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ENGINEERING  MATHEMATICS-I[As per Choice Based Credit System (CBCS) scheme](Effective from the academic year 2015 -2016)SEMESTER - I/II

Subject Code - 15MAT11
IA Marks - 20
Number of Lecture Hours/Week - 04   
Exam Marks - 80
Total Number of Lecture Hours - 50    
Exam Hours - 03

Course Objectives:
To enable the students to apply the knowledge of Mathematics in various engineering fields by making them to learn the following:
•nth derivatives of product of two functions and polar curves.
•Partial derivatives
•Vector calculus
•Reduction formulae of integration; To solve First order differential equations.
•Solution of system of linear equations , quadratic forms.

Module - 1
Differential Calculus -1: determination of nth order derivatives of
Standard functions - Problems. Leibnitz’s theorem (without proof) - problems. 
Polar Curves - angle between the radius vector and tangent, angle between two  curves, Pedal equation of polar curves. Derivative of arc length - Cartesian, Parametric and Polar forms (without proof)  - problems.  Curvature  and Radius of Curvature – Cartesian, Parametric, Polar  and  Pedal  forms
(without proof)  -problems

Module -2
Differential Calculus -2
Taylor’s and Maclaurin’s theorems for function of one variable(statement only)- problems.  Evaluation of Indeterminate forms.
Partial derivatives – Definition and simple problems,  Euler’s theorem(without proof) – problems, total derivatives, partial differentiation of composite functions-problems. Definition and evaluation of Jacobians 
Module – 3
Vector Calculus:
Derivative of  vector valued functions, Velocity,  Acceleration  and related problems, Scalar and Vector point functions. Definition of Gradient, Divergence and Curl-problems. Solenoidal and
Irrotational vector fields. Vector identities - div(ɸA),  curl (ɸA ), curl( grad ɸ),  div(curl A).     
Integral Calculus:
Reduction formulae - (m and n are  positive integers), evaluation of these integrals with standard limits (0 to π/2) and problems.
Differential Equations ;
Solution of first order and first degree differential equations – Exact, reducible to exact and Bernoulli’s differential equations .Orthogonal trajectories in Cartesian and polar form. Simple problems on Newton's law of cooling. 

Linear Algebra
 Rank of a matrix by elementary transformations, solution of system of linear equations - Gauss-elimination method, Gauss –Jordan method and   Gauss-Seidel method 
Eigen values and Eigen vectors, Rayleigh’s power method to find the largest Eigen value and the corresponding Eigen vector. Linear transformation, diagonalisation of a square matrix .
Reduction of  Quadratic form to Canonical form     

Course outcomes:
On completion of this course, students are able to
•Use partial derivatives to calculate rates of change of multivariate functions.
•Analyze position, velocity, and acceleration in two or three dimensions using the calculus of vector valued functions.
•Recognize and solve first-order ordinary differential equations,  Newton’s law of cooling 
•Use matrices techniques for solving systems of linear equations in the different areas of Linear Algebra.
Question paper pattern:
•The question paper will have ten questions. 
•Each full Question consisting of 16 marks
•There will be 2 full questions(with a maximum of  four sub questions) from each module. 
•Each full question will have sub questions covering all the topics under a module. 
•The students will have to answer 5 full questions, selecting one full question from each module.

Text Books:
1. B.S. Grewal, "Higher Engineering Mathematics", Khanna publishers, 42nd edition, 2013.
2. Erwin Kreyszig, "Advanced Engineering MathematicsI,  Wiley, 2013
Reference Books: 
1.       B.V. Ramana, "Higher Engineering M athematics", Tata Mc Graw-Hill, 2006
2.       N.P.Bali and Manish Goyal, "A text book of Engineering mathematics”, Laxmi publications, latest edition. 
3.       H.K. Dass and Er. RajnishVerma, "Higher Engineerig Mathematics", S.Chand publishing, 1st edition, 2011.
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