# General Mathematics

## Objectives

To provide the basis for some working knowledge of mathematical techniques (in the Algebra and Analysis domains) relevant for Biology, Physics and Chemestry problem modelization

## General characterization

##### Code

10692

##### Credits

6.0

##### Responsible teacher

Jorge Orestes Lasbarrères Cerdeira, Paulo José Fernandes Louro Ribeiro Doutor

##### Hours

Weekly - 6

Total - 88

##### Teaching language

Português

### Prerequisites

The student must be familiar with mathematics taught at pre-university level in Portugal (science area).

### Bibliography

Lecture notes covering all the program contents, available for students

Howard Anton and Chris Rorres, Elementary Linear Algebra with Applications (11th Edition), John Wiley & Sons, 2013. ISBN: 978-1-118-43441-3

William F. Trench, Introduction to Real Analysis, Pearson Education, 2013 Faculty Authored and Edited Books & CDs. 7. https://digitalcommons.trinity.edu/mono/7

Cabral, I., Perdigão, C., Saiago, C., Álgebra Linear, (5ª edição), Escolar Editora, 2018.

### Teaching method

In this module there are theoretical and practical classes.

Theoretical lectures consist on the introduction of theoretical concepts, and the presentation of proofs of some results, using blackboard and slides.

Practical lectures consist of solving and discuss some of the proposed exercises.

Students dispose from texts covering all subjects, including lists of suggested exercises and application problems.

Substantial part of the study is done on learner autonomy, using the lecture texts and bibliographic brackets, and with the support of teachers outside the classroom, on pre-established schedules reserved for the students attendence.

### Evaluation method

Rules of evaluation

The student will be excluded of the evaluation if his presences in problem-solving lessons is inferior to 2/3.

The student may be evaluated by two tests with value 10 and will be approved if the two tests sum up (rounded) at least 10. The grade will be the rounded sum of the tests.

The student may also be approved by a final exam if the exam''s grade is at least 10. The grade will be the one attained in the exam (rounded) and any grade in any test will be discarded.

## Subject matter

1 - Linear Algebra

1.1 Matrices. Examples. Matrix operations (transposition, sum, multiplication by a scalar, product) - definition and properties. Invertible matrices.

1.2 Systems of linear equations. Matricial form of a system. Elementary operations on matrices. Hermite matrix. Rank of a matrix. Resolution and discussion of systems.

1.3 Determinants - definition and properties. Relation between determinant and invertibility of a matrix. Applications.

1.4 Eigenvalues and eigenvectors. Definition. Diagonalization of a square matrix and applications.

2 - Integral Calculus in R

2.1 Revisions: notion of derivative of a function; derivation rules (sum, difference, product, quotient, and chain rule). Inverse function theorem. Inverse trigonometric functions and expression of the respective derivatives.

2.2 Indefinite integrals. Integration by parts and by substitution. Integration of rational functions.

2.3 Riemann’s integral: definition and geometric interpretation. Integrability of sectionally continuous functions. Mean value theorem. Fundamental Theorem of Calculus. Integration by parts and integration by substitution. Applications.

2.4 Improper integrals: definition and examples. Applications.

3 - Ordinary Differential Equations

3.1 Concept of Differential Equation. Initial value problems.

3.2 First-order linear differential equations. Application to exponential models.

3.3 Separable variable equations. Application to logistic models.

3.4 Second order linear differential equations of constant coefficients. Applications.