ISCTE-IUL > Education > LETI , LEI , LEI-PL

Calculus II (2 º Sem 2019/2020)

Code:	L0132
Acronym:	L0132
Level:	1st Cycle
Basic:	No
Teaching Language(s):	English, Portuguese
Friendly languages:

Be English-friendly or any other language-friendly means that UC is taught in a language but can either of the following conditions:
1. There are support materials in English / other language;
2. There are exercises, tests and exams in English / other language;
3. There is a possibility to present written or oral work in English / other language.

Semester	ECTS Credits	Theoretical Lesson (T)	Theoretical and Pratical Lesson (TP)	Practical and laboratorial lesson (PL)	Seminary (S)	Field Work (TC)	Training Period (E)	Tutorial Orientation (OT)	Contact Hours	Autonomous Work	Others (O)	Total Load Hours [?]Contact Hours + Autonomous Work
1	6.0	0.0 h/sem	54.0 h/sem	0.0 h/sem	0.0 h/sem	0.0 h/sem	0.0 h/sem	1.0 h/sem	55.0 h/sem	95.0 h/sem	0.0 h/sem	150.0 h/sem


Since year	2017/2018
Pre-requisites	The students should know the fundamental concepts and techniques on differential and integral calculus (one variable), taken from the point of view of its applications in engineering.
Objectives	To learn the main methods on differential and integral calculus - several variables - and on complex analysis that are necessary to the analysis and resolution of engineering problems.
Program	PC1-Multivariable differential calculus 1a)Graphics an level curves. A brief reference to limits and continuity 1b)Partial derivatives and Jacobian matrix. Differentiability and first order Taylor development. Directional derivative. Chain rule 1c)Higher order partial derivatives, Hessian matrix and differential operators PC2-Multiple integrals 2a)Double integral and their properties 2b)Regular domain and calculation of double integrals 2c)Changes of coordinates 2d)Volums PC3-Line and surface integrals 3a)Regular and piecewise regular curves 3b)Parametrical equations and orientation of regular curves and surfaces. Tangent vector and normal vector 3c)Line integral. Green theorem 3d)Surface integral. Stokes theorem PC4-Complex analysis 4a)Analytic functions. Differentiation. Cauchy-Riemann equations 4b)Elementary functions 4c)Integrals of complex functions. Cauchy-Goursat theorem. Cauchy integral formula in various orders.
Evaluation Method	A student must obtain an overall grade of at least 10 (out of 20) in one of the assessment modes: -Continuous assessment: Online Mini-test (10%) per week, 2 written tests (45% each one with minimal grade of 8 values); -Exam assessment: in any of exam seasons (100%). A student that obtained the minimal grade in the first test may nonetheless choose, in the regular exam season, to be evaluated by exam. Pay attention (below) to additional remarks about the Continuous assessment.
Teaching Method	The lectures are of theoretical and practical nature according the following learning methodologies (LM): LM1. Expositional, in order to introduce the theoretical contents LM2. Participative, through the resolution of exercises and practical applications LM3. Self-study, according the autonomous work by the student included in the Class Planning
Observations	The analytical approach of the program contents is completed, whenever possible, with the geometric one. The concepts are still worked in terms of its applications in other scientific areas. ADDITIONAL RULES ABOUT THE CONTINUOUS ASSESSMENT: 1 - There are 10 Online Mini-tests which are performed along the semester using the e-learning platform. Each test has a maximum duration of 30 minutes, is available for a full week and is focused on the course materials lectured in the previous week(s). Missing to an Online Mini-test implies a null grade in the corresponding assessment moment. The final grade in the Online Mini-tests component is computed after discarding the the worst grade within the first 5 tests and the worst grade within the 5 last tests. The grade will be the average of the 8 tests considered. 2 - Eventual problems in the access to the e-learning platform won't change the deadlines of any of the Online Mini-tests. So, the students are therefore encouraged to perform the Online Mini-tests as soon as they are available. 3 - A student is excluded from the Continuous assessment mode (automatically enters the exam assessment mode) if any of the following conditions apply: (i) Missing four or more Online Mini-tests; (ii) Missing or getting a test grade below 7.5 values; (iii) Getting caught in an academic fraud attempt.
Basic Bibliographic	1 - Pires, G. (2012), Cálculo diferencial e integral em Rn, IST Press, Lisboa (disponível na biblioteca ISCTE-IUL) 2 - Cadernos/sebentas fornecidos pelos docentes (via e-learning) 3 - Ferreira, M.A.M. e Amaral, I. (2002), Cálculo Diferencial em Rn, Edições Sílabo, Colecção Matemática, Lisboa (disponível na biblioteca ISCTE-IUL) 4 - Ferreira, M.A.M. e Amaral, I. (1994), Integrais Múltiplos e Equações Diferenciais, Edições Sílabo, Colecção Matemática, Lisboa (disponível na biblioteca ISCTE-IUL) 5- Ablowitz, M.J. and Fokas, A.S. (2003), Complex variables : introduction and applications, Cambridge University Press, Cambridge (disponível na biblioteca ISCTE-IUL)
Complementar Bibliographic	1 - Marks. E.J. (coord.) (2005), Multivariate calculus, John Wiley, New York Multivariate (disponível na biblioteca ISCTE-IUL) 2 - Marsden J. and Weistein, A. (1984), "Calculus II, Springer-Verlag, New York (disponível na biblioteca ISCTE-IUL) 3 - Marsden, J. and Weistein, A. (1984), Calculus III, Springer-Verlag, New York (disponível na biblioteca ISCTE-IUL) 4 - Brown, J.W. and Churchill, R.V. (2004), Complex Variables and Applications, McGraw-Hill, New York