Nirma University Institute of Technology

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1 Nirma University Institute of Technology Department of Mathematics & Humanities Template B. Tech. Electrical Engineering Semester: III Academic Year: Term: Odd 28 Course Code & Name : MA04, Mathematics for Electrical Engineers Credit Details : Lectures-4, Tutorial-1, Practicles-0 Credits-5 Course Co-ordinator : Dr. Sandeep Malhotra Contact No. & sandeep.malhotra@nirmauni.ac.in Office : B100 Visiting Hours : 1:0pm to 2pm (Tuesday & Thursday) Course Faculty 1. Dr. Sandeep Malhotra sandeep.malhotra@nirmauni.ac.in Contact No Office: B100 Visiting Hours: Tuesday -1.0pm to 2.00pm Wednesday - 1.0pm to 2.00pm Course Blog : 2. Ms. Dhara Patel dhara patel@gmail.com Contact No Office: A20 Visiting Hours: Tuesday-1.0pm to 2.00pm Wednesday - 1.0pm to 2.00pm ma04blog.wordpress.com 1. Introduction to Course - Mathematics for Electrical Engineers Modern science and engineering requires high levels of qualitative logic before the act of precise problem formulation can occur. Thus, much is known about a physio electrical problem beforehand, derived from experience or experiment (i.e., empiricism). Most often, a theory evolves only after detailed observation of an event. Thus, the first step in problem formulation is necessarily qualitative. The second step is the bringing together of all applicable physical and electrical information, imperial laws, and rate expressions. At this point, the engineer must make a series of critical decisions about the conversion of mental images to symbols, and at the same time, how detailed Page 1 of 9

2 the model of a system must be. Here, one must classify the real purposes of the modelling effort. Is the model to be used only for explaining trends in the operation of an existing piece of equipment? Is the model to be used for predictive or design purposes? The scope and depth of these early decisions will determine the ultimate complexity of the final mathematical description. The third step requires the setting down of appropriate methods to solve the model. In the limit, as the differential elements shrink, then differential equations arise naturally. Next, the imposed conditions must be addressed, and this aspect must be treated with considerable circumspection. When the problem is fully posed in quantitative terms, an appropriate mathematical solution method is sought out, which finally relates dependent (responding) variables to one or more independent (changing) variables. The final result may be an elementary mathematical formula, or a numerical solution portrayed as an array of numbers. 1.2 Course objective: Calculus aims at the study of advanced modules like Fourier Series, Laplace Transformation, Vector Calculus, and Complex Analysis Fourier Series which is useful to do Network Analysis in Electrical Engineering Differential Equations and its applications to Electric Circuit Analysis 2. Course Learning Outcomes(CLOs): 1. formulate problems in electrical engineering from real life situations 2. develop mathematical model of physical concepts. validate solution to electrical engineering problems. Syllabus: Unit 1: Vector differential calculus Teaching hours: 4 Reorientation (Vector algebra), Differentiation of vectors, Scalar and vector point function, Gradient of a scalar field and directional derivatives, Divergence and curl of vector field, Solenoidal, irrotational and conservative fields. Unit 2: Function of complex variables Teaching hours: 9 Reorientation, Limit continuity and Differentiation of a function of complex variables, Cauchy Riemann equations in Cartesian and polar form, Analytic function, Harmonic functions and orthogonal curves, Application of Cauchy Riemann equations in electrostatic problems, integration of function of complex variables, Cauchy s integral theorem, Cauchy s integral formula. Unit : Laplace Transforms Teaching hours : 12 Motivation, Definition, Linearity property, Laplace transforms of elementary functions, Shifting theorem, Inverse Laplace transforms of derivatives and integrals, Convolution theorem, Application of Laplace transforms in solving ordinary differential equations, Laplace transforms of periodic, Unit step and Impulse functions. Page 2 of 9

