# 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.

## Prerequisites

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

## Course contents summary

1. Curves.

2. Vector valued functions: continuity and limits.

3. Differential calculus in several variables.

4. Multiple integrals.

5. Differential equations.

## 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.

## Recommended readings

Any book of Elements of Mathematical Analysis in more variables.

## Teaching methods

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

## 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.

## Other informations

It is strongly recommended to attend the lessons.