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Note for Applied Mathematics-1 - M-1 By vtu rangers

  • Applied Mathematics-1 - M-1
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  • Visvesvaraya Technological University Regional Center - VTU
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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

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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

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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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ENGINEERING MATHEMATICS-I 15MAT11 SUCCESSIVE DIFFERENTIATION In this lesson, the idea of differential coefficient of a function and its successive derivatives will be discussed. Also, the computation of nth derivatives of some standard functions is presented through typical worked examples. 1. Introduction:- Differential calculus (DC) deals with problem of calculating rates of change. When we have a formula for the distance that a moving body covers as a function of time, DC gives us the formulas for calculating the body‟s velocity and acceleration at any instant. Definition of derivative of a function y = f(x):- x) f ( x) x The derivative of a function y = f(x) is the function f (x) whose value at each x is defined as dy = f (x) = Slope of the line PQ (See Fig.1) dx f (x x) f ( x) = lim -------- (1) x 0 x = lim (Average rate change) Fig.1. Slope of the line PQ is x f (x 0 = Instantaneous rate of change of f at x provided the limit exists. The instantaneous velocity and acceleration of a body (moving along a line) at any instant x is the derivative of its position co-ordinate y = f(x) w.r.t x, i.e., dy Velocity = = f (x) --------- (2) dx And the corresponding acceleration is given by Acceleration d2y dx 2 DEPT OF MATHS, SJBIT f ( x) ---------- (3) Page 5

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