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.
II. To develop and acquire the methods in the theory of continuous and regular functions in several real variables.
Curriculum
teacher profile teaching materials
Definition and convergence criteria.
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.
Programme
0. Number seriesDefinition and convergence criteria.
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, ChierchiaType of delivery of the course
4 hours of frontal teaching 2 of exercises and 2 of tutoring a weekAttendance
course attendance is strongly recommendedType 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
Inductive assemblies; definition of N and induction principle. Definition of Z
and Q; Z is a ring, Q is a field.
Nth roots; rational powers.
Part 2: Theory of Limits
The extended line R*: intervals, neighbourhoods and accumulation points.
Limits of functions in R*.
Comparison theorems.
Lateral limits; limits of monotone functions.
Algebra of limits on R and R*.
Composition limit of functions.
Limits of inverse functions.
Notable limits. The number of Napier. Exponential and trigonometric functions.
Part 3: Continuous functions
Topology of R.
Theorem of existence of zeroes. Bolzano-Weierstrass theorems. Weierstrass’ Theorem.
Uniformly continuous functions.
Part 4: Differentiable functions
Rules of derivation. Derivatives of elementary functions.
Local minima and maxima and elementary theorems on derivatives (Fermat, Rolle, Cauchy,
Lagrange).
Bernoulli-Hopital theorem.
Convexity.
Taylor’s formulae.
Part 5: Riemann integral in R.
The Riemann integral and its fundamental properties.
Integration criteria. Integrability of continuous and monotone functions.
The fundamental theorem of calculus and its applications (integration by parts,
changes of variables in integration). Generalized ("improper") integrals and
related integrability criteria.
McGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte
Testi di esercizi:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010
Programme
Part 1: Axiomatics of R and its main subsets Axiomatic definition of R.Inductive assemblies; definition of N and induction principle. Definition of Z
and Q; Z is a ring, Q is a field.
Nth roots; rational powers.
Part 2: Theory of Limits
The extended line R*: intervals, neighbourhoods and accumulation points.
Limits of functions in R*.
Comparison theorems.
Lateral limits; limits of monotone functions.
Algebra of limits on R and R*.
Composition limit of functions.
Limits of inverse functions.
Notable limits. The number of Napier. Exponential and trigonometric functions.
Part 3: Continuous functions
Topology of R.
Theorem of existence of zeroes. Bolzano-Weierstrass theorems. Weierstrass’ Theorem.
Uniformly continuous functions.
Part 4: Differentiable functions
Rules of derivation. Derivatives of elementary functions.
Local minima and maxima and elementary theorems on derivatives (Fermat, Rolle, Cauchy,
Lagrange).
Bernoulli-Hopital theorem.
Convexity.
Taylor’s formulae.
Part 5: Riemann integral in R.
The Riemann integral and its fundamental properties.
Integration criteria. Integrability of continuous and monotone functions.
The fundamental theorem of calculus and its applications (integration by parts,
changes of variables in integration). Generalized ("improper") integrals and
related integrability criteria.
Core Documentation
Luigi Chierchia: Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su RMcGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte
Testi di esercizi:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010
Type of delivery of the course
Lectures and exercises in class. All the material of the program will be explained in class. The lessons / exercises will include a continuous dialogue with the students: the feedback from the students during the course is a fundamental tool for the success of the course itself. In the event of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) governing the methods of carrying out educational activities will be implemented. In particular, distance learning may be necessary.Attendance
Attendance is optional and the understanding of the text adopted is sufficient for the full use of the course. Of course, attendance is desirable and STRONGLY recommended, as the interaction between teacher and students is a fundamental and unrepeatable teaching tool.Type of evaluation
The evaluation is based on a written test and an oral exam. There are two written tests "in progress" which, in the event of a positive outcome, replace the final written test. Examples of tests of past years will be available on the web site dedicated to the course which will be constantly updated by the teacher. In the case of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) that regulate the methods of student assessment will be implemented. In particular, distance assessments may be necessary and in this case the oral evaluation will be preceded by a preliminary written test which is an integral part of the oral examination. teacher profile teaching materials
Definition and convergence criteria.
