## Mathematics-II for Civil Engineering stream

### Module-1

Integral Calculus:Introduction to Integral Calculus in Civil Engineering applications.Multiple Integrals: Evaluation of double and triple integrals, evaluation of double integrals by change of order of integration, changing into polar coordinates. Applications to find Area and Volume by double integral.Problems.

Beta and Gamma functions: Definitions, properties, relation between Beta and Gamma functions.Problems.

Self-Study: Volume by triple integration, Center of gravity.

Applications: Applications to mathematical quantities (Area, Surface area, Volume), Analysis of probabilistic models.

### Module-2

Vector Calculus:Introduction to Vector Calculus in Civil 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: Heat and mass transfer, oil refinery problems, environmental engineering. Analysis of streamlines, velocity and acceleration of a moving particle.

### Module-3

Partial Differential Equations:Importance of partial differential equations for Civil Engineering applications Formation of PDE's by elimination of arbitrary constants and functions. Solution of nonhomogeneous PDE by direct integration. Homogeneous PDEs involving derivatives with respect to one independent variable only. Solution of Lagrange's linear PDE.Derivation of one-dimensional heat equation and wave equation.

Self-Study: Solution of one-dimensional heat equation and wave equation by the method of separation of variables.

Applications: Design of structures (vibration of rod/membrane)

### Module-4

Numerical Methods -1:Importance of numerical methods for discrete data in the field of Civil Engineering.Solution of algebraic and transcendental equations: Regula-Falsi and Newton-Raphson methods (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.

Applications: Estimating the approximate roots, extremum values, area, volume, and surface area.Finding approximate solutions to civil engineering problems.

### Module-5

Numerical Methods -2:Introduction to various numerical techniques for handling Civil Engineering applications.Numerical Solution of Ordinary Differential Equations (ODE’s): 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 predictor-corrector formula (No derivations of formulae). Problems.

Applications: Finding approximate solutions to ODE related to civil engineering fields.

List of Laboratory experiments

1 Program to compute surface area, volume and centre of gravity

2 Evaluation of improper integrals

3 Finding gradient, divergent, curl and their geometrical interpretation

4 Verification of Green’s theorem

5 Solution of one-dimensional heat equation and wave equation

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’smethod

10 Solution of ODE of first order and first degree by Runge-Kutta 4th order and Milne’s predictor-corrector method

Suggested Learning Resources:

Text Books

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

2. E. Kreyszig: “Advanced Engineering Mathematics”, John Wiley & Sons, 10thEd., 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,3rd Ed., 2016.

3. N.P Bali and Manish Goyal: “A Textbook of Engineering Mathematics” LaxmiPublications, 10thEd., 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.ChandPublication, 3rd Ed.,2014.

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

Web links and Video Lectures (e-Resources):

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

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