# Teaching

#### MATH655 Differential Equations II

This course is a natural sequal to MATH555 Differential Equations 1 with a greater emphasis placed on theoretical issues. Topics include higher order linear ordinary differential equations, systems of first-order linear ordinary differential equations (including the basics of linear algebra), some numerical methods and a stability, and large time behaviour solutions to non-linear ordinary differential equations.

#### MATH555 Differential Equations I

This is a course for juniors and seniors in mathematics and engineering at Wichita State. It serves as an introduction to both first and second-order ordinary differential equations, including integrating factors, separation of variables, the method of variation of parameters, the general theory of initial value problems, power series solutions, and the Laplace transform.

#### MATH344 Calculus III

This is a course for sophomores in mathematics and engineering at Wichita State. It treats the calculus of objects and functions in two and three dimensions. Topics include vector functions, curvature, partial derivatives, directional derivatives, limits of multivariable functions, double and triple integrals, vector fields, surface integrals and flux, the integral theorems of Stokes and Gauss.

#### MATH705 Introduction to Vortex Dynamics

This is a 4th-year Masters degree level course of my design. I will be lecturer for this course in the second semester 2018-2020.

This is an advanced course which will provide an introduction to the mathematics associated with vorticity. Vortices are ubiquitous throughout nature and many fluid dynamical processes arising in physics and engineering make use of different vortex models to facilitate the modelling, and to encapsulate the key features, of their associated flow fields. Three important types of vortex which will be explored in this course are the point vortex, the vortex patch and the hollow vortex.

At the end of the course, students will have an understanding of the introductory concepts in vortex dynamics and will be equipped with the mathematical tools to solve problems involving three types of vortex in different domains.

A basic knowledge of fluid dynamics and complex variable theory is desirable but not essential.

- Euler and vorticity equations
- Biot-Savart law
- Kelvin's circulation theorem
- Helmholtz laws
- Point vortex dynamics
- Equilibria and stability
- Kirchhoff-Routh theory
- Point vortex motion on a sphere
- Distributed vorticity
- Vortex patches
- Hollow vortices.

#### MATH236 Mathematical Methods IIB

I was lecturer for the 'vector calculus' part of this course in semester two 2017.

This course is for second-year mathematics and engineering undergraduates at Macquarie. The course consists of two parts: complex analysis and vector calculus.

The following is an outline of the course:

- complex numbers
- complex functions and analyticity
- Cauchy's Theorem
- singularities
- complex integration
- vector fields
- gradient, divergence and curl
- path integrals
- surface integrals
- integral theorems of vector calculus.

#### MXB202 Advanced Calculus

I was lecturer for this course in semester two 2016.

This course is for second-year mathematics and engineering undergraduates at QUT. The course builds on a typical first-year single variable calculus course by extending calculus to several variables, using ‘div grad and curl’ to study vector fields, and applying the key integral theorems of vector calculus. The topics covered lay the foundations for many advanced undergraduate courses at the third and fourth year level (such as asymptotic analysis, partial differential equations, stochastic processes, fluid dynamics, mathematical biology, etc).

The following is an outline of the course:

- partial differentiation
- multiple integrals
- vector fields
- gradient, divergence and curl
- path integrals
- surface integrals
- Green’s Theorem
- Gauss’ Theorem
- Stokes’ Theorem
- physical applications.