Vector Analysis and Differential Equations | LEB


Biomedical Engineering

Curricular Unit (UC)

Vector Anaysis and Differential Equations

Mandatory  x
Scientific Area MAT Category  

Course category: B - Basic; C - Core Engineering; E - Specialization; P - Complementary.

Year: 1st Semester: 2st ECTS: 7 Total Hours: 180
Contact Hours T: 45 TP: 45 PL: S: OT:4
Professor in charge

 Jorge das Neves Duarte

T - Lectures; TP - Theory and practice; PL - Lab Work; S - Seminar; OT - Tutorial Guidance.

  • Learning outcomes of the curricular unit

    Upon approval in this curricular unit, the student should be able to:

    Understand the basic concepts of limit, continuity and differentiability for scalar and vector fields.

    Solve problems in various contexts involving the chain rule.

    Understand the calculus of multiple integrals, identifying the geometrical representation of the domain and the convenient coordinates to be used.

    Define parametric representations of lines and surfaces and interpret and solve Engineering problems using line and surface integrals.

    Devise models based on scalar and/or vector fields and use spacial reasoning and visualisation in the analysis and solution of problems.

    Show a basic knowledge in the area of ordinary differential equations, including the solution of some 1st order equations and the linear equations of order n with constant coeficients.

    Apply the properties of linear differential equations.

    Choose autonomous and judicious learning strategies.

  • Conteúdos programáticos

    1. Introdução aos campos escalares e vectoriais. Noções topológicas em IRn, de campo escalar e vectorial, domínio, conjunto de nível, gráfico, limite e continuidade.

    2. Cálculo Diferencial em IRn. Derivadas segundo um vector, derivadas parciais de 1ª ordem e superior. Plano tangente e diferenciabilidade para campos escalares. Matriz Jacobiana e derivação da função composta para campos vectoriais.

    3. Cálculo Integral em IRn. Integrais duplos e triplos: definição, propriedades, cálculo, transformações de variáveis. Integrais de linha e de superfície: representação paramétrica de linhas e superfícies, integrais de campos escalares e vectoriais.

    4. Equações Diferenciais Ordinárias. Noção de equação diferencial, ordem, solução geral, problema de valores iniciais. Existência e unicidade de solução. Resolução de algumas equações de 1ª ordem. Aplicações. Propriedades e métodos gerais das equações diferenciais lineares de ordem n. Resolução das equações lineares de coeficientes constantes.

  • Syllabus

    1. Introduction to scalar and vector fields. Notions of topology in IRn, scalar and vector field, domain, level set, graphic, limit an continuity.

    2. Differential Calculus in IRn. Derivatives along vectors, partial derivatives of 1st and higher orders. Tangent plan and differentiability for scalar fields. The jacobian matrix and the chain rule for general vector fields.

    3. Integral Calculus in IRn. Double and triple integrals: definition, properties, evaluation, coordinate transforms. Line and surface integrals: parametric representation of curves and surfaces, integration of scalar and vector fields.

    4. Ordinary differential equations. Notion of differential equation, order, general solution, initial value problem. Existence and uniqueness of solution. Solution of some 1st order equations. Applications. Linear differential equations: general properties and methods. Solution of the linear equations with constant coefficients.

  • Demonstration of the syllabus coherence with the curricular unit's objectives

    Objectives 1 and 2 are met by syllabus chapters 1 and 2.

    The contents and practical examples of chapter 3 correspond to objectives 3 and 4.

    The syllabus chapters 1 to 3, which may be included in the general area of  Analysis in Rn, meet objective 5 particularly well as a consequence of the emphasis placed on the examples in dimension up to n=3.

    Syllabus chapter 4 acounts for  objectives 6 and 7.

    Objective 8 is inherent to the mathematical context of the issues under study and the general orientation that has been set for the curricular unit.

  • Teaching methodologies (including evaluation)

    The two theoretical weekly classes (a total of three ours) are dedicated, through the teacher’s initiative, to the presentation of issues and discussion of examples. The two practical classes ( three ours) are dedicated to finishing the resolution of the set of problems previously scheduled for each week by initiative of the student and with the assistance and ocasional initiative of the teacher,. Additional exercises are proposed that may be further discussed during the complementary period of doubt clarification.

    All relevant materials and information are electronically accessible to the students.

    There are two forms of evaluation: continuous and final. The first includes three tests and takes place during the class period. The student must obtain a minimum of eight values in each of the tests and achieve an average grade of ten values on the three tests to be approved. The final exams include the first, second and special dates.

  • Demonstration of the coherence between the teaching methodologies and the learning outcomes.

    The mathematical nature of this curricular unit requires a teaching approach including a time for formal correction, a time for intuitive interpretation and a time for getting acquainted with the issues under study and consolidating knowledge through practice and application. The separation of theoretical and practical classes aims to establish a weekly-based transition between these moments of the learning process. Emphasis is placed on the first two aspects during the theoretical classes taking place at the beginning of the week and on the third aspect during the following practical classes. This rythmic process also implies the weekly transfer of the initiative from the teacher to the student in agreement with the curricular unit objective number 8. The scheduling of issues and corresponding sets of exercises is essential for this purpose, and allows the reinforcement of the student’s habits of planning and finishing their work  in an effective way. A further reinforcement of this effect is intended by setting a continuous evaluation with a balanced frequency based on three tests, each corresponding to the conclusion of one the main thematic units – Differential Calculus, Integral Calculus and Differential Equations. The final exams complete the spectrum of possible approaches to obtaining approval at the curricular unit.


  • Main Bibliography

    R. Adams, Calculus: a complete course, Adison Wesley, 1999.

    T. Apostol, Cálculo, Ed. Reverté, 1983.

    Acilina Azenha e Maria Amélia Jerónimo, Cálculo Diferencial e Integral em IR e IRn, McGraw-Hill, 1995.

    W. E. Boyce e R. C. DiPrima, Equações Diferenciais Elementares e Problemas de Valor de Contorno, Livro Técnico e Científico, 1998.

    M. Braun, Differential Equations and their Applications, Springer, 1979.