Andrew D. Hwang
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 3h comment How is a field different from a group? The pedagogical issue I had in mind was lower-tech than my comment suggested: A tangent plane to a surface in $3$-space is a two-dimensional vector space consisting of ordered triples (subject to a constraint). (Moreover, the zero vector isn't generally the triple $(0,0,0)$.) I think this seeming dichotomy (are tangent vectors pairs because they're elements of a $2$-dimensional space, or triples?) can be confusing to students at the OP's level. In other words, I find that treating vectors as tuples from the outset creates an incorrect fundamental presumption that must be broken later. 4h comment How does All metric functions on R look like? Could you please clarify what type of answer you're seeking? The space of metrics on the reals inducing the usual topology is infinite-dimensional, so you're certainly not going to get a list; you're probably not even going to get an abstract characterization more useful than the definition. 4h answered Is it possible to convert a divergent series by subtracting a constant? 5h comment why there are no parabolic (on a paraboloid) non-euclidean geometry? Ah, I see. :) I'd thought you were addressing the misperception that a one-sheeted hyperboloid (or a hyperbolic paraboloid) in Euclidean space is a model of hyperbolic geometry. 18h comment why there are no parabolic (on a paraboloid) non-euclidean geometry? (+1) Hyperbolic geometry can be viewed as the geometry of (half of) the two-sheeted hyperboloid $x^{2} + y^{2} - z^{2} = -1$, $z > 0$, a.k.a., the "unit sphere of future-pointing timelike unit vectors" embedded in Minkowski space, however. :) 18h answered Limiting behaviour of a real Möbius sequence 23h answered Estimating how many spheres there are inside of a domain, only knowing one slide 1d answered Looking for a Simple Argument for “Integral Curve Starting at A Singular Point is Constant” 1d comment Estimating how many spheres there are inside of a domain, only knowing one slide This can be done by elementary geometry and arithmetic. All that matters is the total number of slices, and the thickness of each slice compared to the size of the objects you're counting. I'd post an answer, but something about the wording suggests to me this is part of an assignment or test. In any case, the "solution energy" is significantly lower than the "statement energy"; I'm afraid this always raises uncharitable suspicions. Hoping the hints above help direct you toward an easy solution. 1d comment Rules for polynomial fit to n points Your first paragraph sounds like Lagrange interpolation, in which a one-variable polynomial graph passes through specified points, while the example sounds like finding a two-variable polynomial whose zero set contains the specified points. These are different goals. Could you please clarify? Particularly, what form do your data have (are they triples $(x, y, T)$...?), and in what sense do you want $T$ to "fit" your data? 1d comment (Anti-) Holomorphic significance? That's the definition of "holomorphic": $\dd f/\dd \bar{z}^{\beta} = 0$, i.e., "$f$ is constant with respect to $\bar{z}^{\beta}$". 1d comment How is a field different from a group? Not downvoting, but in a fundamental sense, a vector space is not a collection of vectors with coordinates in $F$. To express a vector space in coordinates amounts to picking a basis; the impossibility of doing this naturally is connected to concrete, substantial phenomena in geometry and topology (e.g., the non-existence of a continuous, non-vanishing vector field -- much less an orthonormal frame -- on an even-dimensional sphere). 1d comment How is a field different from a group? (+1) Just a friendly note that for accessibility reasons, you might consider using blue rather than red (or perhaps even better, \oplus or \boxplus (etc.) instead of colours, though I do understand your pedagogical reasons for using $+$ and $\cdot$). :) I'm not colourblind, but the red is a tad hard to read. 1d comment The complement of a union of uncountable many curves in $\mathbb{R^3}$ As HSN's answer suggests, a good approach is to partition some known separating surface into curves. With that in mind, perhaps try to find a family of circles partitioning a separating set, and parametrized by a circle, so that any two circles in the family intersect in exactly one point. (Such an example exists, and is easily visualized.) 1d comment Archimedes' derivation of the spherical cap area formula Welcome to Math.SE! This is interesting information; are you able/willing to add mathematical details, perhaps including diagrams? 1d comment Evaluate the following line integral For the velocity components, take the displacement between the endpoints, not the terminal point; thus $x = 1 + 2t$ and $dx = 2\, dt$. 1d comment Why is $\operatorname{Log}(2z-3i)$ not well defined? Presumably because there's trouble along the half-line $x \leq 0$ and $y = 3/2$, since on this half-line, $2z - 3i$ isn't in the domain of Log. Strictly speaking, the book should have added something about the domain $U$ of $z \mapsto \operatorname{Log}(2z - 3i)$, and said "$\operatorname{Log}(2z - 3i)$ is not well-defined in $U$", since, for example, your function is well-defined on the open unit disk, or the lower half-plane. 1d comment Misinterpretations of Hilbert's Theorem? Largest hyperbolic disk embeddable in Euclidean 3-space? at mathoverflow is probably of interest. :) Looks like my preceding comment is wrong: A Dini-type crocheted surface can be long and thin, and have arbitrarily large hyperbolic radius. (Again, that MO post and its answers and comments don't seem to contradict "there exists a $C^{1}$ embedding, but no $C^{2}$ embedding".) 2d revised Find normal line to hyperbole $\frac{x^2}{3}-\frac{y^2}{2}=1$ edited tags 2d comment Find normal line to hyperbole $\frac{x^2}{3}-\frac{y^2}{2}=1$ Take an arbitrary point on the hyperbola; calculate the slope of its tangent line using elementary calculus, and use the fact that perpendicular lines has negative reciprocal slopes to write the equation of the normal line at the selected point. Does that not to what you want?