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June 20

First fundamental form and curvature

I wanted to ask you for help in better understanding two concepts that leave me a little perplexed regarding the first fundamental form:

1) The first fundamental form was defined to me as follows: given a parameterization ϕ(u,v) , the metric A is found with the dot products like ϕu​⋅ϕv​ (through the 4 permutations of u,v), where the dot products are restricted to the tangent spaces induced by R3. It was then explained that quantities dependent on E,F,G (such as the Christoffel symbols, and therefore the curvature via Gauss's Theorema Egregium since it shows that they depend only on the Christoffel symbols) are intrinsic quantities, meaning they do not depend on how the surface is immersed in R3, but are intrinsic to the object itself.

My confusion revolves around this: if I define the first fundamental form in this way, I note that ϕ(u,v) is the map ϕ:U→R3, so what comes out of this map is precisely the figure I have as a surface in R3R3, that is, the shape it takes. In fact, the map gives me the coordinates (x,y,z). Now, ϕ​ and ϕv​ are the tangent vectors to that figure, so they indeed depend on the shape realized in R3. Then I define the matrix A through the dot products of ϕu​ and ϕv​, so what is intrinsic here? I am using tangent vectors to a figure that has a shape given by ϕ:U→R3, so I would say it indeed depends on the immersion and how the figure is geometrically realized in it.

What seems to be suggested by the explanation is this: if I immerse the "abstract concept" of a sphere in R3, I have different realizable figures. I do not understand this concept given the considerations above: curvature depends only on E,F,G through the Christoffel symbols, but E,F,G depend on the dot products of tangent vectors to a shape of the surface, so on an immersion of it in R3.

2) The second question is this: if I change parameterization, I will have new ϕi'​,i∈u,v which differ from the initial ones, so I will find E',F',G'. However, for an object (let's take the usual sphere), the curvature is fixed and depends only on E,F,G right? Well, if I have changed parameterization and have E',F',G', why wouldn't the curvature change? They are different values. It seems to me that by changing parameterization, the first fundamental form changes and therefore the curvature should change as well (but it shouldn't obviously be so).

Could you help me with these two questions? Thank you. --151.36.108.141 (talk) 15:42, 20 June 2024 (UTC)

E,F,G are relative to a particular tangent space and the values will of course change if one changes the tangent space. One has to get rid of the tangent space dependency to get things like the curvature. The determinant gives a measure of the area of dudv and dividing the determinant of the second fundamental form by that of the first fundamental form gives something independent of dudv - and which in fact is the Gaussian curvature. NadVolum (talk) 15:09, 21 June 2024 (UTC)

June 22

Repeating decimals 1/((k-1)k+1) in base k^2+1 don't include multiples of k in expansion?

I noticed that 1/7 in base that 3,6 & 9 are missing from the expansion in base 10, but it appears that there is a pattern.

• k = 2, 1/3 in base 5 is .13(rep), 2/3 in base 5 is .31 (no digit which is a multiple of 2 appears in the pentary expansion, neither 2 or 4 occur
• k = 3, 1/7 in base 10 is .142857(rep) (no digit which is a multiple of 3 occurs in the decimal expansion, none of 3, 6 or 9 occur
• k = 4, 1/13 (1/C) in base 17 is .153FBD(rep) (not only doesn't any digit which is a multiple of 4 occur (4,8,C,G), it appears that digits which are a multiple of 2 occur.
• k = 5, 1/26 (1/L) in base 26 is .164OJL(rep) (no digit which is a multiple of 5 occurs (5,A,F,K,P)

Any idea for a proof or extension of this? (I can't find an easy calculation in base 37)Naraht (talk) 02:11, 22 June 2024 (UTC) .

