00674 - Mathematics

Learning outcomes

On successful completion of the course, the student will have acquired the basic knowledge of calculus, linear algebra and geometry, essential to describe geological processes and to deal with other courses of the degree programme, especially those related to physics. In particular, the student will be able to represent functions graphically, to apply one-variable and multivariable calculus, to compute solutions of first order differential equations, to perform operations on vectors and matrices, and to solve systems of linear equations and easy geometric problems on the plane and the three-dimensional space.

Course contents

MODULE 1

Set theory and combinatorics

• Set and functions.
• Fondamental counting principles. 
  

Linear algebra

• The geometric vectors: algebraic structure, scalar and vector product.
• Matrices: vector structure and product of matrices; echelon form; definition of rank and calculation techniques; linear transformation associated with a matrix.
• Square matrices: invertible matrices; definition of determinant and calculation techniques.
• Linear systems: matrix notation; Rouché-Capelli theorem and solution techniques for linear systems; parametric and Cartesian representation of subspaces of R^n; structure theorem for linear systems.

Analysis

• Real functions of a real variable: definition, injectivity, surjectivity, monotony; graph of a function; elementary functions (powers, roots, exponentials, logarithms, functions, circular); limits and continuity.
• Differential calculus for real functions of real variable: derivative, growth and decrease, local extremes, study of the graph of a
  function, Taylor's formula.
• Integral calculus for real functions of real variable: primitive, fundamental theorem of integral calculus, integration by substitution and by parts.
  

MODULE 2

Introduction to the course; introduction to descriptive statistics; starting test; means, median, and mode; quantiles.

Dipsersion indices; data distributions; the normal distribution; the standard normal curve and the test Z.

Normality tests (quantiles and Q-Q plot; Shapiro and Wilk’s test).

Introduction to inferential statistics; one-sample Student’s t-test.

Two-sample t-test (paired/unpaired samples); nonparametric tests: Wilcoxon’s and Mann-Whitney tests.

Qualitative variables: chi squared test.

Linear regression and correlation: Pearson’s method, r and R squared; significance of a correlation.

Analysis of variance; one-way and two-way ANOVA; Tukey’s post-hoc test.

Lab class: introduction to R; create an input file; call and explore data; normality and t tests with R; chi squared, regression and ANOVA with R.

 

 

Readings/Bibliography

• Marco Abate. Matematica e Statistica. Le basi per le scienze della vita. Mc Graw Hill.

Teaching methods

Traditional lesson and exercises.

Assessment methods

The exam consists of a written test and an oral exam. More information are available on virtuale.

Teaching tools

All the material is available on Virtuale. [https://virtuale.unibo.it/]

In addition to the usual lessons, a tutor will be available every week to answer questions and to helps solving the exercises.

Office hours

See the website of Riccardo Biagioli