Department Details

Course Name : Calculus and Linear Algebra

Credits : 04

Teaching Hours/week (L-T-P) : 3-2-0

Exam Hrs. : 03

Course Learning Objectives: This course Calculus and Linear Algebra (18MAT11) will enable the students :

      1. To familiarize the important tools of calculus and differential equation that are essential in all branches of engineering.

      2. To develop the knowledge of matrices and linear algebra in a comprehensive manner

Module No

Module Name

Description

Duration

1

Differential Calculus-1

Review of elementary calculus, Polar curves - angle between the radius vector and tangent, angle between two curves, pedal equation. Curvature and radius of curvature- Cartesian and polar forms (without proof). Centre and circle of curvature (formulae only) –applications to evolutes and involutes.

10 Hours

2

Differential Calculus-2

Taylor’s and Maclaurin’s series expansions for one variable (statements only), indeterminate forms- L’Hospital’s rule. Partial differentiation; Total derivatives-differentiation of composite functions. Maxima and minima for a function of two variables; Method of Lagrange multipliers with one subsidiary condition. Applications of maxima and minima with illustrative examples.

Jacobians-Simple problems.

10 Hours

3

Integral Calculus

Multiple integrals: Evaluation of double and triple integrals. Evaluation of double integrals- change of order of integration and changing into polar co-ordinates. Applications to find area, volume and centre of gravity.

Beta and Gamma functions: definitions, Relation between beta and gamma functions and simple problems.

10 Hours

4

Ordinary differential equations(ODE’s)of first order

Exact and reducible to exact differential equations. Bernoulli’s equation. Applications of ODE’s-orthogonal trajectories, Newton’s law of cooling and L-R circuits.

Nonlinear differential equations: Introduction to general and singular solutions; Solvable for p only; Clairaut’s and reducible to Clairaut’s equation only.

10 Hours

5

Elementary Linear Algebra

Rank of a matrix-echelon form. Solution of system of linear equations – consistency. Gauss-elimination method, Gauss –Jordan method and Gauss-Seidel method. Eigen values and eigen vectors- Rayleigh’s power method. Diagonalization of a square matrix of order two.

10 Hours

Course Outcomes (COs):

At the end of this course student will be able to,

        1. Apply the knowledge of calculus to solve problems related to polar curves and its applications in determining the bentness of a curve.

        2. Learn the notion of partial differentiation to calculate rates of change of multivariate functions and solve problems related to composite functions and Jacobians.

        3. Apply the concept of change of order of integration and variables to evaluate multiple integrals and their usage in computing the area and volumes.

        4. Solve first order linear/nonlinear differential equation analytically using standard methods.

        5. Make use of matrix theory for solving system of linear equations and compute eigen values and eigen vectors required for matrix diagonalization process.

Text Books:

      1. B.S. Grewal: Higher Engineering Mathematics, Khanna Publishers, 43rd Ed., 2015.

      2. E. Kreyszig: Advanced Engineering Mathematics, John Wiley & Sons, 10th Ed.(Reprint), 2016.

Reference books:

      1. C.Ray Wylie, Louis C.Barrett : “Advanced Engineering Mathematics", 6th Edition, McGraw-Hill Book Co., New York, 1995.

      2. N.P.Bali and Manish Goyal: A Text Book of Engineering Mathematics, Laxmi Publishers, 7th Ed., 2010.

      3. B.V.Ramana: "Higher Engineering Mathematics" 11th Edition, Tata McGraw-Hill, 2010.

      4. Veerarajan T.,” Engineering Mathematics for First year", Tata McGraw-Hill, 2008.

      5. Thomas G.B. and Finney R.L.”Calculus and Analytical Geometry”9th Edition, Pearson, 2012.

Web links and Video Lectures:

       1. http://nptel.ac.in/courses.php?disciplineID=111

       2. http://www.class-central.com/subject/math(MOOCs)

       3. http://academicearth.org/

Course Name : Advanced Calculus and Numerical Method

Credits : 04

Teaching Hours/week (L-T-P) : 3-2-0

Exam Hrs. : 03

Course Learning Objectives: The purpose of the course 18MAT21 is to facilitate the students with concrete foundation of vector calculus, ordinary and partial differential equations, infinite series and numerical methods enabling them to acquire the knowledge of these mathematical tools.

