Infinitesimal Calculus II

Code:

M112

Acronym:

M112

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2016/2017 - 2S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=218
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	0	Plano de Estudos a partir de 2008	1	-	7,5	-
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	-
MI:EF	46	Plano de Estudos a partir de 2007	1	-	7,5	-

Teaching language

Portuguese

Objectives

Acquiring knowledge of basic concepts, results and techniques of differential and integral calculus on several variables.

Learning outcomes and competences

Problem-solving skills. Theoretical understanding.

The student should master the basic concepts of analysis of  functions in several variables. It is also intended that this unit allows students to work rigorously with concepts that, thus far, have been introduced only in an intuitive way

Working method

Presencial

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

One-variable Calculus and basic concepts of Linear Algebra and geometry.

Program

Programme

• The euclidean vector space R^n. Conic sections and quadratic surfaces in the space. 


• 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

Marsden Jerrold E.; Vector calculus. ISBN: 0-7167-0462-5
Marsden Jerrold; Calculus iii. 2nd ed. ISBN: 0-387-90985-0
Young Eutiquio C.; Vector and tensor analysis. ISBN: 0-8247-6671-7
Carmo Manfredo Perdigão do; Differential geometry of curves and surfaces. ISBN: 0-13-212589-7

Complementary Bibliography

Bers Lipman; Calculus. ISBN: 0-03-089268-6
Marsden Jerrold; Calculus ii. 2nd ed. ISBN: 0-387-90975-3
Spivak Michael; Calculus on manifolds

Comments from the literature

Program given in lectures is the most important"bibliography"

Teaching methods and learning activities

* Lectures: Exposure of the material of the program and resolution of exercises.
* Theorical -Pratical Classes: Resolution, by the students, of the proposed exercises and answering questions about the resolution of problems and proposed work. Theorical complements

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Eligibility for exams

Terms of frequency: If the limit of absences is exceeded the student will not be admited access to examination, either in time or normal use (except for students exempted from frequency)

 

Calculation formula of final grade

Formula Evaluation: There wil be two components of assessment: • Continuous Evaluation (optional): based on test results and itcan be corrected by the assessment practices in the classroom (including level of participation and performance in class) *.

• Final written exam

The evaluation will be done through two tests required and the final exam. Admission to the second test will be conditional upon a minimum grade of 8.0 values. Minimum grade of second test 6 values The tests may replace the exam. The notice of exemption will not necessarily be the arithmetic mean of test scores * The student with a grade exceeding eighteen values in tests or final examination may eventually be subjected to an extra proof.

Special assessment (TE, DA, ...)

According to the General Evaluation Rules

Classification improvement

For students under normal conditions with access to examination and which have succeeded the distributed evaluation( score above or equal to 10/20), the final classification is obtained by the highest ranking achieved in the distributed evaluation and / or examination.