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20810114 -
CALCULUS I
(objectives)
To allow the acquisition of the deductive-logic method and provide basic mathematical tools for the differential and integral calculus. Each topic will be strictly introduced and treated by carrying out, whenever needed, detailed demonstrations and by referring largely to the physical meaning, the geometrical interpretation and the numerical application. A proper methodology combined with a reasonable skill in the use of the concepts and results of the integro-differential calculus, will enable students to face more applicative concepts that will be tackled during the succeeding courses.
Group:
CANALE 1
-
BIASCO LUCA
( syllabus)
The numbers refer to the chapters and paragraphs of the textbook:
Calcolo of P. Marcellini and C. Sbordone.
1) Real numbers and functions
Natural, whole and rational numbers; density of rationals (5). Axioms of real numbers (2). Overview of set theory (4).
The intuitive concept of function (6) and Cartesian representation (7).
Injective, surjective, bijective and invertible functions. Monotonic functions (8).
Absolute value (9). The principle of induction (13).
2) Complements to real numbers
Maximum, minimum, supremum, infimum.
7) Succession limits
Definition and first properties (56.57).
Limited successions (58). Operations with limits (59).
Indefinite forms (60).
Comparative theorems (61). Other properties of succession limits (62).
Notable limits (63). Monotone sequences, the number e (64).
Sequence going to infinity of increasing order (67).
8) Function limits. Continuous functions
Definition of limit and property (71,72,73).
Continuous functions (74). discontinuity (75).
Theorems on continuous functions (76).
9) Additions to the limits
The theorem on monotonic sequences (80).
Extracted successions; the Bolzano-Weierstrass theorem (81).
The Weierstrass theorem (82).
Continuity of monotonic functions and inverse functions (83).
10) Derivatives
Definition and physical meaning (88-89). Operations with derivatives (90).
Derivatives of compound functions and inverse functions (91).
Derivative of elementary functions (92).
Geometric meaning of the derivative: tangent line (93).
11) Applications of derivatives. Function study
Maximum and minimum relative. Fermat's theorem (95).
Theorems of Rolle and Lagrange (96).
Increasing, decreasing, convex and concave functions (97-98).
De l'Hopital theorem (99).
Study of the graph of a function (100).
Taylor's formula: first properties (101).
14) Integration according to Riemann
Definition (117). Properties of the defined integrals (118).
Uniform continuity. Cantor's theorem (119).
Integrability of continuous functions (120).
The theorems of the average (121).
15) Undefined integrals
The fundamental theorem of integral calculus (123).
Primitives (124). The indefinite integral (125). Integration by parts and by substitution
(126,127,128,129).
Improper integrals (132).
16) Taylor's formula
Rest of Peano (135).
Use of Taylor's formula in the calculation of limits (136).
17) Series
Numerical series (141).
Series with positive terms (142).
Geometric series and harmonic series (143.144).
Convergence criteria (145).
Alternate series (146).
Absolute convergence (147).
Taylor series (149).
( reference books)
S. Lang, A First Course in Calculus, Springer Ed.
-
BATTAGLIA LUCA
( syllabus)
The numbers refer to the chapters and paragraphs of the textbook:
Calcolo of P. Marcellini and C. Sbordone.
1) Real numbers and functions
Natural, whole and rational numbers; density of rationals (5). Axioms of real numbers (2). Overview of set theory (4).
The intuitive concept of function (6) and Cartesian representation (7).
Injective, surjective, bijective and invertible functions. Monotonic functions (8).
Absolute value (9). The principle of induction (13).
2) Complements to real numbers
Maximum, minimum, supremum, infimum.
7) Succession limits
Definition and first properties (56.57).
Limited successions (58). Operations with limits (59).
Indefinite forms (60).
Comparative theorems (61). Other properties of succession limits (62).
Notable limits (63). Monotone sequences, the number e (64).
Sequence going to infinity of increasing order (67).
8) Function limits. Continuous functions
Definition of limit and property (71,72,73).
Continuous functions (74). discontinuity (75).
Theorems on continuous functions (76).
9) Additions to the limits
The theorem on monotonic sequences (80).
Extracted successions; the Bolzano-Weierstrass theorem (81).
The Weierstrass theorem (82).
Continuity of monotonic functions and inverse functions (83).
10) Derivatives
Definition and physical meaning (88-89). Operations with derivatives (90).
Derivatives of compound functions and inverse functions (91).
Derivative of elementary functions (92).
Geometric meaning of the derivative: tangent line (93).
