|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]).|