3 Unit 4: Fourier Series Teaching hours: 8 Periodic functions, Dirichlet s conditions, Fourier series, Euler s formulae, Fourier expansion of periodic functions with period 2π, Fourier series of even and odd functions, Fourier series of periodic functions with arbitrary periods, half range Fourier series, Harmonic analysis. Unit 5: Ordinary Differential Equations Teaching hours: 9 Definition, formation, order and degree, Linear differential equations of higher order with constant coefficients, complimentary function, method of undetermined coefficients, method of variation of parameters, Higher order linear differential equations with variables coefficients (Cauchy s and Legendre s equation), Simultaneous linear differential equations, Modeling of Electrical circuit. Unit 6: Partial Differential Equations Teaching hours: 8 Formation of Partial differential equations, Directly Integrable equations, Models of Engineering problems leading to first order partial differential equations. Langrage s equation, Method of separation of variable. Applications to Electrical Engineering. Unit 7: Solution of Transcendental and Algebraic Equations Teaching hours: 9 Newton-Raphson, Bisection, False position, Iteration methods, Convergence of these methods. Solution of System of Linear Equations : Gauss-Seidel and Gauss-Jacobi s methods. Numerical Solutions of Ordinary Differential Equations : Solution of initial value problems: Picard s method, Taylor series method, 4 th order Runge Kutta method. Self Study: Around 10% of the questions will be asked from self study contents. 1 st order Ordinary Differential Equations Short cut methods to solve Higher Order Differential Equations References : 1) Erwin Kreyszig- Advanced Engineering Mathematics (5 th Edition) Publishers: John Wiley ) Peter V. O neil- Advanced Engineering Mathematics (5 th Edition) Publisher : Thmoson Books/Cole, Singapore. ) Dr.B.S.Grewal- Higher Engineering Mathematics (40 th Edition), Khanna Publishers, New Delhi. 4) S. C. Chapra and R. P. Canale, Numerical Methods for Engineers with Programming and Software Applications (6 th Edition), McGraw-Hill Publications. 5) M. K. Jain and S. R. K.. Iyengar, R. K. Jain-Numerical Methods for Scientific & Engineering Computation (4 th Edition), New age International Publication. Page of 9

4 4. Tutorial Plan: Tutorial No Topic Schedule Mapp ed CLO 1 Fourier Series 1 2/07/28 to 4/08/27 CLO2 2 Fourier Series 2 6/08/28 to 1/08/28 CLO1 Higher Order Differential Equation 1 27/08/28 to 04/09/28 CLO2 4 Higher Order Differential Equation 2 04/09/28 to 11/09/28 5 Laplace Transformation 11/09/28 to 18/09/28 CLO2 6 Inverse Laplace Transformation 8/10/28 to 15/10/28 CLO1 7 Vector Calculus 15/10/28 to 22/10/28 8 Partial Differential Equations 22/10/28 to 29/10/28 CLO1 9 Numerical Methods 29/10/28 to 5/11/28 CLO2 10 Complex Analysis 19/11/28 to 26/11/28 5. Component wise Continuous Evaluation & Semester End Examination weightage: Assessment scheme Component weightage Class Test 0% CE Sessional Exam 40% SEE Tutorial Evaluation 0% 5.1 Assessment Policy for Continuous Evaluation (CE) Assessment of Continuous Evaluation comprises of three components. 1. Class Test will be conducted as per academic calendar. It will be conducted online/ offline for the duration of 1 hour and will be of 0 marks. 2. Sessional Exam will be conducted as per academic calendar. It will be conducted offline for the duration of 1 hour and 15 minutes and will be of 40 marks.. There will be 10 tutorials each carrying weightage of 10 marks. At the end of the course total marks obtained out of 100 will be converted according to weightage assigned. Assessment of Tutorials will be carried out based on parameters like timely submission, neat and clean work, originality, involvement of the student, regularity, discipline etc. during the session. Page 4 of 9

