Differential and Integral Calculus Volume II2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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CHAPTER Ⅰ PRELIMINARY REMARKS ON ANALYTICAL GEOMETRY AND VECTOR ANALYSIS1

1．Rectangular Co-ordinates and Vectors1

2．The Area of a Triangle，the Volume of a Tetrahedron the Vector Multiplication of Vectors12

3．Simple Theorems on Determinants of the Second and Third Order19

4．Affine Transformations and the Multiplication of Determinants27

CHAPTER Ⅱ FUNCTIONS OF SEVERAL VARIABLES AND THEIR DERIVATIVES39

1．The Concept of Function in the Case of Several Variables39

2．Continuity44

3．The Derivatives of a Function50

4．The Total Differential of a Function and its Geometrical Meaning59

5．Functions of Functions (Compound Functions) and the Introduction of New Independent Variables69

6．The Mean Value Theorem and Taylor＇s Theorem for Functions of Several Variables78

7．The Application of Vector Methods82

APPENDIX95

1．The Principle of the Point of Accumulation in Several Dimensions and its Applications95

2．The Concept of Limit for Functions of Several Variables101

3．Homogeneous Functions108

CHAPTER Ⅲ DEVELOPMENTS AND APPLICATIONS OF THE DIFFERENTIAL CALCULUS111

1．Implicit Functions111

2．Curves and Surfaces in Implicit Form122

3．Systems of functions，Transformations，and Mappings133

4．Applications159

5．Families of Curves，Families of Surfaces，and their Envelopes169

6．Maxima and Minima183

APPENDIX204

1．Sufficient Conditions for Extreme Values204

2．Singular Points of Plane Curves209

3．Singular Points of Surfaces211

4．Connexion between Euler＇s and Lagrange＇s Representations of the Motion of a Fluid212

5．Tangential Representation of a Closed Curve213

CHAPTER Ⅳ MULTIPLE INTEGRALS215

1．Ordinary Integrals as Functions of a Parameter215

2．The Integral of a Continuous Function over a Region of the Plane or of Space223

3．Reduction of the Multiple Integral to Repeated Single Integrals236

4．Transformation of Multiple Integrals247

5．Improper Integrals256

6．Geometrical Applications264

7．Physical Applications276

APPENDIX287

1．The Existence of the Multiple Integral287

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 Planimeter294

3．Volumes and Areas in Space of any Number of Dimensions298

4．Improper Integrals as Functions of a Parameter307

5．The Fourier Integral318

6．The Eulerian Integrals (Gamma Function)323

7．Differentiation and Integration to fractional Order．Abel＇s Integral Equation339

8．Note on the Definition of the Area of a Curved Surface341

CHAPTER Ⅴ INTEGRATION OVER REGIONS IN SEVERAL DIMENSIONS343

1．Line Integrals343

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 Plane370

4．Surface Integrals374

5．Gauss＇s Theorem and Green＇s Theorem in Space384

6．Stokes＇s Theorem in Space392

7．The Connexion between Differentiation and Integration for Several Variables397

APPENDIX402

1．Remarks on Gauss＇s Theorem and Stokes＇s Theorem402

2．Representation of a Source-free Vector Field as a Curl404

CHAPTER Ⅵ DIFFERENTIAL EQUATIONS412

1．The Differential Equations of the Motion of a Partiele in Three Dimensions412

2．Examples on the Mechanics of a Particle418

3．Further Examples of Differential Equations429

4．Linear Differential Equations438

5．General Remarks on Differential Equations450

6．The Potential of Attracting Charges468

7．Further Examples of Partial Differential Equations481

CHAPTER Ⅶ CALCULUS OF VARIATIONS491

1．Introduction491

2．Euler＇s Differential Equation in the Simplest Case497

3．Generalizations507

CHAPTER Ⅷ FUNCTIONS OF A COMPLEX VARIABLE522

1．Introduction522

2．Foundations of the Theory of Functions of a Complex Variable530

3．The Integration of Analytic Functions537

4．Cauchy＇s Formula and its Applications545

5．Applications to Complex Integration (Contour Integration)554

6．Many-valued Functions and Analytic Extension563

SUPPLEMENT569

Real Numbers and the Concept of Limit569

Miscellaneous Examples587

Summary of Important Theorems and Formulae600

Answers and Hints623

Index679