Analysis II

Code:

M1015

Acronym:

M1015

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2017/2018 - 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:F	71	Official Study Plan	1	-	9	84	243
			2
L:G	4	study plan from 2017/18	2	-	9	84	243
			3
L:Q	1	study plan from 2016/17	3	-	9	84	243
MI:EF	79	study plan from 2017/18	1	-	9	84	243

Teaching language

Suitable for English-speaking students

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 functions of one real variable.

Program

1. Eigenvalues and eigenvectors of an endomorphism of a vector space. Euclidean spaces. DIagonalization of symmetric matrices. Conic sections and quadratic surfaces in the space. Diagonalization of quadratic forms.

2. Differential calculus of vector-valued multivariate functions.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 composition of functions. Inverse and implicit function theorems. Maxima and minima of real-valued multivariate functions.

3. Multiple integrals. Definition of 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.

4. Applications to Physics. Vector fileds; Line integrals; Green's theorem; Rotational and divergence; Surface integrals; Stokes  theorem; Divergence theorem; Solving 2nd order differential equations: Vibrating spring and electrical circuits.

Mandatory literature

James Stewart; Calculus early transcendentals, Cengage, 2016. ISBN: 978-1-285-74155-0 (8th Edition)

Complementary Bibliography

Marsden Jerrold; Calculus iii. 2nd ed. ISBN: 0-387-90985-0
Lang Serge; Calculus of several variables. ISBN: 0-387-96405-3
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

Teaching methods and learning activities

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

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Teste	50,00
Total:	100,00

Eligibility for exams

Unconditional.

Calculation formula of final grade

1) A student may be exempted from the final exam by obtaining a passing grade from the average of the grades of two midterm exams. The student is approved if he/she obtains a grade equal or greater to 10 (average of the two tests)

2) The final exam has two parts corresponding to the division of the topics of the course in the two midterm tests.

3) A student that is already approved by means of test, i.e. by (1) may still do the final exam, and do only one or both of the two parts of the exam. If he delivers one (or both parts) for grading, then the classification obtained substitutes the corresponding grades obtained in the midterm test.

4) The same rule applies to the the second phase of exams or to students that want to increase their classification.

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.