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CHAPTER Ⅰ PRELIMINARY REMARKS ON ANALYTICAL GEOMETRY AND VECTOR ANALYSIS 1

1．Rectangular Co-ordinates and Vectors 1

2．The Area of a Triangle，the Volume of a Tetrahedron the Vector Multiplication of Vectors 12

3．Simple Theorems on Determinants of the Second and Third Order 19

4．Affine Transformations and the Multiplication of Determinants 27

CHAPTER Ⅱ FUNCTIONS OF SEVERAL VARIABLES AND THEIR DERIVATIVES 39

1．The Concept of Function in the Case of Several Variables 39

2．Continuity 44

3．The Derivatives of a Function 50

4．The Total Differential of a Function and its Geometrical Meaning 59

5．Functions of Functions (Compound Functions) and the Introduction of New Independent Variables 69

6．The Mean Value Theorem and Taylor＇s Theorem for Functions of Several Variables 78

7．The Application of Vector Methods 82

APPENDIX 95

1．The Principle of the Point of Accumulation in Several Dimensions and its Applications 95

2．The Concept of Limit for Functions of Several Variables 101

3．Homogeneous Functions 108

CHAPTER Ⅲ DEVELOPMENTS AND APPLICATIONS OF THE DIFFERENTIAL CALCULUS 111

1．Implicit Functions 111

2．Curves and Surfaces in Implicit Form 122

3．Systems of functions，Transformations，and Mappings 133

4．Applications 159

5．Families of Curves，Families of Surfaces，and their Envelopes 169

6．Maxima and Minima 183

APPENDIX 204

1．Sufficient Conditions for Extreme Values 204

2．Singular Points of Plane Curves 209

3．Singular Points of Surfaces 211

4．Connexion between Euler＇s and Lagrange＇s Representations of the Motion of a Fluid 212

5．Tangential Representation of a Closed Curve 213

CHAPTER Ⅳ MULTIPLE INTEGRALS 215

1．Ordinary Integrals as Functions of a Parameter 215

2．The Integral of a Continuous Function over a Region of the Plane or of Space 223

3．Reduction of the Multiple Integral to Repeated Single Integrals 236

4．Transformation of Multiple Integrals 247

5．Improper Integrals 256

6．Geometrical Applications 264

7．Physical Applications 276

APPENDIX 287

1．The Existence of the Multiple Integral 287

2．General Formula for the Area (or Volume) of a Region bounded by Segments of Straight Lines or Plane Areas (Guldin＇s Formula)．The Polar Planimeter 294

3．Volumes and Areas in Space of any Number of Dimensions 298

4．Improper Integrals as Functions of a Parameter 307

5．The Fourier Integral 318

6．The Eulerian Integrals (Gamma Function) 323

7．Differentiation and Integration to fractional Order．Abel＇s Integral Equation 339

8．Note on the Definition of the Area of a Curved Surface 341

CHAPTER Ⅴ INTEGRATION OVER REGIONS IN SEVERAL DIMENSIONS 343

1．Line Integrals 343

2．Connexion between Line Integrals and Double Integrals in the Plane．(The Integral Theorems or Gauss，Stokes，and Green) 359

3．Interpretation and Applications of the Integral Theorems for the Plane 370

4．Surface Integrals 374

5．Gauss＇s Theorem and Green＇s Theorem in Space 384

6．Stokes＇s Theorem in Space 392

7．The Connexion between Differentiation and Integration for Several Variables 397

APPENDIX 402

1．Remarks on Gauss＇s Theorem and Stokes＇s Theorem 402

2．Representation of a Source-free Vector Field as a Curl 404

CHAPTER Ⅵ DIFFERENTIAL EQUATIONS 412

1．The Differential Equations of the Motion of a Partiele in Three Dimensions 412

2．Examples on the Mechanics of a Particle 418

3．Further Examples of Differential Equations 429

4．Linear Differential Equations 438

5．General Remarks on Differential Equations 450

6．The Potential of Attracting Charges 468

7．Further Examples of Partial Differential Equations 481

CHAPTER Ⅶ CALCULUS OF VARIATIONS 491

1．Introduction 491

2．Euler＇s Differential Equation in the Simplest Case 497

3．Generalizations 507

CHAPTER Ⅷ FUNCTIONS OF A COMPLEX VARIABLE 522

1．Introduction 522

2．Foundations of the Theory of Functions of a Complex Variable 530

3．The Integration of Analytic Functions 537

4．Cauchy＇s Formula and its Applications 545

5．Applications to Complex Integration (Contour Integration) 554

6．Many-valued Functions and Analytic Extension 563

SUPPLEMENT 569

Real Numbers and the Concept of Limit 569

Miscellaneous Examples 587

Summary of Important Theorems and Formulae 600

Answers and Hints 623

Index 679