Calculus and Mathematical Analysis

To connect with the subject of "Fundamentos de Matemáticas" and serve for the student to consolidate the mathematical language and to train him/her in the main numerical and analytical methods of differential calculus in several variables, integral calculus in one variable and the resolution of basic differential equations differences. In the practical part, the student will learn to solve numerically the problems posed with the different numerical methods studied.

1. Numerical solution of nonlinear equations: Separation of the roots. Dichotomy method. Newton's method.

2. Basic concepts of functions of several variables: domain, image, contour lines, graph of a function of several variables.

3. Derivation in several variables: partial derivatives, gradient, Jacobian matrix. Calculation of maxima and minima of scalar functions of several variables. Optimization using the method of rapid descent.

4. Numerical resolution of nonlinear and linear systems: Newton's method for nonlinear systems. Fast descent method. LU and Cholesky factorizations. Computational cost analysis.

5. Integral calculus in one variable: indefinite integral. Fundamental theorems of integral calculus. Calculus of primitives. Leibniz's rule.

6. Numerical integration: Trapezius rule and Simpson's rule.

7. First order differential equations: ordinary differential equations. Initial value problem. Differential equations in separate variables. Linear first order differential equations. Numerical solution with Euler's method.

Basic:
THOMAS, G. B. Cálculo. Varias variables.. 13ª ed., México: Pearson, 2015. ISBN: 9789702627357
CHAPRA, S. C. y CANALE, R. P. Métodos numéricos para ingenieros. 7ª ed., México: McGraw-Hill, 2015. ISBN: 9786071512949

Complementary:
ANASTASSIOU, G. A. y MEZEI, R. Numerical Analysis Using Sage. Switzerland, Springer, 2015. ISBN: 978-3-319-16739-8

To contribute to achieve the competences described in the document "Grao de Enxeñaría Informática na USC": G5, G8, G9, G10, TR1, TR2, TR3, FB1 and RI6.
Competences associated to the Mathematics module:
- To expose and argue clearly both hypotheses and methods used in the resolution of problems, and to use a suitable language.
- To develop the ability to analyze the resolution of a problem.
- To have a critical attitude when dealing with different types of solutions.
- To master notation, method and vocabulary for mathematical modeling and case studies.
- Knowledgeable use of the mathematical language.
- To develop capacity for abstraction and formalization using the language of logic to express ideas with precision and rigor.
- To know mathematical techniques that allow to solve problems related to engineering.

The subject is assigned two hours of lecture and two hours of practice per week.

In the expository hours the contents of the subject will be presented using the lecture method. The competences FB1 and CG8 will be worked on. The contents of the subject will be developed following a book that will serve as a guide.
On the occasion of COVID-2019 support material was developed for the subject that is provided in the virtual campus.

The exposition of the contents will be supported by the use of transparencies provided by the teachers and the development of the concepts and exercises on the blackboard.

In the interactive classes, students will learn how to use the scientific software Sage to solve mathematical problems. In these classes we will focus on the generic competences CG5 by modifying in the classes related to numerical calculation the codes provided by the teachers.

Competences RI6, CG9 and CG10 will be worked on in the interactive classes as the teachers take them into account in the design of the tasks to be presented. It is the students who will have to select the problems and apply the methodologies of calculus and numerical analysis learned to solve the problems and present them orally or in writing for evaluation. The time intervals for the deliveries will be established, so they will have to be organized in groups and/or individually to be taken into account for their evaluation.

The competences of type TR1 will be worked fundamentally in the aspect of written communication in task 1, both in the presentation of the problems, as in the resolution and the correction that they make between pairs. The evaluation takes into account the presentation of the tasks that can be in Galician, Spanish or English, if the corresponding group of practices is defined. Tasks two and three will be presented orally in the interactive hours, face-to-face or telematically, where the assessment and interaction of all students attending the group will be encouraged. At the same time, a log of the expository classes will be constructed with the collaboration of the students. Each class will be described by a group of students who will make a summary of the class, giving an account of the number of students present, the agreements reached in that class in relation to deliveries and assignment of exercises, the contents studied and the material used. This log will be corrected by another group of students.

The TR2 type competences are worked implicitly by the dynamics of the classes and especially in tasks two and three where teamwork and the participation of all group members in the presentation of the tasks is encouraged, which implies an assignment of individual work and coordination that is taken into account in the grade.

In the tasks they have to present, the competences of the TR3 group will be worked by having to design the problems that are the object of the same, dedicating a part of the qualification to the presentation of the same. The references of other years that are presented as models motivate the environmental sensitivity, the commitment to entrepreneurship, the quality and rigor of the presentations.

All course material will be available on the Virtual Campus of the USC. On this page it will also be possible to access information on the organization of the subject and to contact the professors and other colleagues to resolve doubts.

Continuous assessment:

* Three group assignments: assignments two and three will be presented in the interactive and tutorial hours. To make the presentations it is necessary to attend the corresponding classes. They will be used for the evaluation of the competences CG5, CG8, CG9, CG10, TR1, TR2, TR3 and RI6 as specified in the methodology.

* The realization of a collaborative log of the subject: each student will participate in the elaboration or correction of the log of a class that the student attends. It will be used for the evaluation of competencies type TR1.

* Satisfactory completion of the continuous evaluation will award up to 40% of the final grade for any evaluation opportunity of the current course. That is, the grade obtained in the continuous assessment will be maintained exactly for the second opportunity and can only be obtained within the dates indicated.

Final exam:
60% of the grade will be obtained in a final written test both in the first evaluation opportunity and in the second one. It will be used for the evaluation of competencies FB1, CG5, CG8, CG9, CG10 and TR1.

The status of “not presented” will be assigned to those who do not attend any of the final tests of the two opportunities.

For cases of fraudulent performance of exercises or tests will be applied as stated in the Regulations for the evaluation of the academic performance of students and grade review.

* Time of study and personal work that a student must dedicate to pass each week:
2 hours theory + 2 hours laboratory + 4 h personal work associated with problem solving and analysis of the theoretical contents.
Total: 12.5 x 8 = 100 hours.

* Preparation of the work to be handed in, 6 hours per work.

Total course: 100 + 10 x 3 = 130 h.

* Final test:
3 h final test + 20 h exam preparation.
Total exam: 23 h.

Total: 153 h

* Attendance to classes with active participation in them.
* Use of the textbook and recommended material.
* Completion of the practices and the necessary exercises corresponding to the different topics to achieve the established objectives.