65965 - Mathematics, Statistics and Physics

Learning outcomes

At the end of the course, the student is able to operate with real numbers, knows algebraic calculation and the fundamental properties of geometric figures. In particular, the student is able to: - use mathematical technical knowledge and tools to a good level; - set up and solve problems; - assimilate new concepts based on previous experience and knowledge. The student learns the main methods and basic tools of the quantitative study of collective phenomena. In particular, the student is able to: - interpret and critically evaluate information of a statistical nature (reading and understanding articles in journals and/or specialized publications); - autonomously produce and process statistical data; - apply some tools of the statistical methodology for the description and quantitative study of phenomena of biological and economic interest. At the end of the course, the student knows the fundamental principles of point mechanics and thermodynamics, in order to describe and interpret the physical phenomena involved in the main processes of the agricultural and agroforestry sector.

Course contents

Mathematics (Module 1)

Introduction. Numerical sets; functions; order relations; powers and logarithms; Analytic geometry; trigonometry: basic trigonometric functions and identities; Limits and continuous functions: neighborhoods and accumulation points; definition of limit and continuous function; sum, product, reciprocal and compound limit theorems; limits of elementary functions and indeterminate forms; comparison theorems; limits of sequences and the Nepero number; notable limits and calculation rules. Differential calculus: differentiable and derived functions; theorem on the derivative of sum, product, ratio, compound and inverse; derivative of elementary functions; higher order derivatives; main theorems of differential calculus: Fermat, mean value, intervals of monotonicity and convexity; function study. Riemann integration: definition of integral over an interval; properties of the integral and of integrable functions; integrals of continuous functions; integral mean theorem; fundamental theorem of calculus; integration methods: by parts, substitution, rational functions; improper integrals.

 

Statistics (Module 2)

1. INTRODUCTION

• Definition and purposes of Statistics. Stages of the statistical method. Structure of Statistics.

2. DESCRIPTIVE STATISTICS

• Description of data: Classification schemes. Numerosity, intensities and frequencies. Graphical representations.
• Measures of central tendency: mean, median, mode.
• Variability of data: Measures of variability. Covariance and correlation.

3. INFERENTIAL STATISTICS

• Probability: Definitions of probability. Axioms of probability. Probability distributions. Normal and standardized distributions.
• Sampling: Concepts of population and sample. Sampling techniques.
• Inference: Sum and sample mean. Parameter estimation. Confidence interval. Hypothesis testing. Chi-square test.
• Analysis of Variance: one-way and two-way ANOVA.
• Regression: Linear regression. Interpolation of data. Least squares criterion. Correlation coefficient.
  

 

Physics (Module 3)

Brief notes on the scientific method. Measurements and systems of units of measurement: physical quantities, symbols and units of measurement; standards and instruments, systems of units of measurement, the SI system, the c.g.s. system, fundamental and derived units. Vectors: vector quantities and scalar quantities; operations with vectors: sum, product of a scalar by a vector, scalar product. Kinematics: kinematics of the material point in one dimension and two dimensions: position and displacement, average and instantaneous velocity, average and instantaneous acceleration; uniform rectilinear motion, uniform circular motion, uniformly accelerated rectilinear motion. Dynamics: concepts of force and examples: gravitational force, weight force, ground reaction force, frictional force. Newton's laws. Inclined plane. Work and potential energy; work-energy principle; kinetic energy; principle of conservation of mechanical energy. Examples: work done by weight force. Thermodynamics: concepts of temperature and heat. The principles of thermodynamics. Specific heat. Examples of thermodynamic transformations: adiabatic, isochoric, isobaric, isotherm, cyclic transformations, free expansion. The concept of entropy and its importance; reversible and irreversible phenomena; the arrow of time. Equation of state of perfect gases. Heat engines; efficiency; Carnot cycle and efficiency of the Carnot heat engine. At the end of each topic some exercises will be discussed and carried out to meditate and better understand the concepts discussed.

Readings/Bibliography

Mathematics (Module 1)

• teacher's notes

• annotated lecture notes

• further references will be provided during lectures

Statistics (Module 2)

• teacher's notes

Physics (Modulo 3)

• teacher's notes

• exercises uploaded on the course virtual space

• Mazzoldi P., Nigro M., Voci C., Elementi di Fisica - Meccanica e Termodinamica. EdiSES

Teaching methods

Theory classes and exercises.

Assessment methods

The final grade of the exam is determined by the weighted average of the evaluations of the single modules that compose the integrated course. Weights are the following:

• Mathematics: 4/10
• Statistics: 3/10
• Physics: 3/10

The assessment methods of the single modules are as follows.

The final examination for Mathematics module consists in a written test and an oral exam. The written test is divided into two parts: a preliminary quiz to verify the learning of the minimum knowledge, and one in which the learning is evaluated in more detail and includes the resolution of exercises. The second part of the written test is accessed only if the preliminary part of the written test is passed. The oral exam is optional, but necessary in order to achieve a grade higher than 28.

The final examination for Statistics module consists in a written test. This test includes closed questions aimed at understanding the students' familiarity with the fundamental terms and concepts of Statistics; open questions are also foreseen which consist in reading and interpreting the results of descriptive and inferential statistical analyses. Duration of the test: 2 hours.

The final examination for Physics consists in a written test plus optional work to be carried out individually and handed in on the day the written test is taken.
The written test lasts 90 minutes and consists of 8 questions: 4 short numerical exercises and 4 open-ended theory questions. Each question is worth 4 points.
The optional work consists of writing concept maps of all the theoretical topics covered in class. The concept maps must meet the following requirements:
(i) be complete, i.e. cover all the topics discussed in class
(ii) be drawn using at least two different colours
(iii) be handwritten on paper
(iv) bear the student's name, surname and badge number on each side.
Concept maps will be assessed during the written examination and can be worth up to 4 points. If the maps do not meet one or more of the above requirements, no marks will be given. A preliminary check of the fulfilment of the requirements is possible by arranging a meeting with the professor.
The final mark for the Physics course is the sum of the marks obtained in the written test and the mark eventually obtained for the concept maps. If the total score exceeds 30 points, the grade is 30 with honours.

The examination committee is constituted by the instructors of the three modules that compose the integrated course.

Teaching tools

Laboratory tools and instruments will occasionally be brought in class.

Office hours

See the website of Mirko Maraldi

See the website of Antonello Pesce

See the website of Cosimo Rota