Matemática Aplicada à Eletrotecnia

Elementary mathematics, differencial calculus and integral calculus.

In the current academic year, no classes of any type will be taught.

– 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.

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.

The assessment may be done in two ways:

• Final assessment:

Final exam, quoted for 20 values. Exam may be done at regular season and at appeal season. Student will be approved if the exam result, rounded, is greater or equal than 10 values.

• Distributed assessment:

Two tests, each one quoted for 10 values:

1. Test 1 will focus on chapters 1 and 2;
2. Test 2 will focus on chapters 3, 4 and 5.

Final result will be given by the summation of the grades of both tests. Student will be approved if final result, rounded, is greater or equal than 10 values and, at least, 30% on each test.

NAO