Infinitesimal Calculus II

Code:

M112

Acronym:

M112

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2013/2014 - 2S

Active?	Yes
Web Page:	http://moodle.up.pt/course/view.php?id=238
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:AST	20	Plano de Estudos a partir de 2008	1	-	7,5	-
L:B	0	Plano de estudos a partir de 2008	3	-	7,5	-
L:CC	0	Plano de estudos de 2008 até 2013/14	2	-	7,5	-
			3
L:F	60	Plano de estudos a partir de 2008	1	-	7,5	-
L:G	0	P.E - estudantes com 1ª matricula anterior a 09/10	3	-	7,5	-
		P.E - estudantes com 1ª matricula em 09/10	3	-	7,5	-
L:M	117	Plano de estudos a partir de 2009	1	-	7,5	-
L:Q	0	Plano de estudos Oficial	3	-	7,5	-
MI:EF	69	Plano de Estudos a partir de 2007	1	-	7,5	-

Teaching language

Portuguese

Objectives

The student should know: to identify the graphs of quadratic equations in two and three real variables; the basic concepts about calculus of parametrized curves in the plane and the space; the fundamental results concerning the analysis of multivariate functions and understand the concepts of partial derivative, gradient vector, local maxima and minima, tangent plane to the graph of functions of two variables being able to determine extreme values of constrained functions; the student should also know the methods of multiple integration and use them to determine areas, volumes, etc, of bounded plane or space regions, using change of variables if necessary.

Learning outcomes and competences

The student should know: to identify the graphs of quadratic equations in two and three real variables; the basic concepts about calculus of parametrized curves in the plane and the space; the fundamental results concerning the analysis of multivariate functions and understand the concepts of partial derivative, gradient vector, local maxima and minima, tangent plane to the graph of functions of two variables being able to determine extreme values of constrained functions; the student should also know the methods of multiple integration and use them to determine areas, volumes, etc, of bounded plane or space regions, using change of variables if necessary.

Working method

Presencial

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

One-variable calculus and Linear Algebra with Analytical Geometry I.

Program

1. Conic sections and quadratic surfaces in the space.
The euclidean vector space R^n. Diagonalization of quadratic forms. Conic sections and quadratic surfaces in the space.

2. Parametrized curves.
Velocity, acceleration, curvature, Frenet frame.

3. 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. Examples. Inverse function theorem. Maxima and minima of real-valued multivariate functions. Second derivative test to find the local extremes. The method of Lagrange multipliers for finding extreme values of constrained functions.  

4. Multiple integrals.
Definition of integral of a multivariate real-valued function over a rectangle and a bounded region. Fubini's theorem.  Calculation of double and triple integrals via iterated integrals. Integration and the change of coordinates. Applications: double integrals in polar coordinates, and triple integrals in cylindrical and spherical coordinates.

Mandatory literature

Lang Serge; Calculus of several variables. ISBN: 0-387-96405-3
Stewart James; Calculus. ISBN: 978-0-495-38273-7

Complementary Bibliography

J.Marsden and A.Weinstein; Calculus. Vol. III. , New York: Springer-Verlag, 1985
R.Larson, R.P.Hostetler and B.H.Edwards; Calculus (8th Edition), Houghton Mifflin Company , 2006
Lima Elon Lages; Curso de análise
Edwards Jr. C. H.; Advanced calculus of several variables. ISBN: 0-12-232550-8

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.

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Participação presencial	0,00
Total:	100,00

Eligibility for exams

No requirements.

Calculation formula of final grade

Final mark at "Época normal"

 Test 1 date: April 1, 2014.

Test 1 classification: Y (out of 10).

 Test 2 date: June 5, 2014.

Test 2 classification: Z (out of 10).

Final mark at "época normal":

(Y + Z)  if (Y + Z) is greater or equal to 10 and the student didn't make the exam at "época normal".

 

Exam classification: X (out of 20);

Final mark at "época normal":

X if the student made the "época normal" exam.

 

Remark 1: The "época normal" exam will consist of two parts, each marking out of 10, in correspondence to the material that have been evaluated at test 1 and 2. 

The student can decide not to make one of the parts of the exam, and then it will be considered the classification obtained by the student at the corresponding test. 

 

Final mark at Época de recurso

Final mark: mark obtained at the exam of "Época de recurso" (out of 20).

Remark 2: The "época de recurso" exam will consist of two parts, each marking out of 10, in correspondence to the material that have been evaluated at test 1 and 2. The student that was not approved at "Época normal" can decide not to make one of the parts of the exam, and then it will be considered the classification obtained by the student at the corresponding test.

 

Remark 3: The student that was approved at "Época normal"  can't use the classification(s) obtained by him at the test(s) if he decides to make the exam of "Época de recurso" for improving his final mark at the course.
 

Remark 4: At the "Época de recurso" exam, it is not possible to use the classification obtained  at any of the two parts of the"Época normal" exam. 

Final marks above 17

The students that have marks above 17 (at the two tests, or at the "Época normal" exam, or at the"Época de recurso" exam) will have to do an additional written test, at date after the "Época normal" exam.

Special assessment (TE, DA, ...)

Any student asking for an exam because of special conditions of his registration will do a written exam, but possibly, only, after an extra written or oral examination, in order to check if the student has a minimum knowledge about the unit so that he can do the special exam.

Classification improvement

The student that was approved at "Época normal"  can't use the classification(s) obtained by him at the test(s) if he decides to make the exam of "Época de recurso" for improving his final mark at the course. 

The student that has been approved at the school year 2012/13 and decides to improve his final mark at the course, he can do that only by making the exam and not by doing the tests.

Observations

Unit jury: Ana Paula Dias e José Carlos Santos.