5 5.2 Assessment Policy for Semester End Examination (SEE) A written examination of hour duration will be conducted for the course as per academic calendar. It will carry 100 marks and marks obtained out of 100 will be converted as per weightage assigned. 6. Lesson Plan (Total Lectures 60) Topics to be covered Hours Mapped Lect.No. CLOs Overview of the Course Ch.1 Fourier Series 08 1,2 Periodic functions, Dirichlet s conditions, Fourier series, Euler s formulae 02 Fourier expansion of periodic functions with period 2 π, 4 Fourier series of even and odd functions 5 Fourier series of discontinuous functions CLO1 6 Change of intervals (0, 2π) to (0, 2l ) and ( -π, π ) to (-l, l ) Half range sine and cosine Series 8 Harmonic Analysis Ch. 2 Ordinary Differential Equations 10 9 Reorientation, Higher order, first degree linear differential equations with constant coefficients 10 Methods to find Complimentary function CLO2 11, 12 Particular integrals using direct integration method 02 1,14 Particular integrals using method of Undetermined Coefficient Particular integral by the method of variation of parameters Class Test 16 Cauchy s and Legendre s homogeneous linear differential equations 17 Simultaneous linear differential equations with constant coefficients CLO2 18 Models for the real world problems and their solutions, Modelling of electric Circuits Ch. Laplace Transform Definition, Laplace transform of elementary functions 20 Linearity property, First Shifting theorem (without proof) 21 Theorems on Multiplication of f(t) by t and division of f(t) by t and Examples based on it. 22 Laplace transforms of the derivative (No example) and integration 2,24 Laplace transforms of periodic, Unit step and Impulse functions inverse Laplace transform and standard forms 26 Method of Using Partial fractions to find inverse Laplace transforms 27 Convolution theorem (without proof) and examples based on it 28 Inverse Laplace transforms of Unit step and Impulse functions 29,0 Application of Laplace transforms in solving ordinary differential equations 02 Ch. 4 Vector Differential Calculus 04 1 Reorientation (Vector algebra), Differentiation of vectors, Scalar CLO2 Page 5 of 9

6 and vector point function 2 Gradient of a scalar field and directional derivatives Divergence and curl of vector field 4 Solenoidal, irrotational and conservative fields. Sessional Exam Ch. 5 Function of Complex Variable 08 5,6 Reorientation, Limit continuity and differentiation of a function of complex variables, 02 7,8 Cauchy Riemann equations in Cartesian and polar form, Analytic 02 function, 9 Harmonic functions and orthogonal curves, 40 Application of Cauchy Riemann equations in electrostatic problems, 41,42 integration of function of complex variables, Cauchy s integral 02 theorem, Cauchy s integral formula. Ch. 6 Partial Differential Equations 09 4 Introduction to Partial differential equations, Formation of Partial differential Equations 44 Directly Integrable equations, Models of Engineering problems leading to first order Partial differential equations 45,46 Lagrange s equation Partial differential equations of the first order & first 02 Degree 47 Applications of Partial differential equations, Method of separation of variables 48,49 Solution of Wave equation by the method of separable variables one-dimensional heat equation 51 Two-dimensional Laplace equation Ch. 7 Numerical Methods Introduction, Solution of algebraic and transcendental equations by Bisection method 5 Solution of algebraic and transcendental equations by Method of False position 54 Solution of algebraic and transcendental equations by Newton- Raphson method 55 Solution of algebraic and transcendental equations by Iteration method, Convergence of the methods discussed. 56,57 Solution of System of Linear Equations : Gauss-Jacobi s and Gauss-Seidel methods. 58,59,60 Numerical Solutions of Ordinary Differential Equations : Picard s method, Taylor series method, 4th order Runge Kutta method. 61 Review of the course, Feedback related to the course, Linkages with advanced course/s in succeeding years CLO1 CLO2 Page 6 of 9

7 7. Mapping of Session Learning Outcomes (SLO) with Course Learning Outcomes (CO) Session No. Session Learning Outcomes: After successful completion of the session, student will be able to CO 1 Overview of the Course 2 Periodic functions, Dirichlet s conditions, Fourier series, Euler s formulae 1 Fourier expansion of periodic functions with period 2 π, 1 4 Fourier series of even and odd functions 1 5 Fourier series of discontinuous functions 1 6 Change of intervals (0, 2π) to (0, 2l ) and ( -π, π ) to (-l, l ) 1 7 Half range sine and cosine Series 1 8 Harmonic Analysis 1 9 Reorientation, Higher order, first degree linear differential equations with constant coefficients 2 & 10 Methods to find Complimentary function 2 & 11, 12 Particular integrals using direct integration method 2 & 1,14 Particular integrals using method of Undetermined Coefficient 2 & 15 Particular integral by the method of variation of parameters 2 & 16 Cauchy s and Legendre s homogeneous linear differential equations 2 & 17 Simultaneous linear differential equations with constant coefficients 2 & 18 Models for the real world problems and their solutions, Modelling of electric Circuits 2 & 19 Definition, Laplace transform of elementary functions 20 Linearity property, First Shifting theorem (without proof) 21 Theorems on Multiplication of f(t) by t and division of f(t) by t and Examples based on it. 22 Laplace transforms of the derivative and integration 2,24 Laplace transforms of periodic, Unit step and Impulse functions 25 Definition of inverse Laplace transform and standard forms 26 Method of Using Partial fractions to find inverse Laplace transforms 27 Convolution theorem (without proof) and examples based on it 28 Inverse Laplace transforms of Unit step and Impulse functions 29,0 Application of Laplace transforms in solving ordinary differential equations 1 Reorientation (Vector algebra), Differentiation of vectors, Scalar and vector point function 2 2 Gradient of a scalar field and directional derivatives 2 Divergence and curl of vector field, 2 4 Solenoidal, irrotational and conservative fields. 2 5,6 Reorientation, Limit continuity and differentiation of a function of complex variables, 1 7,8 Cauchy Riemann equations in Cartesian and polar form, Analytic function, 1 9 Harmonic functions and orthogonal curves, 1 40 Application of Cauchy Riemann equations in electrostatic problems, 1 41,42 integration of function of complex variables, Cauchy s integral theorem, Cauchy s integral formula. 1 Page 7 of 9

