Analysis II

Code:

M1015

Acronym:

M1015

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2022/2023 - 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	84	243
L:CC	0	study plan from 2021/22	2	-	9	84	243
L:EF	102	study plan from 2021/22	1	-	9	84	243
L:F	87	Official Study Plan	1	-	9	84	243
			2
L:G	0	study plan from 2017/18	2	-	9	84	243
			3
L:Q	0	study plan from 2016/17	3	-	9	84	243

Teaching language

Portuguese

Objectives

Learning outcomes and competences

Working method

Presencial

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

Students are expected to have previously acquired basic knowledge of Linear Algebra (matrices, vector spaces) and Calculus of real functions of one real variable.

Program

1. Differentiable curves: velocity and acceleration vectors; arc length; curvature and radius of curvature; Frenet trihedron and torsion for curves in three-dimensional space.

2. Graphs, contour lines and level surfaces. Open and closed subsets, accumulation and isolated points. Limits and continuity. Directional and partial derivatives. Gradient vector of a real valued multivariate function. Derivability. Tangent plane to the graph of a function of two variables. Normal line and tangent hiperplane at a point on the level surface of a function. Jacobian matrix. Derivation of composite functions. Inverse and implicit function theorems.

3. Maxima and minima of real-valued multivariate functions. Constrained extrema: Lagrange multipliers. (Hyper)surfaces in Euclidean space. Orientable surfaces.5

4. Multiple integrals. Definition of (Riemann) integral of a multivariate real-valued function over a bounded region. Fubini's theorem: computation of integrals via iterated integrals. Change of coordinates. Double integrals in polar coordinates, triple integrals in cylindrical and spherical coordinates.

Mandatory literature

James Stewart; Calculus early transcendentals, Cengage, 2016. ISBN: 978-1-285-74155-0 (8th Edition)
Jerrold E. Marsden; Vector calculus. ISBN: 978-1-4292-2404-8

Complementary Bibliography

Marsden Jerrold; Calculus iii. 2nd ed. ISBN: 0-387-90985-0
Lang Serge; Calculus of several variables. ISBN: 0-387-96405-3
Elon Lages Lima; Análise Rea,l volume 2 - Funções de n Variáveis, IMPA, 2014. ISBN: e-ISBN 978-85-244-0381-1
George Arfken Hans Weber Frank E. Harris; Mathematical Methods for Physicists, Academic Press, 2012. ISBN: 9780123846549
Sokolnikoff I. S.; Mathematics of physics and modern engineering
Fleming Wendell; Functions of several variables. ISBN: 0-387-90206-6
P. C. Matthews; Vector calculus, Springer, 2005. ISBN: 3-540-76180-2
Bressoud David M.; Second year calculus. ISBN: 0-387-97606-X
Arfken George B.; Mathematical methods for physicists. ISBN: 0-12-059825-6
W. F. Trench; Introduction to real analysis, Mathematics Commons, 2013 (http://digitalcommons.trinity.edu/mono/7/)
K. Kuttler; Calculus - Theory and Applications, Vol. 2, World Scientific, 2011. ISBN: 978-981-4329-70-5
Michael Spivak; Calculus on manifolds

Teaching methods and learning activities

Lectures where the professor presents the course material and tutorials for the discussion and solution of problems.

Software

Sage

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	159,00
Frequência das aulas	84,00
Total:	243,00

Eligibility for exams

Unconditional.

Calculation formula of final grade

In the normal exam season, the final grade depends 60% on the midterm tests and 40% on the final exam. In the appeal season, the exam is worth 100% of the final grade.

Classification improvement

During the semester, the rules are those indicated in the formula of computation of the final grade. After that, the general relevant rules apply. In particular, students who have been approved in a previous academic year who wish to atempt to improve the grade in the normal exam season must also participate in the midterm tests.

Observations

Students with a grade of at least 18 may be asked to take an additional exam to confirm their grade. Students with a grade of 8 or 9 may also be invited to take an additional exam as an opportunity for approval; as a result, they may obtain a grade of 10 or keep the previous grade.