## Engineering Mathematics II

#### Instructor: Dr. Davide Batic (davide.batic@uwimona.edu.jm)

**Office:** 04 Mathematics Building

**Office hours (for Sem. I 2011/12):** Monday 1-3pm, Wednesday 1-3 pm, or by appointment.

### Course goals

The objective of this course is to enhance your mathematical and computational knowledge and ability to deploy these skills effectively in the solution of problems, principally in the area of engineering. As you can see from the course outline, we will cover a large amount of material. This should give you some warning of the fact that this will not be a course to relax at; though this does not mean that you will not be able to enjoy it. This course will make you aware about the importance and symbiosis between mathematics and applied sciences such as physics or engineering. Remeber that the fluency with the mathematical tools you achieve in this course will be an essential weapon in your future career.

### Syllabus

The following is the plan for the course. Some topics might be covered at slightly different times

**Second order linear differential equations**: power series solution, regular and singular points, Legendre equation, Bessel equation and their solutions;**Laplace transform**: improper integrals, definition of the Laplace transform, Laplace transform of elementary functions, periodic functions, convolution theorem, application to initial value problems and integral equations;**Fourier transform**: Fourier series, even and odd functions, half range expressions, applications to ordinary differential equations;**Vector calculus**: scalar and vector fields, vector calculus, curves in three-dimensional Euclidean space, arc length, curvature and torsion, directional derivatives, gradient, divergence and curl of a vector field, line integrals, surface integrals, Green's and Stoke's theorem, divergence theorem;**Partial differential equations**: classification, method of separation of variables, the one-dimensional wave equation, the heat equation, the diffusion equation, Laplace equation in spherical and cylindrical coordinates and its relation to the Legendre and Bessel equations;

### Textbooks and class lecture notes

The textbooks for this course are

- Bird, J.,
*Higher Engineering Mathematics*, Elsevier, 2006, Fifth Edition; - Hassani, S.,
*Mathematical Methods for Students of Physics and Related Fields*, Springer Verlag, 2009; - Bayin, S.,
*Mathematical Methods in Science and Engineering,*Wiley Interscience, 2006; - Byron, F. W., Fuller, R. W.,
*Mathematics of Classical and Quantum Physics,*Dover, 1992;

**Lecture notes**

Review of power series: convergent and absolutely convergent power series, tests for absolute convergence, radius of convergence, subtraction, addition, multiplication and division of power series, differentiation, Taylor series and analytic functions.

Standard form of second order linear differential equations, ordinary and singular points, power series solution of second order linear differential equations around a regular point, recurrence relation, gymnastics in shifting the index of summation.

Fuchs theorem, general considerations on the convergence radius of series solutions for the Legendre and Bessel equations around an ordinary point, elementary and special functions, the Legendre equation: solutions around x=0, Legendre polynomials, Rodrigues formula and orthogonality property.

Singular points, regular singular points, method of Frobenius, the indicial equation and the exponents at the singularity, singularities at infinity, classification of the point at infinity for the Legendre and Bessel equations.

Bessel equation of order ν, Bessel functions of fractional order, Bessel function of order zero of the first kind, Bessel function of order ν of the first kind and its asymptotic behavior for large x, Gamma function and Bessel function of arbitrary order.

Review of even and odd functions, hyperbolic sine and cosine, orthogonal families of functions, Fourier approximation, trigonometric Fourier series and coefficients formulae, piecewise continuous functions, example of a Fourier series of a piecewise continuous function.

General existence theorem for Fourier series, example of a Fourier series of a periodic piecewise continuous function, cosine and sine series.

Review of partial derivatives, classification of linear second order partial differential equations (PDEs) into elliptic, parabolic and hyperbolic PDEs; examples of parabolic equations: the one-dimensional heat equation, the Schroedinger equation, the linear transport equation and the Kolmogorov/Fokker-Planck equation.

Sturm-Liouville boundary value problem, homogeneous unmixed boundary conditions, eigenvalues and eigenfunctions.

