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CHAPTER IV.

• THE DIFFERENTIAL AND INTEGRAL CALCULUS    120
• Its two fundamental Divisions    120
• Their Relations to each Other    121
• 1. 1. Use of the Differential Calculus as preparatory to that of the Integral    123
  2. 2. Employment of the Differential Calculus alone    125
  3. 3. Employment of the Integral Calculus alone    125
  4. 1. Three Classes of Questions hence resulting    126
• The Differential Calculus    127
• 1. Two Cases: Explicit and Implicit Functions    127
  2. 1. Two sub-Cases: a single Variable or several    129
     2. Two other Cases: Functions separate or combined    130
  3. Reduction of all to the Differentiation of the ten elementary Functions    131
  4. Transformation of derived Functions for new Variables    132
  5. Different Orders of Differentiation    133
  6. Analytical Applications    133
• The Integral Calculus    135
• 1. Its fundamental Division: Explicit and Implicit Functions    135
  2. Subdivisions: a single Variable or several    136
  3. Calculus of partial Differences    137
  4. Another Subdivision: different Orders of Differentiation    138
  5. Another equivalent Distinction    140
  6. Quadratures    142
  7. 1. Integration of Transcendental Functions    143
     2. Integration by Parts    143
     3. Integration of Algebraic Functions    143
  8. Singular Solutions