65934 - Principles of Mathematics 2

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 (ODE) and systems of ordinary differential equations. 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

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. Maxima, minima and critical points. Review of eigenvalues, eigenvectors and positive/negative definite matrices.

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.

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. Introduction to compartment models.

Systems of two 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.

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. Likelihood functions. Estimators of maximum likelihood. 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. Operative characteristic curve. Unilateral and bilateral tests. 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. Tests on two normal population with same mean.

Linear regression. Esteem of the regression parameters. The least square method.

Readings/Bibliography

Teaching material is provided by the instructor. The material includes:

• all the slides used in class;
• exercises sheets (with solutions to exercises)

Recommended textbooks:

• (under definition)

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. At the end of each main subject, other entirely solved exercised are proposed.

Assessment methods

Written test with exercises of the same type and difficulty level as those presented in class

Teaching tools

Notebook and/or tablet PC and projector

Office hours

See the website of Andrea Mentrelli