Infinitesimal Analysis

Code:

M2028

Acronym:

M2028

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2022/2023 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=4204
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Geospatial Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:EG	15	The study plan from 2019	2	-	6	56	162

Teaching language

Portuguese

Objectives

Vector Analysis in curve domains. Line and surface integrals. Integral theorems of Vector Analysis.  
Inverse function theorem, implicit function theorem and its main applications. 
Introduction to methods of solving ordinary differential equations with special emphasis on equations and systems of linear differential equations.

Learning outcomes and competences

Problem-solving skills.
Theoretical understanding

Working method

Presencial

Program

1 - Line integrals and surface integrals

Paths in Rⁿ; line integral of a scalar function; vector fields; work of a vector field along a path; conservative fields and gradient fields; Green theorem; principle of energy conservation; differential forms; parametrizatiion and geometry of surfaces; surface integrals; surface areas; integral of a scalar function on a surface; surface orientation; vector field flux on a surface; rotational and divergence operators; Stokes' theorem; Divergence (Gauss) theorem.

2 - Inverse function theorem; implicit function theorem; implicit differentiation.

3 - Some physical processes described by differential equations; motion of a particle in free fall; oscillations; linearization around an equilibrium point; processes of exponential growth and decay; hyperbolic functions as a solution of differential equations.
First-order differential equations: separable differential equations; homogeneous differential equations and first-order linear differential equations; linear differential equations with constant and variable coefficients.

Mandatory literature

Marsden Jerrold; Calculus ii. 2nd ed. ISBN: 0-387-90975-3
Marsden Jerrold; Calculus iii. 2nd ed. ISBN: 0-387-90985-0

Complementary Bibliography

Morris W. Hirsch; Differential equations, dynamical systems, and linear algebra. ISBN: 0-12-349550
Braun M.; Differential equations and their applications. ISBN: 0-387-90114-0

Comments from the literature

All the supporting material made available for lectures is the most important "bibliography".

Teaching methods and learning activities

There are two types of lessons: lectures and pratical classes.  There are also practical lessons, where exercises and related problems are solved. All resources are available for students at the unit’s web page.

Lectures:
Presentation of the syllabus content, where examples are given to illustrate the explained concepts.
Resolution of some illustrative exercises and proposed work to be done in pratical classes.

Pratical classes:
Resolution of the proposed exercises.
Answering questions about the resolution of problems and proposed work.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	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 requirements.

Calculation formula of final grade

Continuous evaluation without final exam

Continuous evaluation is based on two test results. The tests will be done  at the following days:

• November, 11 (room 1.22, 9-11 am);
• December, 16 (room 1.22, 9-11 am).

The final score will be the sum of the two test scores.

Any student can choose not to submit to continuous evaluation and obtain the final classification performing the examination in the second examination period (Época de Recurso).

In any case, a student with a final grade ≥ 16.5 may eventually be subjected to an extra oral or written test.

All registered students are admitted, without restrictions, to the tests and exams.

Examinations or Special Assignments

Special assessment (TE, DA, ...)

According to the General Evaluation Rules.

Classification improvement

The general evaluation rules apply.