Mathematical Analysis II

Code:

EMG0007

Acronym:

AM II

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2019/2020 - 2S

Active?	Yes
Responsible unit:	Department of Civil and Georesources Engineering
Course/CS Responsible:	Bachelor in Mining and Geo-Environmental Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
LCEEMG	30	Plano de estudos oficial a partir de 2008/09	1	-	6	56	162
MIEA	58	Syllabus since 2006/07	1	-	6	56	162

Teaching language

Portuguese

Objectives

OBJECTIVES:
To acquire theoretical and practical knowledge on the differential and integral calculus of scalar functions and vector functions of one or several variables and on some of their applications in engineering. To acquire an elementary knowledge in the resolution of differential equations. To develop competences in the areas of formulation, identification and modelling of engineering problems. To develop creative and critical thinking and for solving engineering problems.

Learning outcomes and competences

LEARNING OUTCOMES: After this course, the students are able to:

Obtain partial and directional derivatives for scalar fields and know how to construct the gradient vector. Calculate partial derivatives of functions of several variables (either composite or implicit); Locate extreme values of functions. Calculate the divergence and the rotational of a vector field.

Calculate double and triple integrals (in rectangular, cylindrical and spherical coordinates) and understand its practical applications (calculation of area, volume, and centroid); Represent curves and surfaces in the 3D space; Understand Vector Analysis, calculate line and surface integrals and understand its practical applications (calculation of work done by a force and the flux of a vector field).

To solve first order ordinary differential equations and linear equations of second order. To formulate and implement simple mathematical models describing physical-chemical phenomena.

Working method

Presencial

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

Basic knowledge of Mathematical Analysis I and Algebra

Program

1. Differential Calculus in Rⁿ.
   Real valued functions. Graph and level set. Properties of the continuous functions and properties of the derivative. Gradients and directional derivatives. Partial derivatives. Extrema.
   Vector Valued functions. Matrix of partial derivatives. Divergence and Curl of a vector field.


2. Modelling and Differential Equations.
   First order Differential equations.
   Second order linear equations.


3. Integral Calculus in Rⁿ.
   Double and triple integrals. The change of variables.
   Line integrals.
   Parametrized surfaces. Integrals over surfaces.


4. Some applications using mathematical models describing physical-chemical phenomena. (Transverse module).

Mandatory literature

Marsden Jerrold E.; Vector calculus. ISBN: 0-7167-0462-5
Edwards Charles Henry; Differential Equations. ISBN: 0-13-067337-4

Complementary Bibliography

C. Henry Edwards, David E. Penney; Calculus. ISBN: 0-13-095006-8

Teaching methods and learning activities

Exposition of theoretical concepts and resolution of practical examples during classes. Interaction with students will be emphasized. Small evaluation tests will take place along the semester, in order to stimulate and monitor the process of acquisition of knowledge and skills by the students along that time period. This curricular unit is inserted in the Moodle platform, in order to enhance the discussion among all participants. In this platform, all students have access to every issue provided by the teachers and may strengthen their concepts by solving self-evaluation activities. Students will be encouraged to use software (Maxima).

Software

Maxima, a Computer Algebra System

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation	Weight (%)
Exame	70,00
Teste	30,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	56,00
Total:	162,00

Eligibility for exams

Students cannot exceed the absence limit according to the General Evaluation Rules in force.

Calculation formula of final grade

Final mark (CF) is calculated according to:

CF = maxima {EX; AD}

where,

EX – Fianl exam mark

AD = (20% x MT+10% x MDL) + 70% x EX

MT - mark of the mini test;

MDL- average of the classifications obtained in 4 Moodle activities (M1, M2, M3 and M4):

MDL=(M1+M2+M3+M4)/4

All marks are quoted on a  0 to 20 scale.

Examinations or Special Assignments

Not foreseen.

Internship work/project

Not foreseen.

Special assessment (TE, DA, ...)

According to the general evaluation rules.

Classification improvement

Final Exam.

Observations

Due to the COVID-19 situation, the calculation formula of final grade is going to suffer the following change:

Final mark (CF) is calculated according to:

CF = maxima {EX; AD}

where,

EX – Fianl exam mark

AD = (10% x MT+10% x MDL) + 80% x EX

TM - mark of the Moodle Test;

MDL- average of the classifications obtained in 4 Moodle activities (M1, M2, M3 and M4):

MDL=(M1+M2+M3+M4)/4

All marks are quoted on a  0 to 20 scale.