Calculus and Differential Equations

About

Calculus is concerned with relating quantities, which vary in a non-linear way. It is used extensively in science and engineering (like velocity, acceleration, and current in a circuit etc., for which the function may be linear or non-linear.). If a system (involving independent and dependent variables) is dynamic,then study of calculus helps to analyze the behavior of the parameters. The study of calculus and its applications to Science and Industrial problems is the most significant factor in the development of modern science.Most of the engineering problems are modelled as differential equations, starting from Newton’s laws of motion, Newton’s laws of cooling, bending of beams, motion of the planets, the spread of a disease, the oscillations of a suspension bridge, population dynamics, wave equations, heat equations etc. Therefore, study of differential equations is essential for most of the engineering applications.

Course Co-ordinator: Dr. Amit Mishra Contact No.: 079-30642521, E-mail: amitmishra@nirmauni.ac.in

Course Faculty

1. Dr. Amit Mishra Email: amitmishra@nirmauni.ac.in Contact No. 079-30642521 Office: PG 109 Visiting Hours:  (Monday –Friday) 1.30pm to2.00pm Odd Saturdays: 11.00am to12.00am	2. Dr. Kunal Pathak. Email: kunal.pathak@nirmauni.ac.in Contact No. 079-30642215 Office: B100 Visiting Hours: (Monday –Friday)  -1.30pm to 2.00pm Odd Saturdays: 11.00am to12.00am	3.  Dr. Motilal Pangrahi. Email: motilal.pangrahi@nirmauni.ac.in Contact No. 079-30642521 Office: A100 Visiting Hours: (Monday –Friday)  -1.30pm to 2.00pm Odd Saturdays: 11.00am to12.00am
4.Dr. Sandeep Malhotra Email: sandeep.malhotra@nirmauni.ac.in Contact No. 079-30642215 Office: B100 Visiting Hours: (Monday –Friday)  -1.30pm to 2.00pm Odd Saturdays: 11.00am to12.00am	5. Dr. Vijay Yadav Email: vijay.yadav@nirmauni.ac.in Contact No.079-30642142 Office: A203 Visiting Hours: (Monday –Friday) 1.30pm to2.00pm Odd Saturdays: 11.00am to12.00am	6. Dr. Dhirn Pandit Email: dhiren.pandit@nirmauni.ac.in Contact No. 079-30642142 Office: A203 Visiting Hours: (Monday –Friday) -1.30pm to 2.00pm Odd Saturdays: —11.00am to12.00am
7. Dr. Bijal Yeolekar Email: Bijal.yeolekar@nirmauni.ac.in Contact No. 079-30642142 Office: A203 Visiting Hours: (Monday –Friday) -1.30pm to 2.00pm Odd Saturdays: —11.00am to12.00am	8. Dr. Amisha Patel Email: amisha.patel@nirmauni.ac.in Contact No. 079-30642142 Office: A203 Visiting Hours: (Monday –Friday) -1.30pm to 2.00pm Odd Saturdays: —11.00am to12.00am

Course Learning Outcomes

After successful completion of the course, a student will be able to-

1. Apply differential and integral calculus to solve engineering problems,
2. Use of power series to solve differential equations appearing in engineering field,
3. Deal with functions of several variables those are essential in engineering.

Lesson Plan

Lecture No	Topic	Mapped CO
1	Overview of the course, Discussion on Course Policy, Course Blog, Importance of the course, Evaluation, Linkages of the course with other course’s and Professional relevance
2	Introduction of improper integrals	1
3	Evaluation of improper integrals	1
4	Beta and Gamma functions and their properties	1
5	Applications of definite integrals to evaluate surface areas	1,3
6	Applications of definite integrals to evaluate volumes of revolutions-1	1,3
7	Applications of definite integrals to evaluate volumes of revolutions-2	1,3
8	Double integration in Cartesian coordinates	1,3
9	Change of order of integration	1,3
10	Double integral in polar coordinates	1,3
11	Change of variable in double integral (Cartesian to polar)	1,3
12	Area as double integral (In Cartesian and polar coordinates)	1,3
13	Triple Integral	1,3
14	Volume by double Integrals	1,3
15	Volume by double Integrals	1,3
	Class Test
16	Center of mass and Gravity	1,3
17	Concept of Series and its convergence	2
18	Test of convergence by using Comparison test	2
19	De’ Alembert’s Ratio test	2
20	Cauchy’s root test and Integral test	2
21	Alternating series and Leibnitz’s test	2
22	Power series, Taylor’s and Maclaurin’s series.	2
23	Series for exponential, trigonometric and logarithmic functions	2
24	Concept of limit and continuity of functions of several variables	1,3
25	Partial derivatives: First order and Higher order	1,3
26	Total derivative, chain rule of composite functions	1,3
27	Euler’s theorem for homogeneous function	1,3
28	Taylor’s series in two variables, examples	1,3
	Sessional Examination
29	Tangent plane and normal line to a surface	1,3
30	Maxima, minima and saddle points	1,3
31	Lagrange’s Method of multiplier	1,3
32	Introduction to ordinary differential equation	2
33	Cauchy Euler Equation	2
34	Power series method for solution of second order ODE	2
35	Examples on Power series method for solution of second order ODE	2
36	Legendre polynomials	2
37	Frobenius Method for solution of second order homogeneous differential equation	2
38	Examples on Frobenius Method for solution of second order homogeneous differential equation	2
39	Bessel functions and series solution of Bessel’s equation	2
40	Bessel’s functions of the first kind	2
41	Properties of Bessel functions	2
42	Introduction of partial differential equation- Linear and non linear	1,2,3
43	Solution of first order non-linear PDE	1,2,3
44	Lagrange’s Linear equation	1,2,3
45	Review of the course, Feedback related to the course, Linkages with advanced course/s in succeeding years.

Syllabus for Class Test

1. Improper Integrals- Gama and Beta Functions
2. Surface area and volume of solid of revolution
3. Multiple integrals except center of mass and gravity