About eigensteve Expertise Any questions about introductory or advanced calculus, real or complex analysis, linear algebra, or ordinary differential equations are fair game. I can answer questions about specific calculations, derivations, proofs, and physical applications. Your questions are as good for me as they are for you!
Experience I use linear algebra and differential equations almost every day for my research on modeling unsteady aerodynamics. In particular, my research experience includes numerical integration of trajectories, stability analysis of fluid flow fields, numerical computation of Lyapunov exponents, and more.
Organizations SIAM
AIP
APS
AIAA
IEEE
Publications AIAA Aerospace Sciences Meeting and Exhibit [2008, 2009].
IEEE Photovoltaic Specialist [2009].
For these publications, see http://carlsbad.princeton.edu/~steve/papers/
Education/Credentials I earned my B.S. in Mathematics from Caltech in 2006. I am currently a PhD. Candidate in Mechanical and Aerospace Engineering at Princeton. I expect to graduate in 2011.
Awards and Honors Athena-Feron Scholarship Award for Excellence in Mathematical Coursework [2007],
Princeton MAE Second Year Fellowship for Research Excellence [2007],
Gordon Wu Fellowship [2006-2010],
Caltech Summer Undergraduate Research Fellowship [2003-2005].
For some reason or another I can't seem to get the idea of epsilon-delta definitions of limits into my head, specifically multi-variable epsilon-delta proofs.
As an example, one of the problems in my book is...
"Use the e-d definition of limit to show that Lim(x,y)->(0,0) y/(x^2 + 1) = 0."
I understand that I need to relate |f(x)-L| < e to
0 < sqrt(x^2 + y^2) < d, but how to progress further is unclear to me; however, I hope you can help.
Answer Hi Robert,
You have the right idea, that you need to relate |f(x)-L|<e to 0 < sqrt(x^2+y^2) < d.
This is tricky though, because you don't want to deal with sqrt(x^2+y^2). The first step to find a way to put a delta bound on x and y individually.
The way I think of it is like this: If sqrt(x^2+y^2)<d, this is the equation for a filled circle of possible points. This circle has radius d, and it fits inside the filled square of possible points defined by |x|<d and |y|<d. So, if we can prove that |f(x)-L|<e as long as |x|<d and |y|<d, then it follows that sqrt(x^2+y^2)<d. It is much easier to deal with individual bounds on x and y.
Now we are in a good position to solve the problem. The delta-epsilon proof in words, goes like this:
"For all e>0, there exists some d>0 so that |f(x)-L|<e for all x,y satisfying sqrt(x^2+y^2)<d"
Choose an e, and solve for d!
|y/(x^2+1)-0|<e
At this point, we could try to find the exact d needed to keep the expression bounded by e, but we just need to find /some/ d. Just looking at the expression, if |x|<e and |y|<e, we have the desired result. So we can choose our delta equal to our epsilon, and the problem is solved.
