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Advanced mathematics for engineers

Description

This is a course of mathematical methods for beginning graduate and senior engineering students. The topics presented in this course pretend to be and appetizer for the student, allowing her to self-study engineering systems that involve mathematical models not covered in basic engineering subjects.

Objectives

At the end of the course the student should be able to use mathematical models to solve engineering problems. In particular, the students should

  • Understand different vector operators in several coordinate systems.

  • Solve second order ordinary differential equations.

  • Apply mathematical methods to solve important boundary value problems: Laplace, Poisson, Wave and Heat equations.

  • Identify types of equations and decide the method to solve it.

  • Identify the origin of some special functions and understand some of their property.

Methodology

Lectures, Examples, recommended reading. The course is divided in 5 units, emphasizing the concepts. For this reason is suggested the use of a Computer Algebra System (CAS) such as Maple, Maxima or SymPy. The instructor will show examples using SymPy. There will be assignments for each unit that will help to reinforce the understanding of the topics.

Contents

Linear Algebra Review (1 week)

It is suggested to watch the series of videos "Essence of Linear Algebra".

  1. Linear transformations

  2. Vector spaces and bases

  3. Eigenvalue and Eigenvector problems

Vector calculus and coordinates (2 weeks)

  1. Vectors and tensors

  2. Coordinate Systems

  3. Line, surface and volume differentials

  4. Differential operators

  5. Vector identities

  6. Integral theorems

Ordinary differential equations (5 weeks)

  1. First order differential equations

  2. Systems of differential equations

  3. Power series solutions

  4. Frobenius method

  5. Laplace transform method

  6. Qualitative methods for non-linear systems

Orthogonal bases and Fourier analysis (3 weeks)

  1. Discrete bases

  2. Continuous bases

  3. Fourier Series

  4. Fourier Integrals

Partial differential equations (5 weeks)

  1. Classification of partial differential equations

  2. Common equations

    1. Poisson equation

    2. Diffusion equation

    3. Wave equation

    4. Navier-Cauchy equation

  3. Separation of variables

    1. Sturm-Liouville problems
    2. Bessel functions

  4. Ritz method

  5. Weighted residual methods

Textbook

The textbookd for the course are “Física Matemática” by Alonso Sepúlveda, and “Advanced Engineering Mathematics” by Erwin Kreyszig.

Evaluation

  • Assignments 30%

  • 2 Midterms 40%

  • Final project 30%

Pre-requisites

  • Linear Algebra.

  • Differential equations.

  • Vector calculus.

References

  1. SymPy Development Team. Sympy’s documentation., 2016.

  2. Grant Sanderson. Essence of linear algebra., 2016.

  3. Erwin Kreyszig. Advanced engineering mathematics. John Wiley & Sons, 2010.

  4. Antonio Velasco and Ruben Sánchez. Curso Básico de Álgebra Lineal (Spanish). Comex, 1980.

  5. Alonso Sepulveda Soto. Fı́sica matemática (Spanish). Ciencia y Tecnologı́a. Universidad de Antioquia, 2009.

  6. FWJ Olver, DW Lozier, RF Boisvert, and CW Clark. NIST digital library of mathematical functions.. NIST, 2010.

  7. Louis Leithold. The calculus. New York, USA: Harper and Row Publishers, 7 edition, 1995.

  8. H. Hochstadt. Differential equations: a modern approach. Courier Dover Publications, 1975.

  9. Stanley J Farlow. Partial differential equations for scientists and engineers. Courier Corporation, 2012.