There seems to be a typo in the k = 5 case; (k-1)k+1=21, not 26, and 1/26 base 26 is just .1. It might be easier to use a different notation for large bases, say by inserting a comma between each digit and leaving the digits in base 10. So 1/21 = .1,6,4,24,19,21(rep). In this notation 1/31 = (base 37) .1,7,5,35,29,31. I think you can probably see what's going on by looking at k=10 and k=100: 1/91 = (base 101) .1,11,9,99,89,91(rep), and 1/9901 = (base 10001) .1,101,99,9999,9899,9901(rep). The pattern is 1/(k²-k+1) = (base k²+1) .1,k+1,k-1,k²-1,k²-k-1,k²-k+1, which shouldn't be too hard to verify. There is almost certainly python code to compute fractions in large bases to do further experimentation. k²-k+1 and k²+1 are both cyclotomic polynomials, and there are probably similar patterns when you look at 1/Phi_m(k) base Phi_n(k) for various m and n. --RDBury (talk) 15:15, 22 June 2024 (UTC)

PS. The fact that the denominator is a cyclotomic polynomial is the part of what makes this work, the other part isn't that the base is a cyclotomic polynomial in k, but that it's equal to ±k mod the denominator. In this case the denominator is Φ₆(k) and the denominator is Φ₆(k)+k. A generalization is that if P(x) is congruent to x mod Φ_n(x) (taken as polynomials in x) then for a 1/Φ_n(k) has period dividing n base P(k). For example Φ₁₀(x)=x⁴-x³+x²-x+1 and P(x)=x⁸+x⁴+x²+1 is congruent to x mod Φ₁₀(x). So the generalization states that Φ₁₀(k) has period dividing 10 base P(k). For k=10 we get 1/9091 = (base 100010101) .11001,110010,1100100,11001001,9999909,99999099,99900090,98910000,89009099,90010191(rep) and this indeed has period 10. If P(x) is congruent to -x mod Φ_n(x) the rule is that the period divides 2n. For n=3, k=3, 1/13 = .076923(rep) base 10; for k=4, 1/21 = .0,13,12,16,3,4(rep) base 17; for k=10, 1/111=.0,91,90,100,9,10; and in general 1/(k²+k+1) = .0,k²-k+1,k²-k,k²,k-1,k(rep). There are probably additional tweaks and generalizations you can make on this. --RDBury (talk) 16:03, 23 June 2024 (UTC)

thank you for all of this, my noticing of the missing "k" was a small part of what could actually be found.Naraht (talk) 16:48, 25 June 2024 (UTC)

June 25

Defining a function involving a limit

If you are defining g(x), why is it written

$\lim _{x\to \pi }{\sqrt {x^{3}f(x)}}=g(x)\,$,

and not

$g(x)=\lim _{x\to \pi }{\sqrt {x^{3}f(x)}}\,$,

or are both ways acceptable?

I mean you usually don't say ${\sqrt {x^{3}f(x)}}=g(x)\,$, you say $g(x)={\sqrt {x^{3}f(x)}}\,$, right? (Yes, g(x) is purely nonsensical in meaning, for demonstrative purposes only :)

173.14.155.129 (talk) 23:20, 25 June 2024 (UTC)

It is definitely more conventional to place the entity being defined in the lhs of the equation, but as long as it is clear which is the definiendum and which the definiens, there is nothing logically wrong with the swapped version – it is merely highly unusual and therefore may put the reader on the wrong foot. This is independent of whether the definiens involves a limit. By the way, the value of the expression $\lim _{x\to \pi }{\sqrt {x^{3}f(x)}},$ if defined, does not depend on $x.$ If function $f$ is continuous at $\pi ,$ it is equal to ${\sqrt {\pi ^{3}f(\pi )}}.$

 

When asked to determine

$\lim _{~\epsilon \to 0^{+}}(1+\epsilon x)^{\frac {1}{\epsilon }},$

I'd present the answer in the form

$\lim _{~\epsilon \to 0^{+}}(1+\epsilon x)^{\frac {1}{\epsilon }}=e^{x},$

but when asked to define the exponential function, I might give

$\exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}$

as one of several definitions, and then I'd present it in this order.  --Lambiam 00:02, 26 June 2024 (UTC)

Like I said, that was just a bogus example (I'm actually thinking of a segmemt that changes as it grows, so that the midpoint value of the whole is different than the midpoint value calculated "locally", with an infinitesimal length), but you answered the question.

173.14.155.129 (talk) 01:03, 26 June 2024 (UTC)

Quick side note that equality in the sense of definition - as opposed to equality in the sense of identity - is often denoted with a colons-equal := (as a single character, ≔), in which case I believe the function $g(x)$ being defined would have to go on the left. GalacticShoe (talk) 05:41, 26 June 2024 (UTC)