Format No. QSP/7.1/01.F01 (B) Issue No.04 Rev. No 5 Dated: June 2, 2015 ________________________________________________________________ UNIVERSITY OF PETROLEUM & ENERGY STUDIES College of Engineering Studies Dehradun COURSE PLAN Programme
: B. Tech
Course
: MATHEMATICS-II
Subject Code : MATH-102 No. of credits : 4 Semester
: II
Session
: Jan 2016 - June 2016
Batch
: 2015-2019
Prepared by
: Dr. Nitin Uniyal, Dr. Vipin Kumar, Dr. Pradeep Malik, Dr. Sanoj Kumar and Dr. Anupam Bhandari
Email
: (nuniyal, vipin, pmalik, sanoj.kumar, abhandari)@upes.ac.in
Approved By ___________________________ HOD UPES Campus
___________________________ Associate Dean Tel : +91-135-2770137
“Energy Acres”
Fax : +91 135- 27760904
P.O. Bidholi, Via Prem Nagar, Dehradun
Website : www.upes.ac.in
COURSE PLAN A. PREREQUISITE: a. Basic concepts of Mathematics taught up to B. Tech Semester-I level. b. Basic concepts of differential equations and its solution. c. Basic knowledge of differentiation and integration rules. d. Basic knowledge of Mean, Median and Mode of data. B. PROGRAM OUTCOMES (POs) for B. Tech: PO1. An ability to apply knowledge of mathematics, science, and engineering PO2. An ability to design and conduct experiments as well as to analyze and interpret data PO3. An ability to design a system, component, or process to meet desired needs within realistic constraints such as economic, environmental, social, political, ethical, health and safety, manufacturability, and sustainability PO4. An ability to function on multidisciplinary teams PO5. An ability to identify, formulates, and solves engineering problems PO6. An understanding of professional and ethical responsibility PO7. An ability to communicate effectively PO8. The broad education necessary to understand the impact of engineering solutions in a global, economic, environmental, and societal context PO9. Recognition of the need for and an ability to engage in life-long learning PO10. Knowledge of contemporary issues PO11. An ability to use the techniques, skills, and modern engineering tools necessary for engineering practice
C. COURSE OUTCOMES FOR MATHEMATICS-II: At the end of this course student should be able to CO1. Develop insight into the concept of Integral transformations (Laplace and Fourier Transforms) and their applicability in solving various equations. CO2. Understand the dynamical behavior of real world systems by the concept of differential equations, their formulation, solution, physical interpretation and applications in various engineering disciplines. This includes the study of various techniques to solve first and second order differential equations with constant and variable coefficients. CO3. Discuss the fundamental concepts of probability and statistics from an engineering perspective emphasizing mainly on applications. CO4. Work with the fundamental differential operators of vector calculus, compute integrals over a variety of regions of space, understand the relation between line and surface integrals, surface and volume integrals, use the integral theorems to move from one type of integral to another, and applications to various physical problems. CO5. Develop technical writing skills of students by means of practical assignments bridging mathematical theory and engineering applications. Table: Correlation of POs v/s COs PO/CO CO1 CO2 CO3 CO4 CO5 1. WEAK
PO1 2 3 3 3 3
PO2 -
PO3 PO4 PO5 3 3 3 3 3
PO6 -
2. MODERATE
PO7 -
PO8 PO9 PO10 -
PO11 2 2 2 2 2
3. STRONG
D. PEDAGOGY The course will be taught using lecture method. The concepts will be adequately illustrated with examples to make applications of theoretical concepts clear. Students will be required to sole relevant problems.
E. COURSE COMPLETION PLAN Total Class room sessions Total Tests Total Assignment
43 02 04
One Session =60 minutes
F. EVALUATION & GRADING Students will be evaluated based on the following 3 stages: 5.1 Internal Assessment 30% 5.2 Mid-term Examination 20% 5.3 End term Examination 50% F1. INTERNAL ASSESSMENT: WEIGHTAGE – 30%
Internal Assessment shall be done based on the following: Sl. No.
Description
% of Weightage out of 30%
1
Common Class Tests
40%
2
Assignments/Tutorials (Problems/Presentations)
40%
3
Attendance and Discipline in the class
20%
F2. Internal Assessment Record Sheet (including Mid Term Examination marks) will be
displayed online at the end of semester i.e. last week of regular classroom teaching. F3. CLASS TESTS: Two Common Class Tests based on descriptive type theoretical &
numerical questions based on objective type questions will be held; one common class test at least ten days before the Mid Term Examination and second common class test at least ten days before the End Term Examination. Those who do not appear in test examinations shall lose their marks. The marks obtained by the students will be displayed on Black-Board a week before the start of Mid Term and End Term Examinations respectively.