11) Applications of derivatives. Function study
Maximum and minimum relative. Fermat's theorem (95).
Theorems of Rolle and Lagrange (96).
Increasing, decreasing, convex and concave functions (97-98).
De l'Hopital theorem (99).
Study of the graph of a function (100).
Taylor's formula: first properties (101).
14) Integration according to Riemann
Definition (117). Properties of the defined integrals (118).
Uniform continuity. Cantor's theorem (119).
Integrability of continuous functions (120).
The theorems of the average (121).
15) Undefined integrals
The fundamental theorem of integral calculus (123).
Primitives (124). The indefinite integral (125). Integration by parts and by substitution
(126,127,128,129).
Improper integrals (132).
16) Taylor's formula
Rest of Peano (135).
Use of Taylor's formula in the calculation of limits (136).
17) Series
Numerical series (141).
Series with positive terms (142).
Geometric series and harmonic series (143.144).
Convergence criteria (145).
Alternate series (146).
Absolute convergence (147).
Taylor series (149).
( reference books)
S. Lang, A First Course in Calculus, Springer Ed.
Group:
CANALE 2
-
BIASCO LUCA
( syllabus)
The numbers refer to the chapters and paragraphs of the textbook:
Calcolo of P. Marcellini and C. Sbordone.
1) Real numbers and functions
Natural, whole and rational numbers; density of rationals (5). Axioms of real numbers (2). Overview of set theory (4).
The intuitive concept of function (6) and Cartesian representation (7).
Injective, surjective, bijective and invertible functions. Monotonic functions (8).
Absolute value (9). The principle of induction (13).
2) Complements to real numbers
Maximum, minimum, supremum, infimum.
7) Succession limits
Definition and first properties (56.57).
Limited successions (58). Operations with limits (59).
Indefinite forms (60).
Comparative theorems (61). Other properties of succession limits (62).
Notable limits (63). Monotone sequences, the number e (64).
Sequence going to infinity of increasing order (67).
8) Function limits. Continuous functions
Definition of limit and property (71,72,73).
Continuous functions (74). discontinuity (75).
Theorems on continuous functions (76).
9) Additions to the limits
The theorem on monotonic sequences (80).
Extracted successions; the Bolzano-Weierstrass theorem (81).
The Weierstrass theorem (82).
Continuity of monotonic functions and inverse functions (83).
10) Derivatives
Definition and physical meaning (88-89). Operations with derivatives (90).
Derivatives of compound functions and inverse functions (91).
Derivative of elementary functions (92).
Geometric meaning of the derivative: tangent line (93).
11) Applications of derivatives. Function study
Maximum and minimum relative. Fermat's theorem (95).
Theorems of Rolle and Lagrange (96).
Increasing, decreasing, convex and concave functions (97-98).
De l'Hopital theorem (99).
Study of the graph of a function (100).
Taylor's formula: first properties (101).
14) Integration according to Riemann
Definition (117). Properties of the defined integrals (118).
Uniform continuity. Cantor's theorem (119).
Integrability of continuous functions (120).
The theorems of the average (121).
15) Undefined integrals
The fundamental theorem of integral calculus (123).
Primitives (124). The indefinite integral (125). Integration by parts and by substitution
(126,127,128,129).
Improper integrals (132).
16) Taylor's formula
Rest of Peano (135).
Use of Taylor's formula in the calculation of limits (136).
17) Series
Numerical series (141).
Series with positive terms (142).
Geometric series and harmonic series (143.144).
Convergence criteria (145).
Alternate series (146).
Absolute convergence (147).
Taylor series (149).
( reference books)
S. Lang, A First Course in Calculus, Springer Ed.
-
BATTAGLIA LUCA
( syllabus)
The numbers refer to the chapters and paragraphs of the textbook:
Calcolo of P. Marcellini and C. Sbordone.
1) Real numbers and functions
Natural, whole and rational numbers; density of rationals (5). Axioms of real numbers (2). Overview of set theory (4).
The intuitive concept of function (6) and Cartesian representation (7).
Injective, surjective, bijective and invertible functions. Monotonic functions (8).
Absolute value (9). The principle of induction (13).
2) Complements to real numbers
Maximum, minimum, supremum, infimum.
7) Succession limits
Definition and first properties (56.57).
Limited successions (58). Operations with limits (59).
Indefinite forms (60).
Comparative theorems (61). Other properties of succession limits (62).
Notable limits (63). Monotone sequences, the number e (64).
Sequence going to infinity of increasing order (67).
