OBJECTIVES :
This course is designed to cover topics such as Matrix Algebra, Vector Calculus, Complex
Analysis and Laplace Transform. Matrix Algebra is one of the powerful tools to handle practical
problems arising in the field of engineering. Vector calculus can be widely used for modelling the
various laws of physics. The various methods of complex analysis and Laplace transforms can
be used for efficiently solving the problems that occur in various branches of engineering
disciplines.
UNIT I MATRICES 12
Eigenvalues and Eigenvectors of a real matrix – Characteristic equation – Properties of Eigenvalues
and Eigenvectors – Cayley-Hamilton theorem – Diagonalization of matrices – Reduction of a
quadratic form to canonical form by orthogonal transformation – Nature of quadratic forms.
UNIT II VECTOR CALCULUS 12
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 III ANALYTIC FUNCTIONS 12
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 1 2 z z w z c, cz, , - Bilinear transformation.
UNIT IV COMPLEX INTEGRATION 12
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 V LAPLACE TRANSFORMS 12
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.
TOTAL: 60 PERIODS
OUTCOMES :
After successfully completing the course, the student will have a good understanding of the following topics and their applications:
- Eigenvalues and eigenvectors, diagonalization of a matrix, Symmetric matrices, Positive definite matrices and similar matrices.
- Gradient, divergence and curl of a vector point function and related identities. Evaluation of line, surface and volume integrals using Gauss, Stokes and Green’s theorems and their verification.
- Analytic functions, conformal mapping and complex integration.
- Laplace transform and inverse transform of simple functions, properties, various relatedtheorems and application to differential equations with constant coefficients.
TEXT BOOKS :
1. Grewal B.S., “Higher Engineering Mathematics”, Khanna Publishers, New Delhi,
43rd Edition, 2014.
2. Kreyszig Erwin, "Advanced Engineering Mathematics ", John Wiley and Sons,
10th Edition, New Delhi, 2016.
REFERENCES :
1. Bali N., Goyal M. and Watkins C., “Advanced Engineering Mathematics”, Firewall
Media (An imprint of Lakshmi Publications Pvt., Ltd.,), New Delhi, 7th Edition, 2009.
2. Jain R.K. and Iyengar S.R.K., “ Advanced Engineering Mathematics ”, Narosa
Publications, New Delhi , 3rd Edition, 2007.
3. O’Neil, P.V. “Advanced Engineering Mathematics”, Cengage Learning India
Pvt., Ltd, New Delhi, 2007.
4. Sastry, S.S, “Engineering Mathematics", Vol. I & II, PHI Learning Pvt. Ltd,
4th Edition, New Delhi, 2014.
5. Wylie, R.C. and Barrett, L.C., “Advanced Engineering Mathematics “Tata McGraw Hill
Education Pvt. Ltd, 6th Edition, New Delhi, 2012.
No comments:
Post a Comment
GIVE RIGHT REVIEW