PHYSICAL METHODS APPLIED TO EARTH SCIENCES

The aim of the course is to provide adequate knowledge and skills in the field of mathematical methods applied to physics, as a tool for the treatment and modeling of geological and geophysical problems, as well as and basic knowledge of programming for the development of codes for the analysis of data acquired in the field of geology and geophysics.

At the end of the course, the student will have acquired inductive and deductive reasoning skills, will be able to schematize a phenomenon in terms of physical quantities, will be able to critically deal with the studied subjects, set a problem and solve it with analytical methods, taking care of them, with due rigor, both mathematical and physical aspects. The student will apply the scientific method to the study of natural phenomena and will be able to critically evaluate analogies and differences between physical systems and the methodologies to be used. He will also be able to use the appropriate mathematical tools, with due rigor, in the modeling of geophysical problems. The student will acquire the ability to develop codes for the analysis of data acquired in the field of geology and geophysics.

In particular, among the learning objectives, the following skills will be assessed: 

- knowledge and understanding

- applying knowledge and understanding

- making judgements - communication skills

Frontal lessons and exercises.

If the course is taught in mixed or distance mode, the necessary changes may be introduced to comply with the planned program and reported in the syllabus.

Information for students with disabilities and/or Specific Learning Disorders:

To guarantee equal opportunities and in compliance with the laws in force, interested students can request a personal interview in order to plan any compensatory and/or dispensatory measures, based on the educational objectives and specific needs.

It is also possible to contact the CInAP (Centre for Active and Participatory Integration - Services for Disabilities and/or DSA) contact professor of our Department, prof. George De Guidi.

1) DIFFERENTIAL AND INTEGRAL CALCULUS 

Functions of several variables. Limits and continuity, partial derivatives, differential and differentiable functions, higher order derivatives and Schwartz's lemma. Taylor series. Extremes. Integral calculus for functions of one variable. Riemann integral. Fundamental theorem of integral calculus. Remarkable integrals. Improper integrals. Integral calculus for functions of several variables. Vector differential calculus. Differential operators: gradient, divergence, curl and Laplacian. 

2) NUMERICAL SERIES AND SERIES OF FUNCTIONS, FOURIER ANALYSIS. 

Number series. Convergence and absolute convergence. General theorems on numerical series. Convergence test of series with positive terms, series with alternating signs. Harmonic series, geometric series. Function series. Punctual and uniform convergence. Power series. Taylor and MacLaurin series. Fourier series and their convergence. Examples and applications: square and triangular wave. Review of complex numbers. Fourier transforms. Spectral analysis of a signal. 

3) PROBABILITy

Samplings. Binomial coefficients. Conditional probability. Independent and mutually independent events. Bayes theorem. Random variables. Probability mass function. Expectation values and variances of Bernoulli, binomial, geometric and Poisson. Rare events and radioactive decay. Two or more random variables. Joint and marginal distribution. Independent variables. Covariance and correlation coefficient. Conditional expectation values. Applications to the random walk. Continuous random variables. Cumulative distribution and probability density. Uniform, exponential and normal distributions. Central limit theorem.

4) APPLICATION 1: UNIVARIATE AND BIVARIATE STATISTICS

Fundamentals of Matlab programming: variables, indexing, operators, matrix operations, scripts, functions, 2D plot, 3D plot, flow control (if statement, for loop, while loop). Empirical distributions, measure of central tendency and dispersion, correlation coefficients, linear regression, estimation of regression coefficients by bootstrap, jackknife and cross-validation.

5) APPLICATION 2: TIME SERIES ANALYSIS

Creating signals in time domain, spectral analysis, spectral analysis of non-stationary signals (Short Time Fourier Transform, Wavelet Power Spectrum), comparison between signals via cross wavelet spectrum and wavelet coherence, interpolation in one dimension.

The exam consists of an oral test of about 30 minutes aimed at ascertaining the level of knowledge and understanding achieved by the student on the theoretical contents indicated in the program, and the ability to write code in Matlab environment. Students will be able to start the exam by presenting a topic of their choice.

The verification of learning can also be carried out online, should the conditions require it.

What is the differential of a function? 

What are the critical points for a function of two variables. 

Give the definition of integral according to Riemann 

Calculate the primitive of a given function 

What is an improper integral 

What is the geometric series? How can we prove its convergence? And to what value does it converge? 

Talk about Fourier series and Taylor series. 

How is a Fourier transform defined and what is it used for? 

Give the definition of conditional probability. 

Prove Bayes' theorem. 

Calculate expectations and variances for Bernoulli, binomial, geometric and Poisson distributions.

What is meant by empirical distribution?

What are mean, median, mode and standard deviation and how are they calculated?

Talk about the main correlation coefficients.

Talk about spectral analysis.

What are the main differences between Short-Time Fourier Transform and Wavelet Power Spectrum?