**Mathematical Methods book PDF Free download By SM Yousaf **

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## Content

**PREFACE**

**CHAPTER 1**

**TRIGONOMETRY 1**

**Section I**

**Complex Numbers 1**

**The Argand’s Diagram 7**

**EXERCISE 1.1 20**

**De Moivre’s Theorem 21**

**Roots of a Complex Number 31**

**EXERCISE 1.2 35**

**Basic Elementary Functions 37**

**EXERCISE 1.3 41**

**Logarithmic Function 43**

**Inverse Hyperbolic Functions 45**

**Inverse Trigonometric Functions48**

**Complex Powers 50**

**EXERCISE 1.4 53**

**Summation of Series 54**

**EXERCISE 1.5 59**

**CHAPTER 2**

**GROUPS 61**

**Definition and Examples**

**Properties of Groups 65**

**EXERCISE 2.1 69**

**Subgroups 71**

**Cyclic Groups 75**

**Cosets – Lagrange’s Theorem 77**

**EXERCISE 2.2 80**

**Permutations 82**

**Cycles 87**

**Transpositions 90**

**Order of a Permutation**

**EXERCISE 2.3**

**Rings and Fields 98**

**EXERCISE 2.4 105**

**CHAPTER 3**

**MATRICES 120**

**Introduction 107**

**Algebra of Matrices 110**

**Partitioning of Matrices**

**EXERCISE 3.1 126**

**Inverse of a Matrix 130**

**Elementary Row Operations**

**Elementary Column Operations**

**EXERCISE 3.2 144**

**CHAPTER 4 SYSTEMS OF LINEAR EQUATIONS 149**

**Preliminaries 149**

**Equivalent Fractions 150**

**CHAPTER 4 SYSTEMS OF LINEAR EQUATIONS 149**

**Preliminaries 149**

**Equivalent Equations 150**

**Gaussian Elimination Method 152**

**Gauss-Jordan Elimination Method 158**

**Consistency Criterion**

**Network Flow Problems 170**

**EXERCISE 4 174**

**CHAPTER 5 DETERMINANTS**

**Determinant of a Square Matrix**

**Axiomatic Definition of a Determinant 185**

**Determinant as Sum of Products of Elements**

**Determinant of the Transpose**

**An Algorithm to Evaluate Det A 195**

**Determinants and Inverses of Matrices**

**EXERCISE 5.1 209**

**Miscellaneous Results 216**

**EXERCISE 5.2 219**

**CHAPTER 6**

**VECTOR SPACES**

**223**

**Definition and Examples 223**

**Subspaces 230**

**Linear Combinations and Spanning Sets**

**233**

**EXERCISE 6.1 236**

**Linear Dependence and Basis 239**

**EXERCISE 6.2 251**

**Row and Column Spaces 253**

**Rank-Alternative Method 255**

**Proofs of Theorems in Chapter 4**

**Linear Transformations 264**

**EXERCISE 6.3 272**

**Matrix of a Linear Transformation 274**

**EXERCISE 6.4 279**

**CHAPTER 7**

**INNER PRODUCT SPACES 281**

**Definitions and Examples 281**

**Orthogonality 286**

**EXERCISE 7.1 294**

**Orthogonal Matrices 295**

**EXERCISE 7.2 299**

**Eigenvalues and Eigenvectors**

**EXERCISE 7.3 306**

**Similar Matrices 307**

**Symmetric Matrices 310**

**Diagonalization of Matrices**

**EXERCISE 7.4 323**

**CHAPTER 8**

**INFINITE SERIES**

**Sequences 325**

**EXERCISE 8.1 330**

**Infinite Series 331**

**The Basis Comparison Test 339**

**The Limit Comparison Test**

**The Integral Test 343**

**EXERCISE 8.2 347**

**The Ratio Test 349**

**Cauchy’s Root Test 352**

**EXERCISE 8.3 354**

**Alternating Series 356**

**Absolute and Conditional Convergence 358**

**EXERCISE 8.4 362**

**Power Series 363**

**Convergence of Power Series 364**

**Differentiation and Integration of Power Series 370**

**Addition and Multiplication of Power Series 373**

**EXERCISE 8.5 375**

**CHAPTER 9**

**FIRST-ORDER DIFFERENTIAL**

**EQUATIONS 377**

**D.E. and Their Classifications377**

**Formation of Differential Equations 380**

**Initial and Boundary Conditions 381**

**EXERCISE 9.1 384**

**Separable Equations 386**

**EXERCISE 9.2 388**

**Homogeneous Equations 388**

**D.E. Reducible to Homogeneous Form**

**EXERCISE 9.3 393**

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