Calculus II

Calculus II (280 - 15065)
Study: Bachelor in Energy Engineering
Semester 2/Spring Semester
1st Year Course/Lower Division

Please note: this course is cross-listed under the majority of engineering departments. Students should select the course from the department that best fits their area of study.

STUDENTS ARE EXPECTED TO HAVE COMPLETED:

Calculus I
Linear Algebra

Competences and Skills that will be Acquired and Learning Results:

The student must be able to state, solve and understand, from a matematical point of view, problems related to Industrial Technologies. First of all, a comprehensive approach to Euclidean spaces with a special emphasis in the 3D case as well as their most relevant subsets will be done. He must handle the main properties of functions in several variables related to continuity, differentiability and integrability both in the scalar and vector cases. The study of problems related to optimization with and without constraints constitutes a nice application of Taylor formula and local extrema.
Iterated integrals on domains as well as the integration on lines and surfaces will provide the basic background for the analysis of areas and volumes as well as the computation of some characteristics of rigid solids.

Description of Contents/Course Description:

Chapter 1. n-dimensional Euclidean Space.Open, closed, compact and connected subsets.
Chapter 2. Functions of several variables. Limits and continuity.
Chapter 3. Partial and directional derivatives. Differentiability. Gradient vector.Jacobian matrix.
Chapter 4. Chain rule. Higher order derivatives.
Chapter 5. Taylor formula. Local extrema. Extremum problems with constraints. Lagrange multipliers.
Chapter 6. Integration in R^n.Iterated integrals. Fubini,s Theorem. Applications.
Chapter 7. Line integrals. Conservative fields.
Chapter 8. Green Theorem
Chapter 9. Surfaces in R^3.
Chapter 10. Surface integrals.
Chapter 11. Stokes and Gauss Theorems.

Learning Activities and Methodology:

The learning activities will be focused on
1.- Magistral sessions devoted to the presentation of the basic concepts and results of every chapter as well as come exercises. The theoretical background will be supported by the basic monographs listed in the bibliography.
2.- Problem sessions. Here we will solve questions and problems proposed in the magistral classes as well as individual homeworks in order to allow the self asessment of the students.
3.- Two partial tests concerning Differential Calculos (Chapters 1-5) and Integral Calculus (Chapters 6-11).
4.- Final exam.
5.- Tutorial activities for small teams of 5-6 students.

Assessment System:

The assessment system will be focused on
1.- Two partial tests with a maximum grading of 3 points each.
2.- Four homeworks with a maximum grading of 1 point each.
Thus, the grading for the assessment taking into account the work during the semester will be of 10 points at most.
In order to have the final grading, the weight of this activity will be 40%.

Basic Bibliography:

B.P. DEMIDOVICH. Problemas de Análisis Matemático,. Editorial Paraninfo . 1991
J. E . MARSDEN A. J. TROMBA,. Vector Calculus. Freemann. 2012
S. L. SALAS, E. HILLE,. Calculus:One and several variables. Wiley. 1999

Additional Bibliography:

R. C. WREDE, M. R. SPIEGEL. Schaum's Outline of Avanced Calculus. Editorial Mc-Graw Hill. 2005
R. G. BARTLE. The Elements of Real Analysis,. Editorial Wiley International. 1976
T. APOSTOL. Calculus, Volume 2. John Wiley& Sons. 1969