. "Finite element method and Its applications"@en . . "6" . "Modern science has many approaches to creating a quantitative mathematical model for any system. One of them is considered the finite element method, which is based on the creation of behavioural differences (infinitely) of its elements, based on the assumption of the relationship between the main elements that are able to provide a complete description of this system.\n\nOutcome:\nAble to solve plane and axially symmetrical tasks of thermal conductivity and the ones of theory of elasticity with the help of ELCUT. - Practical work. Control work\r\nAble to solve plane tasks of theory of elasticity and the ones of fracture mechanics with the help of FRANC. - Practical work. Control work\r\nAble to execute an analysis of strength of a specific 3D object with the help of Mechanical Desktop. - Practical work. Control work\r\nTo be proficient in general questions of theory of the finite element method. - Exam." . . "Presential"@en . "TRUE" . . "Calculus I"@en . . "6" . "no data" . . "Presential"@en . "TRUE" . . "Calculus II"@en . . "9" . "no data" . . "Presential"@en . "TRUE" . . "Calculus II and complements of calculus"@en . . "12" . "Description is not available" . . "Presential"@en . "TRUE" . . "Variational calculus"@en . . "6" . "Learning outcomes\n\n\nThe course is aimed at providing fundamental notions on the optimization problems, by defining particularly the general problem of Lagrange and those of Maier and Bolza. At the end of the course the students must be able to apply the basic knowledge learned on variational calculus to structural optimization problems." . . "Presential"@en . "FALSE" . . "Calculus I (fundamentals)"@en . . "6" . "Obligatory base module 1 \nLearning outcomes\nAfter passing this course the student:\n1. knows definitions of matrix, inverse matrix; is able to multiplicate matrices and find inverse matrix;\n2. knows properties of determinants, is able to evaluate determinants;\n3. is able to solve systems of linear equations;\n4. is able to find vectors dot product, cross product in space and knows their applications;\n5. is able to construct equation formulas for lines and planes in space, find their position;\n6. knows the definition of the function and general classes of functions and their graphs;\n7. is familiar with the main properties of the limit, is able to find liimits; knows, what is continous function;\n8. is familiar with the notions of the derivative, the differential; is able to find derivatives and differentials; knows the most important applications of derivatives;\n9. knows the l'Hospital rule and knows how to use it;\n10. knows how to investigate functions, when they are decreasing/increasing;\n11. knows, what is antiderivates and how to find them in simpler case;\n12. knows, what is definite integral and how to calculate it;\n13. knows applications of definite integral;\n14. knows, what are improper integrals with infinite limits of integration and how to find them;\n15. chooses suitable mathematical conceptions (formulae) for problems, and applies these conceptions while solving problems.\nBrief description of content\nLinearalgebra. Vectors in space, their dot and vector product. Lines and planes in space. Functions, limits and continuity. Derivatives and applications of derivatives. Antiderivatives. Definite integrals and their applications, improper integral with infinite limits of integration." . . "Presential"@en . "TRUE" . . "Calculus in the mathematical and"@en . . "no data" . "no data" . . "Presential"@en . "TRUE" . . "Calculus of several variables"@en . . "no data" . "no data" . . "Presential"@en . "FALSE" . . "Differential calculus"@en . . "no data" . "no data" . . "Presential"@en . "FALSE" . . "Partial differential equations"@en . . "4" . "First order linear and quasilinear PDEs.\n\nSecond order linear PDEs, classiicaion.\n\nParabolic and hyperbolic iniial value problems.\n\nHilbert spaces, Fourier series, linear operators.\n\nEllipic boundary value problems.\n\nEigenvalue problems, separaion of variables. Fourier series expansion of the soluion.\n\nGreen’s funcion, spherical funcions.\n\nParabolic and hyperbolic iniial- boundary value problems.\n\nFourier transform, wavelets." . . "Presential"@en . "TRUE" . . "Calculus"@en . . "7" . "1. to convey and reinforce the knowledge on real number sequences, functions of one variable, the\n constant e, one-variable differential and integral calculus, definite and improper integrals, and their\n application,\n 2. to acquire thorough understanding of basic concepts and computational processes, and to master skills of using them, 3. to acquire the skill of correct mathematical reasoning and inference. After completing his course the students will be able to: \n 1. establish the convergence of sequences and evaluate limits of basic types of sequences;\n 2. establish the limits of functions and known basic types of functions;\n 3. evaluate derivatives of elementary functions, know basic rules of differentiation