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.
Programme
0. Number seriesDefinition and convergence criteria.
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, ChierchiaType of delivery of the course
4 hours of frontal teaching 2 of exercises and 2 of tutoring a weekAttendance
course attendance is strongly recommendedType 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
Inductive assemblies; definition of N and induction principle. Definition of Z
and Q; Z is a ring, Q is a field.
Nth roots; rational powers.
Part 2: Theory of Limits
The extended line R*: intervals, neighbourhoods and accumulation points.
Limits of functions in R*.
Comparison theorems.
Lateral limits; limits of monotone functions.
Algebra of limits on R and R*.
Composition limit of functions.
Limits of inverse functions.
Notable limits. The number of Napier. Exponential and trigonometric functions.
Part 3: Continuous functions
Topology of R.
Theorem of existence of zeroes. Bolzano-Weierstrass theorems. Weierstrass’ Theorem.
Uniformly continuous functions.
Part 4: Differentiable functions
Rules of derivation. Derivatives of elementary functions.
Local minima and maxima and elementary theorems on derivatives (Fermat, Rolle, Cauchy,
Lagrange).
Bernoulli-Hopital theorem.
Convexity.
Taylor’s formulae.
Part 5: Riemann integral in R.
The Riemann integral and its fundamental properties.
Integration criteria. Integrability of continuous and monotone functions.
The fundamental theorem of calculus and its applications (integration by parts,
changes of variables in integration). Generalized ("improper") integrals and
related integrability criteria.
McGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte
Testi di esercizi:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010
Programme
Part 1: Axiomatics of R and its main subsets Axiomatic definition of R.Inductive assemblies; definition of N and induction principle. Definition of Z
and Q; Z is a ring, Q is a field.
Nth roots; rational powers.
Part 2: Theory of Limits
The extended line R*: intervals, neighbourhoods and accumulation points.
Limits of functions in R*.
Comparison theorems.
Lateral limits; limits of monotone functions.
Algebra of limits on R and R*.
Composition limit of functions.
Limits of inverse functions.
Notable limits. The number of Napier. Exponential and trigonometric functions.
Part 3: Continuous functions
Topology of R.
Theorem of existence of zeroes. Bolzano-Weierstrass theorems. Weierstrass’ Theorem.
Uniformly continuous functions.
Part 4: Differentiable functions
Rules of derivation. Derivatives of elementary functions.
Local minima and maxima and elementary theorems on derivatives (Fermat, Rolle, Cauchy,
Lagrange).
Bernoulli-Hopital theorem.
Convexity.
Taylor’s formulae.
Part 5: Riemann integral in R.
The Riemann integral and its fundamental properties.
Integration criteria. Integrability of continuous and monotone functions.
The fundamental theorem of calculus and its applications (integration by parts,
changes of variables in integration). Generalized ("improper") integrals and
related integrability criteria.
Core Documentation
Luigi Chierchia: Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su RMcGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte
Testi di esercizi:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010
Type of delivery of the course
Lectures and exercises in class. All the material of the program will be explained in class. The lessons / exercises will include a continuous dialogue with the students: the feedback from the students during the course is a fundamental tool for the success of the course itself. In the event of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) governing the methods of carrying out educational activities will be implemented. In particular, distance learning may be necessary.Attendance
Attendance is optional and the understanding of the text adopted is sufficient for the full use of the course. Of course, attendance is desirable and STRONGLY recommended, as the interaction between teacher and students is a fundamental and unrepeatable teaching tool.Type of evaluation
The evaluation is based on a written test and an oral exam. There are two written tests "in progress" which, in the event of a positive outcome, replace the final written test. Examples of tests of past years will be available on the web site dedicated to the course which will be constantly updated by the teacher. In the case of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) that regulate the methods of student assessment will be implemented. In particular, distance assessments may be necessary and in this case the oral evaluation will be preceded by a preliminary written test which is an integral part of the oral examination.