# CALCULUS 2

## 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 the analysis in more variables, and he should be able to grasp how these enter in the solution to 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 will be used in a more applied context.

Making judgements:

The student should be able to evaluate coherence and correctness of the results obtained by himself or offered him.

Communication skills:

The student should be able to communicate in a clear and precise way, also in a context broader than mere calculus.

Knowledge and understanding:

At the end of this course the student should know the essential definitions and results of the analysis in more variables, and he should be able to grasp how these enter in the solution to 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 will be used in a more applied context.

Making judgements:

The student should be able to evaluate coherence and correctness of the results obtained by himself or offered him.

Communication skills:

The student should be able to communicate in a clear and precise way, also in a context broader than mere calculus.

## Prerequisites

Passing both the exams of Mathematical Analysis 1 and Geometry is mandatory.

Any book of Elements of Mathematical Analysis 2.

## Course contents summary

1. Curves.

2. Vector valued functions: continuity and limits.

3. Differential calculus in several variables.

4. Multiple integrals.

5. 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-Integration along curves.

5-Riemann integral for functions of vector variable.

6-Linear ordinary differential equations with continuous coefficients.

## Course contents

1. Curves.

Preliminaries (linear algebra; polar coordinates and planar rotations). Parametric curves (velocity and acceleration; regular curves). Curve in polar coordinates. Length of a curve (curves in polar form; cartesian curves; piecewise regular curves). Repamareterization (cylindrical helix). Curvature and torsion.

2. Vector valued functions: continuity and limits.

Real functions of two real variables (level sets). Polar, spherical, and cylindrical coordinates (vector valued functions). Elemens of topology. Continuous functions (intermediate values theorem; Lipschitz and unuiformly continuous functions; distance function to a set). Integral of a function along a curve (curvilinear integral of functions and of vector fields; integral on the oriented boundary of planar sets). Quadratic forms (Sylvester criterion). Limits of functions.

3. Differential calculus in several variables.

Partial derivates (Jacobian matrix; a relevant counterexample; directional derivatives). Differentiable functions (infinitesimals and Taylor expansions; diﬀerential; maximum slope direction; total differential theorem; vector valued differentiable functions). Operations with partial derivatives. Higher order derivatives (Schwarz theorem; Taylor formula). Local extremes (Fermat theorem; nature of stationary points; sufficient conditions in two and three dimensions). Contrained extremes (Lagrange multipliers method in two variables). Surfaces in Euclidean space (Lagrange multipliers method in three variables). Potentials and curvilinear integrals (curl-free vector fields).

4. Multiple integrals.

Integral over a rectangular box (reduction formulas for double integrals). Integration on a normal set. Change of variables (polar coordinates; an improper integral; implicit transformations). Integrals in three dimensions (integration by wires, integration by layers; change of variables).

5. Diﬀerential equations.

Preliminar examples. Cauchy problem for equations and systems. Existence, extendability, and uniqueness of solutions. First order differential equations (first order linear equations; separable variable equations; Bernoulli equations). Second order linear equations with constant coeﬃcients (variation of constants). Linear systems with constant coefficients.

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-Integration along curves.

4.1 Parametric curves.

4.2 Length of a curve.

4.3 First and second type integrals.

5-Riemann integral for functions of vector variable.

5.1 Definition of Riemann integrable for function defined on a bounded regular n-dimensional set and properties of the integral.

5.2 Theorem of reduction of multiple integrals.

5.3 Theorem of the change of variables in multiple integrals.

6-Linear ordinary differential equations with continuous coefficients.

6.1 Theorem of characterization of the solutions of ordinary differential linear equations with continuous coefficients of order n.

6.2 Theorem of existence and uniqueness of the solution of the Cauchy problem.

6.3 Method for finding n linearly independent solutions of the homogeneous equation with constant coefficients.

6.4 Method for finding a particular solution of the non homogeneous equation.

## Recommended readings

Any book of Elements of Mathematical Analysis in more variables.

Any book of Elements of Mathematical Analysis 2.

## Teaching methods

Live streaming lectures, integrated by course notes. Live streaming exercise activities.

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

No test is expected during the course.

There will be a final written exam with both closed and free answers, lasting three hours. After passing the written exam an oral colloquium is mandatory: it concerns a discussion of the exercises, of the related theoretical results, and possibly the proof of one of the main results.

No test is expected during the course.

There will be a final written exam with both closed and free answers, lasting three hours. After passing the written exam an oral colloquium with a discussion of the exercises and of the related theoretical results is mandatory.

## Other informations

It is strongly recommended to attend the lessons.

It is strongly recommended to attend the lessons.