The natural, rational, real, complex numbers. Numerical sequences. Real functions of a real variable: differential calculus. Integration. Basic elements of Linear Algebra.
Course Content - Last names M-Z
The natural, rational, real, complex numbers. Numerical sequences. Real functions of a real variable: differential calculus. Integration. Basic elements of Linear Algebra.
Aim of the course is to provide the students with basic instruments in order to deal with technical-scientific teachings characterizing the Laurea degree in Chemistry. In particular, the object of the course consists in giving the knowledge and the comprehension of the mathematical language besides the ability of making use of the essential techniques of Linear Algebra and Calculus.
Learning Objectives - Last names M-Z
Aim of the course is to provide the students with basic instruments in order to deal with technical-scientific teachings characterizing the Laurea degree in Chemistry. In particular, the object of the course consists in giving the knowledge and the comprehension of the mathematical language besides the ability of making use of the essential techniques of Linear Algebra and Calculus.
Prerequisites - Last names A-L
None
Prerequisites - Last names M-Z
None
Teaching Methods - Last names A-L
Total number of hours for Lectures: 52
Total number of hours for Laboratory: 20
Teaching Methods - Last names M-Z
Total number of hours for Lectures: 52 Total number of hours for Laboratory: 20
Type of Assessment - Last names A-L
The final exam is aimed to evaluate the acquisition of the concepts and of the abilities through a written and a successive oral exam.
The written exam lasts 2 or 3 hours and requires to solve exercises analogous to those assigned and solved in class. The student can consult an A4 sheet where he has handwritten formulas that he reputes useful. The student cannot consult notes or books.
The oral exam consist of a conversation with the teacher aimed to evaluate his knowledge and understanding of the concepts and of the functions presented in the course.
Several examples of written exams assigned in the past years are available in the moodle page dedicated to the course.
Type of Assessment - Last names M-Z
The final exam is aimed to evaluate the acquisition of the concepts and of the abilities through a written and a successive oral exam. The written exam lasts 2 or 3 hours and requires to solve exercises analogous to those assigned and solved in class. The student can consult an A4 sheet where he has handwritten formulas that he reputes useful. The student cannot consult notes or books. The oral exam consist of a conversation with the teacher aimed to evaluate his knowledge and understanding of the concepts and of the functions presented in the course. Several examples of written exams assigned in the past years are available in the moodle page dedicated to the course.
Course program - Last names A-L
Real and complex numbers. Supremum, De Moivre formulas, complex roots and the fundamental theorem of algebra.
Numerical sequences. Limits, existence of the limit for monotonous sequences, known limits and the number e.
Real functions of a real variable. Properties and graphs of elementary functions.
Definition of limit of function. History and example of derivative calculation. Definition in terms of epsilon and delta in the case of finite value and finite point. Continuous functions. Intermediate value Theorem . Absolute maximum and minimum in an interval. Weierstrass theorem.
Concept of derivative and its meaning. Calculation of derivatives. Fermat and Lagrange Theorems. Criterion of monotonicity. Second derivative. Convex and concave functions. Linear approximation, meaning of small-o. Polynomial approximation, Taylor's formulas.
Integral as Riemann's sum limit. Integral mean theorem. Introduction to the methods for evaluating an integral. Primitive and indefinite integral. First fundamental theorem of calculus. Integration by parts and bysubstitution. Integration of rational functions. Method of midpoints for approximate calculation. Integral functions. Second fundamental theorem of the calculus.
Vector spaces. Linear combination and linear independence of vectors. Base and dimension. Vector space of matrices and operations on them. Determinant and rank. Linear systems. Elementary operations on them and equivalent systems. Gauss reduction method. Rouché-Capelli’s Theorem. Second definition of rank. Cramer's Theorem. Linear applications between vector spaces, their representation through a matrix. Image and nucleus. Diagonalizable matrices. Eigenvalues and eigenvectors. Algebraic and geometric multiplicity of an eigenvalue.
Course program - Last names M-Z
Real and complex numbers. Supremum, De Moivre formulas, complex roots and the fundamental theorem of algebra.
Numerical sequences. Limits, existence of the limit for monotonous sequences, known limits and the number e.
Real functions of a real variable. Properties and graphs of elementary functions.
Definition of limit of function. History and example of derivative calculation. Definition in terms of epsilon and delta in the case of finite value and finite point. Continuous functions. Intermediate value Theorem . Absolute maximum and minimum in an interval. Weierstrass theorem.
Concept of derivative and its meaning. Calculation of derivatives. Fermat and Lagrange Theorems. Criterion of monotonicity. Second derivative. Convex and concave functions. Linear approximation, meaning of small-o. Polynomial approximation, Taylor's formulas.
Integral as Riemann's sum limit. Integral mean theorem. Introduction to the methods for evaluating an integral. Primitive and indefinite integral. First fundamental theorem of calculus. Integration by parts and by substitution. Integration of rational functions. Method of midpoints for approximate calculation. Integral functions. Second fundamental theorem of the calculus.
Vector spaces. Linear combination and linear independence of vectors. Base and dimension. Vector space of matrices and operations on them. Determinant and rank. Linear systems. Elementary operations on them and equivalent systems. Gauss reduction method. Rouché-Capelli’s Theorem. Second definition of rank. Cramer's Theorem. Linear applications between vector spaces, their representation through a matrix. Image and nucleus. Diagonalizable matrices. Eigenvalues and eigenvectors. Algebraic and geometric multiplicity of an eigenvalue.