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Help With Choosing Classes 
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#1
Mar112, 04:59 PM

P: 15

Hi PhysicsForum!
Fall 2012 semester will be my last semester as an undergrad in Math and Statistics and I was planning on taking a handful of graduate level classes. I'm planning on applying to grad school in Applied Math and concentrating on differential equations, fluid dynamics, or stochastic processes. I wanted an opinion on the classes I would be choosing. I am for sure going to take Numerical Fluid Dynamics: Addresses numerical techniques for solving linear and nonlinear differential equations in initial value fluid flow problems. Students receive a thorough background in the principles used to evaluate numerical methods, the ability to critically interpret these methods as presented in the literature, and in particular, the practical application of these techniques in modeling multidimensional flow on highperformance computers. Temporal and directional splitting, finite differencing/volume methods, and adaptive nesting will be discussed. and Measure Theory: Lebesgue measure on the real line; integration and differentiation of real valued functions of a real variable; and additional topics at discretion of instructor. But I'm not sure what else to take. I want to take two more classes, but I don't know if I should lean towards theory (i.e. Differential Equations or Complex Analysis) or computing (Numerical Methods), especially since I want to do applied math in grad school. Here are the course descriptions for the classes I'm trying to decide between. Graduate ODEs: An introduction to the study of dynamical systems. Considers continuous and discrete dynamical systems at a sophisticated level: differential equations, flows and maps on Euclidean space and other manifolds. Emphasis will be placed on the fundamental theoretical concepts and the interaction between the geometry and topology of manifolds and global flows. Discrete dynamics includes Bernoulli shifts, elementary Anosov diffeomorphisms and surfaces of sections of flows. Bifurcation phenomena in both continuous and discrete dynamics will be studied. Graduate PDEs: Basic introduction to the study of partial differential equations; topics include: the Cauchy problem, powerseries methods, characteristics, classification, canonical forms, wellposed problems, Riemann's method for hyperbolic equations, the Goursat problem, the wave equation, SturmLiouville problems and separation of variables, Fourier series, the heat equation, integral transforms, Laplace's equation, harmonic functions, potential theory, the Dirichlet and Neumann problems, and Green's functions. Graduate Complex Analysis: Topics include the Cauchy theory, harmonic functions, entire and meromorphic functions, and the Riemann mapping theorem. or Numerical Analysis: Linear system solvers, optimization techniques, interpolation and approximation of functions, solving systems of nonlinear equations, eigenvalue problems, least squares, and quadrature; numerical handling of ordinary and partial differential equations. TL;DR: What is more important in preparation for applied math PhD programs, math theory or computing knowledge. Will I get enough computing knowledge from the Numerical Fluid Dynamics class? Which two other classes should I choose. 


#2
Mar112, 05:06 PM

Mentor
P: 18,333

I don't think you can ever go wrong with a differential equations class. Especially since you mention you would like to study this subject in grad school. Stochastic processes also uses differential equations a lot (or at least something very similar to it). So I would study this.
Whether to study ODE's or PDE's, I don't know. I'm leaning towards PDE's here, but I'm not sure how useful it will be for you later... 


#3
Mar112, 06:14 PM

P: 15




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