Here is a geometric picture of how symmetries can aid in the solution of certain ODEs. I'll begin with some technical discussion and conclude with some qualitative remarks.
Let M denote the phase space. This is the manifold where solution curves to the ODE are contained. Let X:M\rightarrow...
In several textbooks I've read that treat classical thermodynamics, there is a theorem due to Carnot that is commonly stated:
"No engine operating between two given temperatures is more efficient than a Carnot engine"
(incidentally, this is the statement in Huang's book)
In these same...
If
f:U\rightarrow\mathbb{R}
where U is an open neighborhood of g(x), then the derivative at g(x) can also be defined by f^\prime(g(x))=\text{lim}_{b\rightarrow g(x)} \frac{f(b)-f(g(x))}{b-g(x)}. Now let h(b)=\frac{f(b)-f(g(x))}{b-g(x)}. By assumption, \text{lim}_{b\rightarrow g(x)} h(b)...
As for the third line, it's certainly not the correct application of the definition of the derivative. However, I don't think it's hard to show that if h(b)->c as b->f(u), then h(f(a))->c as a->u provided f is continuous. So I think that the limit in the third line might still be evaluated...
I know that if f(x)->a and g(x)->b as x->y, then f(x)g(x)->ab as x->y. So is the problem with the second line that the existence of the limit we are interested in hasn't been established?
I stumbled upon this document that discusses the single variable chain rule:
http://math.rice.edu/~cjd/chainrule.pdf
At the bottom, there is an incorrect proof of the validity of the chain rule, but the author does not cite why the proof is wrong. I'm wondering if the problem is...
Thanks for the suggestions. The first of the four that have been suggested that I took a look at was Kardar (the ocw lecture notes, to be precise). I really like the connection he makes early on between information entropy and estimating probability distributions.
Hi all,
I'm looking for a rigorous (in a mathematical sense) treatment of statistical thermodynamics. I'm at the tail end of a class on stat thermo that used the book by Bowley and Sanchez. This book is not what I'm looking for. Does anyone have any suggestions?
Hi,
I'm studying for the final exam in my first course in topology. I'm currently recalling as many theorems as I can and trying to prove them without referring to a text or notes. I think I have a proof that the closed interval [0,1] is connected, but it's different than what I have in my...
I've actually got the second of the two Rudin books you mentioned sitting on my hard drive. I skimmed the first few sections of the first chapter on measure theory, and while I don't find this material to be over my head, I think I'm looking for the material in the other of his books you...
I realize this may be kind of strange, but as it turns out, I've experienced a good introduction to topology (at the level of Munkres) before taking a rigorous class on analysis. With a month-long winter break in my near future, I'm wondering if anyone could suggest a text on real analysis that...