F4. ASSIGNMENTS: After completion of each unit or in the mid of the unit, there will be
home assignments based on theory and numerical problems. Those who fail to submit the assignments by the due date shall lose their marks. F5. GENERAL DISCIPLINE: Based on student’s regularity, punctuality, sincerity and behavior in the class. The marks obtained by the students will be displayed on Black-Board at the end of semester. F6. MID TERM EXAMINATION:
WEIGHTAGE – 20%
Mid Term examination shall be Two Hours duration and shall be a combination of Short and Long theory Questions. Date of showing Mid Term Examination Answer Sheets: Within a week after completion of Mid Term examination. F7. END TERM EXAMINATION:
WEIGHTAGE – 50%
End Term Examination shall be Three Hours duration and shall be a combination of Short and Long theory/numerical Questions. Date of showing End Term Examination Answer Sheets: Within three week after completion of End Term examination. F8. GRADING:
The overall marks obtained at the end of the semester comprising all the above three mentioned shall be converted to a grade. G. COURSE DELIVERY PLAN Topics/Subtopics
Unit 1 Ordinary Differential Equations
No. of Sessions
Course Outcomes addressed
Assignments/ Tests
9
CO2, CO5
Assignment 1
CO1, CO5
Assignment 2
CO4, CO5
Assignment 3
CO3, CO5
Assignment 4
Unit 2 Integral Transform
11
Unit 3 Vectors
9
Unit 4 Statistics
14
S.N.
Unit
Contents 1. Linear Differential Equations with Constant Coefficients
Unit I 1.
Ordinary
2. Cauchy-Euler Differential Equations 3. Solution of Second Order Differential Equations (when a part of
Differential complementary function is known, by reduction to Normal Form, by Equations
changing the Independent Variable and by Variation of Parameters)
1. Laplace Transform 2.Unit Step Function and Dirac-Delta Function 3. Periodic Functions Unit II 2. Integral Transform
4. Differentiation and Integration of Laplace Transform 5. Inverse Laplace Transform 6. Convolution Theorem 7. Solution of Linear Differential Equations 8. Fourier Transform 1. Differentiation of vector valued functions and applications
Unit III
2. Gradient, Divergence, Curl 3. Integration of vector valued functions: Line, Surface and Volume
3. Vectors
Integrals 4. Applications of Green’s, Gauss divergence and Stokes Theorems 1. Random Variable: Discrete and Continuous 2. Probability mass and Probability density Functions 3. Moments, Skewness and Kurtosis
Unit IV 4. Statistics
4. Moment Generating Functions and their properties 5. Binomial, Poisson and Normal Distributions 6. Correlation: Carl-Pearson coefficient and Spearman Brown’s Rank correlation 7. Linear Regression 8. Chi Square Test
H. DETAILED SEESSION PLAN Topics
# Lectures
References
Pedagogy
UNIT I: ORDINARY DIFFERENTIAL EQUATIONS 1. Solution of Linear Differential equation with
L1
constant coefficients 2. Particular integral for non-homogeneous
L2-L3
Linear Differential equation 3. Cauchy-Euler Differential equation
L4
4. Solution of LDE of type: 𝑦 ′′ (𝑥) + 𝑃(𝑥)𝑦 ′ (𝑥) + 𝑄(𝑥)𝑦(𝑥) = 𝑅(𝑥):
a. When a part of C.F. is known
L5
b. Reduction to normal form
L6
c. Changing the independent variable
L7
d. Method of variation of parameters.
L8-L9
Ref- 1,2,3
Assignment – 1
Text- 1,2,3
Class test - 1
UNIT II: INTEGRAL TRANSFORMS 1. Laplace transform and sufficient condition of
L10
existence: Piecewise continuous function and growth restriction. 2. Evaluation of 𝑓(𝑡)
𝐿{𝑓(𝑡)}, 𝐿{𝑒 𝑎𝑡 𝑓(𝑡)}, 𝐿{𝑡 𝑛 𝑓(𝑡)}, 𝐿 { 𝑡 𝑛 }
L11
where 𝑓(𝑡) is an elementary function. 3. Unit Step function and Dirac delta function and their Laplace
L12 Ref -1,2,3
transforms and their properties.
