# CALCULUS 2

## Learning outcomes of the course unit

Knowledge and understanding of the basic symbols and vocabulary concerning functions of several variables, and of the relevant theorems in the field.

Ability to apply the knowledge acquired to solve simple practical problems, and to understand the mathematical machinery employed in non-mathematical courses.

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

Mathematical analysis 1, Geometry and linear algebra

Any book of Elements of Mathematical Analysis 2.

## Course contents summary

Elements of differential calculus in several variables and 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-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

The list with the suggested textbooks will be updated later on the web page of the teacher (http://www2.unipr.it/~lorluc99/index.html) and shown during the first lecture of the course.

Any book of Elements of Mathematical Analysis 2.

## Teaching methods

Frontal lectures with exemples (2/3 of the course), exercises (1/3 of the course)

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

Final exam, with easy theoretical/practical questions and 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

It is strongly recommended to attend the lessons.