## Mathematics-I for Civil Engineering stream

### Module-1

Calculus: Introduction to polar coordinates and curvature relating to Civil engineering. Polar coordinates, Polar curves, angle between the radius vector and the tangent, and angle between two curves. Pedal equations. Curvature and Radius of curvature - Cartesian, Parametric, Polar andPedal forms. Problems.

Self-study: Center and circle of curvature, evolutes and involutes.

Applications:Structural design and paths, Strength of materials, Elasticity.

### Module-2

Series Expansion and Multivariable Calculus: Introduction to series expansion and partial differentiation in the field of Civil 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 withsingle constraint.

Applications: Computation of stress and strain, Errors and approximations, Estimating the critical points and extreme values.

### Module-3

Ordinary Differential Equations (ODEs) of First Order: Introduction to first-order ordinary differential equations pertaining to the applications for Civil engineering.Linear and Bernoulli’s differential equations. Exact and reducible to exact differential equations -Integrating factors on 1/𝑁(𝜕𝑀/𝜕𝑦 −𝜕𝑁/𝜕𝑥) 𝑎𝑛𝑑 1/𝑀(𝜕𝑁/𝜕𝑥 −𝜕𝑀/𝜕𝑦). Orthogonal trajectories and 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 in Civil Engineering problems like bending of the beam, whirling of shaft,solution of non-linear ODE by the method of solvable for x and y.

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

### Module-4

Ordinary Differential Equations of Higher Order: Importance of higher-order ordinary differential equations in Civil engineering applications.Higher-order linear ODEs with constant coefficients - Inverse differential operator, method of variation of parameters, Cauchy’s and Legendre’s homogeneous differential equations -Problems.

Self-Study: Formulation and solution of Cantilever beam. Finding the solution by the method of undetermined coefficients.

Applications: Oscillations of a spring, Transmission lines, Highway engineering.

### Module-5

Linear Algebra:Introduction of linear algebra related to Civil engineering applications. Elementary row transformationofa 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 linear equations by Gauss-Jacobi iterative method. Inverse of a square matrix by Cayley- Hamilton theorem.

Applications: Structural Analysis, Balancing equations.

List of Laboratory experiments:

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 a differential equation of oscillations of a spring/deflection of a beam with different loads

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 the Rayleigh power method.

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, 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” 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, 7thEd., 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.

Web links and Video Lectures (e-Resources):

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

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