subject name : mathematics-ii · 2020. 7. 17. · lecture plan (theory subject) : b. tech. –ii...
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GALGOTIAS COLLEGE OF ENGINEERING & TECHNOLOGY
1, Knowledge Park-II, Greater Noida
Department of Applied Science (Mathematics)
STUDY MATERIAL FILE
Subject Name : Mathematics-II Subject Code : KAS-203
Course Coordinator Dr. Dushyant Kumar
(Associate Professor, Mathematics)
Faculty Members
Dr. Padam Singh Tomar (Asst. Professor, Mathematics)
Dr. Vinay Gautam (Asst. Professor, Mathematics)
Syllabus (as prescribed by affiliating university, i.e. AKTU) KAS-203: Mathematics II
L T P 3 1 0
Module 1: Ordinary Differential Equation of Higher Order [10]
CO-1 Apply the mathematical concepts for solving differential equations
Linear differential equation of nth order with constant coefficients, Simultaneous linear differential equations, Second order linear differential equations with variable coefficients, Solution by changing independent variable, Reduction of order, Normal form, Method of variation of parameters, Cauchy-Euler equation, Series solutions (Frobenius Method).
Module 2: Multivariable Calculus-II [08]
CO-2 Apply the concept of definite integral for evaluating surface areas and volumes
Improper integrals, Beta & Gama function and their properties, Dirichlet’s integral and its applications, Application of definite integrals to evaluate surface areas and volume of revolutions.
Module 3: Sequences and Series [08]
CO-3 Apply the concept of convergence in sequence and series and evaluate the expansion in terms of sine and cosine
Definition of Sequence and series with examples, Convergence of sequence and series, Tests for convergence of series, (Ratio test, D’ Alembert’s test, Raabe’s test). Fourier series, Half range Fourier sine and cosine series.
Module 4: Complex Variable – Differentiation [08]
CO-4 Apply the working methods of complex functions to find analytic functions
Limit, Continuity and differentiability, Functions of complex variable, Analytic functions, Cauchy- Riemann equations (Cartesian and Polar form), Harmonic function, Method to find Analytic functions, Conformal mapping, Mobius transformation and their properties
Module 5: Complex Variable –Integration [08]
CO-5 Expand Taylor’s series and Laurent’s series for complex function and evaluation of definite integrals
Complex integrals, Contour integrals, Cauchy- Goursat theorem, Cauchy integral formula, Taylor’s series, Laurent’s series, Liouvilles’s theorem, Singularities, Classification of Singularities, zeros of analytic functions, Residues, Methods of finding residues, Cauchy Residue theorem, Evaluation of real integrals of the types.
Text Books:-
1. B. V. Ramana, Higher Engineering Mathematics, Tata McGraw-Hill Publishing Company Ltd., 2008.
2. B. S. Grewal, Higher Engineering Mathematics, Khanna Publisher, 2005.
3. R. K. Jain & S. R. K. Iyenger , Advance Engineering Mathematics , Narosa Publishing -House, 2002.
Reference Books:-
1. E. Kreyszig, Advance Engineering Mathematics, John Wiley & Sons, 2005.
2. Peter V. O’Neil, Advance Engineering Mathematics, Thomson (Cengage) Learning, 2007.
3. Maurice D. Weir, Joel Hass, Frank R.Giordano, Thomas, Calculus, Eleventh Edition, Pearson.
4. G.B Thomas, R L Finney, Calculus and Analytical Geometry, Ninth Edition Pearson, 2002.
5. James Ward Brown and Ruel V Churchill, Fourier Series and Boundary Value Problems, 8th Edition-Tata McGraw-Hill
6. D. Poole , Linear Algebra : A Modern Introduction, 2nd Edition, Brooks/Cole, 2005.
7. Veerarajan T., Engineering Mathematics for first year, Tata McGraw-Hill, New Delhi, 2008.
8. Charles E Roberts Jr, Ordinary Diffrential Equations, Application, Model and Computing, CRC Press T&F Group.
9. Ray Wylie C and Louis C Barret, Advanced Engineering Mathematics, 6th Edition, Tata McGraw-Hill.
10. James Ward Brown and Ruel V Churchill, Complex Variable and Applications, 8th Edition, Tata McGraw-Hill.
11. P. Sivaramakrishna Das and C. Vijayakumari, Engineering Mathematics, 1st Edition, Pearson India Education Services Pvt. Ltd.
