Università Roma Tre

1. Functions of n real variables
Vector spaces. Scalar product (Cauchy-Schwarz inequality), norm, distance,
standard topology, compactness in Rn. Continuous functions from Rn to Rm. Continuity and uniform continuity. Weierstrass theorem.
Definitions of partial and directional derivative, differentiable functions, gradient.
Total Differential Theorem Schwarz's Lemma, Proposition 5.24. Ck functions, chain rule. Hessian matrix.
Taylor's formula at the second order. Maximum and minimum stationary points Positive definite matrices.
Hessian and study of the nature of stationary points;
Differentiable functions from Rn to Rm; Jacobian matrix. Jacobian matrix of the
composition.

2. Normed spaces and Banach spaces
Examples. Convergent and Cauchy sequences. Equivalent standards. Equivalence of the norms in Rn. The space of
continuous functions with the sup norm of a Banach space. Matrix exponential.
Ordinary differential equations with constant coefficients. Neumann series
The fixed point theorem in Banach spaces.

3. Implicit functions
The theorem of implicit functions Teo. 7.1 (with Proposition 7.4 and the Theorem of
Inverse Function).
Constrained maximums and minimums, Lagrange multipliers (Prop. 7.9).

4. Ordinary differential equations
Examples: equations with separable variables, linear systems with constant coefficients
(solution with the matrix exponential), one-dimensional conservative systems.
Existence and uniqueness theorem (Theo 8.8).
Lipschitzian dependence on initial data Prop. 8.10.
The set of solutions of a system of linear differential equations of order n
forms an n-dimensional vector space (see paragraph 8.5). Wronskian, variation of constants.

Chierchia Analisi Matematica II

Giusti Analisi Matematica II