Contents Preface ........................................................................................................................ (vii)
UNIT - 1
| Linear Systems of Equations
1.1 Introduction ............................................................................................................... 2 1.1.1 Matrices - Fundamental Concepts ............................................................ 2 1.1.2 Notation ..................................................................................................... 2 1.1.3 Rectangular Matrix ................................................................................... 3 1.1.4 Square Matrix ........................................................................................... 3 1.1.5 Row Matrix ............................................................................................... 3 1.1.6 Column Matrix .......................................................................................... 4 1.1.7 Null (zero) Matrix ..................................................................................... 4 1.1.8 Principal Diagonal ..................................................................................... 4 1.1.9 Unit Matrix or Identity Matrix .................................................................. 4 1.1.10 Diagonal Matrix ........................................................................................ 5 1.1.11 Scalar Matrix ............................................................................................ 5 1.1.12 Triangular Matrices ................................................................................... 5 1.1.13 Equality of Matrices .................................................................................. 5 1.1.14 Trace of a Square Matrix.......................................................................... 6 1.1.15 Transpose of a Matrix ............................................................................... 6 1.1.16 Symmetric Matrix ..................................................................................... 7 1.1.17 Skew-Symmetric Matrix ........................................................................... 7 1.1.18(1) Complex Matrix ......................................................................................... 7 1.1.18(2) Conjugate of a Matrix ............................................................................... 8 1.1.19 Transposed Conjugate of a Matrix ............................................................ 8 1.1.20 Hermitian Matrix ....................................................................................... 8 1.1.21 Skew-Hermitian Matrix ............................................................................ 9 (ix)
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1.1.22 Addition and Substraction of Matrices ..................................................... 9 1.1.23 Multiplication of Matrices ....................................................................... 10 1.1.24 Idempotent Matrix .................................................................................. 11 1.1.25 Nilpotent Matrix ...................................................................................... 11 1.1.26 Periodic Matrices .................................................................................... 12 1.1.27 Involutary Matrix .................................................................................... 12 1.1.28 Unitary Matrix ........................................................................................ 12 1.1.29 Orthogonal Matrix ................................................................................... 12 1.1.30 Properties of an Orthogonal Matrix ........................................................ 14 1.2 Rank of a Matrix ..................................................................................................... 21 1.2.1 Submatrix ................................................................................................ 21 1.2.2 Minors of Matrix ..................................................................................... 21 1.2.3 Rank of a Matrix ..................................................................................... 22 1.2.4 Calculation of the rank of a given matrix ................................................ 22 1.2.5 Zero rows and non-zero rows ................................................................. 24 1.2.6 Elementary Transformations or Elementary Operations ........................ 25 1.2.7 Elementary Matrices ............................................................................... 25 1.2.8 Equivalent Matrices ................................................................................ 25 1.2.9 Methods to determine the Rank of a Matrix .......................................... 25 1.2.10 Echelon form of a matrix ........................................................................ 26 1.2.11 Normal Form or Canonical Form ............................................................ 26 1.3 Inverse of a Square Matrix-Gauss Jardan Method ............................................... 41 1.3.1 Singular and non-singular matrices ........................................................ 41 1.3.2 Inverse of a square matrix ..................................................................... 42 1.3.3 Computation of inverse of a given matrix by Gauss-Jordan method ..... 42 1.4 Simultaneous Linear Equations ............................................................................... 47 1.4.1 Matrix form of a system of nonhomogeneous Linear Equations ............ 47 1.4.2 Consistency and Inconsistency ............................................................... 48 1.4.3 Augmented matrix (A/B) ........................................................................ 49 1.4.4 Condition for the consistency of (1) ....................................................... 49 1.4.5 Working Rules ......................................................................................... 49 1.4.6 Some Other Methods of Solving Non-homogenous Linear Equations ... 50 1.4.7 Gauss Elimination Method ...................................................................... 51 1.4.8 LU Decomposition .................................................................................. 52 1.4.9 LU Decomposition from Guass Elimination ........................................... 54 1.4.10 Tri-diagonal Matrix .................................................................................. 55 1.4.11 Solution of Tri-diagonal Systems ............................................................. 56 1.4.12 Homogeneous Linear equations .............................................................. 57 1.5 Gauss-Siedel Iterative Method ................................................................................ 89
