Syllabus Information

1. General information

The basic mathematical methods are part of the many tools that all professionals of Architecture must have to solve the problems that arise in their work. The learning results include precisely the mastery of not only theoretical but also practical techniques, which allow the direct application of the methods considered in the subject to real problems, with realistic calculation methods that are incorporated in effective and proven softwarepackages.

It is therefore fundamental in the correct formation of an Architect and Engineer to obtain the learning results covered by this subject .

The final goal is for the student to integrate the basic knowledge of this subject in all kinds of aspects related to Technical Architecture, so that they serve as a basis for other subjects and at the same time acquire techniques that enable their professional development.}

These approaches and goals are aligned with the following Sustainable Development Goals (SDGs) of the United Nations Agenda 2030 (https://www.un.org/sustainabledevelopment/es/)so that the acquisition of the learning results of the subject provides training and competence to contribute to some extent to their achievement.

Goal 4: Quality education

2. Learning results

The student, in order to pass this subject, must demonstrate the following results:

• Ability to apply data processing and analysis techniques.
• Know the concepts, applications and fundamental results of Differential and Integral Calculus.
• Understand the concepts of unidimensional and multidimensional variables.
• Know the techniques of integration and estimation.
• Ability to prepare, understand and critique reports based on analyses developed with numerical, differential, integral and matrix calculations.
• Know how to solve mathematical problems that may arise in Engineering, using correctly the acquired knowledge of Differential and Integral Calculus and Linear Algebra
• Understand the difficulty of solving certain mathematical problems in an exact way and is capable of resorting to the application of numerical approximation methods in their solution
• Capable of posing and rigorously solving problems related to his/her speciality in Engineering, selecting critically the most appropriate methods and mathematical theoretical results.
• Understand the impossibility of solving real problems manually, and is able to implement and solve them with mathematicalsoftware of symbolic calculation.

Posses the skills of logical-deductive thinking and handles a mathematical language that allows him/her to model problems of Building Engineering

3. Syllabus

Introduction to WXMaxima and review of real functions of real variables

Limits and Continuity

• Limits, indeterminacies, equivalences
• Continuity and discontinuity of functions
• Classical Theorems
• Bisection method

Referral

• Derivative and tangent line, properties
• Chain rule
• Implicit function derivative, inverse function and function in parametrics
• Newton's method
• Classical theorems: Rolle, average value, L'Hôpital
• Taylor's limited developments
• Interpolation and numerical derivation
• Monotonicity, maxima and minima, concavity and convexity

Integration

• Riemann integral and its basic properties
• Calculation of primitives
• Fundamental theorems of calculus
• Improper integrals
• Geometric applications
• Numerical quadrature methods

Systems of linear equations

• Groups, rings, bodies
• Systems of linear equations: elementary operations
• Gaussian elimination and rank of a matrix
• Characterization theorem for linear systems (Rouché-Frobenius)
• Determinants
• Numerical Gaussian elimination, condition number
• LU, QR and Cholesky decompositions
• Iterative methods

Vector spaces with scalar product

• Linear independence, dimension and basis
• Subspaces
• Scalar product
• Distances, angles and orthogonality
• Orthogonal systems and subspaces
• Projectors and optimal approximation theorem

Diagonalization

• Eigenvalues and eigenvectors
• Spectral decomposition and matrix functions
• Normal matrices
• Numerical calculation of eigenvalues
• Compatible matrices
• Decomposition into singular values

4. Academic activities

The activities to be developed in the subject are the following

• Theoretical classes, in which the fundamental concepts that constitute the basic body of knowledge that must be  learned in order to achieve the learning results listed below are presented. The theoretical concepts are  complemented by detailed examples that illustrate how they work in a concrete context.
• Practical classes, in which problems are proposed to be solved using the methods and concepts previously considered . Discussion, participation, cooperation and reflection are encouraged in these classes.
• Assessment sessions, in which students are submitted to written tests on certain well-specified parts of the subject matter covered, or they of the subject matter covered, or they publicly present the work done in groups proposed in the previous activity
• Personal work, in which students dedicate time outside of class to study the concepts taught in class, solve problems analogous and/or complementary to those considered in class.
• Global assessment test, which consists of a written test of the whole subject. There are two global tests, one for each official call, and both take place after the end of the classes and when the rest of the activities have been concluded and assessed

Key dates will be announced well in advance during the term. There are two types:

• Evaluative milestones associated with the progressive assessment system, in which one of the activities described above is developed.  These dates are fixed at the beginning of the term by the teacher, and can be modified with prior notice if the development of the calendar so requires.
• Official calls, in which any student can take the global assessment test on the entire subject . These dates are set at the beginning of the school year by the school administration.

The dates of the final tests will be officially published at https://eupla.unizar.es/asuntos-academicos/examenes

5. Assessment system

At the beginning of the subject the student will choose one of the following two assessment methodologies:

• A continuous assessment system, which will be carried out throughout the entire teaching period.
• An overall assessment test, reflecting the achievement of the learning results, at the end of the teaching period.

Continuous evaluation system:

• Written tests: Individual exercises remain a reliable way to know if the student has the capacity to apply the methods under consideration. Two exams are distributed throughout the semester, each covering different parts of the syllabus, although they cannot always be mutually exclusive due to the nature of Mathematics.

The written tests comprise 80% of the total grade, divided into two tests with values of 40% and 40%. A minimum score of 3 on each written test is required to continue with the continuous assessment

• Participation controls: Some classes of problems are complemented with the elaboration of exercises analogous to those considered to be submitted for assessment, similar to the previous tests but focused on more concrete and lower value problems. In this way, the collaboration of the students is assessed, both among themselves and with the class discourse, and their involvement in the previous activities that lead to the resolution of these controls. The participation controls comprise 20% of the total grade, distributed in four controls with equal values. Students will be able to pass the subject by progressive assessment if the arithmetic average of of the written tests and the participation controls is a 5.

Global written assessment test: In each of the two official calls, a global test can be taken at assessment, which consists of a global written test comprising 100%. Thus, if a student has not been able to pass the written tests and the controls, he/she can opt through this test to achieve the highest grade. All students have right to this global test.