Discussion of the boundary and initial conditions for the heat equation, solution of the heat equation by separation of variables, example.

One-dimensional wave equation, string oscillations, solution of the wave equation by separation of variables.

Laplace equation in Cartesian coordinates, spherical coordinates, Laplace equation in spherical coordinates, solution by separation of variables: radial. polar and azimuthal equations; reduction of the polar equation to a Legendre equation, application to electrostatics.

Cylindrical coordinates, Laplace equation in cylindrical coordinates, solution by separation of variables, azimuthal symmetry, application to electrostatics.

Definition of the Laplace transform, linearity, examples of Laplace transforms of elementary functions, Laplace transform of a step function, shifting formula, examples.

Laplace transform of derivatives, application of the Laplace transform to second order linear non homogeneous differential equations, inverse Laplace transform. Table of most important Laplace transforms.

Laplace transform of the Heaviside function, Laplace transform of periodic functions, applications to ODEs with discontinuous forcing functions, convolution theorem, applications to initial value problems and integral equations.

Scalar and vector fields, vector calculus, curves in three-dimensional Euclidean space, arc length, curvature and torsion.

Directional derivative and gradient of a scalar function.

Examples of vector fields, curl, divergence and gradient operators, conservative fields, line integrals and Green's theorem.

Parametric representation of the surface of a sphere, area formula of a parametric surface, surface integrals, oriented surfaces and unit normal vector for a sphere of assigned radius in parametric form, surface integrals of vector fields, Gauss law, Stokes theorem, divergence theorem and Green's identity.

**Study tips**

The best advice I can give you is to read the material in each lecture very carefully, tracing the logical steps leading to important results. As a second step make sure you can reproduce these logical steps as well as all the relevant examples without looking at the lecture notes. Remember that a single logical step that you have taken by yourself is much more important than following passively other people's logic. Finally, do as many problems at the end of each chapter as your devotion and dedication to the subject allows! If you get stuck on something or are confused by a particular concept, you are strongly encouraged to come to my office hours. I will be happy to discuss it with you. However, the more thought you have put into it beforehand, the more productive the discussion is likely to be.

**Homework **

Homework is due every second Thursday **in the secretary room, ground floor of the department of mathematics** and will be posted here below every Thursday starting Thursday September 15. For each homework you have two weeks time. *Homework turned in after the time it is due will not be accepted*. Problems will be taken from the practice problem section that you find at the end of the class lecture notes. Solutions to the corresponding homework will be posted as well.

**First incourse test**

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

**Second incourse test**

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

**Homework 1 (35 marks)**

Lect. I: 2 (a),(b) (

**6 marks**); 4(a) (

**4 marks**);

Lect. II: 1 (a),(b) (

**6 marks**); 2 (b) (

**3 marks**); 3 (a) (

**6 marks**); 4 (

**10 marks**);

**Homework 2 (38 marks)**

Lect. III: 4 (

**10 marks**); 7 (

**6 marks**);

Lect. IV: 1 (c) (

**2 marks**); 4 (

**10 marks**);

Lect. V: 3 (

**10 marks**)

**Homework 3 (50 marks)**

Lec. VI: 4 (

**5 marks**); 8 (

**10 marks**);

**Lec. VII: 3 (**

**7 marks**);

**Lec. VIII: 2 (**

**8 marks**);

**Lec. IX**

**:**3 (

**10 marks**);

Lec. X: 3 (

**10 marks**);

**Homework 4 (45 marks)**

Lec. XI: 1 (

**10 marks**); 2 (

**10 marks**);

Lec. XII: 1 (

**10 marks**);

Lec. XVII: 13(a) (

**5 marks**); 22 (

**10 marks**)

**Homework 5 (50 marks)**

Lec. XVIII: 2 (a) (

**5 marks**); 5 (a) (

**5 marks**); 9 (

**10 marks**);

Lec. XIX: 5 (

**10 marks**); 8 (a) (

**5 marks**);

Lec. XX: 1 (

**5 marks**); 10 (

**10 marks**)