Table of Contents
Preface vii
1 Vector Algebra I: Scalars and Vectors 1
1.1 Scalars and Vectors 1
1.2 Addition of Vectors 4
1.2.1 Sum of Two Vectors: Geometrical Addition 4
1.3 Subtraction of Vectors 6
1.4 Components and Projection of a Vector 7
1.5 Component Representation in Coordinate Systems 9
1.5.1 Position Vector 9
1.5.2 Unit Vectors 10
1.5.3 Component Representation of a Vector 11
1.5.4 Representation of the Sum of Two Vectors in Terms of Their Components 12
1.5.5 Subtraction of Vectors in Terms of their Components 13
1.6 Multiplication of a Vector by a Scalar 14
1.7 Magnitude of a Vector 35
2 Vector Algebra II: Scalar and Vector Products 23
2.1 Scalar Product 23
2.1.1 Application: Equation of a Line and a Plane 26
2.1.2 Special Cases 26
2.1.3 Commutative and Distributive Laws 27
2.1.4 Scalar Product in Terms of the Components of the Vectors 27
2.2 Vector Product 30
2.2.1 Torque 30
2.2.2 Torque as a Vector 31
2.2.3 Definition of the Vector Product 32
2.2.4 Special Cases 33
2.2.5 Anti-Commutative Law for Vector Products 33
2.2.6 Components of the Vector Product 34
3 Functions 39
3.1 The Mathematical Concept of Functions and its Meaning in Physics and Engineering 39
3.1.1 Introduction 39
3.1.2 The Concept of a Function 40
3.2 Graphical Representation of Functions 42
3.2.1 Coordinate System, Position Vector 42
3.2.2 The Linear Function: The Straight Line 43
3.2.3 Graph Plotting 44
3.3 Quadratic Equations 47
3.4 Parametric Changes of Functions and Their Graphs 49
3.5 Inverse Functions 50
3.6 Trigonometric or Circular Functions 52
3.6.1 Unit Circle 52
3.6.2 Sine Function 53
3.6.3 Cosine Function58
3.6.4 Relationships Between the Sine and Cosine Functions 59
3.6.5 Tangent and Cotangent 61
3.6.6 Addition Formulae 62
3.7 Inverse Trigonometric Functions 64
3.8 Function of a Function (Composition) 66
4 Exponential, Logarithmic and Hyperbolic Functions 69
4.1 Powers, Exponential Function 69
4.1.1 Powers 69
4.1.2 Laws of Indices or Exponents 70
4.1.3 Binomial Theorem 71
4.1.4 Exponential Function 71
4.2 Logarithm, Logarithmic Function 74
4.2.1 Logarithm 74
4.2.2 Operations with Logarithms 76
4.2.3 Logarithmic Functions 77
4.3 Hyperbolic Functions and Inverse Hyperbolic Functions 78
4.3.1 Hyperbolic Functions 78
4.3.2 Inverse Hyperbolic Functions 81
5 Differential Calculus 85
5.1 Sequences and Limits 85
5.1.1 The Concept of Sequence 85
5.1.2 Limit of a Sequence 86
5.1.3 Limit of a Function 89
5.1.4 Examples for the Practical Determination of Limits 89
5.2 Continuity 91
5.3 Series 92
5.3.1 Geometric Series 93
5.4 Differentiation of a Function 94
5.4.1 Gradient or Slope of a Line 94
5.4.2 Gradient of an Arbitrary Curve 95
5.4.3 Derivative of a Function 97
5.4.4 Physical Application: Velocity 98
5.4.5 The Differential 99
5.5 Calculating Differential Coefficients 100
5.5.1 Derivatives of Power Functions; Constant Factors 101
5.5.2 Rules for Differentiation 102
5.5.3 Differentiation of Fundamental Functions 106
5.6 Higher Derivatives 112
5.7 Extreme Values and Points of Inflexion; Curve Sketching 113
5.7.1 Maximum and Minimum Values of a Function 113
5.7.2 Further Remarks on Points of Inflexion (Contraflexure) 117
5.7.3 Curve Sketching 118
5.8 Applications of Differential Calculus 121
5.8.1 Extreme Values 121
5.8.2 Increments 122
5.8.3 Curvature 123
5.8.4 Determination of Limits by Differentiation: L'Hôpital's Rule 125