Text- 1,2,3,
4. 𝐿{𝑓(𝑡)}, where 𝑓(𝑡) is periodic. 𝑡
5. 𝐿{𝑓 (𝑛) (𝑡)}, 𝐿 {∫0 𝑓(𝑡)𝑑𝑡} , Initial and final value theorems
L13 L14
Assignment – 2
6. Evaluation of integrals using Laplace
L15
transforms. 7. Inverse Laplace transform using
L16
Shifting theorems, Heaviside’s expansion formula 8. Convolution theorem and its
L17
applications 9. Solution of Linear Differential
L18-L19
Equation using Laplace transform 10. Fourier transform
L20 UNIT III: VECTORS
1. Scalar and vector fields,
L21
Differentiation of vector valued function. 2. Gradient of scalar function, divergence and
L22-L23
curl of a vector valued function. 3. Line integral and path
L24
independence of conservative field 4. Surface integral
L25
5. Volume integral
L26
6. Green’s theorem in a plane
L27
7. Stokes’ s theorem
L28
8. Gauss’s divergence theorem
L29 UNIT IV: STATISTICS
1. Random Variable: Discrete and
L30
Continuous 2. Probability mass and Probability
L31-L32
density Functions 3. a. Moments about mean, origin and
L33
Ref- 1,2,3
Assignment – 3
Text- 1,2,3
Class test -2
arbitrary point. b. Skewness and Kurtosis c. Moment generating function and
L34 L35-L36
its properties
Text -1,2,3,4
Assignment -4
5. Probability distributions: a. Binomial distribution
L37
b. Poisson distribution
L38
c. Normal distribution
L39
6. Correlation: Carl-Pearson coefficient
L40-L41
and Spearman Brown’s Rank correlation 7. Linear Regression
L42
8. Chi-square test
L43
I. SUGGESTED READINGS: I1. TEXT BOOK:
1. Jain, R. K., Iyengar, S. R. K., "Advanced Engineering Mathematics", 3e, Narosa Publications, 2. Kreyszig, Erwin., "Advanced Engineering Mathematics", 9e, Wiley Publications, 2006 3. Ramana, B. V., "Higher Engineering Mathematics", Tata McGraw Hill publications, 2007 4. Miller, I. and Miller, M., “John E. Freund’s Mathematical Statistics and applications” 7e Pearson, 2003. I2. REFERRENCE BOOKS:
1. Stewart, James, “Calculus Early Transcendentals”, Cengage Learning, 2013. 2. Jeffery, Alan, “Advanced Engineering Mathematics”, Academic Press, 2005. 3. Greenberg, Michael, “Advanced Engineering Mathematics”, Pearson, 2013.
GUIDELINES
Cell Phones and other Electronic Communication Devices: Cell phones and other electronic communication devices (such as Blackberries/Laptops) are not permitted in classes during Tests or the Mid/Final Examination. Such devices MUST be turned off in the class room. E-Mail and online learning tool: Each student in the class should have an e-mail id and a pass word to access the Black-Board system regularly. Regularly, important information – Date of conducting class tests, guest lectures, via online learning tool. The best way to arrange meetings with us or ask specific questions is by email and prior appointment. All the assignments preferably should be uploaded on online learning tool. Various research papers/reference material will be mailed/uploaded on online learning platform time to time. Attendance: Students are required to have minimum attendance of 75% in each subject. Students with less than said percentage shall NOT be allowed to appear in the end semester examination. Course outcome assessment: To assess the fulfilment of course outcomes two different approaches have been decided. Degree of fulfillment of course outcomes will be assessed in different ways through direct assessment and indirect assessment. In Direct Assessment, it is measured through tests, assignment, Mid-term and/or End-term examinations. It is suggested that each examination is designed in such a way that it can address one or two outcomes (depending upon the course completion). Indirect assessment is done through the student survey which needs to be designed by the faculty (sample format is given below) and it shall be conducted towards the end of course completion. The evaluation of the achievement of the Course Outcomes shall be done by analyzing the inputs received through Direct and Indirect Assessments and then corrective actions suggested for further improvement. Passing criterion: Student has to secure minimum 40% marks of the “highest marks in the class scored by a student in that subject (in that class/group class)” individually in both the ‘End-Semester examination’ and ‘Total Marks’ in order to pass in that paper.
Passing Criterion for B. Tech: minimum 40% of the highest marks in the class
Passing Criterion for M. Tech: minimum 40% of the highest marks in the class
Sample format for Indirect Assessment of Course outcomes NAME: ENROLLMENT NO: SAP ID: COURSE: PROGRAM:
Please rate the following aspects of course outcomes of Mathematics II. Use the scale 1-4* Sl. No. 1
1 CO1. Develop insight into the concept of Integral transformations
(Laplace and Fourier Transforms) and their applicability in solving various equations. 2
3 4
5
CO2. Understand the dynamical behavior of real world systems by the concept of differential equations, their formulation, solution, physical interpretation and applications in various engineering disciplines. This includes the study of various techniques to solve first and second order differential equations with constant and variable coefficients. CO3. Discuss the fundamental concepts of probability and statistics from an engineering perspective emphasizing mainly on applications. CO4. Work with the fundamental differential operators of vector calculus, compute integrals over a variety of regions of space, understand the relation between line and surface integrals, surface and volume integrals, use the integral theorems to move from one type of integral to another, and applications to various physical problems. CO5. Develop technical writing skills of students by means of practical assignments bridging mathematical theory and engineering applications.
*
1
Below Average
3
Good
2
Average
4
Very Good
2
3
4