and apply derivatives\n in evaluations approximate values of expressions, tangent lines, finding the limits of undetermined\n expressions, finding local extrema of a function and drawing it’s graph;\n 4. calculate the indefinite integrals of elementary functions." . . "Presential"@en . "TRUE" . . "Calculus 2"@en . . "5" . "no data" . . "Presential"@en . "TRUE" . . "Calculus 3"@en . . "3" . "no data" . . "Presential"@en . "TRUE" . . "Calculus"@en . . "no data" . "no data" . . "no data"@en . "TRUE" . . "Adjustment calculus"@en . . "2" . "Estimation theory. Types of errors and uncertainties. Geodetic network as a series of points and measurements. Elements of algorithms. Solving of numerical problems with use of Matlab/Octave programming environment." . . "Presential"@en . "TRUE" . . "Adjustment calculus 2"@en . . "3" . "Methods of adjustment and uncertainty estimation. Parametric and conditional methods. Linearization on non linear problems. Mixed methods of adjustment. Adjustment of correlated measurements.\r\nAdjustment of classic geodetic levelling and planar networks. Solving numerical problems with use of Matlab/Octave enivronment." . . "Presential"@en . "TRUE" . . "Finite element method 1"@en . . "4" . "no data" . . "Presential"@en . "TRUE" . . "Finite element method 2"@en . . "2" . "no data" . . "Presential"@en . "TRUE" . . "Calculus"@en . . "6" . "Objectives and Contextualisation\nReach a sufficient level in the calculation of a variable to deal with phenomena and solve the mathematical problems raised in engineering that can be described in these terms.\n\nSupport the parts of the other subjects of the degree that require mastery of real functions of a variable. Achieve a sufficient level in the use of complex numbers and above all in trigonometry.\n\n\nCompetences\nElectronic Engineering for Telecommunication\nCommunication\nDevelop personal attitude.\nDevelop personal work habits.\nDevelop thinking habits.\nLearn new methods and technologies, building on basic technological knowledge, to be able to adapt to new situations.\nWork in a team.\nTelecommunication Systems Engineering\nCommunication\nDevelop personal attitude.\nDevelop personal work habits.\nDevelop thinking habits.\nLearn new methods and technologies, building on basic technological knowledge, to be able to adapt to new situations.\nWork in a team.\nLearning Outcomes\nApply, in the problems that arise in engineering, knowledge about linear algebra, geometry, differential geometry, differential and integral calculus, differential and partial derivative equations, numerical methods, numerical algorithms, statistics and optimisation.\nApply, to the problems that arise in engineering, knowledge of linear algebra, geometry, differential geometry, differential and integral calculus, differential and partial derivative equations, numerical methods, numerical algorithms, statistics and optimisation.\nCommunicate efficiently, orally and in writing, knowledge, results and skills, both professionally and to non-expert audiences.\nDevelop curiosity and creativity.\nDevelop scientific thinking.\nDevelop the capacity for analysis and synthesis.\nManage available time and resources.\nManage available time and resources. Work in an organised manner.\nPrevent and solve problems.\nResolve the mathematical problems that can arise in engineering.\nWork autonomously.\nWork cooperatively.\nWork in an organised manner.\n\nContent\n1. Complex numbers.\n\n1.1 Trigonometric functions. Addition formulae. Identities. Trigonometric inverse functions.\n\n1.2 Trigonometric equations.\n\n1.3 Complex numbers. Sum, product and the invers. Square roots. Second degree equations.\n\n1.4 Module and argument. Euler's formula.\n\n1.5 Polynomials, roots and factorization. Fundamental theorem of Algebra.\n\n2. Continuity\n\n2.1 Continuity and limits.\n\n2.2. Fundamental theorems of continuous functions. Exponential and logarithmic functions.\n\n3. Differential calculus.\n\n3.1 Derivatives of functions. Algebraic rules of derivation. Chain rule. Derived of the inverse.\n\n3.2 Mean value theorem and consequences. Intervals of monotony.\n\n3.3 Relative and absolute extremes. Optimization.\n\n3.4 Calculation of limits using derivation.\n\n3.5 Taylor's formula.\n\n4. Integral Calculus.\n\n4.1 Notion of Riemann integral.\n\n4.2 Fundamental Theorem of Calculus. Barrow's theorem.\n\n4.3 Calculation of primitives.\n\n4.4 Applications of integrals (part in seminars).\n\n5. Differential equations.\n\n5.1 Notion of differential equation.\n\n5.2 Solving the equations of separate variables.\n\n5.3 First order linear equations.\n\n5.4 Second order linear with constant coefficients.\n\n5.5 Examples of applications of the differential equations." . . "Presential"@en . "TRUE" . . "Partial differential equations a"@en . . "3.00" . "Course Contents I: (Wi3150TU) Introduction. Types of second order equations. Initial and initial boundary value problems. Fourier series. Quasilinear, first order partial differential equations. Waves and reflections of waves. Separation of variables. Sturm-Liouville\nproblems. Parabolic, elliptic and hyperbolic equations. Maximum principle. Diffusion and heat transport problems. Lectures (3\nECTS).