Subject: Calculus III

1 - The primary objective of this course unit is to advance basic Calculus to a higher level by introducing two subjects: Differential Equations and Complex Analysis. To achieve this goal, the theory of Series is also taught, given its impact on both of these added disciplines. Consequently, the other objectives are:
2 - Study the fundamentals of the theory of Series, along with their applications to concrete problems encountered in various fields of Engineering.
3 - Study the foundations of the theory of Differential Equations, including theoretical applications that enable the development of research and scientific communication skills across various engineering fields.
4 - Compare and understand the extension of results from Real-variable Calculus to Complex-variable Calculus, examining in detail the concepts of limit, continuity, differentiability, and integrability within the complex domain.

1 - Numerical sequences and series 1.1 Sequences: definitions and theorems toward limit calculations 1.2 Series: basic concepts of series in engineering. Geometric series 1.3 General convergence criterion 1.4 Series with non-negative terms: convergence criteria 1.5 Harmonic series of order p (or Dirichlet series) 1.6 Series of absolute values: Cauchy and d'Alembert criteria 1.7 Alternating series: Leibniz criterion
2 - Series of functions 2.1 Power series 2.2 Radius and interval of convergence 2.3 Representation of functions as power series 2.4 Differentiation and integration of power series 2.5 Polynomial approximation of functions. Taylor's formula 2.6 Taylor and MacLaurin series
3 - Modeling physical phenomena with differential equations 3.1 Definition and classification of differential equations (DE) 3.2 Directly integrable equations 3.3 Solving equations using indefinite integrals 3.4 Solving equations using definite integrals
4 - Introduction to the study of differential equations 4.1 Differential equation for an exponential function 4.2 General solution and initial value problem (IVP) 4.3 Separable differential equations 4.4 Geometric interpretation of a differential equation 4.5 Differential equations involving homogeneous functions
5 - Exact differential equations (EDE) 5.1 Definitions 5.2 Integrating factor 5.3 Initial value problem 5.4 Solving EDEs using definite integrals 5.5 Applications of EDEs to real-valued functions of two variables
6 - First-order linear differential equations 6.1 Classification of ordinary differential equations (ODEs) 6.2 Separation of variables method for a homogeneous equation 6.3 Initial value problem for a non-homogeneous ODE 6.4 Integrating factor method for a non-homogeneous equation 6.5 Variation of parameters method for a non-homogeneous equation 6.6 Differential equations with piecewise continuous functions
7 - Second-order linear differential equations 7.1 Complex number algebra 7.2 Homogeneous equations with constant coefficients 7.3 The Wronskian and Abel's theorem 7.4 Non-homogeneous equations: method of undetermined coefficients
8 - Systems of first-order differential equations 8.1 General concept of a system of differential equations 8.2 Transformation of a DE into a system of first-order DEs 8.3 Transformation of a system of first-order DEs into a DE 8.4 Review of matrices, eigenvalues, and eigenvectors 8.5 Homogeneous systems of equations with constant coefficients 8.6 Complex and repeated eigenvalues 8.7 Non-homogeneous systems of first-order linear equations
9 - Concept of partial differential equations 9.1 General notion of partial differential equations (PDEs) 9.2 Solution of a first-order PDE with constant coefficients 9.3 Solution of a linear second-order PDE with constant coefficients 9.4 Equations involving only second-order partial derivatives 9.5 Method of separation of variables. Superposition of solutions.
10 - Complex numbers 10.1 Essential algebra of complex numbers 10.2 Geometric representation of complex numbers. Polar coordinates 10.3 Euler's formula 10.4 Exponential form of complex numbers 10.5 Products and powers in exponential form 10.6 Arguments of products and quotients 10.7 Roots of complex numbers
11 - Specific engineering considerations on complex numbers 11.1 Inverse Euler's formula. De Moivre's formula 11.2 Function of a complex argument 11.3 Logarithmic function
12 - General study of complex analysis 12.1 Limits and continuity 12.2 Differentiable, analytic, and entire functions 12.3 Cauchy-Riemann equations 12.4 Harmonic function
13 - Integration in the complex plane 13.1 Parametrization of curves 13.2 General calculation of line integrals 13.3 Power theorem 13.4 Cauchy's theorem 13.5 Simply connected regions 13.6 Cauchy's integral formulas
14 - Series, singularities, and residues in the complex plane 14.1 Taylor and MacLaurin series 14.2 Basic power series 14.3 Radius of convergence and operations with series 14.4 Laurent series and singularities 14.5 Residue theorem and its applications

Boyce W.E., DiPrima R.C., Meade D.B., , 2017 , Boyce's Elementary Differential Equations and Boundary Value Problems, Global Edition , John Wiley & Sons
Zill D.G., , 2013 , A First Course in Differential Equations: With Modeling Applications, 10th ed., , Brooks/Cole, USA: Cengage Learning
Nagle R.K., Saff E.B., Snider A.D., , 2011 , Fundamentals of Differential Equations and Boundary Value Problems, 6th ed., , Pearson Addison Wesley,
Simmons G.F., , 2017. , Differential Equations with Applications and Historical Notes, 3rd ed., , USA: CRC Press
Howell K.B., , 2020 , Ordinary Differential Equations: An Introduction to the Fundamentals, 2nd ed., , USA: CRC Press,
Brown J.W., Churchill R.V., , 2014 , Complex Variables and Applications, 9th ed., , McGraw-Hill
Ablowitz M.J., Fokas, A.S., , 2021 , Introduction to Complex Variables and Applications , Cambridge University Press
Stewart J , 2016 , Calculus: Early Transcendentals, 8th ed , Brooks/Cole: Cengage Learning

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Classes consist of both theoretical and theory-practice parts. The theoretical component employs an expository approach, accompanied by problems that link the tools being developed with concepts studied in specialized curricular units. Theory-practice classes involve the oral and written presentation of the curriculum content, encouraging students to apply and consolidate their theoretical knowledge by solving assigned exercises. Combining theoretical and theory-practice classes ensures rigorous and comprehensive coverage of the syllabus topics. These topics are introduced in response to practical engineering situations and problems, thereby enhancing student motivation and fostering a better understanding of concepts and results. Solving exercises during class illustrates the practical application of the concepts and tools studied while simultaneously deepening theoretical knowledge. Assessment is based on two midterm tests (each weighted at 50%), to be completed individually during the regular assessment period. This allows students to monitor their performance more effectively throughout the semester. However, the theoretical/theory-practice component, assessed via the two midterm tests during the regular period, can be retaken during the Supplementary and Special assessment periods through a Comprehensive Final Exam covering the entire syllabus. Nevertheless, each instructor responsible for teaching and assessment may define an alternative assessment method, provided that it fully complies with the University of Madeira´s Student Learning Assessment Regulation currently in force.