Analysis II
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2022/2023 - 2S ![Requerida a integração com o Moodle Ícone do Moodle](/fcup/pt/imagens/MoodleIcon)
Cycles of Study/Courses
Teaching language
Portuguese
Objectives
To introduce the concepts and basic results of Vector Analysis.
Learning outcomes and competences
Upon successful course completion, the student should know: 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)
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.