple applications.

## Unit – III Ordinary Differential Equations (ODE) and Applications (18+6)

Solution of higher order linear ODE with constant coefficients and solution of second order ODE by the method of variation of parameters – Cauchy"s and Legendre"s linear equations - Simultaneous first order linear equations with constant coefficients.Formulation and solution of ODE related to Simple harmonic motion, mechanical and electrical oscillatory circuits.

## Unit – IV Analytic Functions and Complex Integration (18+6)

Functions of a complex variable – Analytic function – Necessary conditions – Cauchy-Riemann equations – Sufficient conditions (excluding proof) – Harmonic and orthogonal properties of analytic function – Harmonic conjugate – Construction of Analytic functions - Conformal mapping: w=z+c, cz, 1/z, and bilinear transformation.Complex integration-Statement and application of Cauchy's integral theorem and integral formula – Taylor and Laurent expansions – singular points – Residues - Residue theorem. Application of residues to evaluate real integrals-Unit circle and semicircular contours (excluding poles on boundaries)

## Unit – V Laplace Transform (16+6)

Laplace Transform of elementary functions – Basic properties – Derivatives and integrals of transforms – Transforms of derivatives and integrals – Transforms of unit step function and impulse function – Transform of periodic functions.Inverse Laplace Transform – Convolution theorem – Solution of linear ODE of second order with constant coefficients and first order simultaneous equations with constant coefficients using Laplace transformation.

L+T=90+30 Total=120 Periods

## TEXT BOOK

1. Bali.N.P and Manish Goyal, "A Textbook of Engineering Mathematics", 7th Edition, Laxmi Publications(p) Ltd. (2007)## REFERENCES

1. Grewal B.S, "Higher Engineering Mathematics", 39th Edition, Khanna Publishers, Delhi, (2007)2. Ramana.B.V., "Higher Engineering Mathematics", Tata Mc-Graw Hill Publishing Company Limited, New Delhi (2007)

3. Glyn James, "Advanced Modern Engineering Mathematics", 3rd Edition-Pearson Education (2007).

4. Jain R.K, and Iyengar S.R.K, "Advanced Engineering Mathematics", 3rd Edition-Narosa Publishing House Pvt.Ltd (2007)

5. Erwin Kreyszig, "Advanced Engineering Mathematics", 7th Edition-Wiley India (2007).

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