Mathematical Analysis II

Code:

L.EGI006

Acronym:

AM II

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 2S

Active?	Yes
Web Page:	https://sigarra.up.pt/feup/pt/ucurr_geral.ficha_uc_view?pv_ocorrencia_id=483725
Responsible unit:	Mathematics Section
Course/CS Responsible:	Bachelor in Industrial Engineering and Management

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EGI	119	Syllabus	1	-	6	52	162

Teaching language

Portuguese

Objectives

Development of the reasoning capacity of students and knowledge of essential mathematical concepts. Students should get solid theoretical and practical formation on the main concepts and results of differential and integral calculus of several variables.

Learning outcomes and competences

At the end of the semester students should be able to: 

1 - Obtain parametric representation of curves in Rn and the corresponding tangent and normal vector. Determine the length and the curvature of a curve.  Calculate line integrals along curves.
2 - Discuss the continuity of scalar functions of several variables.
3 - Calculate partial and directional derivatives of scalar and vector fields and build the gradient vector and the jacobian matrix.
4 - Calculate the derivatives of composite functions of scalar  and vector fields, as well as of the implicitly defined functions.
5 - Obtain maximum and minimum of unconstrained functions of two variables and maximum and minimum of constrained functions, with one or two constraints, by the method of Lagrange multipliers.
6 - Calculate double integrals over limited regions of R2, either in Cartesian or polar coordinates.
7 - Calculate triple integrals over limited regions of R3, get the volume of these regions by integration and change triple integrals from Cartesian coordinates into cylindrical and spherical ones.

Working method

Presencial

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

Knowledge acquired in course of Mathematical Analysis I.

Program

I –Vector-valued functions of one real variable; parametric equations of a line in Rn. Limits, continuity, differentiation and integration of vector-valued functions. Applications to the geometry of curves in R2 and R3: tangent, principal normal and binormal unitary vectors.  Arc length and curvature of curves.

II - Introduction to surfaces in R3: quadric, cylindrical and revolution surfaces. General notions for real-valued functions of n variables: domain and graph. Vector-valued functions of n variables. Introductory topological notions on Rn. Limits and continuity for scalar and vector-valued functions of n variables.

III - Differentiation: partial and directional derivatives; gradient vector; partial derivatives of higher order; total derivative or Fréchet’s derivative and differentiability of a scalar function of n variables. Applications of the gradient: tangent plane and maximum of a directional derivative. Differentiability of vector-valued functions of n variables–Jacobian matrix. Properties of the derivative; different cases of the chain rule. Functions defined implicitly; implicit function theorem and implicit differentiation. Taylor’s formula for scalar functions of n variables. Extrema of scalar functions of n variables; constrained extrema and Lagrange multipliers.

IV - Line integrals for scalar and vector functions. Definition and properties. Differential formulation of some line integrals. Application to the work concept.


V - Double integrals: over a rectangle and over more general regions in R2. Properties and geometric interpretation of double integrals. Fubini’s theorem – changing the order of integration. Applications of double integrals to the computation of average values, center of mass and moment of inertia. Changing variables in double integrals; double integrals in polar coordinates. Triple integrals over rectangular parallelepiped and more general regions in R3 . Properties and geometric interpretation of triple integrals. Fubini’s theorem – changing the order of integration for triple integrals. Changing variables: triple integrals in cylindrical and spherical coordinates.

General change of variables in multiple (double) integrals.

Mandatory literature

Larson, Hostetler & Edwards; Cálculo, McGraw-Hill Interamericana , 2006. ISBN: 85-86804-56-8
C.C. António; Apontamentos de Análise Matemática II, AEFEUP. ISBN: 2017

Complementary Bibliography

Teresa Arede; Análise Matemática II - Apontamentos Teóricos, Efeitos Gráficos - AEFEUP, 2019
Apostol, Tom M.; Calculus, N. ISBN: 84-291-5001-3
Marsden, Jerrold E.; Vector Calculus, N. ISBN: 0-7167-1856-1

Comments from the literature

Análise Matemática II - Apontamentos Teóricos e Práticos
Carlos C. António

Published in  L.EGI006 webpage in SIGARRA

Teaching methods and learning activities

In the theoretical classes the proposed subjects are introduced in presentation mode. The concepts and methodologies are exemplified by solving exercises. Clarification of doubts during classes by interaction.
In the practical classes (face-to-face) exercises made available in the contents of the curricular unit in the SIGARRA  are proposed and solved.
Clarification of doubts during classes and by email.

keywords

Physical sciences > Mathematics

Evaluation Type

Distributed 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	97,00
Frequência das aulas	65,00
Total:	162,00

Eligibility for exams

To obtain attendance, students must attend 75% of the theoretical and theoretical-practical classes planned for the course.

Calculation formula of final grade

Final mark will be based on the following components:
- 1st Test (T1)- It will cover the first half of the program of the course. Students will be informed about the date; if the student reaches a minimum mark of 7 out 20 it might account for 50% of the final mark according to the rules described below.
- 2nd Test (T2) – Only for students who reached a minimum mark of 7 in the first test. It will cover the rest of the program of the course; if the student reaches a minimum mark of 7 out 20 it might account for 50% of the final mark according to the rules described below.
- Final Exam (FE)- Simultaneous to the second test; it is for students who did not reach a minimum mark of 7 in the first test plus for students that although they reached the minimum mark of 7 in the first test decide to do the whole exam instead of the second test. This exam will cover all the program of the course.
- Recurso Exam (RE)- For students who did not reach a minimum mark in both tests or who did not reach a passing grade in both tests or exam EF. It is also for students who want to improve their mark. It will cover all the program of the course.

Final Mark (FM) calculation: FM=( T1+ T2)/2 only if T1>=7 and T2>=7 or FM = FE or FM= RE.

Examinations or Special Assignments

Not applicable.

Internship work/project

Not applicable.

Special assessment (TE, DA, ...)

This evaluation will exclusively be the result of the written exam according to Evaluation Rules of FEUP.

Classification improvement

Classification improvement will exclusively be the result of the written exam according to Evaluation Rules of FEUP. This exam will be made only in next epochs after the conclusion of distributed evaluation process.

Observations

Not applicable.