Universitat Internacional de Catalunya

Mathematics

Mathematics
6
7975
1
First semester
FB
Propedeutic Introductory Module
Mathematics
Main language of instruction: English

Other languages of instruction: Spanish

Teaching staff


The class schedule is:

Monday from 12.00 to 14.00

Tuesday from 12.00 to 14.00

If more time is needed, the student will request an appointment for the teachers by e-mail.

Introduction

This subject is related exclusively to learning the tools that students need for their application in the field of architecture.

It is intended that the student finishes the course using mathematics as a work tool related to real problems that they might encounter in the future, especially for the calculation of structures.

The 2022 - 2023 academic year is fully face-to-face.

The classes will be participatory and practical with the aim of increasing the student's ability to work in facets close to what will be her professional life.

The resolution of all the practical exercises proposed to the student will be important, which will be corrected, as soon as possible, by the teacher either through the Moodle platform or in person.

Pre-course requirements

The skills required are:

  • Operations with and without fractions.
  • Inequalities                                                          
  • Systems of equations
  • Areas, perimeters and volumes
  • Logarithmic relationships
  • Trigonometry (sin, cos, tan)
  • Vector representation in a plane
  • Representation of figures in space
  • Operations with vectors
  • Operations with matrices and determinants
  • Derivatives and application
  • Graphing
  • Integrals and their area of application.

Objectives

The main objective of these courses is to acquire knowledge of how to use the tools needed to address solving architectural problems.

Competencies

  • 07 - To acquire adequate knowledge and apply it to the principles of general mechanics, statics, mass geometry, vector fields and force lines of architecture and urban planning
  • 08 - To aquire adequate knowledge and apply it to the principles of thermodynamics, acoustics and optics of architecture and urban planning.
  • 09 - To acquire adequate knowledge and apply it to the principles of fluid mechanics, hydraulics, electricity and electromagnetism in architecture and urban planning.
  • 11 - To acquire knowledge and apply it to numerical calculation, analytic and differential geometry and algebraic methods.

Learning outcomes

  • Perform vector operations for application to the design of structures.
  • Understanding the concepts of a linear combination of vectors and linear dependence.
  • Understanding the classical concepts of vector spaces and their applications.
  • Understanding the concepts of scalar product, norm and orthogonality in vector spaces.
  • Understanding the concepts of vectors and eigenvalues of a matrix and its application to the diagonalization of matrices.
  • Knowing how to relate linear transformation matrix transformations with issues specific to systems of linear equations.
  • Understanding the definition of the various matrix operations and their application to linear transformations and systems of linear equations.
  • Understanding the concept of the staggered and reduced echelon form of a matrix.
  • Understanding the notion of inductive determinant.
  • Knowing the properties of determinants and their applications.
  • Understanding the subspaces associated with a matrix and their relation to linear transformations and systems of linear equations.
  • Understanding the concept of the inductive factor.
  • Knowing the properties of determinants and their applications.
  • Understanding the notion of a system of linear equations.
  • Knowing how to identify each element of a linear system with a matrix-standardized method.
  • Understanding and interpreting the concept of the solution set of a linear system.
  • Knowing how to handle with ease the calculation of partial derivatives using different rules in an existing chain.
  • Dominate the calculation of partial derivatives.
  • Knowing how to calculate with ease domains and images of real functions
  • Learn to study all the concepts necessary for the representation of a function.
  • Understanding the concept of primitive function.
  • Learn compute fluently primitive functions by selecting the most appropriate method.
  • Knowing how to calculate definite integrals
  • Knowing how to calculate with ease double and triple integrals using iterated integrations.

Syllabus

1. Representation of functions in space (2D and 3D)

  • Meaning of the function of a real variable
  • Continuity of a function (types of discontinuities in a function)
  • Asymptotes
  • Symmetry of functions
  • Mean Value Theorem
  • Calculation of roots (cutting the x-axes), Rolle's Theorem, Newton’s Theorem and the tangent
  • Criteria bypass
  • Application of the derivative of a function: maximum, minimum, inflection points, concavity and convexity, waxing and waning, etc.
  • Brief notions of partial derivatives
  • The meaning of geometric features.
  • Optimization

2. Integral calculus.

  • The meaning of analytical geometry
  • Definite and indefinite integrals
  • Properties of integrals
  • Integral Calculus
  • Double Integrals. Meaning and calculation. Application in the calculation of areas. Variable change (polar) to simplify the calculation of surfaces
  • Implementation in architecture (structure, surface plots, bulk of buildings, budgets etc.)

