07169 - Fundamentals of Mathematics II

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. Review of the Taylor 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 of functions 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 fluid 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 and median 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

The teaching material needed to prepare for the exam will be provided by the teacher (slides and exercise sheets related to the topics covered). This teaching material will be uploaded to the Virtual platform 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

The knowledge and skills to be acquired are verified through a written test. The test consists of exercises on each of the two parts of the course, as well as one or two open questions of a more theoretical nature. The maximum score that can be obtained for the exercises in each of the two parts is 16, so that the maximum score for the entire written test is 32. The score for completely incorrect or absent answers is zero, no negative scores are foreseen. 

The time available to the student for the written test is 2 hours. During the test, the use of support material such as textbooks, notes, summary sheets, diagrams, cell phones (even with calculator apps), computer media is not permitted. The use of a simple personal calculator is allowed.

The maximum score that can be obtained by providing all correct and complete answers is therefore 30 cum laude, corresponding to a score of 31 or 32 obtained in the written test.

The exam is considered passed with a minimum score of 18, including at least 6 in each of the two parts.

All exercises are comparable (by type and difficulty level) with those done during the classroom exercises and with the supplementary exercises made available by the teacher during the course by uploading them to the Virtual platform.

Class attendance is very important in the learning process and strongly recommended, but it does not influence the evaluation process in any way. 

There are no papers to be delivered before the exam date.

To participate in the exam, it is necessary to have passed the Istituzioni di Matematica I exam and it is necessary to register via AlmaEsami for the chosen exam session, by the scheduled deadlines. 

The student who is registered on the AlmaEsami exam list on the day registration closes but does not show up to take the exam or notifies the teacher or unregisters on AlmaEsami, or the student who during the written exam expresses the desire to withdraw, requesting the professor to write RT on the sheets submitted, will be given the grade RT (withdrawn).

The student who submits his/her written test for correction without declaring the desire to withdraw and receives an insufficient grade (grade lower than 18/30) will be given the grade RE (rejected).

Teaching tools

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

Students with specific learning disorders (SLD) or temporary or permanent disabilities: please, contact the responsible University office (https://site.unibo.it/studenti-con-disabilita-e-dsa/it) in good time: it will be their responsibility to propose to the interested students acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the professor, who will assess the appropriateness of the adjustments also in relation to the teaching objectives of the course. 

Office hours

See the website of Silvia Tozza