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1.6 Finding Current in a Electrical Circuit .................................................................... 99 1.6.1 Current in multi-loop circuit ................................................................... 100 Summary ...................................................................................................................... 110 Solved University Questions (JNTU) ....................................................................... 113 Objective Type Questions ......................................................................................... 158
UNIT - 2
| Eigen Values – Eigen Vectors and Quadratic Forms
2.1 Eigen Values and Eigen Vectors ........................................................................... 170 2.1.1 Matrix polynomial .................................................................................. 170 2.1.2 Characteristic matrix ............................................................................. 170 2.1.3 Characteristic equation of A ................................................................. 170 2.1.4 Eigen Values .......................................................................................... 171 2.1.5 Eigen vectors ........................................................................................ 171 2.1.6 Properties of Eigen Values and Eigen Vectors ..................................... 171 2.2 Cayley-Hamilton Theorem .................................................................................... 188 2.2.1 Cayley-Hamilton Theorem .................................................................... 188 2.3 Diagonalisation of a Matrix ................................................................................... 197 2.3.1 Diagonal matrix ..................................................................................... 197 2.3.2 Computations with Diagonal Matrices .................................................. 197 2.3.3 Definition ............................................................................................... 198 2.3.4 Theorem ................................................................................................ 198 2.3.5 Working rule to diagonalise a square matrix A .................................... 201 2.3.6 Computation of positive integral powers of an n × n matrix 'A' ........... 202 2.4 The reader is advised to study the definitions and properties of various types of matrices given in 1.1.9 to 1.1.21 and 1.1.24 to 1.1.29 before studying the following properties of eigen values and vectors. ................................................. 212 2.5 Properties of Eigen Values of Orthogonal, Hermitian, Skew-hermitian and Unitary Matrices .................................................................. 212 2.6 Linear Transformation .......................................................................................... 215 2.6.1 Non-singular Transformation ................................................................ 217 2.6.2 Orthogonal Transformation ................................................................... 217 2.7 Quadratic Forms ................................................................................................... 218 2.7.1 Quadratic Form ..................................................................................... 218 2.7.2 Matrix Representation of a Quadratic Form ........................................ 218 2.7.3 Rank of a quadratic form ...................................................................... 219
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2.7.4 2.7.5 2.7.6 2.7.7 2.7.8 2.7.9 2.7.10 2.7.11
Linear Transformation of a Quadratic Form ........................................ 223 Canonical form or Normal form of a quadratic form ........................... 223 Reduction of a given quadratic form to a canonical form .................... 223 Nature of a quadratic form q = X'AX with the help of principal minors of A ............................................................................. 225 The Nature of a quadratic form can also be discussed by simply examining the eigen values of its matrix ............................................... 226 Sylvester's Law..................................................................................... 226 Methods of reduction of a given quadratic form to canonical form ..... 227 Orthogonal reduction of a quadratic form ............................................ 233
2.8 Free Vibration of a Two Mass System ................................................................. 251 Summary ...................................................................................................................... 258 Solved University Questions (JNTU) ....................................................................... 263 Objective Type Questions ......................................................................................... 313
UNIT - 3 | Multiple Integrals 3.1 Curve Tracing ....................................................................................................... 318 3.1.1 Tracing of curves (polar form) .............................................................. 345 3.1.2 Tracing of curves when the equation is given in parametric form ....... 360 3.2 Riemann Sums ...................................................................................................... 370 3.2.1 Lower and upper Riemann integrals ..................................................... 371 3.2.2 Riemann integral ................................................................................... 371 3.3 Lengths of Plane Curves (Rectification) .............................................................. 371 3.4 Areas ..................................................................................................................... 386 3.5 Volume and Surface of Solids of Revolution ........................................................ 398 3.6 Area of the Surface of Revolution ........................................................................ 407 3.7 Multiple Integrals .................................................................................................. 415 3.8 Change of Order of Integration (Change of Variables) ........................................ 425 3.8.1 Changing from Cartesion to Polar co-ordinates ................................... 425 3.8.2 Change of order of integration .............................................................. 428 3.8.3 Triple integration ................................................................................... 432 3.9 Moment of Inertia ................................................................................................. 443 Summary ...................................................................................................................... 447 Solved University Questions (JNTU) ....................................................................... 450 Objective Type Questions ......................................................................................... 493
