ENGINEERING MATHEMATICS

Size: px

Start display at page:

Download "ENGINEERING MATHEMATICS"

Brittney Parker
5 years ago
Views:

2 A TEXTBOOK OF ENGINEERING MATHEMATICS For B.Sc. (Engg.), B.E., B. Tech., M.E. and Equivalent Professional Examinations By N.P. BALI Formerly Principal S.B. College, Gurgaon Haryana Dr. MANISH GOYAL M.Sc. (Mathematics), Ph.D., CSIR-NET Associate Professor Department of Mathematics Institute of Applied Sciences & Humanities G.L.A. University, Mathura, U.P. LAXMI PUBLICATIONS (P) LTD BANGALORE CHENNAI COCHIN GUWAHATI HYDERABAD JALANDHAR KOLKATA LUCKNOW MUMBAI RANCHI NEW DELHI BOSTON, USA

3 Copyright 2014 by Laxmi Publications Pvt. Ltd. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Published by: LAXMI PUBLICATIONS (P) LTD 113, Golden House, Daryaganj, New Delhi Phone: Fax: info@laxmipublications.com Price: ` Only. First Edition : 1996, Sixth Edition : 2004, Seventh Edition : 2007, Reprint : 2008, 2009, 2010, Eighth Edition : 2011, Ninth Edition : 2014 OFFICES Bangalore Jalandhar Chennai Kolkata Cochin , Lucknow Guwahati , Mumbai , Hyderabad Ranchi EEM ATB ENGG MATH-BAL C Typeset at: Excellent Graphics, Delhi. Printed at:

5 CONTENTS 1. Complex Numbers Real Numbers Basic Properties of Real Numbers Complex Numbers Conjugate Complex Numbers Geometrical Representation of Complex Numbers Properties of Complex Numbers Standard Form of a Complex Number Effect of Rotation, in the Anti-clockwise Direction, Through an Angle on the Complex Number De Moivre s Theorem Roots of a Complex Number Exponential Function of a Complex Variable Circular Functions of a Complex Variable Trigonometrical Formulae for Complex Quantities Logarithms of Complex Numbers The General Exponential Function Hyperbolic Functions Formulae of Hyperbolic Functions Inverse Hyperbolic Functions C + is Method of Summation Theory of Equations and Curve Fitting Polynomial Zero Polynomial Equality of Two Polynomials Complete and Incomplete Polynomials Zero of a Polynomial Division Algorithm Polynomial Equation Root of an Equation Synthetic Division Fundamental Theorem of Algebra Multiplication of Roots Diminishing and Increasing the Roots Removal of Terms ( v )

6 ( vi ) Reciprocal Equations Sum of the Integral Powers of the Roots and Symmetric Functions Symmetric Functions of the Roots Descarte s Rule of Signs Cardon s Method Irreducible Case of Cardon s Solution Descarte s Method Ferrari s Solution of the Biquadratic Curve Fitting Graphical Method Method of Group Averages Equations Involving Three Constants Principle of Least Squares Method of Moments Matrices Definitions (Matrices) Addition of Matrices Multiplication of a Matrix by a Scalar Properties of Matrix Addition Matrix Multiplication Properties of Matrix Multiplication Transpose of a Matrix Properties of Transpose of a Matrix Symmetric Matrix Skew-symmetric Matrix (or Anti-symmetric Matrix) Every Square Matrix can Uniquely be Expressed as the Sum of a Symmetric Matrix and a Skew-symmetric Matrix Orthogonal Matrix For any Two Orthogonal Matrices A and B, Show that AB is an Orthogonal Matrix Adjoint of a Square Matrix Singular and Non-singular Matrices Inverse (or Reciprocal) of a Square Matrix The Inverse of a Square Matrix, if it Exists, is Unique Theorem : The Necessary and Sufficient Condition for a Square Matrix A to Possess Inverse is that A 0 (i.e., A is Non-singular) If A is Invertible, Then so is A 1 and (A 1 ) 1 = A If A and B be Two Non-singular Square Matrices of the Same Order, then (AB) 1 = B 1 A If A is a Non-singular Square Matrix, then so is A and (A ) 1 = (A 1 ) If A and B are Two Non-singular Square Matrices of the Same Order, then adj(ab) = (adj B) (adj A)

