Università Roma Tre

The numbers of the paragraphs and the theorems refer to the book by Chierchia
(unless otherwise specified).

1) Functions of $ n $ real variables

Vector spaces.
Scalar product (Cauchy-Schwarz inequality), norm, distance, standard topology, compactness in $ \ mathbb R ^ n $
(paragraphs 5.1, 5.2, 5.3).

Continuous functions from $ \ mathbb R ^ n $ to $ \ mathbb R ^ m $ (paragraph 5.5): Prop 5.8 : $ f $ is continuous
in $ x $ if and only if for each sequence $ x_k $ tending to $ x $ even $ f (x_k) $ tends to $ f (x) $; iii) if $ K $ is compact and $ f $ is continuous then $ f (K) $ is
compact; v) sums, products, quotients and compositions of continuous functions are continuous).
Weierstrass Teo 5.11 theorem, a continuous function on a compact is uniformly continuous Teo 5.12.

Paragraph 5.6,
definitions of partial and directional derivative, differentiable functions, gradient, Prop.5.21:
a differentiable function is continuous and has all the directional derivatives. Prop 5.22 ('' of the total differential ''). Schwarz's Lemma, Prop. 5.24.
Functions $ C ^ k $, chain rule (Prop. 5.30). Hessian Matrix.

Taylor's formula for the second order. Maximum and minimum stationary points (Def. 5.43) Positive definite matrices.

Prop. 5.44: the points of maximum or minimum are critical points; the critical points in which the Hessian matrix is ​​defined as positive (negative)
they are points of minimum (maximum); the critical points in which the Hessian matrix has a positive eigenvalue and a negative one are saddles.


Functions that can be differentiated from $ \ mathbb R ^ n $ to $ \ mathbb R ^ m $; Jacobian Matrix. Jacobian matrix of the composition.

2) Normed spaces and Banach spaces


Examples (numbers 1,2,6,7,9 of section 6.1). Convergent and Cauchy sequences (Def 6.2). Equivalent standards (Def 6.4). Equivalence of the rules in $ R ^ n $.
The space of continuous functions with the sup norm is a Banach space. Exponential matrix . Ordinary differential equations with constant coefficients (Oss. 6.8). Neumann series (Oss. 6.9).

The fixed point theorem in Banach spaces. 6:10

3) Implicit functions

 The theorem of implicit functions Teo. 7.1 (with Prop. 7.4 and the Theorem of the Inverse Function).

Maxima and minima on subsets, Lagrange multipliers (Prop. 7.9).


4) Ordinary differential equations

 Examples: equations with separable variables, linear systems with constant coefficients
 (solution with matrix exponential),
 one-dimensional conservative systems.
 
 Theorem of existence and uniqueness (Teo 8.8).

 Lipschitzian dependence from initial data Prop. 8.10.

 The set of solutions of a system of linear differential equations of order $ n $ forms a vectorial space $ n $ -dimensional
(see paragraph 8.5).

5) Curves in $ \ mathbb R ^ n $


From Chapter 15 of Giusti’s book: regular curves,
tangent versor, equivalent curves.
Length of a curve, curvilinear integrals.

Introduction to Calculus and Analysis 2
Richard Courant, ‎ Fritz John