Calculus II

Code:

M1003

Acronym:

M1003

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 2S

Active?	Yes
Web Page:	https://moodle2324.up.pt/course/view.php?id=2206
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Computer Science

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	48	162
L:CC	152	study plan from 2021/22	1	-	6	48	162
			2
L:F	0	Official Study Plan	2	-	6	48	162
L:G	1	study plan from 2017/18	2	-	6	48	162
			3
L:IACD	124	study plan from 2021/22	1	-	6	48	162
L:Q	0	study plan from 2016/17	3	-	6	48	162

Teaching language

Portuguese

Objectives

Acquisition of the basic knowledge and skills of Differential and Integral Calculus in several real variables.

Learning outcomes and competences

Understanding and ability to make use of the concepts and results covered in the syllabus, namely through the resolution of exercises of practical nature.

Working method

Presencial

Program

1. Parametrized curves in R^n:
Velocity, acceleration, curvature and torsion.

2. Differential calculus of vector-valued multivariate functions:
Graphs of real-valued functions of two variables, contour lines of functions of two variables and level surfaces of functions of three variables. Open and closed subsets of R^n. Accumulation point and isolated point. Limits and continuity of functions. Directional derivatives and partial derivatives. Derivative function at a point of a real-valued multivariate function. Gradient vector and derivability. Tangent plane to the graph of a function of two variables. Interpretation of the gradient vector. Normal line and tangent hiperplane at a point on the level surface of a function. Higher order derivatives. Derivative function at a point  of a vector-valued multivariate function. Jacobian matrix. Derivation of composition of functions. Maxima and minima of real-valued multivariate functions. Second derivative test to find the local extremes.

3. Multivariable integrals:
Definition of integral of a multivariate real-valued function over a rectangle and a bounded region. Fubini's theorem.  Change of coordinates.  

Mandatory literature

Stewart James; Cálculo. ISBN: 85-221-0479-4 (Vol. I)
Adams e Essex; Calculus - A Complete Course, 2010. ISBN: 978-0-321-54928-0 (7th edition)

Complementary Bibliography

Jerrold E. Marsden; Vector calculus. ISBN: 978-1-4292-2404-8

Comments from the literature

Althought there is some mandatory bibliography, the material given in classes is very importante. The notation to be use in the course and the results that can be used are the ones given in the theoretical classes.

Teaching methods and learning activities

Lectures and classes: The contents of the syllabus are presented in the lectures, where examples are given to illustrate the concepts. There are also practical lessons, where exercises and related problems are solved. All resources are available for students at the unit’s web page.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Teste	60,00
Exame	40,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	114,00
Frequência das aulas	48,00
Total:	162,00

Eligibility for exams

Attendance is not required.

Calculation formula of final grade

1. On the Normal Exam period (Época Normal), the total mark is the sum of the marks on each of 3 tests:

Test 1: This test is marked for a total of 6 points and it will take place at a date agreed upon with the students. 

Test 2: This test is marked for a total of 6 points and it will take place at a date agreed upon with the students. 

Test 3: This test is marked for a total of 8 points and it will take place on the date set for the exam of the Normal Exam period.

2. On the Makeup Exam Period (Época de Recurso) the total mark is obtained on an exam set for a total of 20 points.

The exam will be divided into 3 parts, allowing the students who have not yet been approved in this class to use the score they have previously obtained on one or more of the 3 tests from the Normal Exam period (Época Normal) in the corresponding part of this exam. In any of the parts, the grade which will be considered is the maximum of the grade obtained on the exam and the grade possibly obtained on the corresponding test in the Normal Exam period (Época Normal).

Special assessment (TE, DA, ...)

Any type of special student evaluation may take one of the following forms: exclusively an oral examination; an oral examination plus a written examination, the student being required to pass both of them; only a written examination. The option for one of them is the sole responsibility of the professors in charge of the course.

Observations

Article 13 of General Regulations for Student Evaluation at the levels of First Cycle, Integrated Masters, and Second Cycle at U.Porto: "Fraud committed during an exam, in any form, implies the annulment of the exam and the communication to the statutorily competent organ for possible disciplinary action."

Any student may be required to take an oral examination should there be any doubts concerning his/her performance on the assessment pieces.

Jury:
Samuel António de Sousa Dias Lopes
Maria do Rosário Machado Lema Sinde Pinto