7 ( vii ) Elementary Transformations (or Operations) Elementary Matrices The Following Theorems on the Effect of E-operations on Matrices Hold Good Inverse of Matrix by E-operations (Gauss-jordan Method) Rank of a Matrix Solution of a System of Linear Equations Vectors Linear Dependence and Linear Independence of Vectors Linear Transformations Orthogonal Transformation Complex Matrices Characteristic Equation Eigen Vectors Cayley Hamilton Theorem Reduction of a Matrix to Diagonal Form Quadratic Forms Linear Transformation of a Quadratic Form Canonical Form Index and Signature of the Quadratic Form Definite, Semi-definite and Indefinite Real Quadratic Forms Law-of-inertia of Quadratic Form Reduction to Canonical Form by Orthogonal Transformation Analytical Solid Geometry Introduction Co-ordinate Axes and Co-ordinate Planes Co-ordinates of a Point Distance between Two Points Section Formula Centroid of a Triangle Tetrahedron Centroid of a Tetrahedron Angle between Two Skew (or Non-coplanar) Lines Direction Cosines of a Line A Useful Result Relation between Direction Cosines Direction Ratios of a Line Direction Ratios of the Line Joining Two Points Angle between Two Lines Find the Angle between Two Lines whose Direction Ratios are a 1, b 1, c 1 and a 2, b 2, c 2. Deduce the Condition for Perpendicularity and Parallelism of Two Lines

8 ( viii ) Projection To Prove that the Projection of the Join of two Points (x 1, y 1, z 1 ), (x 2, y 2, z 2 ) on a Line whose Direction Cosines are l, m, n is l(x 2 x 1 ) + m(y 2 y 1 ) + n(z 2 z 1 ) The Plane General Equation of First Degree in x, y, z Represents a Plane Intercept Form Normal Form Three Point Form (a) Angle between Two Planes (b) Perpendicular Distance of a Point from a Plane Any Plane Through the Intersection of Two Given Planes Planes Bisecting the Angles between Two Planes Projection on a Plane Theorem General Form Symmetrical Form Reduction of the General Equations to the Symmetrical Form Perpendicular Distance Formula To Find the Point of Intersection of the Line x x 1 y y1 z z1 l m n with the plane ax + by + cz + d = The Conditions that the Line x x 1 y y1 z z1 may be Parallel to l m n the Plane ax + by + cz + d = 0 are al + bm + cn = 0 and ax 1 + by 1 + cz 1 + d The Conditions that the Line x x 1 y y1 z z may Lie in the Plane l m n ax + by + cz + d = 0 are al + bm + cn = 0 and ax 1 + by 1 + cz 1 + d = The Condition for the Line x x 1 y y1 z z to be Perpendicular l m n to the Plane ax + by + cz + d = Angle between a Line and a Plane Any Plane Through a Given Line To Find the Condition that the Two Lines x x 1 y y1 z z1, l1 m1 n1 x x2 y y2 = z z 2 l2 m n 2 2 may Intersect (or May be Coplanar) and to Find the Equation of the Plane in which they Lie Shortest Distance between Two Lines

9 ( ix ) Magnitude and Equations of Shortest Distance Intersection of Three Planes Definition (The Sphere) Equations of a Sphere in Different Forms Touching Spheres Four-point Form Diameter Form Section of a Sphere by a Plane Intersection of Two Spheres Equations of a Circle Any Sphere Through a Given Circle Great Circle Definition of the Tangent Plane Equation of the Tangent Plane at a Point Angle of Intersection of Two Spheres Condition of Orthogonality of Two Spheres Definition (The Cone) Equation of the Cone with Vertex at the Origin The Direction Cosines (or Direction Ratios) of a Generator of a Cone Satisfy the Equation of the Cone whose Vertex is the Origin Quadric Cone Through the Axes Right Circular Cone To Find the Equation to the Cone whose Vertex is the Point (,, ) and Base the Conic F(x, y) = ax 2 + by 2 + 2hxy + 2fy + 2gx + c = 0, z = Enveloping Cone Angle between Two Lines in which a Plane Through the Vertex Cuts a Cone Definitions (The Cylinder) To Find the Equation to the Cylinder whose Generators are Parallel to the Line x y z and Intersect the Curve l m n Equation of Right Circular Cylinder Enveloping Cylinder Definition (The Conicoids) Succesive and Partial Differentiation Successive Differentiation Calculation of n th Order Derivatives Use of Partial Fractions Leibnitz Theorem Determination of the Value of The n th Derivative of a Function at x = Function of Two Variables