5.9 Further Methods for Calculating Differential Coefficients 127
5.9.1 Implicit Functions and their Derivatives 127
5.9.2 Logarithmic Differentiation 128
5.10 Parametric Functions and their Derivatives 129
5.10.1 Parametric Form of an Equation 129
5.10.2 Derivatives of Parametric Functions 133
6 Integral Calculus 145
6.1 The Primitive Function 145
6.1.1 Fundamental Problem of Integral Calculus 145
6.2 The Area Problem: The Definite Integral 147
6.3 Fundamental Theorem of the Differential and Integral Calculus 149
6.4 The Definite Integral 153
6.4.1 Calculation of Definite Integrals from Indefinite Integrals 153
6.4.2 Examples of Definite Integrals 156
6.5 Methods of Integration 159
6.5.1 Principle of Verification 159
6.5.2 Standard Integrals 159
6.5.3 Constant Factor and the Sum of Functions 160
6.5.4 Integration by Parts: Product of Two Functions 161
6.5.5 Integration by Substitution 164
6.5.6 Substitution in Particular Cases 166
6.5.7 Integration by Partial Fractions 170
6.6 Rules for Solving Definite Integrals 175
6.7 Mean Value Theorem 178
6.8 Improper Integrals 179
6.9 Line Integrals 181
7 Applications of Integration 191
7.1 Areas 191
7.1.1 Areas for Parametric Functions 194
7.1.2 Areas in Polar Coordinates 195
7.1.3 Areas of Closed Curves 197
7.2 Lengths of Curves 198
7.2.1 Lengths of Curves in Polar Coordinates 201
7.3 Surface Area and Volume of a Solid of Revolution 202
7.4 Applications to Mechanics 208
7.4.1 Basic Concepts of Mechanics 208
7.4.2 Center of Mass and Centroid 208
7.4.3 The Theorems of Pappus 211
7.4.4 Moments of Inertia; Second Moment of Area 213
8 Taylor Series and Power Series 227
8.1 Introduction 227
8.2 Expansion of a Function in a Power Series 228
8.3 Interval of Convergence of Power Series 232
8.4 Approximate Values of Functions 233
8.5 Expansion of a Function f(x) at an Arbitrary Position 235
8.6 Applications of Series 237
8.6.1 Polynomials as Approximations 237
8.6.2 Integration of Functions when Expressed as Power Series 240
8.6.3 Expansion in a Series by Integrating 242
9 Complex Numbers 247
9.1 Definition and Properties of Complex Numbers 247
9.1.1 Imaginary Numbers 247
9.1.2 Complex Numbers 248
9.1.3 Fields of Application 248
9.1.4 Operations with Complex Numbers 249
9.2 Graphical Representation of Complex Numbers 250
9.2.1 Gauss Complex Number Plane: Argand Diagram 250
9.2.2 Polar Form of a Complex Number 251
9.3 Exponential Form of Complex Numbers 254
9.3.1 Euler's Formula 254
9.3.2 Exponential Form of the Sine and Cosine Functions 255
9.3.3 Complex Numbers as Powers 255
9.3.4 Multiplication and Division in Exponential Form 258
9.3.5 Raising to a Power, Exponential Form 259
9.3.6 Periodicity of rej&alhpa; 259
9.3.7 Transformation of a Complex Number From One Form into Another 260
9.4 Operations with Complex Numbers Expressed in Polar Form 261
9.4.1 Multiplication and Division 261
9.4.2 Raising to a Power 263
9.4.3 Roots of a Complex Number 263
10 Differential Equations 273
10.1 Concept and Classification of Differential Equations 273
10.2 Preliminary Remarks 277
10.3 General Solution of First- and Second-Order DEs with Constant Coefficients 279
10.3.1 Homogeneous Linear DE 279
10.3.2 Non-Homogeneous Linear DE 285
10.4 Boundary Value Problems 291
10.4.1 First-Order DEs 291
10.4.2 Second-Order DEs 291
10.5 Some Applications of DEs 293
10.5.1 Radioactive Decay 293
