Analysis II

Code:

M1015

Acronym:

M1015

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 2S

Active?	Yes
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:B	0	Official Study Plan	3	-	9	72	243
L:CC	0	study plan from 2021/22	2	-	9	72	243
L:EF	64	study plan from 2021/22	1	-	9	72	243
L:F	39	Official Study Plan	1	-	9	72	243
			2
L:G	0	study plan from 2017/18	2	-	9	72	243
			3
L:Q	0	study plan from 2016/17	3	-	9	72	243

Teaching language

Portuguese

Objectives

The student should know the basic concepts about calculus of parametrized curves in the plane and the space, the fundamental results concerning the analysis of multivariate functions and the methods of multiple integration. 

Learning outcomes and competences

The student should know: 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.

Program

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

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.  

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

Jerrold Marsden; Calculus iii. 2nd ed. ISBN: 0-387-90985-0
James Stewart; Calculus. ISBN: 978-1-305-27237-8
Serge Lang; Calculus of several variables. ISBN: 0-387-96405-3

Complementary Bibliography

Jerrold Marsden; Calculus ii. 2nd ed. ISBN: 0-387-90975-3

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 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	171,00
Frequência das aulas	72,00
Total:	243,00

Eligibility for exams

No requirements.

Calculation formula of final grade

Continuous evaluation without final exam

Continuous evaluation is based on two test results. The tests will be done at dates that will be indicated later.

The final score will be the sum of the test scores.

Any student can choose not to submit to continuous evaluation and obtain the final classification performing the examination in the second examination period (Época de Recurso).

In any case, a student with a final grade ≥ 16.5 may eventually be subjected to an extra oral or written test.

All registered students are admitted, without restrictions, to the tests and exams.

Special assessment (TE, DA, ...)

According to the General Evaluation Rules.

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 general evaluation rules apply.