# CALCULUS 2

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

## Learning outcomes of the course unit

Knowledge and understanding.

Students must achieve thorough conceptual understanding of the theoretical foundations of the topics of the course, as well as computational fluency.

Applying knowledge and understanding.

Students must be able to apply the forementioned notions to solve medium level problems and to understand how the forementioned notions can be used for solving problems in a more applied context.

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, precise and complete way mathematical statements in the field of study, also in a broader context than mere calculus.

## Prerequisites

Analisi matematica I e Analisi Matematica 2 1° modulo

## Course contents summary

Multivariable differential and integral calculus.

Paths in R^n and linear differential forms.

Sequences and series of functions.

Ordinary differential equations and systems.

## Course contents

1) Preliminaries of linear algebra and topology.

Linear algebra and geometry: Vectors spaces. Norms and scalar products. Cauchy--Schwarz inequality. Linear mappings and matrices. Eigenvalues and diagonalization of symmetric matrices. Quadratic forms.

Metric and euclidean spaces: Interior, cluster and boundary points. Open and closed sets. Sequences and complete metric spaces. Compact and connected sets. Continuous functions. Bounded linear operators.

Lipschitz functions and contraction principle.

2) Multivariable differential calculus.

Limits and continuity: Limits of multivariable functions. Continuous functions.

Differentiable functions: Directional and partial derivatives. Differentiable functions. The gradient and its geometrical meaning. Tangent planes, tangent and normal vectors. Chain rule. Functions of

class C^1. The inverse function theorem. Diffeomorphisms and changes of variables.

Functions of class C^k: Higher order differentiable functions. Functions of class C^k. Schwartz's theorem. Taylor's formula (Peano's and Lagrange's reminders).

Optimization of multivariable functions: Local and global maxima and minima, saddle points. Necessary and sufficient conditions for optimality.

Manifolds in R^N:

The implicit function theorem. Manifolds in R^N. Lagrange's multiplier.

3) Multiple integrals.

Measure theory: Volume of N-rectangles. Measurable sets and Jordan measure. Negligible sets and characterization of measurable sets. Sufficient conditions for measurability.

Integration: Integration of bounded functions over measurable sets. Properties of the integral. Iterated integrals. Change of variables formula for linear maps. Jacobian of linear maps. Change of variable formula for multiple integrals. Polar and cylindrical coordinates.

4) Linear differential forms.

Paths in R^N: general properties, paths equivalences. Path lenght.

Differential forms: definition, integrals of linear differential forms over paths. Closed forms, exact forms and their primitives. Equivalent conditions for a form to be exact in terms of integrals over paths. Differential forms defined on star-shaped or simply connected open sets.

5) Sequences and series of functions.

Pointwise and uniform convergence. Limits exchange under uniform convergence. Dini Theorem for monotone sequences. Total convergence of series. Power series and Taylor series. Theorems about convergence of sequences of derivatives and of primitives.

6) Differential equations.

Ordinary differential equations and systems. The Cauchy problem: local existence and uniqueness. Extensions of solutions and existence in the large. Qualitative study of solutions. Continuous dependence on the initial data. Explicit solutions of some classes of first order differential equations.

Linear equations and systems with constant coefficients. The exponential of a matrix. The evolution operator for 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).

W. Fleming: Functions of several variables, Springer, New York (1977);

W. Rudin: Principles of Mathematical Analysis, McGraw--Hill, New York (1976);

J. L. Taylor: Foundations of analysis, American Mathematical Society, Providence RI (2012).

## Teaching methods

The teaching consists in frontal lessons where both theoretical and applicable aspects are expounded. The exercises are developed with the collaboration of the students and are programmed in order that they can solve autonomously the problems arising from the theoretical lessons.

## Assessment methods and criteria

The final exam consists in a written test followed by an oral discussion, provided the written test is successful.

The written test consists in some open questions. To succeed, the students should demonstrate to have sufficient knowledge of the subjects of the course as well as calculus skills.

The oral test consists of a discussion about the written test, and in the verification of the level of comprehension of the theoretical parts of the course.

The test will be considered sufficient if the average of the marks of the written and oral tests is greater or equal to 18/30.