Continuous evaluation (1ª opportunity): The continuous evaluation will consist into several multiple-choice test and several practical test. The continuous evaluation represents the 60% of the total of the final evaluation. The final examination results 40%.

Second opportunity: there will be a modification of the criteria and the continuous evaluation will represent the 50% of the final evaluation and the final examination the other 50%. The results of the tests of continuous evaluation at the earliest opportunity will be moved for the second opportunity adapting them.

It will award the qualification of NOT TAKEN to the student that only participate in activities of evaluation that have a weight less than 20% of the final qualification, with independence of the qualification obtained.

Opportunity in Advance: The final qualification of the student that request this opportunity will be the obtained in the face-to-face examination valued on 10 points.

Students part time (or with dispenses of attendence): it will be evaluated according to the same norms that the rest of students.

Conditions for making of the examinations and tests:

1) Face-to-face: During the realisation of the examinations will not be able to have access to any device that allow the communication with the outside and/or the storage of information. It will be able to deny the entrance to the classroom with this type of devices. The student will be able to use a scientific calculator non graphic and non programmable. They will not admit the examinations written with pencil. The students will have to identify by means of DNI or equivalent for making the tests of evaluation.

2) Non-attendence: In the case of telematic tests, students will not be able to keep contact with other people and will be able to request that it activates his camera (or the one of his mobile) and identifies by means of his DNI or equivalent.

Virtual platform: it will use the platform Moodle of the UDC (http://moodle.udc.es) and the platform Microsoft Teams (for communication with the students).

Learning outcomes	Study programme competences
Identify the notable sets of a subset of IRn.	A8 A11
Understand the basic concepts of the euclidean space IRn.	A8 A11
Determine if a set is open, closed, bounded, compact and convex.	A8 A11
Understand the concept of function of several variables.	A8 A11
Draw the level set of a function of two variables.	A8 A11
Understand the concept of continuous function.	A8 A11
Determine if a function is continuous or not.	A8 A11
Recognize a linear function.	A8 A11
Recognize a quadratic form.	A8 A11
Classify a quadratic form by examining the signs of the principal minors.	A8 A11
Classify a constrained quadratic form.	A8 A11
Calculate and interpret partial derivatives and elasticities.	A4 A8 A11	B1 B2 B5 B10	C1 C7
Find the Taylor polynomial of a function.	A8 A11
Calculate the partial derivatives of a compounded function.	A8 A11
Use the existence theorem to analyze if a equation defines an implicit real function.	A8 A11
Find the partial derivatives and elasticities of an implicit function, and interpret them.	A8 A11
Analyze the concavity/convexity of a function.	A8 A11
Formulate mathematical programming problems.	A3 A4 A6 A8 A9 A10 A11	B1 B2 B3 B4 B5 B10	C1 C4 C5 C6 C7 C8
Distinguish between local and global optima.	A8 A11
Graphically solving an optimization problem	A8 A11	B3
Analyze the existence of global optima using the Weierstrass theorem.	A8 A11
Find the critical points of a function of several variables.	A8 A11
Classify the critical points using the second-order conditions.	A8 A11
Determine the local or global character of the optima of an unconstrained problem.	A8 A11
Formulate economic problems as mathematical programs with equality constraints.	A8 A11
Find the critical points of a mathematical program with equality constraints.	A8 A11
Classify the critical points and interpret the Lagrange multipliers.	A8 A11
Determine the local or global character of the optima of an equality-constrained problem.	A8 A11
Know the structure and basic properties of a linear program.	A8 A11
Formulate simple economic problems as linear programs.	A3 A4 A8 A11 A12	B1 B2 B3 B4 B5 B10	C1 C4 C6 C7 C8
Solve linear programs by the simplex algorithm.	A3 A4 A6 A8 A9 A11	B1 B2 B3 B4 B5 B10	C1 C4 C5 C6 C7 C8

Topic	Sub-topic
1. The euclidean space IRn.	The vector space IRn. Inner product. Norm. Distance. Interior, closure, isolated, limit and boundary points. Open and closed sets. Compact sets.
2. Functions of several variables.	Basic concepts. Graphical representation of real functions. Level sets. Limit of a function at a point. Continuity. Linear functions. Quadratic forms. Classification. Constrained quadratic forms.
3. Derivatives of functions of several variables.	Partial derivatives. Partial derivatives of higher order. Class one function Chain's Rule. Taylor's theorem. Implicit function theorem.
4. Convexity of sets and functions.	Convex sets. Properties. Convex functions. Properties. Characterization of twice continuously differentiable convex functions.
5. Introduction to mathematical programming.	Formulation of a mathematical program. Local and global optima. Graphic solving. Basic Theorems in optimization.
6. Unconstrained optimization.	First-order necessary conditions. Second-order conditions. The convex case. Sensitivity analysis.
7. Equality-constrained optimization	Formulation. First-order necessary conditions: the Lagrange theorem. Second-order conditions. The convex case. Sensitivity analysis.
8. Linear programming.	Formulation of linear programs. Basic feasible solutions. Fundamental theorems. The simplex algorithm.