\nII: (Wi3151TU) Boundary value problems. Delta functions and distributions. Greens function for heat, wave and Laplace\nequations. Fourier and Laplace transform methods. Waves in R2 and in R3. Vibrations of membranes. Bessel functions. Shock\nwaves. Lectures and Maple practical work (3 ECTS).\nStudy Goals Many mathematical--physical problems can be formulated using partial differential equations. Therefore it is important to be\nable to both interpret and solve this type of equations. At the end of the course the student\n1- is able to formulate various physical problems (wave--equation, heat--equation, transport--equations) in terms of partial\ndifferential equations.\n2- has knowledge and understanding of various mathematical techniques which are necessary to solve these problems (Fourier--\nseries, method of separation of variables, Sturm-Liouville problems, Greens' functions, Fourier- and Laplace transformations)\nand is able to apply these techniques to (simple) problems.\n3- is able to interpret the solutions obtained and is able to place them in (a physical) context." . . "Presential"@en . "TRUE" . . "Partial differential equations b"@en . . "3.00" . "no data" . . "Presential"@en . "FALSE" . . "Non-linear differential equations"@en . . "6.00" . "no data" . . "Presential"@en . "FALSE" . . "Calculus I"@en . . "9.5" . "Description in Bulgarian" . . "Presential"@en . "TRUE" . . "Calculus II"@en . . "7.0" . "Description in Bulgarian" . . "Presential"@en . "TRUE" . . "Calculus"@en . . "20.0" . "#### Prerequisites\n\n* Normally, A level Mathematics at grade A or better and AS level Further Mathematics at grade A or better, or equivalent.\n\n#### Corequisites\n\n* Linear Algebra I (MATH1071)\n\n#### Excluded Combination of Modules\n\n* Calculus I (Maths Hons) (MATH1081), Linear Algebra I (Maths Hons) (MATH1091), Mathematics for Engineers and Scientists (MATH1551), Single Mathematics A (MATH1561), Single Mathematics B (MATH1571) may not be taken with or after this module.\n\n#### Aims\n\n* This module is designed to follow on from, and reinforce, A level mathematics.\n* It will present students with a wide range of mathematics ideas in preparation for more demanding material later.\n* Aim: to introduce crucial basic concepts and important mathematical techniques.\n\n#### Content\n\n* A range of topics are treated each at an elementary level to give a foundation of basic definitions, theorems and computational techniques.\n* A rigorous approach is expected.\n* Elementary functions of a real variable.\n* Limits, continuity, differentiation and integration.\n* Ordinary Differential Equations.\n* Taylor series and Fourier series.\n* Calculus of functions of many variables\n* Partial differential equations and method of separation of variables\n* Fourier transforms\n\n#### Learning Outcomes\n\nSubject-specific Knowledge:\n\n* By the end of the module students will: be able to solve a range of predictable or less predictable problems in Calculus,\n* have an awareness of the basic concepts of theoretical mathematics in Calculus,\n* have a broad knowledge, and a basic understanding and working knowledge of each of the subtopics,\n* have gained confidence in approaching and applying calculus to novel problems.\n\nSubject-specific Skills:\n\n* Students will have enhanced skills in the following areas: modelling, spatial awareness, abstract reasoning and numeracy.\n\nKey Skills:\n\n#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module\n\n* Lectures demonstrate what is required to be learned and the application of the theory to practical examples.\n* Tutorials provide active engagement and feedback to the learning process.\n* Weekly homework problems provide formative assessment to guide students in the development of their knowledge and skills. They also aid the development of students' awareness of the required standards of rigour.\n* Initial diagnostic testing and associated supplementary support classes fill in gaps related to the wide variety of syllabuses available at Mathematics A-level, and provide extra support to the course.\n* The examination provides a final assessment of the achievement of the student.\n\nMore details at: https://apps.dur.ac.uk/faculty.handbook/2023/UG/module/MATH1061" . . "Presential"@en . "FALSE" . . "Partial differential equations"@en . . "6.0" . "https://sigarra.up.pt/fcup/en/ucurr_geral.ficha_uc_view?pv_ocorrencia_id=502169" . . "Presential"@en . "FALSE" . . "Differential and integral calculus I"@en . . "6.0" . "Prerequisites\nNot applicable.\n\nObjectives\nMaster concepts and techniques of differentiable and integral calculus in one variable. Develop analytic thinking, creativity and innovation capacity, through the application of those concepts and techniques in different contexts.