Mathematical Analysis 2

Code:

L.EC006

Acronym:

AM2

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2025/2026 - 2S

Active?	Yes
Web Page:	http://moodle.up.pt
Responsible unit:	Department of Civil and Georesources Engineering
Course/CS Responsible:	Bachelor in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EC	261	Syllabus	1	-	6	52	162

Teaching language

Portuguese

Objectives

The main objective of this course is to acquire theoretical and practical knowledge in differential and integral calculus of real and vector functions of one or more variables, and some of their applications to physics.
The aim is for the content taught to be useful in providing students with the solid mathematical foundations that are essential in other specific courses in various areas of engineering. 

Learning outcomes and competences

Students who pass the course should be able to recognize the usefulness of the mathematical concepts and results taught in different areas of Physics, with special emphasis on Fluid Mechanics.
The multivariate differential and integral calculus taught will allow the student to learn better the subjects taught in other curricular units.

In particular, differential calculus will allow us to analyse the local behaviour of scalar functions (such as determining their local extrema) and understand that the most important equations in mathematical physics are written using partial derivatives.
As for integral calculus, emphasis will be placed on interpreting double or triple integrals as averages, which makes it easier to understand the concept of geometric centres (centres of mass or centres of gravity).
The classical theorems of analysis are interpreted in the light of fluid mechanics, so that students will understand their usefulness in deducing the properties and equations of fluids.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

The knowledge of differential and integral calculus acquired in the Mathematical Analysis 1 course and the knowledge of linear algebra acquired in the Algebra course, both taught in the 1st Year, 1st Semester of the LEC, are considered essential for attending this course.

Program

1. Differential Calculus
1.1 Partial Derivatives
1.2 Implicit Differentiation
1.3 Extremes
1.4 Derivative Matrix of Vector Functions
1.5 Gradient Operator

2. Integral Calculus
2.1 Double Integrals
2.2 Triple Integrals
2.3 Change of Variables Theorem

3. Differential Geometry
3.1 Curves
3.2 Surfaces

3. Vector Integral Calculus
3.1 Integration of vector fields on curves and Green's theorem
3.2 Integration of vector fields on surfaces and Stokes' theorem
3.3 Integration of vector fields on surfaces and Gauss' theorem

Mandatory literature

Jerrold E. Marsden, Anthony J. Tromba; Vector calculus. ISBN: 0-7167-1856-1

Complementary Bibliography

Jerrold E. Marsden, Anthony J. Tromba; Study Guide for Vector Calculus. ISBN: 0-7167-1980-0
Acilina Azenha, Maria Amélia Jerónimo; Elementos de cálculo diferencial e integral em IR e IRn. ISBN: 972-8298-03-X

Teaching methods and learning activities

Theoretical classes will teach mathematical concepts and results, always complemented by examples. When possible, physical interpretations of concepts and practical applications of results will be given.
Technological resources will be used to aid geometric visualization (especially in the three-dimensional case) and perform calculations.

Practical classes will allow students to explore problem-solving and exercises with the support of a member of teaching staff. The list of exercises will be made available for each class.

Software

Wx-maxima

keywords

Physical sciences > Mathematics

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	110,00
Frequência das aulas	52,00
Total:	162,00

Eligibility for exams

Approval of the course unit implies compliance with the attendance requirement, considering that a student complies with this requirement if, having been regularly enrolled, they do not exceed the limit of absences corresponding to 25% of the classes scheduled for each type. In addition to the cases provided for in the FEUP rules in force, students who have obtained a final grade of 6 or higher in the course in the immediately preceding academic year are exempt from the attendance requirement.

Calculation formula of final grade

The assessment will be distributed without a final exam and consists of two written tests (compulsory).

Assessment components:

CT1 = grade in the first test

CT2 = grade in the second test

The final grade (CF) will be:

CF = 0,50×CT1 + 0,50×CT2

 

The resit exam will be divided into two parts, whose grades, GR1 and GR2, both scored out of 20, will allow students to improve their grades in CT1 and/or CT2 as follows:

 

(A) If the student wishes to improve only one of the parts, they must hand in their exam paper and leave the room at the end of the designated time (and never before) and their final grade after the resit exam (CFR) is given by:

 

CFR = max{0,50xCT1 ; 0,50xCR1} + 0,50xCT2 (if the first part is handed in),

or

CFR = 0,50xCT1 + max{0,50xCT2 ; 0,50xCR2} (if the second part is handed in);

 

(B) If the student wishes to improve both parts, they must take the entire examination and their final grade is given by:

 

CFR = max{CF ; 0,50xCR1 + 0,50xCR2} .

Observations

During any assessment, the possession of any electronic device (e.g., cell phones, tablets, headphones, smartwatches, etc.) is strictly prohibited, with the exception of those expressly indicated by the teaching staff (e.g., calculators).
It is the student's responsibility to anticipate this situation before the start of the assessment.