Calculus II

It is recommended some knowledge of subjects of year 1 first semester mathematics (Mathematical Analysis I and Linear Algebra).

By acquiring knowledge of Mathematical Analysis II (AM2), the student will develop among others the competencies of abstraction, deduction, demonstration, algorithmic and programming of methods associated with the subjects of the AM2 with a computational component.

It is intended that the student realizes the importance of Mathematics and its structuring role as a basic science and tool to support a logical and structured reasoning indispensable to the areas of Computer Engineering. Also, show that it is impossible to solve and properly code a poorly formulated problem, that has ambiguity in its terms or for which it is not possible to collect the necessary data.

In addition to classes (theoretical, theoretical-practice and laboratory practices), where students are constantly called to participation and thus are not at all and only exhibition classes, students are also encouraged and even evaluated regarding their activity in distance participation through e-learning platforms, thus AM2 includes in itself a component of b-learning.

Throughout the semester, learning and evaluation activities (programming and other work) are proposed. In the evaluation tests some questions will be programming on the subjects of the AM2 program.

It is intended that the student realizes the importance of Mathematics and its structuring role as a basic science and tool to support a logical and structured reasoning that is indispensable in the areas of Engineering. Also, show that it is impossible to solve and properly code a poorly formulated problem, that has ambiguity in its terms or for which it is not possible to collect the necessary data.

Differential equations (ED) are essential tools in the mathematical modeling of many engineering problems. Some will be studied analytical, numerically and computationally approached in parallel by teaching programming language of Matlab.

The student, will develop abstraction, demonstration, application, algorithmic and programming skills associated with the topics of the curricular unit and other related subjects: Differential Equations (ED), Systems of Differential Equations (SED); Mathematical modeling of engineering problems translated by ED and SED; Numerical methods for ED, SED, Polynomial Interpolation, Derivatives and Numerical Integration; Differential Calculation in IR^n; Multiple Integrals; 3D Representation and Visualization, Coordinate Systems: Cartesian, polar, cylindrical and spherical; Basic and fundamental programming in Matlab and Programming of Graphical Interfaces – App in Matlab.

1. Ordinary Differential Equations (EDO)
Definition, properties, directional fields and graphs of general and particular solutions. First order Differential Equations (ED): ED of Separable Variables, firt order Linear ED and others to be solved computationally through functions of type dsolve (differential solve). Order n Linear Differential Equations with constant coefficients: Homogeneous and Complete Equations; Wronskian, Fundamental Solution System, Characteristic Equation and Method of variation of arbitrary constants. Differential equations as mathematical models of description/modeling of systems or physical phenomena. Initial value issues. Computational treatment in CAS and Matlab programs.

2. Linear Differential Equation Systems
Linear ED reduction algorithm of order 2 for a 2-ED system of 1st order. Application problems (dynamic systems), mathematical modeling, analytical and numerical resolution.

3. Differential Calculus in IR^n
Brief topological notions in IRn. Functions of several variables: domains, boundaries, continuity, partial derivatives, straight lines and tangent planes, increments and differentials, composite function derivative, directional derivative, gradient, simple and conditioned extremes. Representation of level curves and domains in 2D, 3D visualization of simple and composite surfaces and their computational treatment in CAS programs and Matlab.

4. Multiple Integrals
Double and triple integral. Coordinate systems: cartesian, polar, cylindrical and spherical. Applications: calculation of areas, volumes of solids and center of mass. Representation and visualization of solids and their computational treatment using CAS programs and Matlab.

5. Numerical Methods
5.1 Polynomial Interpolation: Polynomials. Definition of polynomial interpolator. Newton interpolator of divided differences. Linear, quadratic and cubic interpolation (Spline). Extrapolation. Two-dimensional interpolation. Adjustment of curves: Linear approximation (Least Squares Method); Polynomial approach (polyfit).
5.2 Numerical methods for EDO and EDO Systems: Euler’s methods and Runge-Kutta’s methods.
5.3 Numerical derivation: Finite differences formulas in 2 and 3 points: progressive, regressive and centered.
5.4 Numerical integration: trapezoids’ rule and Simpson’s rule applied to double integrals.
6. Programming in Matlab

Throughout the semester essential and fundamental training will be given in Matlab programming. Exploration and use of Matlab’s Symbolic Math Toolbox. Programming numerical methods. Graphical interfaces in Matlab – Apps in Matlab. Creation of applications for automatic resolution of derivatives and integrals and resolution of applied problems mathematically modeled by ED or ED Systems.

NAO

– Zill, D. A first course in differential equations with modeling applications. Thomson Learning. ISBN 0-534-37999-0. [3-11-53 (ISEC) – 11973]
– Burden, Richard L. &  J. Douglas. Numerical Analysis. Pws-Kent Publishing Company. ISBN 0-53491-585-X. [3-4-119 (ISEC) – 05133]
– Stephen J. Chapman (2019). Matlab programming for Engineers. 6th ed, Boston [etc.]. ISBN 978-0-357-03039-4. [1A-1-455 (ISEC) – 18818]
– Ross, S. Diffential Equations, McGraw Hill. ISBN 0-471-81450-4. [3-11-6 (ISEC) – 05176]
– I.E. Kreyszig. Advanced Engineering Mathematics. 7a. Edic., J. Wiley. ISBN 0-471-85824-2. [3-7-22 (ISEC) – 05590]
– Moler, Cleve. Numerical Computing with MATLAB, Mathworks. ISBN 0-89871-560-1. [3-4-23 (ISEC) – 13278]
– Grossman, Stanley I. Calculus. Sauders College Publishing. ISBN 0-03-096420-2. [3-2-183 (ISEC) – 08073]
– Stewart, J. Cálculo. Pioneira. Thomson Learning. ISBN 85-221-0236-8. [3-2-118 (ISEC)  – 11713]

Complementary bibliography
– Fausett, Lauren V. Apllied Numerical Analysis Using Matlab, Prentice Hall
– Simmons, George F. Krantz, Steven G., Equações Diferenciais. Mc Graw Hill
– Correia, Arménio A. S. (2014). Sebenta de Análise Matemática II. ISEC.
– Gouveia, M.L & Rosa. P. (2006). Apontamentos das Aulas Teóricas – Capítulo 1 EDO. ISEC.