6,791 reputation
2628
bio website supermath.info
location Virgina USA
age 35
visits member for 1 year, 8 months
seen 17 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.


Sep
15
comment Deriving equations of motion in spherical coordinates
I think after your edit the spherical unit vectors are fine... I'm just telling you how to derive them via vector calculus. Others prefer to take a geometric approach. That is especially appropriate if this is from a physics course. That said, look at the answer given. Austen's advice is very good. I'm not sure what you mean by differentiation w.r.t. $\theta$. Perhaps, a partial derivative, but mostly, for this problem, I would think about $d/dt$ and keep in mind the spherical frame has a time-dependence but... the cartesian does not. That's the coordinate way to work it out.
Sep
15
comment Deriving equations of motion in spherical coordinates
When we write $\hat{A}$ in physics-type notation this indicates the vector has length one. This beautiful notation allows us to express any nonzero vector $\vec{A}$ as the product of a magnitude and direction; $\vec{A} = A\hat{A}$. This formula also tells you how to calculate $\hat{A}$. To find $\hat{u}$ for a curvelinear coordinate we can calculate $\nabla u = \langle u_x,u_y,u_z \rangle$ and then normalize it to length one by dividing by $| \nabla u |$. For the spherical radius the gradient already has length one, but for $\phi$ some normalization is needed.
Sep
15
comment Deriving equations of motion in spherical coordinates
flag on the play. Your "unit" vector in the $\phi$-direction doesn't have length one... I continue reading, but this is troubling to my hat-customs as a former physics major. See page 383 of supermath.info/math231_CurlDivNoncartesian_375_384.pdf
Sep
15
comment Function and dependent variable are represented by the same symbol?
If it's wrong then I don't wanna be right.
Sep
15
awarded  Investor
Sep
15
comment Linear Algebra Question on Equivalent Systems
you must look at math.stackexchange.com/questions/46050/… I think it is exactly your question!
Sep
15
comment multilinearform over finite dimensional vector space is continuous
This is of course the standard technique (as far as I know), but I wonder, is there any other way to obtain this result?
Sep
15
comment Examples of overly wordy theorem.
You can find dozens of nineteenth century calculus texts available as pdfs for free. They're full of this stuff.
Sep
15
comment Origin of the terminology “connected algebra”
You know definitions have been wisely made when it is possible to find arguments merely by the labels affixed to objects.
Sep
13
awarded  Quorum
Sep
13
comment Suppose that $U$ is a subspace of $V$. What is $U+U$?
@Kevin Driscoll to be clear, trivial in the sense of linear algebra. Perhaps not trivial in terms of geometry, the beauty of linear algebra is manifest in the wealth of cases this proposition covers (see answer by Clive Newstead and discussion by MJD).
Sep
13
comment Suppose that $U$ is a subspace of $V$. What is $U+U$?
yes. the problem is trivial.
Sep
13
comment Suppose that $U$ is a subspace of $V$. What is $U+U$?
I would assume it is $\{ x+y \ | \ x \in U, \ y \in U \}$ which is clearly $U$ again here.
Sep
11
comment Linear Algebra- Matrix derivative
These matrices... they are functions of what? How many independent variables are there? Does x11 mean $x_{11}$?
Sep
11
comment Linear Algebra- Proof of trace property
@Kiana you should notice that $\text{trace}(A) = \sum_{i=1}^n a_{ii}$. It's a shame this notation is not promoted more in the introductory texts. Use of $\Sigma$ notation is very helpful in understanding general properties. Imho.
Sep
10
comment What is 'target manifold'?
@Poli Tolstov perhaps en.wikipedia.org/wiki/Sigma_model would be helpful to start searching.
Sep
10
comment What is 'target manifold'?
it's something to shoot values at.
Sep
9
revised showing that a local diffeomorphism is a local isometry using first fundamental form
added 60 characters in body
Sep
9
comment showing that a local diffeomorphism is a local isometry using first fundamental form
but, to prove smoothness of a map, if we have smoothness with respect to one local coordinate representative then it follows that all others are likewise smooth. This follows because they can be reached by composition of transition functions which are themselves smooth. I suspect something similar is true here, but I don't have all the definitions in front of me here.
Sep
9
accepted Concerning Carathéodory's criteria of differentiability and a proof that differentiable implies continuous