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CALCULUS AND DIFFERENTIAL EQUATIONS (21MAT11)

CALCULUS AND DIFFERENTIAL EQUATIONS

Course Code: 21MAT11
CIE Marks: 50
Teaching Hours/Week (L:T:P:S): 2:2:0:0
SEE Marks: 50
Total Hours of Pedagogy: 40
Total Marks: 100
Credits:03
Exam Hours:03

Course objectives: The goal of the course Calculus and Differential Equations - 21MAT11 is

 To facilitate the students with a concrete foundation of differential calculus
 To solve the first and higher-order ordinary differential equations enabling them to acquire the knowledge of these mathematical tools.
 To develop the knowledge of matrices and linear algebra in a comprehensive manner.

Teaching-Learning Process (General Instructions):

These are sample Strategies, which teachers can use to accelerate the attainment of the various course outcomes.
1. In addition to the traditional lecture method, different types of innovative teaching methods may be adopted so that the delivered lessons shall develop students’ theoretical and applied mathematical skills.
2. State the need for Mathematics with Engineering Studies and Provide real-life examples.
3. Support and guide the students for self–study. 
4. You will also be responsible for assigning homework, grading assignments and quizzes, and documenting students' progress. 
5. Encourage the students for group learning to improve their creative and analytical skills.
6. Show short related video lectures in the following ways:
● As an introduction to new topics (pre-lecture activity).
● As a revision of topics (post-lecture activity).
● As additional examples (post-lecture activity).
● As an additional material of challenging topics (pre-and post-lecture activity).
● As a model solution of some exercises (post-lecture activity).

Module-1: 

Differential Calculus - 1 Polar curves, angle between the radius vector and the tangent, angle between two curves. Pedal equations. Curvature and Radius of curvature - Cartesian, Parametric, Polar and Pedal forms. Problems. Self-study: Center and circle of curvature, evolutes and involutes. (RBT Levels: L1, L2 and L3 )
Teaching-Learning Process Chalk and talk method / Power Point Presentation


Module-2:

 Differential Calculus - 2 Taylor’s and Maclaurin’s series expansion for one variable (Statement only) – problems. Indeterminate forms-L’Hospital’s rule. Partial differentiation, total derivative-differentiation of composite functions. Jacobian and problems. Maxima and minima for a function of two variables. Problems. Self-study: Euler’s Theorem and problems. Method of Lagrange undetermined multipliers with single constraint. (RBT Levels: L1, L2 and L3)
Teaching-Learning Process Chalk and talk method / PowerPoint Presentation

Module-3: 

Ordinary Differential Equations (ODE’s) of first order  Linear and Bernoulli’s differential equations. Exact and reducible to exact differential equations. Applications of ODE’s-Orthogonal trajectories, Newton’s law of cooling. Nonlinear differential equations: Introduction to general and singular solutions; Solvable for p only; Clairaut’s equations, reducible to Clairaut’s equations. Problems. Self-Study: Applications of ODE’s: L-R circuits. Solvable for x and y. (RBT Levels: L1, L2 and L3)
Teaching-Learning Process: Chalk and talk method / PowerPoint Presentation

Module-4:

Ordinary Differential Equations of higher order Higher-order linear ODE’s with constant coefficients - Inverse differential operator, method of variation of parameters, Cauchy’s and Legendre homogeneous differential equations. Problems. Self-Study: Applications to oscillations of a spring and L-C-R circuits. (RBT Levels: L1, L2 and L3)
Teaching-Learning Process: Chalk and talk method / Power Point Presentation


Module-5:

Linear Algebra Elementary row transformation of a matrix, Rank of a matrix. Consistency and Solution of system of linear equations; Gauss-elimination method, Gauss-Jordan method and Approximate solution by Gauss-Seidel method. Eigenvalues and Eigenvectors-Rayleigh’s power method to find the dominant Eigenvalue and Eigenvector. Self-Study: Solution of system of equations by Gauss-Jacobi iterative method. Inverse of a square matrix by Cayley- Hamilton theorem. (RBT Levels: L1, L2 and L3).
Teaching-Learning Process: Chalk and talk method / Power Point Presentation

Course outcomes (Course Skills Set)

After successfully completing the course, the student will be able to understand the topics.
 Apply the knowledge of calculus to solve problems related to polar curves and its applications in determining the bentness of a curve.
 Learn the notion of partial differentiation to calculate rate of change of multivariate functions and solve problems related to composite functions and Jacobian.
 Solve first-order linear/nonlinear ordinary differential equations analytically using standard methods.
 Demonstrate various models through higher order differential equations and solve such linear ordinary differential equations.
 Test the consistency of a system of linear equations and to solve them by direct and iterative methods.

