BMATM101 Mathematics for MES-I syllabus for Mechanical Stream Physics Group

Calculus

Introduction to polar coordinates and curvature relating to Mechanical engineering.

Polar coordinates, 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.

Applications: Applied Mechanics, Strength of Materials, Elasticity.

(RBT Levels: L1, L2 and L3)

Series Expansion and Multivariable Calculus

Introduction to series expansion and partial differentiation in the field of Mechanical engineering applications.

Taylor’s and Maclaurin’s series expansion for one variable (Statement only) – problems. Indeterminate forms - L’Hospital’s rule, Problems. 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’s undetermined multipliers with a single constraint.

Applications: Computation of stress and strain, Errors and approximations in manufacturing process, Estimating the critical points and extreme values, vector calculus.

(RBT Levels: L1, L2 and L3)

Ordinary Differential Equations (ODEs) of First Order

Introduction to first-order ordinary differential equations pertaining to the applications for Mechanical engineering.

Linear and Bernoulli’s differential equations. Exact and reducible to exact differential equationsIntegrating factors on 1⁄ 𝑁 ( 𝜕𝑀⁄ 𝜕𝑦 − 𝜕𝑁⁄ 𝜕𝑥) 𝑎𝑛𝑑 1⁄ 𝑀 ( 𝜕𝑁⁄ 𝜕𝑥 − 𝜕𝑀 ⁄𝜕𝑦). 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 ODEs: L-R circuits. Solvable for x and y.

Applications: Rate of Growth or Decay, Conduction of heat.

(RBT Levels: L1, L2 and L3)

Ordinary Differential Equations of Higher Order

Importance of higher-order ordinary differential equations in Mechanical engineering applications.

Higher-order linear ODEs with constant coefficients - Inverse differential operator, method of variation of parameters, Cauchy’s and Legendre homogeneous differential equations - Problems.

Self-Study: Formulation and solution of oscillations of a spring. Finding the solution by the method of undetermined coefficients.

Applications: Applications to oscillations of a spring, Mechanical systems and Transmission lines.

(RBT Levels: L1, L2 and L3)

Linear Algebra

Introduction of linear algebra related to Mechanical engineering applications.

Elementary row transformation of a matrix, Rank of a matrix. Consistency and solution of a 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 a system of equations by Gauss-Jacobi iterative method. Inverse of a square matrix by Cayley- Hamilton theorem.

Applications of Linear Algebra: Network Analysis, Balancing equations. (RBT Levels: L1, L2 and L3)

List of Laboratory experiments (2 hours/week per batch/ batch strength 15) 10 lab sessions + 1 repetition class + 1 Lab Assessment

1 2D plots for Cartesian and polar curves

2 Finding angle between polar curves, curvature and radius of curvature of a given curve

3 Finding partial derivatives and Jacobian

4 Applications to Maxima and Minima of two variables

5 Solution of first-order ordinary differential equation and plotting the solution curves

6 Solutions of Second-order ordinary differential equations with initial/ boundary conditions

7 Solution of differential equation of oscillations of spring with various load

8 Numerical solution of system of linear equations, test for consistency and graphical representation

9 Solution of system of linear equations using Gauss-Seidel iteration

10 Compute eigenvalues and eigenvectors and find the largest and smallest eigenvalue by Rayleigh power method. Suggested software’s: Mathematica/MatLab/Python/Scilab

Course outcome (Course Skill Set)

At the end of the course the student will be able to:

CO1 Apply the knowledge of calculus to solve problems related to polar curves.

CO2 Learn the notion of partial differentiation to compute rate of change of multivariate functions.

CO3 Analyze the solution of linear and non-linear ordinary differential equations. CO4 make use of matrix theory for solving the system of linear equations and compute eigenvalues and eigenvectors. CO5 familiarize with modern mathematical tools namely MATHEMATICA/ MATLAB/ PYTHON/SCILAB

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 out of 50).
• The minimum passing mark for the SEE is 35% of the maximum marks (18 marks out of 50).
• 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 total of the CIE (Continuous Internal Evaluation) and SEE (Semester End Examination) taken together.

