Problems of Advanced Engineering Mathematics

Sinopse

This book of problems, designed as a practical tool for engineering students, is based on the author’s experience of about 40 years teaching Mathematical Analysis at the University of Porto, Faculty of Engineering. It includes three chapters focused on differential equations, which have wide applications in the field of engineering, as well as a chapter introducing Laplace transforms and its application to ordinary differential equations. The final three chapters cover problems of line and surface integrals, as well as Fourier series.

Índice

INTRODUCTION

CHAPTER 1
FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS
1.1 Separable differential equations
1.2 Homogeneous differential equations
1.2.1 Equations reducible to homogeneous equations
1.3 Orthogonal trajectories
1.4 Exact differential equations. Integrating factor
1.4.1 Integrating factor
1.5 Linear differential equations
1.5.1 Bernoulli equation
1.5.2 Riccati equation
1.6 Lagrange and Clairaut equations
1.6.1 Lagrange equation
1.6.2 Clairaut equation

CHAPTER 2
HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS
2.1 Reducing the order of differential equations
2.2 Linear differential equations of order n
2.2.1 Solutions of the homogeneous and non-homogeneous equations. Main theorems
2.2.2 Homogeneous linear differential equations with constant coefficients
2.2.3 Non-homogeneous linear differential equations with constant coefficients
2.3 Euler equations

CHAPTER 3
SYSTEMS OF LINEAR DIFERENTIAL EQUATIONS
3.2 Systems of homogeneous linear ordinary differential equations with constant coefficients. Euler´s method
3.2 Systems of non-homogeneous linear ordinary differential equations with constant coefficients

CHAPTER 4
LAPLACE TRANSFORM
4.1 Definition, existence, and properties of the Laplace transform
4.2 Laplace transform of the derivative
4.3 Inverse Laplace transform and application to differential equations
4.4 The shifting theorems
4.5 Laplace transform of Dirac Delta function
4.6 Laplace transform of the integral
4.7 The differentiation and integration of transforms
4.8 Convolution theorem

CHAPTER 5
LINE INTEGRAL
5.1 Definition of line integral
5.2 General properties of the line integral
5.3 Fundamental theorems for line integrals
5.4 Green´s theorem
5.5 Line Integral with respect to arc-length

CHAPTER 6
SURFACE INTEGRAL
2.1 Fundamental vector product and area of a surface
2.2 Applications of surface integral to mass geometry
2.3 Flux across a surface
2.4 Gauss’ theorem and Stokes’ theorem
2.5 Pappus’ theorem

CHAPTER 7
FOURIER SERIES
7.1 Fourier series convergence
7.2 Euler Formulas
7.3 Fourier series for even or odd functions
7.4 Alternative form of the Fourier series
7.5 Trigonometric polynomial and quadratic error
7.6 Partial differential equations by the method of separation of variables and Fourier series
7.6.1 Vibrating string equation
7.6.2 Integration of the vibrating string equation by the method of separation of variables

BIBLIOGRAPHY

Sobre o Autor

Luísa Madureira, was born at Porto and graduated in Mathematics in 1984 at University of Porto. She is a Professor in the Mechanical Engineering Department at the University of Porto where she teaches Mathematical Analysis courses to students in Mechanical Engineering, Industrial and Management Engineering and Computing Science Engineering. She completed her PhD in Mechanical Engineering in 1996 and has published several papers in international journals and also two student-oriented books, Problemas de Equações Diferenciais Ordinárias e Transformads de Laplace and Problemas de Análise Matemática para Engenharia.