\n\nProgram\nReal numbers: algebraic, order and supremum axioms. Natural numbers and mathematical induction. Sequences: the concept of limit; applications. Real functions of one real variable: limits and continuity; elementary functions. Global properties of continuous functions: intermediate value and Weierstrass theorems. The concept of derivative. Derivatives of elementary functions. Rolle, Lagrange and Cauchy theorems. L'Hôpital's rule. Derivatives of higher order. Inverse functions. Primitives: parts, substitution, rational functions. Riemann's integral. Fundamental Theorem of Calculus. Barrow's rule. Applications: calculation of areas; definition of functions (ex.: logarithm, error and gamma functions); examples of separable differential equations of the form f(y) y’(t) = g(t). Taylor's polynomial. Numerical series. Convergence criteria. Simple and absolute convergence. Power series, convergence radius. Taylor series: definition, examples and convergence.\n\nEvaluation Methodology\nExam/tests, possibly with minimum grade, complemented with continuous evaluation components and oral evaluation for grades above 17 (out of 20).\n\nCross-Competence Component\nThe UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.\n\nLaboratorial Component\nNot applicable.\n\nProgramming and Computing Component\nNot applicable.\n\nMore information at: https://fenix.tecnico.ulisboa.pt/cursos/lerc/disciplina-curricular/845953938489997" . . "Presential"@en . "TRUE" . . "Differential and integral calculus II"@en . . "6.0" . "Prerequisites\nLinear Algebra and Differential and Integral Calculus I.\n\nObjectives\nMaster the differential and integral calculus of scalar and vector valued functions of several real variables and multiple and line integrals, including the fundamental theorems of calculus for line and double integrals, and geometric and physical applications.\n\nProgram\nBasic topological notions in R^n, sequences. Scalar and vector fields. Limits and continuity. Differentiability and gradient. Applications. Intermediate value theorem. C^k functions, Schwarz lemma. Extremal and sadle points of scalar fields. Weierstrass theorem, Taylor's formula, Hessian matrix, Lagrange multipliers. Inverse and inplicit function theorems. Applications. Multiple integrals and applications. Curves, paths and line integrals. Applications. Fundamental theorem of calculus for line integrals and applications. Greens's theorem and applications. Gradient vector fields of scalar fields.\n\nEvaluation Methodology\nExam/tests, possibly with minimum grade, complemented with continuous evaluation components and oral evaluation for grades above 17 (out of 20).\n\nCross-Competence Component\nThe UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.\n\nLaboratorial Component\nNot applicable.\n\nProgramming and Computing Component\nNot applicable.\n\nMore information at: https://fenix.tecnico.ulisboa.pt/cursos/lerc/disciplina-curricular/845953938489998" . . "Presential"@en . "TRUE" . . "Differential and integral calculus III"@en . . "6.0" . "Prerequisites\nDifferential and Integral Calculus II\n\nObjectives\nMaster of: - Resolution of elementary ordinary differential equations; resolution of linear differential equations and systems of linear differential equations. - Existence, uniqueness and continuous dependence of solutions of ordinary differential equations. - Gauss and Stokes theorems, general properties of the divergence and curl of vector fields, and applications. - Resolution of elementary linear partial differential equations of 1st and 2nd order. - General properties and convergence of Fourier series, Fourier transform and applications.\n\nProgram\nOrdinary Differential Equations (ODEs): examples of solvable 1st order ODEs, integration factors; existence, uniqueness and continuous dependence of solutions of systems of 1st order ODEs; variation of constants formula; ODEs of order > 1; Laplace transform and applications to ODEs. Gauss and Stokes Theorems and introduction to Partial Differential Equations (PDEs): surfaces in R^3; surface integrals of scalar and vector fields; Gauss and Stokes Theorems; divergence and curl of vector fields; derivation of the continuity, wave, heat, Laplace and Poisson differential equations. PDEs and Fourier series: linear 1st order PDEs; wave, heat, Laplace and Poisson equations; trigonometric Fourier series; solutions of wave, heat, Laplace and Poisson equations, via separation of variables and Fourier series; Fourier transform and applications.\n\nEvaluation Methodology\nExam/tests, possibly with minimum grade, complemented with continuous evaluation components and oral evaluation for grades above 17 (out of 20).\n\nCross-Competence Component\nThe UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.\b\n\nLaboratorial Component\nNot applicable.\n\nProgramming and Computing Component\nNot applicable.\n\n\nMore information at: https://fenix.tecnico.ulisboa.pt/cursos/lerc/disciplina-curricular/845953938489999" . . "Presential"@en . "TRUE" . . "Calculus"@en . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .