Calculus III

Code:

L.EQ011

Acronym:

AMIII

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2022/2023 - 1S

Active?	Yes
Responsible unit:	Department of Chemical and Biological Engineering
Course/CS Responsible:	Bachelor in Chemical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EQ	80	Syllabus	2	-	6	58,5	162

Teaching language

Portuguese

Objectives

Background:

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 competencies in the areas of formulation, identification, and modeling of engineering problems.

- Development of creative and critical thinking for solving engineering problems.

Previous knowledge:

Basic knowledge of mathematical analysis and algebra: integration and differentiation, complex numbers, matrix algebra, eigenvalues, and eigenvectors. Basic knowledge of 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 to solve differential equations numerically.

Working method

Presencial

Program

I) Analytical resolution of 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 the 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 method. Analysis of error in numerical 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

Exposure to theoretical concepts and discussion of practical examples of application during classes. Maximization of interaction with students during classes. The numerical methods component will be entirely taught in computer rooms. Mini evaluation tests will be performed throughout the semester to stimulate and monitor the students' acquisition of knowledge and competencies during this period.

Software

Scilab

keywords

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	103,50
Frequência das aulas	58,50
Total:	162,00

Eligibility for exams

Registered students who obtain a minimum mark of 6 in the distributed assessment component (AD) will obtain attendance.

Students without attendance in the current year or without attendance in previous years will not be able to attend the Normal Season exam.

Students who enter directly to the 2nd Cycle of Studies and do not obtain equivalence in this CU are exempt from the previous restriction, obtaining frequency by attending the exam.

Students who do not obtain a minimum mark of 6 in the distributed assessment component may attend the exam in the second call, thus obtaining frequency for the CU.

Calculation formula of final grade

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

AD (distributed evaluation) = 1/3 x (2 best marks in the three analytical math mini-tests + mark in the numerical methods mini-test)

EF = mark in the final exam (80% analytical math component and 20% numerical methods component)

Where:

Conditions for course approval:

• A minimum mark of 6 on distributed evaluation component (AD).
• A minimum mark of 6 on each of the two parts of the final exam or of the second call exam (analytical math and numerical methods).

The completion of the mini-tests is compulsory for students without previous attendance. In case of missing a mini-test, its repetition on a date to be determined will only be possible after evaluation of the justification duly substantiated.

Students who enter directly to the 2nd Study Cycle and do not obtain equivalence in this CU may choose not to take the mini-tests.

The distributed assessment mark of previous years will not be maintained. The taking of the mini-tests is optional for students with previous attendance. If they wish to take them, students must inform the teacher in the first week of classes, thus being bound to the new distributed assessment.

Students with previous attendance who choose not to take the mini-tests will have the exam grade as their final mark.

Classification improvement

The classification improvement test (components of the distributed assessment and final examination) will take place in the examination of the second call. The final classification of the students who have taken the mini-tests will consist of the result of the formula above or the exam mark, according to the option that gives a better result. For the remaining students, the final classification will consist of the exam grade.