65934 - Principles of Mathematics 2

Academic Year 2022/2023

  • Docente: Silvia Tozza
  • Credits: 8
  • SSD: MAT/05
  • Language: Italian
  • Teaching Mode: Traditional lectures
  • Campus: Ravenna
  • Corso: First cycle degree programme (L) in Environmental Sciences (cod. 8011)

Learning outcomes

At the end of the course, the student gains familiarity with the basic tools of calculus of multiple variables, and knows how to handle or solve some types of Ordinary Differential Equations (ODEs) and systems of ODEs. In addition, the student is familiar with the basis of probability and statistics and knows how to tackle problems involving statistics and random variables.

Course contents

First part of the course

Series. Brief review of the Taylor series and brief introduction to the Fourier series.

Functions of more than one variable. Domain and Range. Plot of a function and contour lines for functions of two variables. Contour surfaces for functions of three variables. Limits and continuity. Polynomial and rational functions.

Partial derivatives. Definitions and geometrical interpretation of partial derivatives. Differentiable functions, tangent planes and Taylor expansion offunctions of two variables. Directional derivatives. Gradient. Critical points. Maxima and minima. Saddle points. Recall of determinant and trace of a matrix. Review of eigenvalues, eigenvectors and positive/negative definite matrices. Definition of the Hessian matrix. Test on a Hessian matrix to classify the critical points.

Vector fields. Conservative fields and potential. Criteria for existence of the potential of a vector field. Technique for obtaining the potential of a conservative field. Line integrals of vector fields. Divergence and curl, and their physical interpretation. Partial, total, and material time derivative. Overview of the continuity equation and of the Eulerian/Lagrangian approaches in fuid-dynamics.

Ordinary differential equations. Exponential growth and decay. First order equations. Initial value problems. Verification of the solutions. Equations with separable variables. First order linear equations and general technique for their solutions. The logistic growth model. Omogeneous linear equations of the second order. Introduction to the second order non homogeneous differential equations.

One-compartment models. Introduction to one-compartment models; Relative rate and absolute rate; uptake rate and downtake rate.

Systems of ordinary differential equations. Reduction method and matrix method. Trajectories. Equilibrium points. Stability of the origin. Systems of two non-linear equations: linearization in the neighborhood of an equilibrium point and discussion on the stability. The logistic growth model for the coexistence of two species. The Lotka-Volterra predator/prey model.

 

Second part of the course

Descriptive statistics. Introduction to statistics. Samples and populations. Graphical representation of collected data. Mean value, median and mode of a sample. Quartiles and percentiles. Variance and standard deviation of a sample. Coefficients of asymmetry. Box-plot. Bivariate data sets, scatter diagram. Correlation. Linear regression - least squares method.

Theory of probability. Sample space and events. Incompatible events. The Kolmogorov axioms. Equally probable spaces. Partitions. Conditional probability and the Bayes theorem (introduction). Random variables. Probability distribution functions. Cumulative distribution function. Probability density function, Mean value, median and mode of a random variable. Bernoulli experiment and binomial distribution. Poisson, exponential, normal (and standard normal) distributions. Chi-square and t-Student distributions.

Distribution of sample statistics. Introduction to the inferential statistics. Mean value and variance of the sample mean. The central limit theorem. Correction of continuity. Mean of the sample variance. Joint distribution of the mean and the variance of a sample.

Parametric estimation. Estimators and esteems. Unilateral and bilateral confidence intervals.

Statistical hypothesis testing. Null and alternate hypothesis. First type and second type errors. Significativity level of a test and p-value. Critical region. Unilateral and bilateral tests. Statistical hypothesis testing on the mean of a normal population with known variance (Z test). Test on non-normal populations. Tests on populations with unknown mean and variance (t-test). Statistical hypothesis testing on the mean of a normal population with unknown variance.  

Readings/Bibliography

Teaching material provided by the teacher (slides and sheets of exercises related to all the topics covered during the course).

For further deepening, a list of auxiliary texts for possible consultation is provided below:

  • J. Stewart, Calculus - Early Trascendentals, 8th Edition, Cencage Learning, 2016
  • C. D. Pagani, S. Salsa, Analisi Matematica 2, II Edizione, Zanichelli, 2016
  • S. M. Ross, Probabilità e Statistica per l'Ingegneria e le Scienze, II Edizione, Apogeo (2008)

Teaching methods

During lectures, the focus is primarily on the applications of mathematics and statistics, with special emphasis on environmental sciences.

Topics are presented along with many examples and exercises.

Assessment methods

Written test composed by one set of exercises on the first part of the course, and one set of exercises on the second part of the course, in addition to one or two questions concerning more theoretical aspects of the course. The total time available for the written test is two hours.

All exercises are comparable (by type and level of difficulty) with those carried out during classroom exercises and with the supplementary exercises made available by the teacher during the course.

Teaching tools

Slides and other material provided in electronic format (exercise sheets, etc.)

Office hours

See the website of Silvia Tozza