Differential and Integral Calculus

Volume 2

Chapter III Developments and Applications of the Differential Calculus

3.1 IMPLICIT FUNCTIONS

 

3.4 APPLICATIONS

 
3.1.1 General Remarks   3.4.1 Applications to the Theory of Surfaces
3.1.2 Geometrical Interpretation   3.4.2 Conformal Representation in General  
3.1.3 The Theorem of Implicit Functions   Exercises 3.4  
3.1.4 Examples      
3.1.5 The Theorem of Implicit Functions for more than Two Independent Variables  

3.5 FAMILIES OF CURVES, FAMILIES OF SURFACES AND THEIR ENVELOPES

  
3.1.6 Proof of the Existence and Continuity of Implicit Functions   3.5.1 General Remarks  
Exercises 3.1   3.5.2 Envelopes of One-Parameter families of Curves

3.2 CURVES AND SURFACES IN IMPLICIT FORM

   3.5.3 Examples  
3.2.1 Plane Curves in Implicit Form   Exercises 3.5  
3.2.2 Singular Points of Curves  

3.6 MAXIMA AND MINIMA

  
Exercises 3.2   3.6.1 Necessary Conditions  

3.3 SYSTEMS OF FUNCTIONS, TRANSFORMATlONS AND MAPPINGS

   3.6.2 Examples  
3.3.1 General Remarks   3.6.3 Maxima and Minima with Subsidiary Conditions  
3.3.2 Introduction of New Curvilinear Co-ordinates   3.6.4. Proof of the Method of Undetermined Multipliers in the Simplest Case  
3.3.3 Extension to More than Two Independent Variables   3.6.5 Generalization of the Method of Undetermined Multipliers
3.3.4 Differentiation Formulae for the Inverse Functions   3.6.6 Examples
3.3.5 Resolution and Combination of Mappings and Transfomations   Exercises 3.6  
3.3.6 General Theorem on the Inversion of Transfomations and Systems of Implicit Functions  

Appendix to Chapter III
3.3.7 Non-independent Functions   A3.1 Sufficient Conditions for Extreme Values
3.3.8 Concluding Remarks   A3.2 Singular Points of PlaneCurves  
Exercises 3.3   A3.3 Singular points of surfaces
    A3.4 Link between Euler's and Lagrange's Representations of the Motion of a Fluid  
    A3.5 Tangential Representation of a Closed Curve