Base Knowledge
Elementary mathematics, differencial calculus and integral calculus.
Teaching Methodologies
Expository method will be used to explain theoretical subjects and to analyse and solved problems. On theoretical classes, theoretical subjects and application examples will be presented. On practical classes, exercises will be discussed, solved and analysed. On laboratory classes, exercises will be discussed and analysed, using mathematical software.
Emphasis will be placed on the critical analysis of the selection criteria for mathematical techniques and consequent results.
In Moodle and InforEstudante platforms, will be available all the necessary documents for this course unit: textbook, exercise book, forms, math tables, educational videos, exams from past years and online tests.
Learning Results
– Solve, analytically, order n linear ordinary differential equations and apply it to solve problems in the Electrical Engineering context;
– Perform the Laplace transform of continuous and piecewise continuous functions;
– Analyse the converge of a numerical series, determine the interval of converge of a power series and its summation and determine Fourier series and its summation;
– Perform differential and integral calculus for real-valued function of n and apply it solve problems in the in the engineering context.
Program
1. Linear homogeneous differential equations of order n and constant coefficients.
1.1 Linear homogeneous differential equations of order n and constant coefficients.
– System of fundamental solutions. Wronskian.
– General solution.
– The differential operator. Characteristic equation.
1.2 Linear complete differential equations of order n and constant coefficients.
– General integral.
– Undetermined coefficients method.
– Variation of constants method.
2. Laplace transform.
2.1 Introduction.
– Definition.
– Sufficient conditions for the existence of Laplace transform.
2.2 Properties of Laplace Transform.
– Linearity.
– Differentiation in time domain.
– Shifting in s-domain.
– Differentiation and integration in s-domain.
– Integration in time domain.
– Convolution of functions. Using convolution for Laplace transform.
– Laplace transform of periodic function.
– Heaviside step function. Time shifting.
2.3 Table of Laplace transform.
2.4 Inverse Laplace transform.
2.5 Applications to linear differential equations of order n and constant coefficients.
3. Series.
3.1 Numerical series.
– Definition. Sequence of partial sums. Convergence.
– Geometric and Mengoli series.
– Necessary condition for converge of a series.
– Integral test.
– Dirichlet series.
– Direct comparison test and limit comparison test.
– D’Alembert and Cauchy criteria.
– Absolutely convergent and conditional convergence series.
– Alternating series. Leibniz criterion.
3.2 Power series.
– Definition. Interval and radius of converge.
– Representation of functions using power series.
– Differentiation and integration of power series.
– Taylor and MacLaurin series.
– Function expansions using power series. Uniqueness of power series expansions.
3.3 Fourier series.
– Fourier’s theorem.
– Even and odd functions. Sine and cosine series.
4. Real functions of several variables and their derivatives
4.1
– Conics and quadric surfaces.
– Topology concepts.
4.2 Real functions of several real variables.
– Domain.
– Graph.
– Contours.
– Limits.
– Continuity.
– Partial derivatives.
– Schwarz theorem.
– Chain rule.
– Directional derivative.
– Gradient vector and its applications;
– Free and conditional extrema.
5. Multiple integrals
5.1Double integrals.
– Definition and geometric interpretation.
– Calculus of a double integral in cartesian and polar coordinates.
– Applications.
5.2 Triple integrals.
– Definition and geometric interpretation.
– Calculus of a triple integral in Cartesian, cylindrical and spherical coordinates.
– Applications.
Curricular Unit Teachers
Internship(s)
NAO
Bibliography
– Branco, J.R. (2022). Applied Mathematics to Electronics (new edition), Coimbra: Coimbra Institute of Engineering;
– Branco, J.R. (2022). Applied Mathematics to Electronics – Exercise book, Coimbra: Coimbra Institute of Engineering;
– Kreyszig, E. (1999). Advanced Engineering Mathematics (8th ed), New York: John Wiley & Sons (ISEC’s library: 3-7-22);
– Ross, S. (1984). Differential Equations (3rd ed), New York: John Wiley (ISEC’s library 3-11-6, 3-11-7);
– Zill, D.G. (2001). A first course in differential equations with modelling applications (7th ed), Pacic Grove, CA: Brooks/Cole (ISEC’s library: 3-11-53);
– Zill, D.G. & Cullen, M.R. (2009). Differential equations with boundary-value problem (7th ed), Australia: Brooks/Cole Cengage Learning (ISEC’s library: 3-11-68).