# CALCULUS 2

CALCULUS 2 PART. 1 (6 credits) | CALCULUS 2 PART. 2 (9 credits) |

## Learning outcomes of the course unit

Knowledge and understanding: At the end of this course the student should know the essential definitions and results of analysis in several variables, sequences and series of functions, ordinary differential equations (ODE), implicit functions and multivariable integral calculus, and he should be able to grasp how these enter in the solution of problems. Applying knowledge and understanding: The student should be able to apply the forementioned notions to solve medium level problems, and to understand how they relate to concepts seen in different courses. Making judgements: The student should be able to evaluate coherence and correctness of the proofs he gives during the written test. Communication skills: The student should be able to communicate in a clear and precise way, suitable for a scientist-to-be in an intermediate stage of his formation.

## Prerequisites

Analyis for functions of one real variable; linear geometry; linear algebra.

## Course contents summary

Normed and metric spaces. Limits and continuity of functions of several real variables. Curves. Differential calculus for functions of several real variables. Implicit Function Theorem and consequences. Multiple integrals. Sequences and function series. Potential functions and differential forms. Ordinary Differential Equations.

Full proof is provided for most statements.

## Course contents

Norms, distances, equivalent norms and equivalent distances.

Limits and continuity of functions of several real variables.

Regular curves, simple curves, equivalences among curves, paths, unit tangent vector to regular paths, curve lenghts, integrals of continuous functions along paths; work of a field along a path..

Differential calculus for functions of several real variables: directional derivatives and their geometric meaning, partial derivatives, gradients, differentiation rules, tangent hyperplanes and their geometric meanings, Schwarz Theorem, Taylor formula, quadratic forms, local maxima and minima.

Implicit Function Theorem, Inverse Function Theorem, smooth surfaces, Lagrange Theorem. Multiple integrals: definitions, reduction theorem, changes of variables. Integral in many dimensions. Integral on surfaces.

Function sequences. Power series: radius of convergence; uniform convergence, continuity and integration. Trigonometric series; Fourier series; convergence of Fourier series; Fourier coefficients and regularity.The Cauchy problem. Regularity. Existence and uniqueness of solutions. Continuous dependence on data. Integration of some ordinary differential equations. Systems of First Order Linear Differential Equations. Qualitative behavior of solutions of ode’s.Scalar and vector potential in star-shaped domains. Closed and exact differential forms. Paths and integration of differential forms.

## Recommended readings

Lectures closely follow

E. Acerbi e G. Buttazzo, Secondo corso di analisi matematica. Pitagora Bologna (2016).

One may use instead any good book on analysis in several variables, as e.g.

G. Prodi: Lezioni di Analisi Matematica II. ETS Pisa (1974)

W. Fleming: Functions of several variables. Second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977.

## Teaching methods

Lectures are held in the classroom, encompassing both theoretical and applied aspects. Moreover, exercises are solved by students with the guidance of the teacher, so as to verify the degree of comprehension and knowledge of the students.

## Assessment methods and criteria

The final exam consist of a written and an oral session. Students are admitted to the oral sessions only if they pass the written examination. In the written examination 4 open questions are asked. The students should exhibit calculus skills and mastery of different subjects taught in the course. Marks are given to each question, according to theoretical correctness, precision of execution, precision of exposition.

The oral examination consists of a discussion about the written examination and of questions to verify the level of comprehension of the theoretical parts of the course.