10 ( x ) 5.7. Continuity Partial Derivatives of First Order Partial Derivatives of Higher Order Homogeneous Functions Euler s Theorem on Homogeneous Functions If u is a Homogeneous Function of Degree n in x and y, Deductions From Euler s Theorem Composite Functions Differentiation of Composite Functions Taylor s Theorem for a Function of Two Variables Jacobians Definitions Properties of Jacobians (Chain Rules) Theorem Jacobian of Implicit Functions Functional Relationship Approximation of Errors Maxima and Minima of Functions of Two Variables Conditions for F(x, y) to be Maximum or Minimum Rule to Find The Extreme Values of a Function z = f(x, y) Conditions for f(x, y, z) to be Maximum or Minimum Lagrange s Method of Undetermined Multipliers Geometrical Meaning of Partial Derivatives Tangent Plane and Normal to a Surface Differentiation under Integral Sign Multiple Integrals Double Integrals Evaluation of Double Integrals Evaluation of Double Integrals in Polar Co-ordinates Change of Order of Integration Triple Integrals Change of Variables Area by Double Integration Volume as a Double Integral Volume as a Triple Integral Volumes of Solids of Revolution Calculation of Mass Centre of Gravity (c.g.) Centre of Pressure Moment of Inertia

11 ( xi ) Product of Inertia Principal Axes Vector Calculus Vector Functions Derivative of a Vector Function with respect to a Scalar General Rules for Differentiation Derivative of a Constant Vector Derivative of a Vector Function in terms of its Components If d F F () t has a Constant Magnitude, then F. = dt 7.7. If F F () t has a Constant Direction, then F d = dt 7.8. Geometrical Interpretation of dr dt Velocity and Acceleration Scalar and Vector Fields Gradient of a Scalar Field Geometrical Interpretation of Gradient Directional Derivative Properties of Gradient Divergence of a Vector Point Function Curl of a Vector Point Function Physical Interpretation of Divergence Physical Interpretation of Curl Properties of Divergence and Curl Repeated Operations by Integration of Vector Functions Line Integrals Circulation Work Done by a Force Surface Integrals Volume Integrals Gauss Divergence Theorem (Relation between Surface and Volume Integrals) Green s Theorem in the Plane Stoke s Theorem (Relation between Line and Surface Integrals) Curvilinear Co-ordinates Definitions Unit Vectors in Curvilinear System Arc Length and Volume Element

12 ( xii ) 8.4. Gradient in Orthogonal Curvilinear Co-ordinates Divergence in Orthogonal Curvilinear Co-ordinates Curl in Orthogonal Curvilinear Co-ordinates Laplacian in Terms Of Orthogonal Curvilinear Co-ordinates Special Curvilinear Co-ordinate Systems Some More Special Curvilinear Co-ordinate Systems Infinite Series Sequence Real Sequence Range of a Sequence Constant Sequence Bounded and Unbounded Sequences Convergent, Divergent and Oscillating Sequences Monotonic Sequences Limit of a Sequence Every Convergent Sequence is Bounded Convergence of Monotonic Sequences Infinite Series Series of Positive Terms Alternating Series Partial Sums Behaviour of an Infinite Series Absolute Convergence of a Series Every Absolutely Convergent Series is Convergent Uniform Convergence of Series of Functions Fourier Series Periodic Functions Fourier Series Euler s Formulae Dirichlet s Conditions Fourier Series for Discontinuous Functions Change of Interval Half Range Series Fourier Series of Different Waveforms Parseval s Identity Root Mean Square Value (r.m.s. Value) Complex Form of Fourier Series Practical Harmonic Analysis

