Lecturers
Ruggero Bandiera and Giacomo Cherubini.
Important Announcements
There will be no class on Thursday, October 8.
Calendar
Teaching starts on September 23 and ends on December 22, 2026. There is no teaching in the week October 26-30. See also the official timetable here.
Schedule
Wednesday 13-16 Aula 3, Thursday 13-15 Aula 3. (Access to the rooms is from Via Tiburtina 205).
References
Students are strongly encouraged to take notes during classes. There will be no additional lecture notes from us. As a reference, the following is a good text:
George B. Thomas, Calculus early transcendentals.
There are also many open access lecture notes online covering part of the material of our course and sometimes more, such as:
https://home.iitm.ac.in/asingh/papers/classnotes-ma1101.pdf
https://www.math.hkust.edu.hk/~machas/vector-calculus-for-engineers.pdf
Additional material can be found at last year's Moodle webpage by Prof. Alla.
Syllabus overview
Domains in R^2, interior points and boundary points. Functions of two variables, limits, continuity, Cartesian coordinates and polar coordinates, functions with a parameter. Derivability and differentiability, gradient, tangent plane, directional derivatives, second order derivatives, Hessian. Maximum and minimum of a function of two variables, necessary conditions, stationary points, local max/min, global max/min. Characterisation of stationary points in terms of the Hessian, saddle points, criteria in terms of the eigenvalues of the Hessian. Curves in R^n. Parametric and Cartesian equations. Closed curves, simple curves, regular curves. Tanget vector. Length of a curve. Line integrals. Maxima and Minima in compact domains. Double integrals, normal domains. Change of variables in double integrals, Jacobian. Triple integrals, volumes, cylindrical coordinates, spherical coordinates.
Course Journal and exercises
09.23. Overview of the course and recap of Calculus 1. Domain of a function of one variable. Definition of continuous functions in a point. Examples of continuous and discontinuous functions. Elementary functions and special limits involving elementary functions. Functions that can be extended by continuity. Definition of derivative of a function in a point. Derivatives of elementary functions. Product rule and chain rule. Examples. Integrals of a function of one variable. Integration by parts. Integration by change of variable. Examples.
09.24. Domain of a function of two variables. Definition of interior point of a set in R^2. Definition of boundary point of a set in R^2. Definition of level curve. Examples of domain and boundary of functions of two variables. Introduction to limits in two variables. Example of a function of two variables with different limit value according to the direction (therefore, such a function does not admit limit).
09.30. Definition of limit of a function of two variables. Properties of limits. Examples where the limit exists. Examples where the limit does not exist. Proof of the limit by means of inequalities in Cartesian coordinates. Definition of polar coordinates. Calculation of limits in polar coordinates. Examples. Combination of special limits for elementary functions and polar coordinates. Definition of continuous functions. Examples. Examples with parameters.
10.01. Definition of partial derivatives. Definition of gradient. Examples. Existence of partial derivatives at a point do not imply continuity. Defintion of differentiability. Comparison with the one-dimensional case. Examples. Definition of tangent plane.
Exercises. Exercises October 1st.
10.07. Sufficient conditions for differentiability. Directional derivatives.
10.08. --No class--
10.14. Recap on gradient, tangent plane and directional derivatives. Gradient as direction of maximal growth. Definition of maximum, minimum and stationary point. Theorem: local max/min are stationary points. Second partial derivatives. Definition of Hessian matrix. Schwarz's theorem on the symmetry of the Hessian matrix. Example where Schwarz's theorem does not apply.
10.15. Theorem: classification of local max/min according to the Hessian. Definition of saddle points. Definite and indefinite matrices. Generalization to functions of several variables. Example in three variables.
10.21.
Exercises. Exercises October 21st.
10.22.
Exams
There will be a written exam. You need 18/30 to pass. If you get 25 or higher at the written exam, you must also come to an oral exam to confirm the grade.
Exams will be in January and February. There will be no midterms.