8) Function limits. Continuous functions
Definition of limit and property (71,72,73).
Continuous functions (74). discontinuity (75).
Theorems on continuous functions (76).
9) Additions to the limits
The theorem on monotonic sequences (80).
Extracted successions; the Bolzano-Weierstrass theorem (81).
The Weierstrass theorem (82).
Continuity of monotonic functions and inverse functions (83).
10) Derivatives
Definition and physical meaning (88-89). Operations with derivatives (90).
Derivatives of compound functions and inverse functions (91).
Derivative of elementary functions (92).
Geometric meaning of the derivative: tangent line (93).
11) Applications of derivatives. Function study
Maximum and minimum relative. Fermat's theorem (95).
Theorems of Rolle and Lagrange (96).
Increasing, decreasing, convex and concave functions (97-98).
De l'Hopital theorem (99).
Study of the graph of a function (100).
Taylor's formula: first properties (101).
14) Integration according to Riemann
Definition (117). Properties of the defined integrals (118).
Uniform continuity. Cantor's theorem (119).
Integrability of continuous functions (120).
The theorems of the average (121).
15) Undefined integrals
The fundamental theorem of integral calculus (123).
Primitives (124). The indefinite integral (125). Integration by parts and by substitution
(126,127,128,129).
Improper integrals (132).
16) Taylor's formula
Rest of Peano (135).
Use of Taylor's formula in the calculation of limits (136).
17) Series
Numerical series (141).
Series with positive terms (142).
Geometric series and harmonic series (143.144).
Convergence criteria (145).
Alternate series (146).
Absolute convergence (147).
Taylor series (149).
( reference books)
S. Lang, A First Course in Calculus, Springer Ed.
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12
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MAT/05
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108
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-
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-
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-
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Basic compulsory activities
|
ITA |
|
20810115 -
FUNDAMENTALS OF INFORMATICS AND LINEAR ALGEBRA
(objectives)
The course teaches the basic of computer science for the automated solution of engineering problems, including algorithms design. The course also aims to provide an introduction to those aspects of linear algebra and geometry needed in science and engineering. Numerical applications will concern algorthms to manipulate data structures typical of linear algebra and solve related computational problems.
Group:
CANALE 1
-
GASPARETTI FABIO
( syllabus)
Computer Architecture, Operative System, Problem, Algorithm, Software, Representation of information, Software compiling and execution, Python development environment, Languages - syntax and semantics, Types and expressions, Functions in Python, Conditional instructions, Repetitive instructions, Strings, Dictionaries, Tuples and Matrix in Python, Sorting algorithms, Specification and correctness of software, Computational complexity, Files and Exceptions, Linear Algebra, Linear equations and sets, Matrices in Algebra, Determinant, Reverse matrix, Rank of a matrix, Gauss, Algebraic functions, Vector spaces, Generators, Bases, Operations between subspaces, Affine spaces, Homomorphism, Image, Kernel, Logic
Course calendar will be essentially organized in two similar-length parts: Linear Algebra and Foundation of Computer science.
( reference books)
“Think Python: How to Think Like a
Computer Scientist” di Allen B. Downey
(O’Reilly Media, 2012) – 1st edition
http://www.greenteapress.com/thinkpython/thinkpython.html
“Geometria” di G. Accascina e V. Monti http://www.dmmm.uniroma1.it/accascinamonti/geogest/Geometria.pdf
Group:
CANALE 2
-
GASPARETTI FABIO
( syllabus)
Computer Architecture, Operative System, Problem, Algorithm, Software, Representation of information, Software compiling and execution, Python development environment, Languages - syntax and semantics, Types and expressions, Functions in Python, Conditional instructions, Repetitive instructions, Strings, Dictionaries, Tuples and Matrix in Python, Sorting algorithms, Specification and correctness of software, Computational complexity, Files and Exceptions, Linear Algebra, Linear equations and sets, Matrices in Algebra, Determinant, Reverse matrix, Rank of a matrix, Gauss, Algebraic functions, Vector spaces, Generators, Bases, Operations between subspaces, Affine spaces, Homomorphism, Image, Kernel, Logic
Course calendar will be essentially organized in two similar-length parts: Linear Algebra and Foundation of Computer science.
