# CALCULUS 2

## Learning outcomes of the course unit

Knowledge and understanding.

At the end of the lectures, students should have acquired knowledge and understanding of the basic results of multivariable calculus, ordinary differential equations, curves in Euclidean n-dimensional spaces and of Peano-Jordan measure theory and integration.

Applying knowledge and understanding.

By means of the classroom exercises students learn how to apply the theoretical knowledges to solve concrete problems, such as optimization problems. models from the applied sciences which lead to solving ordinary differential equation or to solving particular integrals for functions of several variables.

Making judgements.

Students must be able to evaluate coherence and correctness of results obtained by themselves or by others.

Communication skills.

Students must be able to communicate in a clear and precise way mathematical statements in the field of study, also in a context broader than mere calculus. Through the front lectures and the assistance of the teacher, the students acquire the specific and appropriate scientific vocabulary.

Learning skills.

The student, who has attended the course, is able to deepen autonomously his/her knowledge of multivariate calculus, of the measure theory and integration for multivariate functions, starting from the basic and fundamental knowledges provided by the course. He/She will be also able to consult specialized textbooks, even outside the topics illustrated during the lectures. This to facilitate the entry in the labour market as well as a second-level study in a field which requires good mathematical skills.

Knowledge and understanding:

At the end of the course the student will acquire the ability to understand the concept of limit and continuity for functions of several variables, basic knowledge of differential and integral calculus of several variables, and the theory of resolution of linear ordinary differential equations of order n with constant and continuous coefficients.

Applying knowledge and understanding:

With the acquired skills, the student will be able to calculate the maximum and minimum points of a smooth function of several variables on an n-dimensional closed and bounded set with smooth boundary, to calculate the volume of an n-dimensional bounded set with smooth boundary, to determine the solution of a Cauchy problem for a linear differential equation of order n with continuous coefficients.

## Prerequisites

Students may take the exams only after they pass the exams of the courses Analisi Matematica 1 and Geometria.

Any book of Elements of Mathematical Analysis 2.

## Course contents summary

The course aims at providing students with the fundamental concepts of multivariable calculus, integration for multivariate functions, curves and (explicitly solvable) ordinary differential equations.

1-Topology on the Euclidean n-dimensional real space.

2-Limit and continuity of vector valued functions of vector variable.

3-Differential calculus for vector valued functions of vector variable.

4-Riemann integral for functions of vector variable.

5-Linear ordinary differential equations with continuous coefficients.

## Course contents

1) Curves.

Oriented curves; simple, closed and smooth curves; tangent vector and tangent straight line; length of a smooth curve, equivalent curves, curvilinear abscissa, line integral.

2) Elements of topology in R^n.

Interior, limit and bundary points; open and closed sets; compact sets, connected sets and convex sets.

3) Multivariable differential calculus.

Limits and continuity: limits for functions of several variables; continuous functions of several variables; Weierstrass' and intermediate value theorems. Multivariable calculus: directional and partial derivatives, differentiability of scalar and vector valued functions, gradient; tangent plane, tangent and normal vectors; chain rule; differentiability of a composite function; functions of class C^1, functions with null gradient. Functions of class C^2: Schwarz's theorem and hessian matrix; second order Taylor's formula; local and global maxima and minima, saddle points; necessary and sufficient conditions for optimization; Lagrange's multipliers; vector fields and line integral;

potentials; irrotational vector fields.

4) Multiple integrals (mainly for functions depending on two or three variables).

Integration: Peano-Jordan measure of sets; definition of integral and of integrable functions; dimensional reduction theorem. Change of variable formula: polar, spherical and cylindrical change of variables.

5) Ordinary Differential Equations.

First order linear differential equations with continuous coefficients, separable equations, linear equations of order n with constant coefficients. Variation-of-the-constants formula.

1-Topology on the Euclidean n-dimensional real space.