8 4 Introduction to Partial differential equations, Formation of Partial differential Equations 2 & 44 Directly Integrable equations, Models of Engineering problems leading to first 2 & order Partial differential equations 45,46 Lagrange s equation Partial differential equations of the first order & first Degree 2 & 47 Applications of Partial differential equations, Method of separation of variables 2 & 48,49 Solution of Wave equation by the method of separable variables 2 & 50 one-dimensional heat equation 2 & 51 Two-dimensional Laplace equation 2 & 52 Introduction, Solution of algebraic and transcendental equations by Bisection method 5 Solution of algebraic and transcendental equations by Method of False position 54 Solution of algebraic and transcendental equations by Newton-Raphson method 55 Solution of algebraic and transcendental equations by Iteration method, Convergence of the methods discussed. 56,57 Solution of System of Linear Equations : Gauss-Jacobi s and Gauss-Seidel methods. 58,59,60 Numerical Solutions of Ordinary Differential Equations : Picard s method, Taylor series method, 4th order Runge Kutta method. 61 Review of the course, Feedback related to the course, Linkages with advanced course/s in succeeding years. 8. Teaching-learning methodology 1. Lectures: Primarily Chalk and Black board will be used to conduct the course. However, where required, Power Point Presentations (PPTs), Video Lectures, Simulations / Animations etc. will be used to enhance the teaching-learning process. 2. Tutorial: Emphasis will be on one to one interaction with students for clearing their doubts and problem solving. 9. Active learning techniques Active learning is a method of learning in which students are actively or experientially involved in the learning process. Following active learning techniques will be adopted for the course. Flipped Class-room (Topics mentioned in self - study))- Flipped classroom is an instructional strategy and a type of blended learning that reverses the traditional learning environment by delivering instructional content, often online, outside of the classroom. It moves activities, including those that may have traditionally been considered homework, into the classroom. In a flipped classroom, students watch online lectures, refer the books or online lecture notes, or carry out derivation at home and present the concepts in the classroom with the guidance of a mentor. Page 8 of 9

9 Muddiest Point- At the end of each session the students are asked share any topic which is not or difficult to understood. For such topics a discussion may be done in the class. Surprise test in the form of Quiz- To ensure the understanding of the students during the class, the surprise test in the form of quiz may be given at any time during the class. 10. Course Material Following course material is uploaded on the course website: ma06appliedmaths.wordpress.com PPTs, Notes, Handouts Tutorials Question bank Useful links (Applications) 11. Course Learning Outcome Attainment Following means will be used to assess attainment of course learning outcomes. Use of formal evaluation components of continuous evaluation, tutorials, laboratory work, semester end examination Informal feedback during course conduction 12. Academic Integrity Statement Students are expected to carry out assigned work under Continuous Evaluation (CE) component and LPW component independently. Copying in any form is not acceptable and will invite strict disciplinary action. Evaluation of corresponding component will be affected proportionately in such cases. Turnitin software will be used to check plagiarism wherever applicable. Academic integrity is expected from students in all components of course assessment. Page 9 of 9

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad

Course Title Course Code INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad - 500 043 CIVIL ENGINEERING COURSE DESCRIPTION MATHEMATICS-II A30006 Course Structure Lectures Tutorials