( reference books)
“Think Python: How to Think Like a
Computer Scientist” di Allen B. Downey
(O’Reilly Media, 2012) – 1st edition
http://www.greenteapress.com/thinkpython/thinkpython.html
“Geometria” di G. Accascina e V. Monti http://www.dmmm.uniroma1.it/accascinamonti/geogest/Geometria.pdf
|
9
|
ING-INF/05
|
81
|
-
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20202021 -
ENGLISH LANGUAGE - PASS/FAIL CERTIFICATE
|
3
|
|
24
|
-
|
-
|
-
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Final examination and foreign language test
|
ITA |
|
20810137 -
PHISICS
(objectives)
The course introduces the scientific method, presents newton's mechanics and the main electric and magnetic phenomena, together with the pertinent laws. The student becomes familiar with the basic models of classical physics and, in particular, with such concepts as physical quantity, field, conservation law. The student is able to apply the above concepts to the solution of simple problems by means of appropriate analytical procedures.
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20810137-1 -
PHISICS MODULE I
(objectives)
The course introduces the scientific method, presents newton's mechanics and the main electric and magnetic phenomena, together with the pertinent laws. The student becomes familiar with the basic models of classical physics and, in particular, with such concepts as physical quantity, field, conservation law. The student is able to apply the above concepts to the solution of simple problems by means of appropriate analytical procedures.
Group:
CANALE 1
-
POMPEO NICOLA
( syllabus)
Introduction
- Physical quantities and units
- Fundamentals on vector algebra
Kinematics of a material point
- Kinematics quantities in the rectilinear motion
- Uniformly accelerated rectilinear motion
- Simple harmonic motion
- Kinematics in 2-D and 3-D
- Motion trajectory
- Tangential and normal components of acceleration
- Parabolic motion
- Circular motion
- Relative motion
Dynamics of a material point
- Principles of Dynamics and Newton's laws
- Momentum and Impulse
- Equilibrium and constraint reaction forces
- Gravitational force
- Weight and motion under gravity
- Forces and motion
- Forces of dry friction
- Inclined plane
- Elastic force and mass-spring system
- Tension force in ropes
- Applications to circular motion
- Viscous force
- Electrical charge and Coulomb force
- Simple pendulum
- Inertial and non-inertial reference frames
- Inertial forces
Work and Energy
- Work and power
- Work of weight, elastic and dry friction forces
- Work-energy theorem. Applications
- Conservative forces. Potential energy
- Central forces
- Gravitational and electrostatic potential energies
- Conservation of mechanical energy. Applications
- Stability conditions for static equilibrium
( reference books)
- P. Mazzoldi, M. Nigro, C. Voci, "Elementi di Fisica. Vol. I: Meccanica - Termodinamica", seconda edizione, Edises, Napoli
(*) Notes on selected arguments are also available on the
course website, under the section complementi
For further studies, the following is suggested:
- R. P. Feynman, "La Fisica di Feynman", volumes 1 and 2
(freely available at http://feynmanlectures.info/)
Group:
CANALE 2
-
POMPEO NICOLA
( syllabus)
Introduction
- Physical quantities and units
- Fundamentals on vector algebra
Kinematics of a material point
- Kinematics quantities in the rectilinear motion
- Uniformly accelerated rectilinear motion
- Simple harmonic motion
- Kinematics in 2-D and 3-D
- Motion trajectory
- Tangential and normal components of acceleration
- Parabolic motion
- Circular motion
- Relative motion
Dynamics of a material point
- Principles of Dynamics and Newton's laws
- Momentum and Impulse
- Equilibrium and constraint reaction forces
- Gravitational force
- Weight and motion under gravity
- Forces and motion
- Forces of dry friction
- Inclined plane
- Elastic force and mass-spring system
- Tension force in ropes
- Applications to circular motion
- Viscous force
- Electrical charge and Coulomb force
- Simple pendulum
- Inertial and non-inertial reference frames
- Inertial forces
Work and Energy
- Work and power
- Work of weight, elastic and dry friction forces
- Work-energy theorem. Applications
- Conservative forces. Potential energy
- Central forces
- Gravitational and electrostatic potential energies
- Conservation of mechanical energy. Applications
- Stability conditions for static equilibrium
( reference books)
- P. Mazzoldi, M. Nigro, C. Voci, "Elementi di Fisica. Vol. I: Meccanica - Termodinamica", seconda edizione, Edises, Napoli
(*) Notes on selected arguments are also available on the
course website, under the section complementi
For further studies, the following is suggested:
- R. P. Feynman, "La Fisica di Feynman", volumes 1 and 2
(freely available at http://feynmanlectures.info/)
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6
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FIS/03
|
-
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-
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-
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-
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Basic compulsory activities
|
ITA |
Università Roma Tre