Differential And Integral Calculus, Vol. 2 (Volume 2)

Differential and Integral Calculus......Page 5
 CONTENTS......Page 9
  1. Rectangular Co-ordinates and Vectors......Page 13
  2. The Area of a Triangle, the Volume of a Tetrahedron, the Vector Multiplication of Vectors......Page 24
  3. Simple Theorems on Determinants of the Second and Third Order......Page 31
  4. Affine Transformations and the Multiplication of Determinants......Page 39
  1. The Concept of Fu nction in the Case of Several Variables......Page 51
  2. Continuity......Page 56
  3. The Derivatives of a Function......Page 62
  4. The Total Differential of a Function and its Geometrical Meaning......Page 71
  5. Functions of Functions (Compound Functions) and the Introduction of New Independent Variables......Page 81
  6. The Mean Value Theorem and Taylor's Theorem for Functions of Several Variables......Page 90
  7. The Application of Vector Methods......Page 94
   1. The Principle of the Point of Accumulation in Several Dimensions and its Applications......Page 107
   2. The Concept of Limit for Functions of Several Variables......Page 113
   3. Homogeneous Functions......Page 120
   1. Implicit Functions......Page 123
  2.  C urves and Surfaces in Implicit Form......Page 134
  3. Systems of Functions, Transformations, and Mappings......Page 145
  4. Applications......Page 171
  5. Families of Curves, Families of Surfaces, and their Envelopes......Page 181
  6. Maxima and Minima......Page 195
   1. Sufficient Conditions for Extreme Values......Page 216
   2. Singular Points of Plane Curves......Page 221
   3. Singular Points of Surfaces......Page 223
   4. Connection between Euler's and Lagrange's Representations of the Motion of a Fluid......Page 224
   5. Tangential Representation of a Closed Curve......Page 225
  1. Ordinary Integrals as Functions of a Parameter......Page 227
  2. The Integral of a Continuous Function over a Region of the Plane or of Space......Page 235
  3. Reduction of the Multiple Integral to Repeated Single Integrals......Page 248
  4. Transformation of Multiple Integrals......Page 259
  5. Improper Integrals......Page 268
  6. Geometrical Applications......Page 276
  7. Physical Applications......Page 288
   1. The Existence of the Multiple Integral......Page 299
   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......Page 306
   3. Volumes and Areas in Space of any Number of Dimensions......Page 310
   4. Improper Integrals as Functions of a Parameter......Page 319
   5. The Fourier Integral......Page 330
   6. The Eulerian Integrals (Gamma Function)......Page 335
   7. Differentiation and Integration to Fractional Order. Abel's Integral Equation......Page 351
   8. Note on the Definition of the Area of a Curved Surface......Page 353
  1. Line Integrals......Page 355
  2. Connexion between Line Integrals and Double Integrals in the Plane. (The Integral Theorems of Gauss,  Stokes, and Green)......Page 371
  3. Interpretation and Applications of the Integral Theorems for the Plane......Page 382
  4. Surface Integrals......Page 386
  5. Gauss's Theorem and Green's Theorem in Space......Page 396
  6. Stokes's Theorem in Space......Page 404
  7. The Connexion between Differentiation and Integration for Several Variables......Page 409
   1. Remarks on Gauss's Theorem and Stokes's Theorem......Page 414
   2. Representation of a Source-free Vector Field as a Curl......Page 416
  1. The Differential Equations of the Motion of a Particle in Three Dimensions......Page 424
  2. Examples on the Mechanics of a Particle......Page 430
  3. Further Examples of Differential Equations......Page 441
  4. Linear Differential Equations......Page 450
  5. General Remarks on Differential Equations......Page 462
  6. The Potential of Attracting Charges......Page 480
  7. Further Examples of Partial Differential Equations......Page 493
  1. Introduction......Page 503
  2. Euler's Differential Equation in the Simplest Case......Page 509
  3. Generalizations......Page 519
  1. Int roduction......Page 534
  2. Foun dations of the Theory of Functions of a Complex Variable......Page 542
  3. The Integration of Analytic Functions......Page 549
  4. Cauchy's Formula and its Applications......Page 557
   5. Applications to Complex Integration (Contour Integration)......Page 566
  6. Many-valued Functions and Analytic Extension......Page 575
  Real Numbers and the Concept of limit......Page 581
  Miscellaneous Examples......Page 675
  Summary of Important Theorems and Formulae......Page 612
  Answers and Hints......Page 635
 Index......Page 691