# CALCULUS 2

## Learning outcomes of the course unit

The knowledge of the methods of n-dimensional Mathematical Analysis.

## Prerequisites

Any book of Elements of Mathematical Analysis 2.

## Course contents summary

Elementary notions of topology.

N-dimensional differential calculus.

N-dimensional integral calculus.

Linear ordinary differential equations of order n.

## Course contents

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

Any book of Calculus II

## Teaching methods

Oral lessons and exercises

## Assessment methods and criteria

Final examination

## Other informations

It is strongly recommended to attend the lessons.