# MATHEMATICAL ANALYSIS C

## Learning outcomes of the course unit

The knowledge and mastery of the methodological aspects and tools of Differential and Integral Calculus for functions of several real variables required to understand the basic concepts and instruments of the technical-applicational discipline in order to be able to use them to interpret and describe Mechanical Engineering problems.

## Prerequisites

Mathematical Analysis AB

## Course contents summary

Plane curves.

Parametrisations, support, derived vectors, scalar velocity, tangent and normal vectors, straight line tangents and normal tangents. Class C1 curves, dotted and regular C1 curves. Length of a curve.

Parametrisations of the edge of subsets of a plane.

Two variable functions

Graph, sections and level curves. Paraboloids, cones, spherical and elliptical surfaces. Functions depending on a single variable and radial functions.

Limits and continuity for functions of two real variables. Elements of topology: interior, cluster and frontier points; open sets and closed sets; open connected sets. Limits for functions of two real variables and their properties. Continuous functions of two real variables and their properties. Weierstrass’s Theorem. Theorem on the existence of zeros.

Differential calculus for functions of two real variables.

Directional derivatives and partial derivatives. C1 functions and their properties. The gradient and its significance.

Differentiable functions. Tangent plane, tangent and normal vectors in the graph of a function. Maximum slope. Derivative of the composition of functions.

Stationary points. C2 functions and the Hessian matrix. Necessary and sufficient conditions for local extreme values. Optimisation of C2 functions.

Ordinary differential equations.

Physics models: population growth, friction spring, fall of a heavy body, electrical circuit with resistor and inductor.

First-order linear ODEs. The Cauchy problem.

n order linear ODEs with constant coefficient. General integral of homogeneous equations. Direct method for the calculation of a particular integral of complete equations.. The Cauchy problem.

Method of variation of constants for second order equations. The particular solution of the complete equation seen as a convolution integral.

Space curves, line integrals, vectorial functions

Space curves. Line integrals of functions on plane and space curves. Properties and applications. Vector valued functions. The Jacob Matrix.

Integral calculus for functions of two real variables.

Double integral construction. Functions of two integrable variables. Geometric significance. Normal domains on the plane. Reduction formulas form double integrals in normal domains. Integrals of Symmetric functions in symmetric domains. Change of variables theorem for double integrals. Regular transformations. Polar coordinates.

Integral calculus for functions of three real variables.

Triple integral construction. Normal domains in three-dimensional space. Reduction formulas for triple integrals: layer and line integration.

Change of variables theorem for triple integrals. Spherical and cylindrical coordinates.

## Recommended readings

Course notes and solutions for the written tests (available at the documents centre).

N. Fusco - P. Marcellini - C. Sbordone, Elementi di Analisi Matematica 2, Liguori Editore, Napoli (2001).

M. Giaquinta - G. Modica, Note di Analisi Matematica. Funzioni di più variabili. Pitagora Editrice Bologna (2006).

## Teaching methods

The course will take the form of classroom lectures and exercise sessions, which will be an integral part of the course.

During the course students are given two written progress tests the results of which contribute to the passing of the written exam. Each written test consists of a first part with multiple-choice questions and score threshold, and a second part with open-answer questions. The oral test, which is usually optional, is at the professor’s discretion for students barely passing the written test.

Students not wishing to sit the progress tests may sit the written and oral examinations in the exam sessions held subsequent to the end of the course.