## 20402076 - AM210 - MATHEMATICAL ANALYSIS 3

I. To acquire a good knowledge of the theory for series and sequences of functions in R.
II. To develop and acquire the methods in the theory of continuous and regular functions in several real variables.

Curriculum

teacher profile | teaching materials

Programme

1. Sequences and series of functions
Pointwise convergence, uniform convergence.
Total convergence of series of functions.
Power series, Fourier series.

2. 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 derivatives, differentiable functions,
gradient, Prop .: a continuous differentiable function and has all the directional derivatives.
Schwarz's Lemma total differential theorem. Functions
Ck, chain rule. Hessian matrix.
Taylor's formula at second order. Maximum and minimum stationary points
Positive definite matrices.
Prop: maximum or minimum points are critical points; the critical points in which the
Hessian matrix is ​​positive (negative) are minimum (maximum) points; the points
critics in which the Hessian matrix has a positive and a negative eigenvalue are saddles.
Functions that can be differentiated from Rn to Rm; Jacobian matrix. Jacobian matrix of the
composition.

3. Normed spaces and Banach spaces
Examples. Converging and Cauchy sequences. Equivalent rules. Equivalence of the norms in Rn. The space of the
continuous functions with the sup norm a Banach space.
The fixed point theorem in Banach spaces.
Implicit and inverse function theorems.

Core Documentation

Analisi Matematica II, Giusti - Analisi Matematica II, Chierchia

Type of delivery of the course

4 hours of frontal teaching 2 of exercises and 2 of tutoring a week

Attendance

course attendance is strongly recommended

Type of evaluation

The written exam focuses on the topics covered in class and tends to verify the ability to solve exercises. It consists of 4 exercises on the topics discussed during the class. Students who pass the two intermediate tests need not take the written exam. The oral exam verifies the ability to present and prove the theorems studied in class and apply them in specific cases.

teacher profile | teaching materials

Programme

1. Sequences and series of functions
Pointwise convergence, uniform convergence.
Total convergence of series of functions.
Power series, Fourier series.

2. 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 derivatives, differentiable functions,
gradient, Prop .: a continuous differentiable function and has all the directional derivatives.
Schwarz's Lemma total differential theorem. Functions
Ck, chain rule. Hessian matrix.
Taylor's formula at second order. Maximum and minimum stationary points
Positive definite matrices.
Prop: maximum or minimum points are critical points; the critical points in which the
Hessian matrix is ​​positive (negative) are minimum (maximum) points; the points
critics in which the Hessian matrix has a positive and a negative eigenvalue are saddles.
Functions that can be differentiated from Rn to Rm; Jacobian matrix. Jacobian matrix of the
composition.

3. Normed spaces and Banach spaces
Examples. Converging and Cauchy sequences. Equivalent rules. Equivalence of the norms in Rn. The space of the
continuous functions with the sup norm a Banach space.
The fixed point theorem in Banach spaces.
Implicit and inverse function theorems.

Core Documentation

Analisi Matematica II, Giusti - Analisi Matematica II, Chierchia

Type of delivery of the course

4 hours of frontal teaching 2 of exercises and 2 of tutoring a week

Attendance

course attendance is strongly recommended

Type of evaluation

The written exam focuses on the topics covered in class and tends to verify the ability to solve exercises. It consists of 4 exercises on the topics discussed during the class. The oral exam verifies the ability to present and prove the theorems studied in class and apply them in specific cases.

teacher profile | teaching materials

Programme

1. Sequences and series of functions
Pointwise convergence, uniform convergence.
Total convergence of series of functions.
Power series, Fourier series.

2. 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 derivatives, differentiable functions,
gradient, Prop .: a continuous differentiable function and has all the directional derivatives.
Schwarz's Lemma total differential theorem. Functions
Ck, chain rule. Hessian matrix.
Taylor's formula at second order. Maximum and minimum stationary points
Positive definite matrices.
Prop: maximum or minimum points are critical points; the critical points in which the
Hessian matrix is ​​positive (negative) are minimum (maximum) points; the points
critics in which the Hessian matrix has a positive and a negative eigenvalue are saddles.
Functions that can be differentiated from Rn to Rm; Jacobian matrix. Jacobian matrix of the
composition.

3. Normed spaces and Banach spaces
Examples. Converging and Cauchy sequences. Equivalent rules. Equivalence of the norms in Rn. The space of the
continuous functions with the sup norm a Banach space.
The fixed point theorem in Banach spaces.
Implicit and inverse function theorems.

Core Documentation

Analisi Matematica II, Giusti - Analisi Matematica II, Chierchia

Type of delivery of the course

4 hours of frontal teaching 2 of exercises and 2 of tutoring a week

Attendance

course attendance is strongly recommended

Type of evaluation

The written exam focuses on the topics covered in class and tends to verify the ability to solve exercises. It consists of 4 exercises on the topics discussed during the class. Students who pass the two intermediate tests need not take the written exam. The oral exam verifies the ability to present and prove the theorems studied in class and apply them in specific cases.

teacher profile | teaching materials

Programme

1. Sequences and series of functions
Pointwise convergence, uniform convergence.
Total convergence of series of functions.
Power series, Fourier series.

2. 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 derivatives, differentiable functions,
gradient, Prop .: a continuous differentiable function and has all the directional derivatives.
Schwarz's Lemma total differential theorem. Functions
Ck, chain rule. Hessian matrix.
Taylor's formula at second order. Maximum and minimum stationary points
Positive definite matrices.
Prop: maximum or minimum points are critical points; the critical points in which the
Hessian matrix is ​​positive (negative) are minimum (maximum) points; the points
critics in which the Hessian matrix has a positive and a negative eigenvalue are saddles.
Functions that can be differentiated from Rn to Rm; Jacobian matrix. Jacobian matrix of the
composition.

3. Normed spaces and Banach spaces
Examples. Converging and Cauchy sequences. Equivalent rules. Equivalence of the norms in Rn. The space of the
continuous functions with the sup norm a Banach space.
The fixed point theorem in Banach spaces.
Implicit and inverse function theorems.

Core Documentation

Analisi Matematica II, Giusti - Analisi Matematica II, Chierchia

Type of delivery of the course

4 hours of frontal teaching 2 of exercises and 2 of tutoring a week

Attendance

course attendance is strongly recommended

Type of evaluation

The written exam focuses on the topics covered in class and tends to verify the ability to solve exercises. It consists of 4 exercises on the topics discussed during the class. The oral exam verifies the ability to present and prove the theorems studied in class and apply them in specific cases.