### MA3452 Syllabus - Vector Calculus And Complex Functions - 2021 Regulation Anna University

## MA3452 Syllabus - Vector Calculus And Complex Functions - 2021 Regulation Anna University

MA3452 | VECTOR CALCULUS AND COMPLEX FUNCTIONS | LTPC |
---|

**3104**

**COURSE OBJECTIVES:**

• 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.

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

• differential equations that model engineering problems.

• 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.

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

• differential equations that model engineering problems.

UNIT I | VECTOR CALCULUS | 9+3 |
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Gradient and directional derivative – Divergence and curl - Vector
identities – Irrotational and Solenoidal vector fields – Line integral
over a plane curve – Surface integral - Area of a curved surface - Volume
integral - Green’s, Gauss divergence and Stoke’s theorems – Verification
and application in evaluating line, surface and volume integrals.

UNIT II | ANALYTIC FUNCTION | 9+3 |
---|

Analytic functions – Necessary and sufficient conditions for analyticity
in Cartesian and polar coordinates - Properties – Harmonic conjugates –
Construction of analytic function - Conformal mapping – Mapping by
functions w z c, az, 1 , z2 - Bilinear transformation.

UNIT III | COMPLEX INTEGRATION | 9+3 |
---|

Line integral - Cauchy’s integral theorem – Cauchy’s integral formula –
Taylor’s and Laurent’s series – Singularities – Residues – Residue theorem
– Application of residue theorem for evaluation of real integrals – Use of
circular contour and semicircular contour.

UNIT IV | LAPLACE TRANSFORMS | 9+3 |
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Existence conditions – Transforms of elementary functions – Transform of
unit step function and unit impulse function – Basic properties – Shifting
theorems -Transforms of derivatives and integrals – Initial and final
value theorems – Inverse transforms – Convolution theorem – Transform of
periodic functions – Application to solution of linear second order
ordinary differential equations with constant coefficients.

UNIT V | ORDINARY DIFFERENTIAL EQUATIONS | 9+3 |
---|

Higher order linear differential equations with constant coefficients -
Method of variation of parameters – Homogenous equation of Euler’s and
Legendre’s type – System of simultaneous linear differential equations
with constant coefficients - Method of undetermined coefficients.

**TOTAL: 60 PERIODS**

**COURSE OUTCOMES: Upon successful completion of the course, students should be able to:**

• Evaluate real and complex integrals using the Cauchy integral formula
and the residue theorem.

• Appreciate how complex methods can be used to prove some important theoretical results.

• Evaluate line, surface and volume integrals in simple coordinate systems.

• Calculate grad, div and curl in Cartesian and other simple coordinate systems, and establish identities connecting these quantities.

• Use Gauss, Stokes and Greens theorems to simplify calculations of integrals and prove simple results.

• Appreciate how complex methods can be used to prove some important theoretical results.

• Evaluate line, surface and volume integrals in simple coordinate systems.

• Calculate grad, div and curl in Cartesian and other simple coordinate systems, and establish identities connecting these quantities.

• Use Gauss, Stokes and Greens theorems to simplify calculations of integrals and prove simple results.

**TEXT BOOKS:**

1. Erwin Kreyszig," Advanced Engineering Mathematics ", John Wiley and
Sons, 10th Edition, New Delhi, 2016.

2. Grewal B.S., “Higher Engineering Mathematics ”, Khanna Publishers, New Delhi, 43rd Edition, 2014.

2. Grewal B.S., “Higher Engineering Mathematics ”, Khanna Publishers, New Delhi, 43rd Edition, 2014.

**REFERENCES:**

1. Sastry, S.S, "Engineering Mathematics", Vol. I & II, PHI Learning
Pvt. Ltd, 4th Edition, New Delhi, 2014.

2. Jain R.K. and Iyengar S.R.K., “ Advanced Engineering Mathematics ”, Narosa Publications, New Delhi , 3rd Edition, 2007.

3. Bali N., Goyal M. and Watkins C., “Advanced Engineering Mathematics ”, Firewall Media (An imprint of Lakshmi Publications Pvt., Ltd.,), New Delhi, 7th Edition, 2009.

4. Peter V. O’Neil, “Advanced Engineering Mathematics”, Cengage Learning India Pvt., Ltd, New Delhi, 2007.

5. Ray Wylie C and Barrett.L.C, "Advanced Engineering Mathematics" Tata McGraw Hill Education Pvt. Ltd, 6th Edition, New Delhi, 2012.

2. Jain R.K. and Iyengar S.R.K., “ Advanced Engineering Mathematics ”, Narosa Publications, New Delhi , 3rd Edition, 2007.

3. Bali N., Goyal M. and Watkins C., “Advanced Engineering Mathematics ”, Firewall Media (An imprint of Lakshmi Publications Pvt., Ltd.,), New Delhi, 7th Edition, 2009.

4. Peter V. O’Neil, “Advanced Engineering Mathematics”, Cengage Learning India Pvt., Ltd, New Delhi, 2007.

5. Ray Wylie C and Barrett.L.C, "Advanced Engineering Mathematics" Tata McGraw Hill Education Pvt. Ltd, 6th Edition, New Delhi, 2012.

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