Module No

Module Name

Description

Duration

1

Vector Calculus

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

Vector Integration: Line integrals, Theorems of Green, Gauss and Stokes (without proof). Applications to work done by a force and flux.

10 Hours

2

Differential Equations of higher order

Second order linear ODE’s with constant coefficients-Inverse differential operators, method of variation of parameters; Cauchy’s and Legendre homogeneous equations. Applications to oscillations of a spring and L-C-R circuits.

10 Hours

3

Partial Differential Equations(PDE’s)

Formation of PDE’s by elimination of arbitrary constants / functions. Solution of non-homogeneous PDE by direct integration. Homogeneous PDEs involving derivative with respect to one independent variable only. Solution of Lagrange’s linear PDE. Derivation of one dimensional heat and wave equations and solutions by the method of separation of variables.

10 Hours

4

Infinite Series

Convergence and divergence of infinite series- Cauchy’s root test and D’Alembert’s ratio test(without proof)- Illustrative examples.

Power series solutions-Series solution of Bessel’s differential equation leading to Jn(x)- Bessel’s function of first kind-orthogonality. Series solution of Legendre’s differential equation leading to Pn(x)-Legendre polynomials. Rodrigue’s formula (without proof), problems.

10 Hours

5

Elementary Numerical Methods

Finite differences. Interpolation/extrapolation using Newton’s forward and backward difference formulae, Newton’s divided difference and Lagrange’s formulae (All formulae without proof). Solution of polynomial and transcendental equations – Newton-Raphson and Regula-Falsi methods( only formulae)- Illustrative examples.

Numerical integration: Simpson’s (1/3)th and (3/8)th rules, Weddle’s rule (without proof ) –Problems.

10 Hours

Course Outcomes: On completion of this course, students are able to:

       1. Solve first order linear/nonlinear differential equations analytically using standard methods.

       2. Explain various physical models through higher order differential equations and solve such linear ordinary differential equations.

       3. Understand a variety of partial differential equations and solution by exact methods/method of separation of variables.

       4. Describe the applications of infinite series and obtain series solution of ordinary differential equations.

       5. Apply the knowledge of numerical methods in the models of various physical and engineering phenomena

Text Books:

      1. B.S. Grewal: Higher Engineering Mathematics, Khanna Publishers, 43rd Ed., 2015.

      2. E. Kreyszig: Advanced Engineering Mathematics, John Wiley & Sons, 10th Ed.(Reprint), 2016.

Reference books:

      1. C.Ray Wylie, Louis C.Barrett : “Advanced Engineering Mathematics", 6th Edition, McGraw-Hill Book Co., New York, 1995.

      2. N.P.Bali and Manish Goyal: A Text Book of Engineering Mathematics, Laxmi Publishers, 7th Ed., 2010.

      3. B.V.Ramana: "Higher Engineering Mathematics" 11th Edition, Tata McGraw-Hill, 2010.

      4. Veerarajan T.,” Engineering Mathematics for First year", Tata McGraw-Hill, 2008.

      5. Thomas G.B. and Finney R.L.”Calculus and Analytical Geometry”9th Edition, Pearson, 2012.

Web links and Video Lectures:

       1. http://nptel.ac.in/courses.php?disciplineID=111

       2. http://www.class-central.com/subject/math(MOOCs)

       3. http://academicearth.org/

Course Name : Engineering Mathematics IV

Credits : 04

Teaching Hours/week (L-T-P) : 4-0-0

Exam Hrs. : 03

Course Objectives: This course will enable students to:

Conversant with numerical methods to solve ordinary differential equations, complex analysis, sampling theory and joint probability distribution and stochastic processes arising in science and engineering.

Module No

Module Name

Description

Duration

1

Numerical Methods-1

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. Milne’s and Adams-Bashforth predictor and corrector methods (No proof).

10 Hours

2

Numerical Methods-2 and Special Function

Numerical Methods: Numerical solution of second order ordinary differential equations, Runge-Kutta method and Milne’s method.

Special Functions: Series solution-Frobenious method. Series solution-Bessel’s differential equation leading to Jn(x)-Bessel’s function of first kind Basic properties and orthogonality.Series solution of Legendre’s differential equation leading to Pn(x)-Legendre polynomials. Rodrigue’s formula, problems.

10 Hours

3

Complex Variables

Complex Variables: Review of a function of a complex variable, limits, continuity, differentiability. Analytic functions-Cauchy-Riemann equations in cartesian and polar forms. Properties and construction of analytic functions. Complex line integrals-Cauchy’s theorem and Cauchy’s integral formula, Residue, poles, Cauchy’s Residue theorem ( without proof) and problems.