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UNIT - 4
| Special Functions
4.1 Gamma Function ................................................................................................... 502 4.1.1 Definition ............................................................................................... 502 4.1.2 Properties of Gamma Function ............................................................. 502 4.2 Beta Function ........................................................................................................ 511 4.2.1 Definition ............................................................................................... 511 4.2.2 Alternate definition to Beta function ..................................................... 512 4.2.3 Relation between Beta and Gamma Functions ..................................... 512 Summary ...................................................................................................................... 530 Solved University Questions ..................................................................................... 530 Objective Type Questions ......................................................................................... 549
UNIT - 5
| Vector Differentiation
5.1 Vector Differentiation ........................................................................................... 554 5.1.1 Vector Point function and vector field .................................................. 554 5.1.2 Differentiation of a vector .................................................................... 554 5.1.3 Application to space curves .................................................................. 555 5.2 Gradient of Scalar Function .................................................................................. 556 5.2.1 The Vector differential operator ‘DEL’ or ‘NABLA’, denoted as ‘ ∇ ’ is defined by .................................................................................. 556 5.2.2 Gradient ................................................................................................. 557 5.2.3 Physical significance of ‘grad φ ’ ......................................................... 557 5.2.4 Directional Derivative ........................................................................... 557 5.2.5 Some basic properties of the gradient .................................................. 558 5.3 The Divergence of a Vector Function .................................................................. 566 5.3.1 Definition ............................................................................................... 566 5.3.2 Physical significance of the divergence ................................................ 566 5.3.3 Some properties of Divergence ............................................................ 567 5.3.4 Solenoidal vectors ................................................................................. 568 5.4 Curl of a Vector Function...................................................................................... 573 5.4.1 If A is a differential vector function, then curl A is defined as, curl A = ∇ × A ..................................................................................... 573 5.4.2 Physical significance of curl ................................................................. 573 5.4.3 Irrotational Vector ................................................................................. 574 5.4.4 Properties .............................................................................................. 574 5.4.5 Conservative vector field ...................................................................... 575
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5.5 Laplacian Operator : ∇ 2 ...................................................................................... 583 5.5.1 Definition ............................................................................................... 583 5.5.2 Vector Identities .................................................................................... 583 5.5.3 Operation of ∇ on product of two functions ....................................... 586 Summary ...................................................................................................................... 592 Solved University Questions (JNTU) ....................................................................... 593 Objective Type Questions ......................................................................................... 604
UNIT - 6
| Vector Integration
6.1 Vector Integration ................................................................................................. 610 6.1.1 Ordinary integration of vectors ............................................................. 610 6.1.2 Line Integrals ........................................................................................ 610 6.1.3 Physical appliations ............................................................................... 611 6.1.4 Theorem ................................................................................................ 612 6.2 Surface Integrals ................................................................................................... 627 6.2.1 Definition ............................................................................................... 627 6.2.2 Definition of surface integral as the limit of a sum ............................... 629 6.2.3 Evaluation of a surface integral ............................................................ 629 6.2.4 Physical interpretation of surface integrals........................................... 630 6.3 Volume Integrals ................................................................................................... 639 6.3.1 Expression of volume integral as the limit of a sum ............................. 640 6.4 Green’s Theorem in the Plane ............................................................................... 646 6.4.1 Green’s Theorem .................................................................................. 646 6.4.2 Vector notation of Green’s theorem ...................................................... 647 6.4.3 Physical interpretation of Green’s theorem .......................................... 648 6.4.4 Application of Green’s theorem to the evaluation of area of a simple closed curve ............................................................... 648 6.5 Gauss Divergence Theorem ................................................................................. 660 6.6 Stoke’s Theorem ................................................................................................... 673 Summary ...................................................................................................................... 689 Solved University Questions (JNTU) ....................................................................... 691