3. Matrices and Determinants 

  • Definition. Types of matrices (according to the shape and the elements). Operations with Matrices. Rank of a matrix (calculation methods)
  • Calculation of determinants. Definition. Properties of determinants. Methods of calculation.
  • Matrix equations

4. Systems of linear equations 

  • Systems of Equations: Incompatible, consistent, determined / undetermined
  • Inequalities
  • Notation matrix equations (simplified matrix / extended)
  • Methods for solving equations
  • Numerically solving system equations
  • Numerically solving system equations

5. Vector algebra 

  • Vectors in free space. Components of a vector (or standard length, direction, sense). Types of vectors
  • Operations with vectors
  • Projection of a vector and angle between vectors
  • Linear dependence and independence of vectors. Linear combination
  • Parallel, perpendicular (orthogonal) and geometric mean vectors

Teaching and learning activities

In person



Different types of methodology are applied depending on the type of educational activity:

Theory sessions to introduce the concepts of protocols and services.

Practical sessions to demonstrate the most important theoretical concepts.

Activities outlined in the University Teaching Platform.

 Each work session and activity is designed to develop skills that students need to acquire during the course.

Key recommendations made to students can be summarized as follows:

Read the presentations before attending the theoretical sessions

Actively participate in the theory sessions

Supplement the topics covered in these sessions with information provided in the recommended reading list

Use tutorial sessions at any time to resolve any questions or problems

Undertake written tests throughout the semester.

Continue the development of practical work according to established criteria.

When the necessary theoretical concepts have been explained, do not delay in putting them to the test

Start undertaking practical work individually

Resolve difficulties with peers

Weekly sessions are proposed as follows:

1. Theoretical sessions (half of the weekly course hours).

Theoretical and instrumental techniques are taught through master classes involving oral and blackboard work, during which students may raise questions, concerns and comments.

2. Hands-on (half of the weekly course hours).

Practical sessions solving exercises and examples presented in class. In addition to these classes weekly assignments will be collected.

3. Sessions

During these sessions, prior to or following the times of the workshops and lessons that each tutor establishes with their assigned group of students, the students may raise any reasonable queries with the teachers that have not been resolved during the remainder of the sessions. Also, during this time, the student may request a specific extension of the recommended reading list, or any other information related to the subject.

TRAINING ACTIVITYCOMPETENCESECTS CREDITS
Class exhibition
07 08 09 11 1,5
Class participation
07 08 09 11 0,5
Clase practice
07 08 09 11 0,5
Tutorials
07 08 09 11 0,5
Individual or group study
07 08 09 11 3,0

Evaluation systems and criteria

In person



The lectures are scheduled on Mondays and Tuesdays from 12 to 14pm.

Class attendance is mandatory, since the Degree in Architecture is face-to-face, therefore 100% of the classes must be attended in order to follow the subject properly and carry out continuous evaluation. If the student does not meet 80% attendance, she will not be able to take the exam in the 1st call.

The students of further years cannot enroll any other subject with the same schedule.

During the semester, we will recollect some exercises/problems that will be the 30% of the final grade. Also, there will be one partial exam.

The final grade will be obtained in the following way:

-The average grade of the collected problems during the semester is 30%.

-The partial exam will count 30% of the final grade.

-The final exam will be the 40% of the final grade.

Bibliography and resources

Calculus. Una y varias variables. Vol I y Vol II. Salas, Hille & Etgen. 4ª Ed. Editorial Reverté, 2002

 M. Piskunov: “Cálculo diferencial e integral” ed. Utecha Noriega

 P. Puig Adam. “Cálculo integral”

 Schaum: “ Problemas de cálculo integral”

 Schaum: “ Problemas de ecuaciones diferenciales”