Reputation
11,785
Top tag
Next privilege 15,000 Rep.
Protect questions
Badges
3 9 36
Newest
 Nice Answer
Impact
~109k people reached

1h
comment coordinate geometry : intersection of a curve with a line
Hint: The line with equation $y = mx + b$ makes angle $\theta = \arctan m$ with the rightward-pointing $x$-axis.
13h
comment Trivial sections of tautological line bundle for $\mathbb{P}_F(V)$.
It's not true that there only exist trivial sections of the tautological bundle; do you have another condition in mind (such as "non-vanishing", or "holomorphic" over the complex field)?
19h
comment Identifying the two-hole torus with an octagon
@LaurentHayez: Modulo mistakes on my part, yes, the two gluings must be topologically equivalent. :) (The gluing in your answer is the "standard" presentation of the two-holed torus, and certainly correct.) The way I read the OP's question, their instructor stated that the "antipodal" gluing gave a two-holed torus, and they wanted to confirm this assertion.
20h
answered Identifying the two-hole torus with an octagon
21h
comment Integrability of dirichlet function in $\mathbb{R}^3$
@RuiCarneiro: Yes, exactly. :) In other words, one should speak of a function being "Riemann integrable on $D \subset \mathbf{R}^{n}$" rather than simply being "integrable" (though in practice the domain and its dimension are implicit).
23h
answered Integrability of dirichlet function in $\mathbb{R}^3$
1d
comment The curvatures of a transformed surface under a similarity transformation
The point of the second comment is that $N' = A \circ N \circ f^{-1}$, so (omitting points of evaluation) $\operatorname{tr}dN' = \frac{1}{r}\operatorname{tr}(A\, dN\, A^{-1}) = \frac{1}{r} \operatorname{tr}(dN)$.
1d
comment The curvatures of a transformed surface under a similarity transformation
Small typographical note: The less-than/greater-than signs in (La)TeX are spaced like binary operators rather than like delimiters, which makes inner products rather difficult to parse visually. \langle and \rangle are preferred; I generally use a macro \newcommand{\Brak}[1]{\left\langle #1 \right\rangle}, and then, e.g., -\Brak{dN_{p}(v), w}. :)
1d
answered The curvatures of a transformed surface under a similarity transformation
Apr
15
answered Why not every homogeneous manifold is parallelizable?
Apr
14
comment Question about parallel displacement on a surface
I haven't carefully checked the calculation, but wouldn't one expect the change in $v$ to depend on $v$? (The endomorphism of the tangent space doesn't depend on $v$, of course, but that looks consistent with your last line.)
Apr
14
comment Circle Geometry
The distance formula coming from the Pythogrean theorem is true even if one or both legs of the "triangle" with corners $(a, b)$ and $(x, y)$ have length zero.
Apr
14
comment Are all continuous one one functions differentiable?
@MarioCarneiro: You're perfectly correct, if course; just emphasizing for posterity that the answer to the OP's "global" question is no, not even locally. :)
Apr
14
comment Key differences between almost complex manifolds and complex manifolds
If you'll forgive a tangential "public service comment" on terminology: It would be more logical if we all spoke of "complex manifolds" (admitting an endomorphism field $J$ with $J^{2} = -I$) and "holomorphic manifolds" (complex manifolds for which $J$ is integrable), the way we already do for vector bundles. But for this changeover to happen, students (and working mathematicians) have to embrace the terminology. :)
Apr
14
answered Möbius strip as a non-trivial principal bundle
Apr
14
comment Unbounded Geodesics and Nonpositive Curvature
Posted the preceding as a comment rather than an answer, since I'm guessing "completeness" was intended...?
Apr
14
comment Unbounded Geodesics and Nonpositive Curvature
1., 2. If $M$ is compact, no geodesic is unbounded, so curvature alone does not guarantee unbounded geodesics. 3. If you assume only simple connectedness, your metric might be incomplete, and there is no guarantee of unbounded geodesics. 4. Non-compactness is not enough for "all"; take a cylinder or similar product example. 5. Non-positive curvature, simple-connectedness, and completeness get you "all unbounded" by the Cartan-Hadamard theorem and the behavior of Jacobi fields.
Apr
14
comment Short differential-form free route to understanding a specific surface integral
@temo: Offhand I can't think of a reference for integrating over hypersurfaces. The material in my answer is "folklore", though if memory serves, Spivak's Calculus on Manifolds has an exercise on the generalized cross product, and it's entirely possible he treats integration over hypersurfaces (perhaps in one or more exercises). Incidentally, I found Spivak to be an especially clear introduction to forms. (Do work all the exercises!) Really, differential forms largely boil down to determinants and index combinatorics; one just has to get past the (substantial!) initial intimidation barrier.
Apr
14
answered Are all continuous one one functions differentiable?
Apr
13
comment How to find $n+1$ equidistant vectors on an $n$-sphere?
@Gaffney: Could you please explain? Each $\tilde{v}_{i}$ has unit length (since the $v_{i}$ have unit length), so all $(n + 1)$ vectors $(\tilde{v}_{i})_{i=1}^{n+1}$ lie on the unit sphere centered at the origin.