13 ( xiii ) 11. Differential Equations of First Order Definitions (Differential Equations) Geometrical Meaning of a Differential Equation of the First Order and First Degree Formation of a Differential Equation Solution of Differential Equations of the First Order and First Degree Variables Separable Form Homogeneous Equations Equations Reducible to Homogeneous Form Linear Differential Equations Equations Reducible to the Linear Form (Bernoulli s Equation) Exact Differential Equations Theorem Equations Reducible to Exact Equations Differential Equations of the First Order and Higher Degree Equations Solvable for p Equations Solvable for y Equations Solvable for x Clairaut s Equation Applications of Differential Equations of First Order Introduction Geometrical Applications Orthogonal Trajectories Working Rule to Find the Equation of Orthogonal Trajectories Physical Applications Application to Electric Circuits Conduction of Heat Rate of Growth or Decay Newton s Law of Cooling Chemical Reactions and Solutions Linear Differential Equations Definitions (Linear Differential Equations) The Operator D Theorems Auxiliary Equation (A.E.) Rules for Finding the Complementary Function The Inverse Operator f ( D ) Rules for Finding the Particular Integral Method of Variation of Parameters to Find P.I

14 13.9. Cauchy s Homogeneous Linear Equation Legendre s Linear Equation Simultaneous Linear Equations with Constant Co-efficients Total Differential Equations Method for Solving Pdx + Qdy + Rdz = Solution of Simultaneous Equations of the Form dx dy dz P Q R 14. Applications of Linear Differential Equations Introduction Simple Harmonic Motion (S.H.M.) Mechanical and Electrical Oscillatory Circuits Simple Pendulum Gain or Loss of Beats Deflection of Beams Boundary Conditions Applications of Simultaneous Linear Differential Equations Special Functions and Series Solution of Differential Equations Gamma Function Reduction Formula for (n) Value of ( ) Beta Function Symmetry of Beta Function i.e., B(m, n) = B(n, m) Relation between Beta and Gamma Functions To Evaluate z /2 p 0 sin q x. cos x dx; p > 1; q > Elliptic Integrals Applications of Elliptic Integrals Error Function Series Solution of Differential Equations Definitions Power Series Solution, When x = 0 is an Ordinary Point of the Equation 2 ( xiv ) d y 2 dx + P(x) dy + Q(x) y = dx Frobenius Method : Series Solution When x = 0 is a Regular Singular Point of the Differential Equation Legendre s Differential Equation Legendre s Function of First kind P n (x) Legendre s Function of Second kind Q n (x)

15 ( xv ) Solution of Legendre s equation Generating Function for P n (x) Rodrigue s Formula Recurrence Relations Beltrami s Result Orthogonality of Legendre Polynomials Laplace s Integral of First Kind Laplace s Integral of Second Kind Cristoffel s Expansion Formula Cristoffel s Summation Formula Expansion of a Function in a Series of Legendre Polynomials (Fourier-Legendre Series) Bessel s Differential Equation Solution of Bessel s Equation Series Representation of Bessel functions Recurrence Relations for J n (x) Generating Function for J n (x) Integral Form of Bessel Function Equations Reducible to Bessel s Equation Modified Bessel s Equation Ber and Bei Functions Orthogonality of Bessel Functions Fourier-bessel Expansion of F(x) Partial Differential Equations Introduction Formation of Partial Differential Equations Definitions Equations Solvable by Direct Integration Linear Partial Differential Equations of the First Order Lagrange s Linear Equation Working Method Non-linear Equations of the First Order (a) Equations of the Form f(p, q) = (b) Equations of the Form z = px + qy + f(p, q) (c) Equations of the Form f (z, p, q) = (d) Equations of the Form f 1 (x, p) = f 2 (y, q) Charpit s Method Homogeneous Linear Equations with Constant Co-efficients Rules for Finding the C.F Rules for Finding the P.I

9788131808320 Author : NP Bali and Dr Manish Goyal Type

16 A Textbook of Engineering Mathematics by NP Bali and Dr Manish Goyal 40% OFF Publisher : Laxmi Publications ISBN : Author : NP Bali and Dr Manish Goyal Type the URL : Get this ebook

A TEXTBOOK APPLIED MECHANICS

A TEXTBOOK APPLIED MECHANICS A TEXTBOOK OF APPLIED MECHANICS A TEXTBOOK OF APPLIED MECHANICS (Including Laboratory Practicals) S.I. UNITS By R.K. RAJPUT M.E. (Heat Power Engg.) Hons. Gold Medallist ; Grad. (Mech. Engg. & Elect. Engg.)