Chapter IV The Differential and Integral Calculus 120
Its two fundamental Divisions 120
Their Relations to each Other 121
1. Use of the Differential Calculus as preparatory to that of the Integral 123 2. Employment of the Differential Calculus alone 125 3. Employment of the Integral Calculus alone 125
Three Classes of Questions hence resulting 126
The Differential Calculus 127
Two Cases: Explicit and Implicit Functions 127
Two sub-Cases: a single Variable or several 129
Two other Cases: Functions separate or combined 130 Reduction of all to the Differentiation of the ten elementary Functions 131 Transformation of derived Functions for new Variables 132 Different Orders of Differentiation 133 Analytical Applications 133
The Integral Calculus 135
Its fundamental Division: Explicit and Implicit Functions 135 Subdivisions: a single Variable or several 136 Calculus of partial Differences 137 Another Subdivision: different Orders of Differentiation 138 Another equivalent Distinction 140 Quadratures 142
Integration of Transcendental Functions 143
Integration by Parts 143
Integration of Algebraic Functions 143 Singular Solutions 144 Definite Integrals 146 Prospects of the Integral Calculus 148