Differential Calculus | Calculus of variations |
Functions of n variables | Functions of infinitely many variables (method of finite differences [Fomi, p.4, l.- 17, l.- 10]) |
Variation of a function | 1. Variation of a functional
defined on a fixed region [Fomi, p.153, (4)]. 2. Variation of a functional defined on a variable region [Fomi, p.173, (95)]. |
All the partial derivatives of a function vanish at an extremum. | The variational derivative of a functional vanishes at every point (Euler’s equation) [Fomi, p. 28, l.- 12-l.- 4] |
Formulas for differentiating sums and products of functions, composite functions, etc. | Analogs are valid [Fomi, p.29, l.11-l.13] |
Method of Lagrange multipliers for finding extrema of functions of several variables. | 1. The isoperimetric problem [Fomi,
p.43, Theorem 1]. 2. Finite subsidiary condition: a limiting case of an isoperimetric problem [Fomi, p.48, l.- 8]. |
A quadratic form is positive definite Û its principal minors are all positive [Fomi, p.127, l.10]. | [Fomi, p.117, Theorem 3; pp.125-129, § 30]. |
Necessary conditions for a minimum [Fomi, p.97, l.- 4]. | Weak minimum : 1. Euler's equation
[Fomi, p.15, Theorem 1]. 2. Legendre's condition [Fomi, p.119, Theorem 1] or
[Akh, p.68, Theorem 2]. 3. Jacobi's condition [Fomi, p.124, Theorem 5]. Strong minimum: Weierstrass’ condition [Akh, p.65, Theorem 1]. |
Sufficient conditions for a minimum [Fomi, l.- 6-l.- 5]. | Sufficient conditions for a weak (strong) minimum [Fomi, p.116, Theorem; p.125, Theorem 6] ([Fomi, p.148, Theorem1]). |