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Calculus II

Code 8541
Year 1
Semester S2
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements There is no entry requirement.
Mode of delivery Presential classes
Work placements Not applicable
Learning outcomes With this Curricular Unit, it is intended that the students acquire basic knowledge of Differential and Integral Calculus of functions of several variables. At the end of this Curricular Unit the student must be able to:
1) Compute limits of functions of several variables;
2) Study the continuity of functions of several variables;
3) Derive functions of several variables;
4) Apply the derivatives to compute the extremes of functions of several variables;
5) Integrate functions of several variables;
6) Use integral calculus to determine areas and volumes.
7) Formulate and solve problems using the differential and integral calculus of functions with several variables
Syllabus 1- Functions of several variables.
1.1 Brief notions of topology in R^n
1.2 Functions from R^n in R^m
1.3 Limits
1.4 Continuity

2- Differential calculus in R^n
2.1 Partial and directional derivatives
2.2 Differentiability of functions of R^n in R^m
2.3 Derivative of a composition of functions
2.4 Derivatives of higher order; Schwarz's Theorem
2.5 Implicit function theorem
2.6 Absolute extremes and local extremes
2.7 Conditional extremes: Lagrange multiplier method

3- Integral calculus in R^n
3.1 Riemann integral
3.2 Properties of integrable functions
3.3 Coordinate change
3.4 Applications

4- Line Integrals
4.1 Paths and lines
4.2 Line integral of a scalar field
4.3 Line integral of a vector field
4.4 Green's Theorem

5. Surface integrals
5.1 Parameterization of surfaces
5.2 Surface integrals of scalar fields; area of a surface
5.3 Surface integrals of a vector field
5.4 Gauss and Stokes theorems
Main Bibliography 1. Stewart, James, "Cálculo", Volume II, 5ª edição Thomson Learning, 2001.
2. Lang, S., "Calculus of Several Variables", Undergraduate Texts in Mathematics, Third Edition, Springer-Verlag,1987.
3. Apostol,T.M., "Calculus",Volume II, John Wiley & Sons, 1968.
4. H. Anton, I. Bivens e S. Davis, Calculus, (Eight Edition), John Wiley & Sons, 2006.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2021-09-06

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