Transformations: Conformal transformations, discussion of transformations: w= _{ } , w=_{ } , w=z+_{ }: and bilinear transformations-problems.

10 Hours

4

Probability Theory

Probability Distributions: Random variables (discrete and continuous),probability mass/density functions. Binomial distribution, Poisson distribution. Exponential and normal distributions, problems

Joint probability distribution: Joint Probability distribution for two discrete random variables, expectation, covariance, correlation coefficient.

10 Hours

5

Sampling Theory

Sampling Theory: Sampling, Sampling distributions, standard error, test of hypothesis for means and proportions, confidence limits for means, student’s t-distribution, Chi-square distribution as a test of goodness of fit.

Stochastic process: Stochastic processes, probability vector, stochastic matrices, fixed points, regular stochastic matrices, Markov chains, higher transition probability-simple problems.

10 Hours

Course Outcomes (COs):

At the end of this course student will be able to

       1. Solve first and second order ordinary differential equations arising in flow problems using single step and multistep numerical methods.

       2. Solve problems of quantum mechanics, hydrodynamics and heat conduction by employing Bessel’s function relating to cylindrical polar coordinate systems and Legendre’s polynomials relating to spherical polar coordinate systems.

       3. Analyze the analyticity, potential fields, residues and poles of complex potentials in field theory and electromagnetic theory and describe conformal and bilinear transformation arising in aero foil theory, fluid flow visualization and image processing.\

       4. Solve problems on probability distributions relating to digital signal processing, information theory and optimization concepts of stability of design and structural engineering.

       5. Draw the validity of the hypothesis proposed for the given sampling distribution in accepting or rejecting the hypothesis and stochastic matrix connected with the multivariable correlation problems for feasible random events. Define transition probability matrix of a Markov chain and solve problems related to discrete parameter random process.

Course Name: Additional Mathematics - II

Credits : 00

Teaching Hours/week (L-T-P) : 3-0-0

Exam Hrs. : 03

Course Objectives: This course will enable students to:

• Understand essential concepts of linear algebra.

• Solve second and higher order differential equations.

• Understand Laplace and inverse Laplace transforms and elementary probability theory.

Module No

Module Name

Description

Duration

1

Linear Algebra

Introduction - rank of matrix by elementary row operations – Echelon form. Consistency of system of linear equations - Gauss elimination method. Eigen values and Eigen vectors of a square matrix. Application of Cayley-Hamilton theorem (without proof) to compute the inverse of a matrix-Examples.

10 Hours

2

Higher order ODE’s

Linear differential equations of second and higher order equations with constant coefficients. Homogeneous /non-homogeneous equations. Inverse differential operators. Solutions of initial value problems. Method of undetermined coefficients and variation of parameters.

10 Hours

3

Laplace transforms

Laplace transforms of elementary functions. Transforms of derivatives and integrals, transforms of periodic function and unit step function- Problems only.

10 Hours

4

Inverse Laplace transforms

Definition of inverse Laplace transforms. Evaluation of Inverse transforms by standard methods. Application to solutions of Linear differential equations and simultaneous differential equations.

10 Hours

5

Probability

Introduction. Sample space and events. Axioms of probability. Addition and multiplication theorems. Conditional probability – illustrative examples. Bayes‘s theorem-examples.

10 Hours

Course Outcomes: On completion of the course, students are able to:

       1. Solve systems of linear equations in the different areas of linear algebra.

       2. Solve second and higher order differential equations occurring in of electrical circuits, damped/un-damped vibrations.

       3. Describe Laplace transforms of standard and periodic functions.

       4. Determine the general/complete solutions to linear ODE using inverse Laplace transforms.

       5. Recall basic concepts of elementary probability theory and, solve problems related to the decision theory, synthesis and optimization of digital circuits..

Text Books:

      1. B.S. Grewal: Higher Engineering Mathematics, Khanna Publishers, 43rd Ed., 2015.

Reference books:

      1. E. Kreyszig: Advanced Engineering Mathematics, John Wiley & Sons, 10th Ed.(Reprint), 2016

      2. N.P.Bali and Manish Goyal: A Text Book of Engineering Mathematics, Laxmi Publishers, 7th Ed., 2010.