# CALCULUS 2

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

## Learning outcomes of the course unit

The course object is to supply the students with knowledge and understanding of sequences and series of functions, ordinary differential equations (ODE), implicit functions and multivariable integral calculus.

Give to them the competence to treat and apply these instruments.

Objects of the course are also : judgement independence, strong written and oral communication skills, learning ability, in accordance with the specific objects of the mathematics degree.

## Prerequisites

Analisi matematica I e Analisi Matematica 2 1° modulo

## Course contents summary

Normed and metric spaces. Norms, distances, equivalent norms and equivalent distances.

Limits and continuity of functions of several real variables.

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.

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

Implicit Function Theorem, Inverse Function Theorem, Lagrange Theorem.

Linear differential forms, integrals along oriented paths, primitives, equivalent conditions for the existence of primitives, primitives on starshaped sets, simply connected sets.

Elementary notions about multiple integrals: definitions, reduction theorem, changes of variables, Gauss-Green formulae in dimension 2.

## Course contents

Multivariable integral calculus. Implicit functions. The inverse function Theorem. Extremum problems with side conditions.

Sequences and series of functions. Pointwise and uniform convergence. Total convergence of series of functions. Power series. Taylor series. Fourier series.

Ordinary differential equations (ODE). The Cauchy problem. Local existence and uniqueness theorem. Extension of the solutions. Continuous dependence of the data. Solutions of some type of first order ODE. Linear equations with constant coefficients. Linear systems.

## Recommended readings

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

M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 2. Zanichelli (2009);

N. Fusco, P. Marcellini, C. Sbordone: Analisi matematica due. Liguori (1996).

## Teaching methods

Academic method. The teaching consists in frontal lessons where both theoretical and applicable aspects are expounded. The exercises are developped with the collaboration of the students and are programmed in order to they can solve independently the problems arising from the theoretical lessons.

## Assessment methods and criteria

The understanding check consists in a final written test and, if it will be positive, in an oral discussion. In both the tests the student has to demonstrate knowledge, comprehension and to be able to connect knowledge and comprehension about sequences and series of functions, implicit functions and multivariable integral calculus.

The written test consists in 3 open questions regarding the above arguments.

The highest mark for each question will be 10, the total amount of the written test will be 30.

The total amount of the oral test is 30.

The marks will be attributed considering : the accuracy of the exposition and the operating methods.

The test will be considered sufficient if the average of the total amount of the written and oral tests is greater or equal to 18.