Lecture Plan (Theory Subject) : B. Tech. –II Sem
Subject Name with Code: Mathematics-II (KAS-203) Total Module : 5
Module No.
Topics to be delivered Lecture Number
1 Definition of Linear differential equations 1
1 Complementary function 2
1 Particular integral 3-4
1 Simultaneous linear differential equations 5
1 Second order differential equations by changing dependent
variables 6
1 Second order differential equations by changing independent
variables 7
1 Second order differential equations by Normal form 8
1 Method of variation of parameters 9
1 Series Solution by Power Series method 10-11
1 Series Solution by Frobenius method 12-13
2 Improper integrals 14-15
2 Beta & Gama function 16
2 Beta & Gama function and their properties 17
2 Dirichlet’s integral 18
2 Dirichlet’s integral and its
applications 19
2 Application of definite integrals 20
3 Definition of Sequence and series with examples 21
3 Convergence of sequence and series 22
3 Ratio test Ratio test, 23
3 D’ Alembert’s test 24
3 Raabe’s test 25
3 Euler’s Formulae, Periodic functions, Even and odd functions 26
3 Fourier Series functions having arbitrary periods 2 27
Module No.
Topics to be delivered Lecture Number
3 Fourier Series functions having arbitrary periods 2C 28
3 Fourier Series functions having arbitrary periods 2C breaking limit 29
3 Half range sine and cosine series 30
4 Limit, Continuity and differentiability 31
4 Functions of complex variable 32
4 Analytic functions, Cauchy- Riemann equations 33-34
4 Harmonic function, Method to find Analytic functions 35
4 Conformal mapping 36
4 Mobius transformation and their properties 37
5 Complex integrals 38
5 Contour integrals 39
5 Cauchy- Goursat theorem 40
5 Cauchy integral formula 41
5 Taylor’s series, Laurent’s series 42
5 Liouvilles’s theorem, Singularities 43-44
5 Classification of Singularities, zeros of analytic functions 45-46
5 Residues, Methods of finding residues,
47
5 Cauchy Residue theorem,
48
5 Evaluation of real integrals
49-50
Revision
51-55
Modul e-1
Course outcome KAS-203.1
Understand the concept of differentiation and apply for solving
differential equations.
Lecture 1
Lecture 2
Lecture 3
Lecture 4(P.I.Continued)
Lecture 5
Lecture 6
Lecture 7
Lecture 8
Lecture 9
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Modul e-2
Course outcome KAS-203.2
Remember the concept of definite integral and apply for evaluating surface
areas and volumes.
Lecture 1
Lecture 2
Modul e-3
Course outcome KAS-203.3
Understand the concept of convergence of sequence and series. Also evaluate
Fourier series.
Lecture 1
Lecture-2
Lecture-3
Lecture-4
Lecture 5
Lecture 6
Lecture 7
Lecture 8
Modul e-4
Course outcome KAS-203.4
Illustrate the working methods of complex functions and apply for finding
analytic functions.
LECTURE -1
LECTUER-2
LECTUER-3
LECTUER-4
Q-1
Q-2 Show that
Q-3
Q-4
Q-5
LECTUER-5
Q-1
Q-2
Q-3
Q-4
LECTUER-6
LECTUER-7
Modul e-5
Course outcome KAS-203.5
Apply the complex functions for finding Taylor’s series, Laurent’s series and
evaluation of definite integrals.
Lecture-1 Line Integrals
Lecture-2
Cauchy Integral Theorem
Deformation
Lecture-3
Cauchy integral Formula
Generalized Cauchy Formula
Lecture-4 Complex Power Series
Taylor Expansion
Lecture-5 Laurent Expansion
Lecture-6
Singularities
Theorem
Zeros of holomorphic functions
Theorem
Lecture-7
Residue Theory
Residue
Calculating Residues
Lecture-8
Integrals of the Form ∫ ( , )
We have
Lecture-9
Integrals of the Form ∫ ( )
