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Applied Mathematics-1

by Vtu Rangers
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ENGINEERING MATHEMATICS-I 15MAT11 SYLLABUS Engineering Mathematics-I Subject Code: 15MAT11 IA Marks: 20 Hours/Week: 04 Exam. Hours: 03 Total Hours: 50 Exam. Marks: 80 Course Objectives To enable students to apply knowledge of Mathematics in various engineering fields by making hem to learn the following: • nth derivatives of product of two functions and polar curves. • Partial derivatives. • Vectors calculus. • Reduction formulae of integration to solve First order differential equations • Solution of system of equations and 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 for 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) and problems. 10hrs Module –2 Differential Calculus -2 Taylor‟s and Maclaurin‟s theorems for function of o ne 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, Jacobians-definition and problems . 10hrs DEPT OF MATHS, SJBIT Page 1
ENGINEERING MATHEMATICS-I 15MAT11 Module –3 Vector Calculus: Derivative of vector valued functions, Velocity, Acceleration and related problems, Scalar and Vector point functions.Definition Gradient, Divergence, Curl- problems . Solenoidal and Irrotational vector fields. Vector identities - div ( F A), curl ( F A),curl (grad F ), div (curl A). 10hrs Module- 4 Integral Calculus: Reduction formulae ∫ sinnx dx ∫cosnx dx ∫sinnxcosmxdx,, (m and n are positive integers), evaluation of these integrals with standard limits (0 to л/2) and problems. Differential Equations: Solution of first order and first degree differential equations – Exact, reducible to exact and Bernoulli‟s differential equations. Applicationsorthogonal trajectories in Cartesian and polar forms. Simple problems on Newton‟s law of cooling. 10hrs Module –5 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. Rayleigh‟s power method to find the largest Eigen value and the corresponding Eigen vector. Linear transformation, diagonalisation of a square matrix, Quadratic forms, reduction to Canonical form 10hrs COURSE OUTCOMES On completion of this course students are able to Use partial derivatives to calculate rates of change of multivariate functions Analyse 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. DEPT OF MATHS, SJBIT Page 2
ENGINEERING MATHEMATICS-I 15MAT11 Engineering Mathematics – I  Module I : Differential Calculus- I…………….3 - 48  Module II : Differential Calculus- II……………49 -81  Module III: Vector Calculus………………........82-104  Module IV : Integral Calculus…………………105-125  Module V : Linear Algebra ……………………126-165 DEPT OF MATHS, SJBIT Page 3
ENGINEERING MATHEMATICS-I 15MAT11 MODULE I DIFFERENTIAL CALCULUS-I CONTENTS: Successive differentiation …………………………………………..3  nth derivatives of some standard functions…………………...7  Leibnitz‟s theorem (without proof)………………………..…16 Polar curves Angle between Polar curves…………………………………….20 Pedal equation for Polar curves………………………………...24 Derivative of arc length………………………………………….28 Radius of Curvature……………………………………………….34 Expression for radius of curvature in case of Cartesian Curve …35 Expression for radius of curvature in case of Parametric Curve………………………………………………………………..36 Expression for radius of curvature in case of Polar Curve……...41 Expression for radius of curvature in case of Pedal Curve……...43 DEPT OF MATHS, SJBIT Page 4

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