8,076 reputation
21036
bio website supermath.info
location Virgina USA
age 36
visits member for 2 years, 5 months
seen 3 hours ago

I'm interested in mathematics which is used to frame physical theory. I guess that means I'm at least a little interested in just about anything.

Currently I'm trying to understand Cartan's Method of moving frames as it applies to various classification questions of low-dimensional geometry.

I also have interest in superanalysis and certain problems of hypercomplex analysis.

More generally, I'm just looking for interested students who want to exceed the status-quo of undergraduate mathematics. Ideally, our interests overlap.


Oct
10
revised John Lee's Intro to Smooth Manifolds Inverse Function Theorem
changed tags to emphasize the source and context of this question
Oct
10
answered John Lee's Intro to Smooth Manifolds Inverse Function Theorem
Oct
10
comment John Lee's Intro to Smooth Manifolds Inverse Function Theorem
I'm looking at page 160 and I see no $F_2$ etc... are you sure?
Oct
10
answered Confused About Trigonometric Substitution
Oct
8
reviewed Approve how to define a circular helix in terms of the Frenet Frame
Oct
8
comment How can you absolutely prove that a function has an infinite power series representation?
we don't always. There are examples of smooth functions which do not admit a Taylor series at a particular point. Fortunately, most of your favorite functions appear as solutions to differential equations at an ordinary point hence, they are analytic, meaning there is a power series for them. For some discussion, about nonanalytics, see math.stackexchange.com/q/189841/36530
Oct
6
comment Why not use two vectors to define a plane instead of a point and a normal vector?
Two linearly independent vectors do give rise to the parametrization of a plane containing point $p$. Simply use $\vec{r}(u,v)=p+u\vec{A}+v\vec{B}$. Sometimes, this is more useful than the Cartesian equation of a plane, it's all about context.
Oct
5
comment stuck on logarithm of derivative of sum $\frac{\partial\mathrm{log}(a+b)}{\partial a}$
I added an answer which expands on the rules I used. I suppose, the question really boils down to how you understand the log to be defined. After that, it's just the chain rule and the definition of partial differentiation.
Oct
5
answered stuck on logarithm of derivative of sum $\frac{\partial\mathrm{log}(a+b)}{\partial a}$
Oct
5
comment stuck on logarithm of derivative of sum $\frac{\partial\mathrm{log}(a+b)}{\partial a}$
$1/(a+b)$ unless $b$ has some implicit dependence on $a$.
Oct
5
comment Complex integration and logarithms
I think the argument matters. It might be equivalent to take limits to the edge of the branch and add the results together, but, ignoring the branch cut will cause problems.
Oct
4
answered Complex integration and logarithms
Oct
4
comment Complex integration and logarithms
Of course, the sign is immaterial as $\log(z^{-1}) = -\log(z)$. It follows you can use the principal logarithm to set-up the inverse tangent. Notice $[1,1+i]$ is in the slit-plane for which the principal log is holomorphic.
Oct
4
comment Complex integration and logarithms
Well, if $z \in [i, i\infty]$ or $z \in [-i , -i \infty]$ then $\frac{1+iz}{1-iz} \in \mathbb{C}-[-\infty,0]$. This is part of Gamelin exercise number 5 of section 1.8 which I happened to spend 20+ minutes working through carefully a week or three ago. I'm not sure I saw an easy way through showing that, but, once you know it you know it. Although, my sign differs in the quotient...
Oct
4
comment how to define a circular helix in terms of the Frenet Frame
Thanks for the added detail, I'll try to sort through it as soon as I have some time to focus on this problem. In any event, I do intend to award the bounty here as your effort on this question is exemplary.
Oct
3
answered help with hyperbolic functions like sinh and tanh
Oct
2
comment how to define a circular helix in terms of the Frenet Frame
Thanks for this, I was waiting for my students to solve this last semester, but, they were lazy. I guess I should revisit it. There was a problem before the one which precipitated this question from Oneill which walked through the construction you mention here I think. Indeed, Problem 7 of section 2.4 precedes problem 9 which is what prompted this question. Problem 7 has the curve you mention here, Def. 4.5 is the definition you mention here also. ( I refer to the revised 2nd ed. of Oneill).
Oct
2
comment how to define a circular helix in terms of the Frenet Frame
The reason I did not accept the answer is that the definition given, while much closer in spirit to what I'm searching for, is still not given in terms of T,N,B of the curve. It may be that my goal is unreasonable and as soon as I'm convinced of that I will accept this answer.
Oct
1
comment The Best of Dover Books (a.k.a the best cheap mathematical texts)
Thanks for the offer, I will probably take you up on that when I start preparing for teaching next fall (maybe next semester or next summer). It is likely I use Edwards again.
Oct
1
comment Evaluating $ \int {e^x \sin (k \pi x) } dx $
this ought to be taught in first calculus courses along with basic complex arithmetic. Nice answer.