1.1 Euclidean scalar product and its properties.

1.2 Euclidean norm, its properties and Schwarz inequality.

1.3 Euclidean distance, its properties and fundamental system of neighborhoods of a point.

1.4 Definition of the interior point of the inner part of a set, of open set and properties of open sets.

1.5 Definition of closed set and properties of closed sets.

1.6 Definition of accumulation point, isolated point, the closure of a set, of boundary point and boundary of a set.

2-Limit and continuity of vector valued functions of vector variable.

2.1 Definition of limit of a sequence of vectors, of limit of a vector valued function of vector variable, uniqueness of the limit, and property of limits.

2.2 Definition of continuity for a vector valued function of vector variable and properties of continuous functions.

2.3 Compact sets, their characterization and Weierstrass theorem.

3-Differential calculus for vector valued functions of vector variable.

3.1 Partial derivatives and directional derivatives.

3.2 Differentiability of real valued functions of vector variable.

3.3 Theorem of the total differential.

3.4 Differentiability of vector valued functions of vector variable.

3.5 Differentiability of composed functions.

3.6 Partial derivatives of higher order and Schwarz theorem.

3.7 Taylor's formula stopped at the second order.

3.8 Stationary points and necessary condition for a point to be a relative minimum or maximum interior point.

3.9 The Hessian matrix and sufficient condition for a point to be minimum (maximum) internal relative.

3.10 Constrained stationary points.

4-Riemann integral for functions of vector variable.

4.1 Definition of Riemann integrable for function defined on a bounded regular n-dimensional set and properties of the integral.

4.2 Theorem of reduction of multiple integrals.

4.3 Theorem of the change of variables in multiple integrals.

5-Linear ordinary differential equations with continuous coefficients.

5.1 Theorem of characterization of the solutions of ordinary differential linear equations with continuous coefficients of order n.

5.2 Theorem of existence and uniqueness of the solution of the Cauchy problem.

5.3 Method for finding n linearly independent solutions of the homogeneous equation with constant coefficients.

5.4 Method for finding a particular solution of the non homogeneous equation.

## Recommended readings

Marino Belloni, Luca Lorenzi: Analisi Matematica 2 - Teoria. Ed. Santa Croce.

Marino Belloni, Luca Lorenzi: Analisi Matematica 2 - Esercizi. Ed. Santa Croce.

Any book of Elements of Mathematical Analysis 2.

## Teaching methods

The course schedules 4 hours of lectures per week plus other two hours per week for additional exercises.

The didactic activities consist of frontal lectures alternating with exercises.

During the lectures, the topics of the course will be presented and discussed in a rigorous way. Much emphasis will be given to the application of the abstract results presented. To this aim, particular importance will be given to exercises, which are the most useful way to make students understand the relevance of the results presented in the theoretical lectures and to learn how they can be applied.

Teaching will consist of lectures conducted by the teacher on the blackboard and in exercises designed to illustrate and apply the theory performed earlier.

## Assessment methods and criteria

The knowledges are verified in a traditional way, through the evalutation of a written test. Students should show that they have achieved knowledges on the subjects illustrated during the lectures. They should also be able to apply the knowledges to concrete situations, solving both multiple choices exercises and more elaborate exercises.

The test contains also a theoretical question which could be the statement of a theorem or the correct definition of a concept illustrated during the lectures.

The exam is passed if the score of the written test is at least 18 points.

The maximum score of the test is 33 points. Students who gain more than 30 points pass the exam with the grade "30 cum laude".

If interested, students who pass the written test (i.e., students with at least 18 points in the written test), may ask the teacher to take also an oral exam. The request should be formulated before the deadline for accepting the grade of the written test on ESSE3. The oral test will focus on all the program of the course and it will consist of both theoretical questions and practical exercises.

No test is expected during the course.

There will be a final written exam with answers free, lasting three hours and divided into three or four computational and theoretical questions. The student may accept the evaluation of the written exam, if it is sufficient or possibly improve it with an oral exam.

## Other informations

Even if not mandatory, it is strongly recommended to attend the lessons.

It is strongly recommended to attend the lessons.