CÀLCUL - Calculus

Transcription

1 Coordinating unit: ETSECCPB - Barcelona School of Civil Engineering Teaching unit: DECA - Department of Civil and Environmental Engineering Academic year: Degree: 2017 BACHELOR'S DEGREE IN GEOLOGICAL ENGINEERING (Syllabus 2010). (Teaching unit Compulsory) ECTS credits: 7,5 Teaching languages: Catalan, Spanish Teaching staff Coordinator: Others: EUSEBIO JARAUTA BRAGULAT EUSEBIO JARAUTA BRAGULAT Opening hours Timetable: Office: Monday from 11 to 13 am and 15:30 to 16:30. You must make an appointment beforehand. By whenever the student wants to use. Degree competences to which the subject contributes Specific: Ability to solve the types of mathematical problems that may arise in engineering. Ability to apply knowledge of: linear algebra; geometry; differential geometry; differential and integral calculus; differential equations and partial derivatives; numerical methods; numerical algorithms; statistics and optimisation Ability to solve ordinary differential equations for application to engineering problems Transversal: 591. EFFICIENT ORAL AND WRITTEN COMMUNICATION - Level 1. Planning oral communication, answering questions properly and writing straightforward texts that are spelt correctly and are grammatically coherent EFFECTIVE USE OF INFORMATI0N RESOURCES - Level 2. Designing and executing a good strategy for advanced searches using specialized information resources, once the various parts of an academic document have been identified and bibliographical references provided. Choosing suitable information based on its relevance and quality SELF-DIRECTED LEARNING - Level 2: Completing set tasks based on the guidelines set by lecturers. Devoting the time needed to complete each task, including personal contributions and expanding on the recommended information sources. Teaching methodology The course runs over 5 hours a week in classes in the classroom. The sessions are devoted to: - Theory (presentation of concepts and basic materials matter with application examples - classroom practice (solving exercises and problems) - Laboratory Practice (performing calculations with the software application subject). It uses material support that is available to students through campus ATENEA. Learning objectives of the subject Students willâ learn to perform differential and integral calculus of several variables and to solve ordinary differential equations. They will also learn to apply these techniques to specific scientific and technical problems and to geological 1 / 5

2 engineering in general. Upon completion of the course, students will be able to: 1. Relate ordinary differential equations to engineering problems and solve them in simple geometric conditions, and conduct analyses such as parametric studies to validate the solutions; 2. Use Fourier series to solve engineering problems; 3. Solve engineering problems requiring minimisation, integration and analysis of functions of several variables. Differential calculus of functions of several variables; Integral calculus of several variables, including integral representation of functions and parameter-dependent integrals; Fourier series and their application to geological engineering problems; Ordinary differential equations and the basic algorithms used to solve them numerically (Euler's method); Existence and uniqueness of solutions, stability * Consolidate own reasoning methodology of applied mathematics and mathematical proof of the theorems and results. * To acquire advanced knowledge of integral calculus of functions of real variable and their applications. * To acquire advanced knowledge of differential and integral calculus of functions vector of a variable and its applications. * To acquire basic knowledge of ordinary differential equations, formulation, interpretation and resolution of basic problems in engineering applications. * To acquire advanced knowledge of differential calculus of functions of several variables and vector their applications. * To acquire advanced knowledge of integral calculus of real functions of several variables and its applications. * Working the concept of mathematical modeling applied to engineering. Study load Total learning time: 187h 30m Hours large group: 44h 23.47% Hours medium group: 24h 12.80% Hours small group: 7h 3.73% Guided activities: 7h 30m 4.00% Self study: 105h 56.00% 2 / 5

3 Content 1. DIFFERENTIAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES Learning time: 57h 35m Theory classes: 12h Practical classes: 9h Laboratory classes: 3h Self study : 33h 35m 1.1 n-dimensional Euclidean space. Basic Topology 1.2 Functions of several variables. Limits. Continuous functions 1.3 Differentiability. Differentiable functions 1.4 The inverse function theorem. The implicit function theorem 1.5 Approximation of functions by polynomials. Taylor formula 1.6 Relative maximum and minimum values of real functions of several variables. Optimization 1.7 Exercises Item Topic 1 Lab 2. INTEGRAL CALCULUS OF REAL FUNCTIONS OF REAL VARIABLE Learning time: 38h 24m Theory classes: 8h Practical classes: 7h Laboratory classes: 1h Self study : 22h 24m 2.1 Antiderivatives of a function 2.2 Integral of a function. Properties 2.3 Numerical integration 2.4 Applications of integrals 2.5 Fourier Series. Applications 2.6 Generalization of the integral: improper integrals 2.7 Exercises item Laboratory item 2 3. LINE INTEGRALS. DOUBLE INTEGRALS. TRIPLE INTEGRALS Learning time: 48h Theory classes: 9h Practical classes: 8h Laboratory classes: 3h Self study : 28h 3.1 Line integrals 3.2 Double integrals 3.3 Triple integrals 3.4 Item 3 Exercises 3.5 Laboratory issue 3 3 / 5

4 4. INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS Learning time: 36h Theory classes: 5h Practical classes: 5h Laboratory classes: 5h Self study : 21h 4.1 Families of curves in the plane. First order differential equations 4.2 Linear equations of first order. Reducible linear equations 4.3 Applications of first order ODE. Mathematical models 4.4 Second order linear differential equations with constant coefficients and 4.5 Exercices Item Laboratory Item 4 Qualification system EP1: 1 exam; theme 1 EP2: 2 exam; items 2, 3 EF: 3 exam 3; item 4 and items 1,2,3 recovery issues ER: reappraisal review; under academic regulations QF = 0.35 * EP * EP * EP3. Exercises are proposed for improving the qualification of the subject. Criteria for re-evaluation qualification and eligibility: Students that failed the ordinary evaluation and have regularly attended all evaluation tests will have the opportunity of carrying out a re-evaluation test during the period specified in the academic calendar. Students who have already passed the test or were qualified as non-attending will not be admitted to the re-evaluation test. The maximum mark for the re-evaluation exam will be five over ten (5.0). The nonattendance of a student to the re-evaluation test, in the date specified will not grant access to further re-evaluation tests. Students unable to attend any of the continuous assessment tests due to certifiable force majeure will be ensured extraordinary evaluation periods. These tests must be authorized by the corresponding Head of Studies, at the request of the professor responsible for the course, and will be carried out within the corresponding academic period. Regulations for carrying out activities The evaluation will be obtained only as a result of the weighted continuous assessment tests and test synthesis. If you can not make any of these cause documental evidence must request explicit authorization to submit to extraordinary final exam. 4 / 5

5 Bibliography Basic: Zill, D.G. Ecuaciones diferenciales con aplicaciones de modelado. 9a ed. México, D.F.: International Thomson, ISBN Estela, M.R.; Saà, J. Cálculo con soporte interactivo en moodle. Madrid: Pearson Educación, ISBN Jarauta, E. Anàlisi matemàtica d'una variable. Fonaments i aplicacions. Barcelona: Edicions UPC, ISBN Complementary: Burgos, J.R. Cálculo Infinitesimal de varias variables. 2a ed. Madrid [etc.]: McGraw-Hill, ISBN Simmons, G.F.. Cálculo y geometría Analítica. 2a ed. Madrid: McGraw-Hill, ISBN / 5