Introduction to Applied Mathematics

Code:

M1027

Acronym:

M1027

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 2S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=955
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:EG	3	The study plan from 2019	3	-	6	56	162
L:M	102	Official Study Plan	1	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

Application of mathematical concepts, namely the ones studied in other first-year courses, to the analytical and numerical treatment of mathematical models in Physics, Biology, Ecology, Economics, Medicine and other fields of knowledge.

Learning outcomes and competences

Ability to solve exercises and mathematical modeling problems.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Prerequisites: Real Analysis I, Linear Algebra and Analytic Geometry I, Mathematical Laboratory (or equivalent courses).
Co-requirements: Real Analysis II.

Prerequisites: basic knowledge of:
Differential Calculus of real functions of a real variable;
 Linear Algebra and Analytic Geometry: linear and affine maps, matrices, determinants.

Co-requirements: basic knowledge of:
Differential Calculus of real functions of several real variables, level curves.

Program

1) Discrete time mathematical modeling with classical examples of application: 

a) modeling in one dimension: discrete dynamical system and its variation, resolution of linear and affine model; fixed points, phase portrait and graph; models in Economics, Biology and Social Sciences; 

b) modeling in dimensions two and three: discrete dynamical system and its variation, resolution of the dynamical system in the linear case; fixed points, phase portrait (in dimension two) and graph; models in Ecology and Epidemiology.

2) Adapting a model to a set of points: transformation into an affine model, graphical method and least square method to determine an affine model. 

3) Continuous time mathematical modeling with classical examples of application: 

a) first order autonomous differential equation (o.d.e.): resolution in the linear and affine case as well cases where an explicit solution can be obtained; continuous time  models in Pharmacy, Physics and Biology; phase portrait of an autonomous o.d.e.: equilibria, monotonicity intervals, concavity; graph of the solutions obtained from the analysis of the phase portrait.

b) conservative systems with one degree of freedom, stability of equilibria, phase portrait on the plane, applications to Physics.

 

 

Mandatory literature

Giordano Frank R.; A first course in mathematical modeling. ISBN: 978-0-495-55877-4
James Stewart; Calculus. ISBN: 978-1-305-27237-8

Complementary Bibliography

Britton Nicholas F.; Essential mathematical biology. ISBN: 1-85233-536-X
Brauer Fred; Mathematical models in population biology and epidemiology. ISBN: 0-387-98902-1
Burghes D. N.; Modelling with differential equations. ISBN: 0-85312-286-5
Arnold V. I.; Equações diferenciais ordinárias
Gregory V. Bard; Sage for Undergraduates, American Mathematical Society, 2015. ISBN: ISBN: 978-1470411114 (http://gregorybard.com/sage_for_undergraduates_color.pdf.zip )
Jaime E. Villate; Dinâmica e Sistemas Dinâmicos, 2019. ISBN: ISBN: 978-972-99396-5-5 (https://def.fe.up.pt/dinamica/index.html)

Teaching methods and learning activities

Theoretical classes: exposition of theory and discussion of examples. Proposed problems to be solved out of class and to be treated in the practical classes.

Practical classes with resolution of exercises and concrete problems.

Software

sagemath
wxmaxima

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	56,00
Total:	162,00

Eligibility for exams

No requisites.

Calculation formula of final grade

All enroled students are admitted to the final exam, scored for 20 values.

Classification improvement