This course is about the basic mathematics that is fundamental and essential component in all streams of undergraduate studies in sciences and engineering. The course consists of topics in complex analysis,numerical analysis, vector calculus and transform techniques with applications to various engineering problems. This course will cover the following main topics.Function of complex variables. Analytic functions. Line integrals in complex plane. Cauchy’s integral theorem, Derivatives of analytic functions. Power series, radius of convergence. Taylor’s and Laurent’s series, zeros and singularities, residue theorem.Iterative method for solution of system of linear equations. Finite differences, interpolation. Numerical integration. Solution of algebraic and transcendental equations.Vector and scalar fields. Limit, continuity, differentiability of vector functions. Directional derivative, gradient, curl, divergence. Line and surface integrals, Green, Gauss and Stokes theorem.Laplace transform and its properties. Laplace Transform of specialfunction. Convolution theorem. Evaluation of integrals by LaplaceTransform. Solution of initial and boundary value problems.Fourier series representation of a function. Fourier sine and cosinetransforms. Fourier Transform. Properties of Fourier Transform.Applications to boundary value problems.

INTENDED AUDIENCE : all branches of science and engineering

PREREQUISITES : Engineering Mathematics - I

INDUSTRY SUPPORT : Nil

Week 1 : Vector and scalar fields. Limit, continuity, differentiability of vector functions. Directional derivative, gradient, curl, divergence

Week 2 : Line and surface integrals, Green, Gauss and Stokes theorem.

Week 3 : Function of complex variables and their properties including continuity and differentiability. Analytic functions and CR equations. Line

integrals in complex plane.

Week 3 : Function of complex variables and their properties including continuity and differentiability. Analytic functions and CR equations. Line

integrals in complex plane.

Week 4 : Cauchy’s integral theorem, Power series, radius of convergence. Taylor’s and Laurent’s series, zeros and singularities, residue theorem.

Week 5 : Iterative method for solution of system of linear equations. Finite differences, interpolation.

Week 6 : Numerical integration. Solution of algebraic and transcendental equations.

Week 7 : Laplace transform and its properties. Laplace Transform of special function.

Week 8 : Convolution theorem. Evaluation of integrals by Laplace Transform. Solution of initial and boundary value problems.

Week 9 : Fourier series & its convergence

Week 10 : Fourier integral representation

Week 11 : Fourier sine and cosine transforms. Fourier Transform. Properties of Fourier Transform.

Week 12 : Applications of Fourier series to boundary value problems.

1 Kreyszig, E. (2010). Advanced Engineering Mathematics, 10th edition. John Wiley & Sons.

2 O’Neil, Peter V. (2011). Advanced Engineering Mathematics, 7th edition. Cengage learning.

3 Colley, S.J. (2012). Vector Calculus, 4 th edition. Pearson Education, Inc.

4 Zill, D.G., Shanahan P.D. (2013). Complex Analysis: A First Course with Applications, 3rd Edition. Jones & Bartlett Learning.

5 Dyke, P.P.G. (2001). An Introduction to Laplace Transforms and Fourier Series. Springer-Verlag London Ltd.

6 Hanna, J.R. and Rowland, J.H. (1990). Fourier Series, Transforms and Boundary Value Problems. Second Edition. Dover Publications, Inc. New York.

7 Pinkus, A. and Zafrany, S. (1997). Fourier Series and Integral Transforms. Cambridge University Press. United Kingdom.

Jitendra Kumar is an Associate Professor at the Department of Mathematics, IIT Kharagpur. He completed his M.Sc. in Industrial Mathematics from IIT Roorkee and Technical University of Kaiserslautern, Germany in 2001 and 2003, respectively. He received his PhD degree in 2006 from Otto-von-Guericke University Magdeburg, Germany. He was Research Associate at the Institute for Analysis and Numerical Mathematics, Otto-von-Guericke University Magdeburg, Germany from 2006 to 2009. Dr. Kumar is the recipient of several recognized awards and fellowships, including Alexander von Humboldt fellowship, DAAD & DGF scholarships. His research interests include Numerical solutions of integro-differential equations, numerical analysis and modelling and simulations of problem in particulate systems.

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams:**24 April 2022** Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

**CRITERIA TO GET A CERTIFICATE**

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

**YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.**

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kharagpur .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams:

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kharagpur .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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