10.5.2 The Harmonic Oscillator 294
10.6 General Linear First-Order DEs 302
10.6.1 Solution by Variation of the Constant 302
10.6.2 A Straightforward Method Involving the Integrating Factor 304
10.7 Some Remarks on General First-Order DEs 306
10.7.1 Bernoulli's Equations 306
10.7.2 Separation of Variables 307
10.7.3 Exact Equations 308
10.7.4 The Integrating Factor - General Case 311
10.8 Simultaneous DEs 313
10.9 Higher-Order DEs Interpreted as Systems of First-Order Simultaneous DEs 317
10.10 Some Advice on Intractable DEs 317
11 Laplace Transforms 321
11.1 Introduction 321
11.2 The Laplace Transform Definition 321
11.3 Laplace Transform of Standard Functions 322
11.4 Solution of Linear DEs with Constant Coefficients 328
11.5 Solution of Simultaneous DEs with Constant Coefficients 330
12 Functions of Several Variables; Partial Differentiation; and Total Differentiation 337
12.1 Introduction 337
12.2 Functions of Several Variables 338
12.2.1 Representing the Surface by Establishing a Table of Z-Values 339
12.2.2 Representing the Surface by Establishing Intersecting Curves 340
12.2.3 Obtaining a Functional Expression for a Given Surface 343
12.3 Partial Differentiation 344
12.3.1 Higher Partial Derivatives 348
12.4 Total Differential 350
12.4.1 Total Differential of Functions 350
12.4.2 Application: Small Tolerances 354
12.4.3 Gradient 356
12.5 Total Derivative 358
12.5.1 Explicit Functions 358
12.5.2 Implicit Functions 360
12.6 Maxima and Minima of Functions of Two or More Variables 361
12.7 Applications: Wave Function and Wave Equation 367
12.7.1 Wave Function 367
12.7.2 Wave Equation 371
13 Multiple Integrals; Coordinate Systems 377
13.1 Multiple Integrals 377
13.2 Multiple Integrals with Constant Limits 379
13.2.1 Decomposition of a Multiple Integral into a Product of Integrals 381
13.3 Multiple Integrals with Variable Limits 382
13.4 Coordinate Systems 386
13.4.1 Polar Coordinates 387
13.4.2 Cylindrical Coordinates 389
13.4.3 Spherical Coordinates 391
13.5 Application: Moments of Inertia of a Solid 395
14 Transformation of Coordinates; Matrices 401
14.1 Introduction 401
14.2 Parallel Shift of Coordinates: Translation 404
14.3 Rotation 407
14.3.1 Rotation in a Plane 407
14.3.2 Successive Rotations 410
14.3.3 Rotations in Three-Dimensional Space 411
14.4 Matrix Algebra 413
14.4.1 Addition and Subtraction of Matrices 415
14.4.2 Multiplication of a Matrix by a Scalar 416
14.4.3 Product of a Matrix and a Vector 416
14.4.4 Multiplication of Two Matrices 417
14.5 Rotations Expressed in Matrix Form 419
14.5.1 Rotation in Two-Dimensional Space 419
14.5.2 Special Rotation in Three-Dimensional Space 420
14.6 Special Matrices 421
14.7 Inverse Matrix 424
15 Sets of Linear Equations; Determinants 429
15.1 Introduction 429
15.2 Sets of Linear Equations 429
15.2.1 Gaussian Elimination: Successive Elimination of Variables 429
15.2.2 Gauss-Jordan Elimination 431
15.2.3 Matrix Notation of Sets of Equations and Determination of the Inverse Matrix 432
15.2.4 Existence of Solutions 435
15.3 Determinants 438
15.3.1 Preliminary Remarks on Determinants 438
15.3.2 Definition and Properties of an n-Row Determinant 439
15.3.3 Rank of a Determinant and Rank of a Matrix 444
15.3.4 Applications of Determinants 445
16 Eigenvalues and Eigenvectors of Real Matrices 451
16.1 Two Case Studies: Eigenvalues of 2 x 2 Matrices 451