Assessment Details (both CIE and SEE)

The weightage of Continuous Internal Evaluation (CIE) is 50% and for Semester End Exam (SEE) is 50%. The minimum passing mark for the CIE is 40% of the maximum marks (20 marks). A student shall be deemed to have satisfied the academic requirements and earned the credits allotted to each subject/ course if the student secures not less than 35% ( 18 Marks out of 50)in the semester-end examination(SEE), and a minimum of 40% (40 marks out of 100) in the sum total of the CIE (Continuous Internal Evaluation) and SEE (Semester End Examination) taken together

Continuous Internal Evaluation:

Three Unit Tests each of 20 Marks (duration 01 hour)
1. First test at the end of 5th week of the semester
2. Second test at the end of the 10th week of the semester
3. Third test at the end of the 15th week of the semester
Two assignments each of 10 Marks
4. First assignment at the end of 4th week of the semester
5. Second assignment at the end of 9th week of the semester
Group discussion/Seminar/quiz any one of three suitably planned to attain the COs and POs for 20 Marks (duration 01 hours)
6. At the end of the 13th week of the semester
The sum of three tests, two assignments, and quiz/seminar/group discussion will be out of 100 marks and will be scaled down to 50 marks
(to have less stressed CIE, the portion of the syllabus should not be common /repeated for any of the methods of the CIE. Each method of CIE should have a different syllabus portion of the course).
CIE methods /question paper is designed to attain the different levels of Bloom’s taxonomy as per the outcome defined for the course.

Semester End Examination:

Theory SEE will be conducted by University as per the scheduled timetable, with common question papers for the subject (duration 03 hours)
1. The question paper will have ten questions. Each question is set for 20 marks.
2. There will be 2 questions from each module. Each of the two questions under a module (with a maximum of 3 sub-questions), should have a mix of topics under that module.
The students have to answer 5 full questions, selecting one full question from each moduleeory SEE will be conducted by University as per the scheduled time table, with common question papers for the subject
1. The question paper will have ten questions. Each question is set for 20 marks.
2. There will be 2 questions from each module. Each of the two questions under a module (with a maximum of 3 sub-questions), should have a mix of topics under that module.
3. The students have to answer 5 full questions, selecting one full question from each module

Suggested Learning Resources:

Text Books

1. B. S. Grewal: “Higher Engineering Mathematics”, Khanna publishers, 44th Ed.2018
2. E. Kreyszig: “Advanced Engineering Mathematics”, John Wiley & Sons, 10th Ed. (Reprint), 2016.


Reference Books

1. V. Ramana: “Higher Engineering Mathematics” McGraw-Hill Education, 11th Ed.
2. Srimanta Pal & Subodh C. Bhunia: “Engineering Mathematics” Oxford University Press, 3rd Reprint, 2016.
3. N.P Bali and Manish Goyal: “A textbook of Engineering Mathematics” Laxmi Publications, Latest edition.
4. C. Ray Wylie, Louis C. Barrett: “Advanced Engineering Mathematics” McGraw – Hill Book Co.Newyork, Latest ed.
5. Gupta C.B, Sing S.R and Mukesh Kumar: “Engineering Mathematic for Semester I and II”, Mc-Graw Hill Education(India) Pvt. Ltd 2015.
6. H.K.Dass and Er. Rajnish Verma: “Higher Engineering Mathematics” S.Chand Publication (2014).
7. James Stewart: “Calculus” Cengage publications, 7th edition, 4th Reprint 2019.

Web links and Video Lectures (e-Resources):

 http://.ac.in/courses.php?disciplineID=111
 http://www.class-central.com/subject/math(MOOCs)
 http://academicearth.org/
 VTU e-Shikshana Program
 VTU EDUSAT Program

Activity-Based Learning (Suggested Activities in Class)/ Practical Based learning

 Quizzes
 Assignments
 Seminars