Continuous Internal Evaluation (CIE):

The CIE marks for the theory component of the IC shall be 30 marks and for the laboratory component 20 Marks. CIE for the theory component of the IC

• Three Tests each of 20 Marks; after the completion of the syllabus of 35-40%, 65-70%, and 90- 100% respectively.
• Two Assignments/two quizzes/ seminars/one field survey and report presentation/one-course project totalling 20 marks. Total Marks scored (test + assignments) out of 80 shall be scaled down to 30 marks CIE for the practical component of the IC  On completion of every experiment/program in the laboratory, the students shall be evaluated and marks shall be awarded on the same day. The 15 marks are for conducting the experiment and preparation of the laboratory record, the other 05 marks shall be for the test conducted at the end of the semester.
• The CIE marks awarded in the case of the Practical component shall be based on the continuous evaluation of the laboratory report. Each experiment report can be evaluated for 10 marks. Marks of all experiments’ write-ups are added and scaled down to 15 marks.
• The laboratory test (duration 03 hours) at the end of the 15th week of the semester/after completion of all the experiments (whichever is early) shall be conducted for 50 marks and scaled down to 05 marks. Scaled-down marks of write-up evaluations and tests added will be CIE marks for the laboratory component of IC/IPCC for 20 marks.
• The minimum marks to be secured in CIE to appear for SEE shall be 12 (40% of maximum marks) in the theory component and 08 (40% of maximum marks) in the practical component. The laboratory component of the IC/IPCC shall be for CIE only. However, in SEE, the questions from the laboratory component shall be included. The maximum of 05 questions is to be set from the practical component of IC/IPCC, the total marks of all questions should not be more than 25 marks. The theory component of the IC shall be for both CIE and SEE.

Semester End Examination (SEE):

Theory SEE will be conducted by University as per the scheduled timetable, with common question papers for the subject (duration 03 hours)

• The question paper shall be set for 100 marks. The medium of the question paper shall be English/Kannada). The duration of SEE is 03 hours.
• The question paper will have 10 questions. Two questions per module. Each question is set for 20 marks. The students have to answer 5 full questions, selecting one full question from each module. The student has to answer for 100 marks and marks scored out of 100 shall be proportionally reduced to 50 marks.
• 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.

Suggested Learning Resources:

Books (Title of the Book/Name of the author/Name of the publisher/Edition and Year)

Text Books

1. B. S. Grewal: “Higher Engineering Mathematics”, Khanna Publishers, 44th Ed., 2021.

2. E. Kreyszig: “Advanced Engineering Mathematics”, John Wiley & Sons, 10th Ed., 2018.

Reference Books

1. V. Ramana: “Higher Engineering Mathematics” McGraw-Hill Education, 11th Ed., 2017

2. Srimanta Pal & Subodh C. Bhunia: “Engineering Mathematics” Oxford University Press, 3 rd Ed., 2016.

3. N.P Bali and Manish Goyal: “A Textbook of Engineering Mathematics” Laxmi Publications, 10th Ed., 2022.

4. C. Ray Wylie, Louis C. Barrett: “Advanced Engineering Mathematics” McGraw – Hill Book Co., New York, 6th Ed., 2017.

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, 3rd Ed., 2014.

7. James Stewart: “Calculus” Cengage Publications, 7th Ed., 2019.

8. David C Lay: “Linear Algebra and its Applications”, Pearson Publishers, 4th Ed., 2018.

9. Gareth Williams: “Linear Algebra with Applications”, Jones Bartlett Publishers Inc., 6th Ed., 2017.

10. Gilbert Strang: “Linear Algebra and its Applications”, Cengage Publications, 4th Ed., 2022.