Analysis
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2023/2024 - 1S
Cycles of Study/Courses
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
Pre-requirements (prior knowledge) and co-requirements (common knowledge)
One and several variables calculus.
Program
1 - First-order differential equations: separable differential equations; homogeneous differential equations and first-order linear differential equations; linear differential equations with constant and variable coefficients. Hyperbolic functions as a solution of differential equations.Differential equations of a higher order. Euler's method. Euler's equations.
2 - Line integrals and surface integralsPaths 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.
3 - Inverse function theorem; implicit function theorem; implicit differentiation.
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 among another one is available for lectures.
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
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 requirements.
Calculation formula of final grade
Score of the final exam.
Examinations or Special Assignments
Special assessment (TE, DA, ...)
According to the General Evaluation Rules.
Any student asking for an exam because of special conditions of his registration will do a written exam, but possibly, only, after an extra written or oral examination, in order to check if the student has a minimum knowledge about the unit so that he can do the special exam.
Classification improvement
The general evaluation rules apply.