This is the third of a sequence of three basic calculus courses. It covers vectors and surfaces in space and the calculus of functions of several variables including partial derivatives and multiple integrals, stokes theorem, and first order differential equations.
Course Syllabus: link
This course presents an introduction to the theory of differential equations from an applied perspective. Topics include linear and nonlinear ordinary differential equations, Laplace transform, and introduction to partial differential equations.
Course Syllabus: link
Students will be exposed to a selection of concepts and algorithms that are used to characterize networks and their properties, along with some specific examples and applications. There are black box tools out there to compute various quantities that show up in network science and to apply established algorithms to given graphs. But in order to use these tools effectively, it is essential for the practitioner to understand the concepts involved and the methods “under the hood”. With deeper understanding, it becomes possible not only to select and use existing tools, but also to adapt these tools and to develop new ones.
This course will provide a broad introduction to mathematical methods typically applied to problems in biology. Models using calculus, ordinary differential equations, partial differential equations, discrete dynamical systems, stochastic dynamics, or a cellular automata framework will be presented and principal methods for their analysis will be described. Computational methods will also be covered, including computing platforms such as XPPAUT. Throughout the course, students will have extensive opportunities to practice the development and analysis of mathematical biology models.
This course will present an overview of various frameworks for modeling and analyzing biological systems, both deterministic (difference equations, ODEs, PDEs) and stochastic (Markov Chains, SDEs) with examples for each framework from areas such as population and disease dynamics. The modeling and analysis methods of this course can be applied to other areas of applications, some of which will be covered if time permits (including regulation of gene expression, mathematical immunology, pattern formation and chemotaxis, and wound healing).
This is the first course in a two-term sequence designed to acquaint students with the fundamental ideas involved in the study of ordinary differential equations. Basic existence and uniqueness of solutions as well as dependence on parameters will be presented. The course will cover linear ODES and the matrix exponential, oscillations via an introduction to Poincare-Bendixson theory for planar systems and to Floquet theory, and Sturm-Liouville problems. Students will also be introduced to geometric concepts such as stability of fixed points and invariance. This first term will provide an excellent introduction to ODE theory for students interested in applied mathematics.