Methodologies / tests	Competencies	Ordinary class hours	Student’s personal work hours	Total hours
Introductory activities	A6 A9 A12 C1	1	0	1
Multiple-choice questions	A10 B2 B3 B4	2	7	9
Mixed objective/subjective test	A10 B2 B3 B4	3	15	18
Guest lecture / keynote speech	A3 A4 A8 A9 A11 A12 B1 B5 C6 C7	15	15	30
Seminar	B10 C4 C5 C8	2	4	6
Practical test:	A8 A11 B1 B2 B3 B4 B5 C1	2	8	10
Problem solving	A6 B1	25	50	75

Personalized attention		1	0	1

(*)The information in the planning table is for guidance only and does not take into account the heterogeneity of the students.

Methodologies	Description
Introductory activities	It will be the presentation of the course (one hour).
Multiple-choice questions	There will be several multiple-choice exams. These exams will have questions with several given answers --only one will be correct-- related to theoretical and practical concepts covered in the course.
Mixed objective/subjective test	At the end of the course, there will be a mixed (theoretical/practical) exam. This exam will take place at the official date determined by the Faculty.
Guest lecture / keynote speech	There will be 15 hours of keynote speech, that will be focused on the exposition of the theoretical contents.
Seminar	They will be several seminars with personalized attention of character essentially practical. These seminars will be virtual through the platform Microsoft Teams.
Practical test:	There will be several practical tests along the term. These tests will consist of one or several questions to which will have to answer by writing and justifying properly the answers.
Problem solving	There will be 25 hours of problem solving classes, which will be focused on the formulation and solving of problems related to the practical contents of the subject.

Methodologies	Competencies	Description	Qualification
Practical test:	A8 A11 B1 B2 B3 B4 B5 C1	There will be several presential exams. It will be valued a good understanding of the concepts, the use of appropriate reasoning, the proper use of mathematical language, and the skills in formulating and solving problems.	40
Mixed objective/subjective test	A10 B2 B3 B4	The final (presential) exam will represent a 40% of the final mark (4 points). It will be valued a good understanding of the concepts, the use of appropriate reasoning, the proper use of mathematical language, and the skills in formulating and solving problems.	40
Multiple-choice questions	A10 B2 B3 B4	There will be several multiple-choice exams.	20

Assessment comments
Continuous evaluation (1ª opportunity): The continuous evaluation will consist into several multiple-choice test and several practical test. The continuous evaluation represents the 60% of the total of the final evaluation. The final examination results 40%. Second opportunity: there will be a modification of the criteria and the continuous evaluation will represent the 50% of the final evaluation and the final examination the other 50%. The results of the tests of continuous evaluation at the earliest opportunity will be moved for the second opportunity adapting them. It will award the qualification of NOT TAKEN to the student that only participate in activities of evaluation that have a weight less than 20% of the final qualification, with independence of the qualification obtained. Opportunity in Advance: The final qualification of the student that request this opportunity will be the obtained in the face-to-face examination valued on 10 points. Students part time (or with dispenses of attendence): it will be evaluated according to the same norms that the rest of students. Conditions for making of the examinations and tests: 1) Face-to-face: During the realisation of the examinations will not be able to have access to any device that allow the communication with the outside and/or the storage of information. It will be able to deny the entrance to the classroom with this type of devices. The student will be able to use a scientific calculator non graphic and non programmable. They will not admit the examinations written with pencil. The students will have to identify by means of DNI or equivalent for making the tests of evaluation. 2) Non-attendence: In the case of telematic tests, students will not be able to keep contact with other people and will be able to request that it activates his camera (or the one of his mobile) and identifies by means of his DNI or equivalent. Virtual platform: it will use the platform Moodle of the UDC (http://moodle.udc.es) and the platform Microsoft Teams (for communication with the students).

Basic	K. Sydsæter, P. J. Hammond y P. Carvajal (2012). Matemáticas para el análisis económico . Madrid, Pearson

Complementary	R. Caballero, S. Calderón, T. P. Galache, A. C. González, Mª. L. Rey y F. Ruiz (2000). Matemáticas aplicadas a la economía y la empresa. 434 ejercicios resueltos y comentados . Madrid, Pirámide E. Minguillón, I. Pérez Grasa y G. Jarne (2004). Matemáticas para la economía. Libro de ejercicios. Álgebra lineal y cálculo diferencial. Madrid, McGraw-Hill I. Pérez Grasa, G. Jarne y E. Minguillón (1997). Matemáticas para la economía: álgebra lineal y cálculo diferencial . Madrid, McGraw-Hill I. Pérez Grasa, G. Jarne y E. Minguillón (2001). Matemáticas para la economía: programación matemática y sistemas dinámicos . Madrid, McGraw-Hill M. Hoy, J. Livernois, C. McKenna, R. Rees y T. Stengos (2001). Mathematics for economics. Cambridge, MA, The MIT Press A. C. Chiang y K. Wainwright (2006). Métodos fundamentales de economía matemática . Madrid, McGraw-Hill R. M. Barbolla, E. Cerdá y P. Sanz (2001). Optimización. Cuestiones, ejercicios y aplicaciones a la economía . Madrid, Prentice Hall

Other comments
It is advisable to have passed Mathematics I. Students must be familiar with the concepts and fundamental results of linear algebra (matrices, determinants and systems of linear equations), and differential calculus in one variable (limit, continuity, derivative, elasticity, optima, convexity).