65965 - Mathematics, Statistics and Physics

Learning outcomes

Mathematics: The students will be able to operate with real numbers, to understand the algebraic calculation and the basic properties of the geometrical figures. The student will be able in general to: use the mathematical knowledge and the mathematical tools; to set and solve problems; to learn new concepts on the base of its experience and the previous knowledge. In Statistics the student will learn the main methods and tools for the qualitative and quantitative inferential and descriptive  analysis. In particular: interpret and critically evaluate the statistical information; produce and process raw data; apply the statistical approach to the natual and social sciences topics related to animal production.

In Physics: the student will learn the meaning of the different units of measure; the meaning of the basic physics terms; the basic principls of physics; the student will then be able to follow the successive courses of physiology, food technology, animal production related buildings and machinery.

Course contents

Module 1 Mathematics

An attempt will be made to answer, collectively, the following questions: what is Mathematics? Can Mathematics teach us anything valuable? During the process of finding an answer to these questions, we will try to overcome the common point of view under which Mathematics is only a matter of solving exercises and doing calculations. During the course:

1. we will highlight the concise character of the mathematical notation (each graphic sign is necessary, no signs are redundant)
2. we will examine in greater depth some of the topics already studied in the previous high-school classes and we will introduce some new upper-level topics
3. we will observe how the apparent complexity and rigour in the definition of a certain object is necessary to extend its field of existence and applicability

In detail, the topics covered will be:

PREPARATORY TOPICS:

• scientific books/papers: how to use and how to cite them
• basic mathematical notation; order of operations
• proportions
• linear and second-order algebraic equations and their solution
• polynomials
• the Cartesian plane and how to draw it
• area and volume of simple plane figures and simple solids
• binary operators and properties of binary operations: associative and commutative properties; neutral, null and opposite elements; examples
• unary operators: linear operators; definition of linearity; examples

  TRIGONOMETRY

• similar triangles; definition of sine, cosine, tangent
• the unit circle and fundamental theorems of Trigonometry
• sine, cosine and tangent values for special angles
• the importance of the cosine in the projection of vectors and of the sine in the evaluation of the area

  REAL-VALUED FUNCTIONS OF REAL VARIABLES

• intervals: open, closed, bounded, unbounded
• Cartesian product of intervals
• definition of a function and examples of functions and non-functions; domain, codomain, image
• injective, surjective and bijective functions; the inverse function and functions composition
• drawing functions on the Cartesian plane
• -odd and even functions; examples

  INTEGRALS – I

• integral as the signed area of a function graph: rectangle algorithm
• integrals properties: domain decomposition; limits swap; integrals of odd and even functions

  LINEAR ALGEBRA

• -vectors as Rⁿ objects; matrices as tabular lists of vectors
• -operations with matrices: sum, multiplication, scalar multiplication
• matrices representative of linear transformations; proof of linearity
• special transformations: scaling; mirroring; identity matrix; null matrix; inverse matrix;

  LINEAR FUNCTIONS

• equation of a straight line: explicit, implicit and parametric forms
• straight lines as linear functions and affine functions; proof of linearity, surjectivity, injectivity; proof of evenness and oddness
• directional cosines
• equations of a straight line and of a plane in R³

  NON-LINEAR FUNCTIONS

• the exponential function; properties of exponents
• the logarithmic function; properties of logarithms; logarithmic and semi-logarithmic scales
• periodic functions and trigonometric functions: sine, arcsine; cosine, arccosine; tangent, arctangent

  LIMITS OF FUNCTIONS

• rigorous definition of the limit of a function; using the definition to determine the limit of simple functions; on continuous functions
• definition of limit at infinity; limit at infinity of simple functions; asymptotic behaviour of polynomials
• -non-existence of the limit; sin(x) at infinity; 1/x at zero
• behaviour of polynomials at zero; definition of the term “asymptotic”
• special limits: (sin x)/x for x à 0; (1-cos x)/x for x à0; (1-cos x)/(1/2 x^2) for x à0
• comparison of the end-behaviour of the functions x, ln(x), exp(x), x^N

  DERIVATIVES

• equation of the secant to a curve; difference quotient; limit of the difference quotient; rigorous definition of a derivative
• tangent equation; linearization of a curve; physical meaning of a derivative
• maximum and minimum values of a function and link with the first derivative of the function; Fermat’s theorem and applicability range
• differential operators and proof of linearity

  INTEGRALS – II

• the integral operator as the inverse of the differential operator
• the fundamental theorems of calculus
• proof of linearity of the integral operator
• physical meaning of the integral
• mean of a function

Module 2 Statistics

Theory 

Introduction: Definition and goal of statistics. Structure of Statistics. Definition of Universe, collective, population, sample. Classification.

Classification schemes: omograde and eterograde classes, numerosity,intensity and frequence. Graphical representations in statistics.

Measures of central tendency: Arithmetic and geometric mean, central value, median and mode.

Measures of statistical dispersion: standard deviation, variance, coefficient of variation.

Probability: Definiiton and assioms of probaility.Binomial and normal distribution.Standardized normal distribution. Theory of errors.

Sampling: general features and meaning.

Statistical inference: sample sum and mean, interval of confidence, Hypothesis test. chi-square test.Covariance correlation and linear regression.analysis of Variance.

Group work:application of descriptive and inferential statistical techniques to simple cases related to the animal production sector

Readings/Bibliography

Module 1 (Mathematics)

• Teacher’s notes
• Internet
• Textbooks and other material provided during the class

1) S. Motta, M.A. Ragusa. Metodi e Modelli Matematici. Libreria CULC, 2011.
2) S. Motta, M.A. Ragusa, A. Scapellato. Metodi e Modelli Matematici. Esercizi e Complementi. Libreria CULC, 2013

Module 2 (Statistics)

Teacher notes and slides

Internet + other material provided during the class

Module 3 (Physics)

Fondamenti di Fisica. Halliday D., Resnick R., Walker J. Casa editrice ambrosiana

Elementi di Fisica. Ageno M. Bollati Boringhieri

Teaching methods

Classroom lessons and group works on case studies

Assessment methods

The final mark of the integrated course will be calculated as an arithmetic mean of the three different modules.

In particular: the modules Mathematics involves a first preliminary written test; if the test is passed an oral examination will follow.

The module Staitstics will involve a written test with both multiple choice quesitons and open quesitons.

Ther module Physics involves and oral examination lasting about 30 mins

  The marks in each module will be valid for 3 years. That is after the first module exam results the student will have three years to finalize the exams for the remaining two modules of  the integrated course

Teaching tools

Mathematics:  papers and slides

Statistics: online and offline statistical software, slides and papers

Phyisics: papers and slides

Office hours

See the website of Cesare Zanasi

See the website of Emanuel Guariglia

See the website of Mirko Maraldi