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Calculus III

Code: EQ0068     Acronym: AM III

Classification Keyword
OFICIAL Physical Sciences (Mathematics)

Instance: 2020/2021 - 1S Ícone do Moodle

Active? Yes
Responsible unit: Department of Chemical Engineering
Course/CS Responsible: Master in Chemical Engineering

Cycles of Study/Courses

Acronym No. of Students Study Plan Curricular Years Credits UCN Credits ECTS Contact hours Total Time
MIEQ 93 Syllabus 2 - 6 63 162

Teaching Staff - Responsibilities

Teacher Responsibility
Fernão Domingos de Montenegro Baptista Malheiro de Magalhães

Teaching - Hours

Lectures: 3,00
Recitations: 1,50
Type Teacher Classes Hour
Lectures Totals 1 3,00
Fernão Domingos de Montenegro Baptista Malheiro de Magalhães 3,00
Recitations Totals 3 4,50
Maria Joana Monteiro de Carvalho Peres 4,50
Mais informaçõesLast updated on 2020-09-22.

Fields changed: Components of Evaluation and Contact Hours, Fórmula de cálculo da classificação final

Teaching language




The theoretical modeling of transient physical-chemical phenomena is an important component of Chemical Engineering, which is set on the construction and resolution of mathematical models based on differential equations.

Specific aims:

- Acquisition of fundamental knowledge in math, namely analytical and numerical resolution of differential equations.

- Development of competences in the areas of formulation, identification and modelling of engineering problems.

- Development of creative and critical thinking and for solving engineering problems. Previous knowledge Basic knowledge on math analysis and algebra: integration and differentiation, complex numbers, matrix algebra, eigen values and eigen vectors. Basic knowledge on programming: programming structures, variable indexation.

Percentual distribution:

Scientific component: 90 %

Technological component: 10 %


Learning outcomes and competences

Capability to solve ordinary differential equations (first and higher orders) using analytical and numerical methods.

Capability to formulate and implement simple mathematical models describing physical-chemical phenomena.

Capability to use computational tools for numerical resolution of differential equations.

Working method



I) Analytical resolution od differential equations

1. Introduction: Definitions. Linearity. General solution and particular solution. Existence and unicity of solution.

2. First order ordinary differential equations: Equations with separable variables. Equations reduced to separable variables by change of variable. Integrating factor method. Exact equations. Bernoulli equations. 

3. Second order linear ordinary differential equations. General solution of homogeneous linear equation. D’Alembert’s method (order reduction). Homogeneous equations with constant coefficients. General solution of non-homogeneous linear equation. Method of indeterminated coefficients. Method of parameter variation.

4. Laplace transforms: Definition and properties. Inverse transform. Unit step function (Heaviside) and unit impulse function (Dirac). Application to resolution of linear ordinary differential equations. 

5. Systems of linear ordinary differential equations: Method of elimination. Laplace transform method. Matrix method. Homogeneous systems. Non-homogeneous systems

6. Introduction to partial differential equations.


II) Numerical resolution of differential equations

Introduction to Scilab programming. Initial value problems. Euler method. Runge-Kutta methodAnalysis of error in mumerical resolution. Resolution of systems of differential equations. Usage of Scilab's ode function. Boundary value problems. Method of finite differences. Numerical stiffness.


Mandatory literature

Farlow, Stanley J.; An introduction to differential equations and their applications. ISBN: 0-07-113315-1
Chapra, Steven C.; Numerical Methods for Engineers. ISBN: 0-07-079984-9

Complementary Bibliography

Kreyszig, Erwin; Advanced Engineering Mathematics. ISBN: 0-471-50729-6

Teaching methods and learning activities

Exposition of theoretical concepts and resolution of practical examples during classes. Interaction with students will be emphasized. The classe dedicated to numerical methods will all take place in computer rooms. Small evaluation tests will take place along the semester, in order to stimulate and monitor the process of aquisition of knowledge and skills by the students along that time period.




Physical sciences > Mathematics > Mathematical analysis > Differential equations
Physical sciences > Mathematics > Applied mathematics > Numerical analysis
Physical sciences > Mathematics > Applied mathematics > Engineering mathematics

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation Weight (%)
Exame 75,00
Teste 25,00
Total: 100,00

Amount of time allocated to each course unit

Designation Time (hours)
Estudo autónomo 0,00
Frequência das aulas 0,00
Trabalho escrito 0,00
Total: 0,00

Eligibility for exams

Previous attendance is attributed to students who obtained a minimum grade of 6 in distributed evaluation.

Students who enter directly to the 2nd Study Cicle and did not obtain equivalence to this Curricular Unit are not subject to this restriction. Previous attendance is then attributed by performing the exam.

Calculation formula of final grade

The final classification (CF) will be computed according to:

First call exam:
CF = 0,8*(0,25*AD + 0,75*EN) + 0,2*TMN
AD = average of the 2 best minitest grades
EN = First call exam 
TMN = Numerical methods test 
Second call exam:
CF = 0,8*MAX(0,25*AD + 0,75*ER, ER) + 0,2*TMN
ER = Second call exam


Conditions for course approval:

  • Minimum grade of 6 on distributed evaluation component (AD).
  • Minimum grade of 6 on each of t etwo parts of the final exam (analytical math and numerical methods).

The mini-tests are mandatory for students without previous attendence of the course. In case of missing a mini-test, its repetition in a later date ill only be possible after evaluation of the properly fundamented  justification.

Students who enter directly to the 2nd Study Cicle and did not obtain equivalence to this Curricular Unit have the option to not do the mini-tests.

The distributed evaluation from previous years is not maintained. Performing the mini-tests is optional for students with previous attendance. In case these students intend to perform the mini-tests, they should inform the teacher during the first week of classes, becoming in this way vinculated to the new distributed evaluation.

Classification improvement

Improvement of classification (distributed evaluation and final exam) can be attempted in the second examination ("recurso"). The computation formula is identical to the one used in final classification described above.

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