BMATE201 Mathematics for EES-II syllabus for EE Stream Physics Group

Vector Calculus
Introduction to Vector Calculus in EC & EE engineering applications.

Vector Differentiation: Scalar and vector fields. Gradient, directional derivative, curl and divergence - physical interpretation, solenoidal and irrotational vector fields. Problems.

Vector Integration:

Line integrals, Surface integrals. Applications to work done by a force and flux. Statement of Green’s theorem and Stoke’s theorem. Problems.

Self-Study: Volume integral and Gauss divergence theorem.

Applications: Conservation of laws, Electrostatics, Analysis of streamlines and electric potentials.

(RBT Levels: L1, L2 and L3)

Vector Space and Linear Transformations

Importance of Vector Space and Linear Transformations in the field of EC & EE engineering applications.

Vector spaces:

Definition and examples, subspace, linear span, Linearly independent and dependent sets, Basis and dimension.

Linear transformations:

Definition and examples, Algebra of transformations, Matrix of a linear transformation. Change of coordinates, Rank and nullity of a linear operator, Rank-Nullity theorem. Inner product spaces and orthogonality.

Self-study: Angles and Projections. Rotation, reflection, contraction and expansion.

Applications: Image processing, AI & ML, Graphs and networks, Computer graphics.

(RBT Levels: L1, L2 and L3)

Laplace Transform

Importance of Laplace Transform for EC & EE engineering applications.

Existence and Uniqueness of Laplace transform (LT), transform of elementary functions, region of convergence. Properties–Linearity, Scaling, t-shift property, s-domain shift, differentiation in the sdomain, division by t, differentiation and integration in the time domain. LT of special functionsperiodic functions (square wave, saw-tooth wave, triangular wave, full & half wave rectifier), Heaviside Unit step function, Unit impulse function.

Inverse Laplace Transforms:

Definition, properties, evaluation using different methods, convolution theorem (without proof), problems, and applications to solve ordinary differential equations.

Self-Study: Verification of convolution theorem.

Applications: Signals and systems, Control systems, LR, CR & LCR circuits.

(RBT Levels: L1, L2 and L3)

Numerical Methods -1

Importance of numerical methods for discrete data in the field of EC & EE engineering applications.

Solution of algebraic and transcendental equations: Regula-Falsi method and Newton-Raphson method (only formulae). Problems. Finite differences, Interpolation using Newton’s forward and backward difference formulae, Newton’s divided difference formula and Lagrange’s interpolation formula (All formulae without proof). Problems.

Numerical integration:

Trapezoidal, Simpson's (1/3)rd and (3/8)th rules (without proof). Problems.

Self-Study: Bisection method, Lagrange’s inverse Interpolation, Weddle's rule.

Applications: Estimating the approximate roots, extremum values, area, volume, and surface area.

(RBT Levels: L1, L2 and L3)

Numerical Methods -2

Introduction to various numerical techniques for handling EC & EE applications.

Numerical Solution of Ordinary Differential Equations (ODEs):

Numerical solution of ordinary differential equations of first order and first degree - Taylor’s series method, Modified Euler’s method, Runge-Kutta method of fourth order and Milne’s predictorcorrector formula (No derivations of formulae). Problems.

Self-Study: Adam-Bashforth method.

Applications: Estimating the approximate solutions of ODE for electric circuits.

(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 Finding gradient, divergent, curl and their geometrical interpretation and Verification of Green’s theorem

2 Computation of basis and dimension for a vector space and Graphical representation of linear transformation

3 Visualization in time and frequency domain of standard functions

4 Computing inverse Laplace transform of standard functions

5 Laplace transform of convolution of two functions

6 Solution of algebraic and transcendental equations by Regula-Falsi and Newton-Raphson method

7 Interpolation/Extrapolation using Newton’s forward and backward difference formula

8 Computation of area under the curve using Trapezoidal, Simpson’s (1/3)rd and (3/8)th rule

9 Solution of ODE of first order and first degree by Taylor’s series and Modified Euler’s method

10 Solution of ODE of first order and first degree by Runge-Kutta 4th order and Milne’s predictor-corrector 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 Understand the applications of vector calculus refer to solenoidal, irrotational vectors, line integral and surface integral.

CO2 Demonstrate the idea of Linear dependence and independence of sets in the vector space, and linear transformation

CO3 To understand the concept of Laplace transform and to solve initial value problems.

CO4 Apply the knowledge of numerical methods in solving physical and engineering phenomena.

CO5 Get 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, 6 th 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, 3 rd Ed., 2014.

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

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