I really am dumb, I'm trying to show that straight lines are minimizers of Euclidean distance, and without using variational methods it seems to be pretty tricky.
I want to introduce the length and distance functions as early as possible (page 1 or 2) and one has to show that the Riemannian distance on the plane is indeed the regular one
To wait until 30 pages in is ridiculous
@BalarkaSen Oh, interesting question: is there a 1-form $\omega$ on $\Bbb R^n$ such that $L(\gamma)=|\int_\gamma\omega|$ is the Euclidean length of a curve $\gamma$, for any $\gamma$?
The kernel of $\omega$ forms an integrable distribution (need to check that) with 1-dimensional integral manifold. Take a curve $c$ in that submanifold. $|\int_c\omega|=0$, but the length is positive.
@0celo7 Wait a tick. $\gamma_1, \gamma_2$ be two curves of the same arclength. Given such a form $\omega$, I can integrate over $\gamma_1$ clockwise, and then over $\gamma_2$ counterclockwise. That's $\int_{\gamma_1}\omega - \int_{\gamma_2}\omega$. By hypothesis that's 0 tho?
@0celo7 I was reading about numerical analysis where the author wrote a theorem relating the relative error and the number of significant figures; the proof was given for three distinct cases; namely $(a) ~m\lt n, ~(b)~m= n$ and $(c) ~m\gt n$
user116211
where $n$ be the number of significant figures, $m$ be the number of correct decimal places.
user116211
While the first one is easy to guess, and the second one is proper fraction, I'm not getting an example for the third case.
1. Is it possible to have something that can be proved to be true, but attempt to prove by contradiction generated no contradiction because the case where it does not hold is not completely excluded from the structure in question?
2. We are lucky to find that the fundemental theorem of calculus holds thus allowing easy computation of integrals. What if we are not so lucky back then and found no fundemental theorem of calculus. How to construct a mathematical system to investigate this alternate possibility?
@Secret When I look at these questions I can only imagine that they are meant to be addressed to someone precisely. A long time ago I thought of calculating integrals in a totally different way. I never developed my ideas that much, and I don't know why, I could have do it (however I think I know how I might like to continue what I already did).
If I don't use @ in my messages and my messages end with a question mark "?" instead of "...?", it means I am asking the whole chat and have no particular person in mind. This often means should I become too deperate and after too many failures to experiment for the answer myself, it will become a MSE question
However no one will share on MSE revolutionary ideas about mathematics that are (possibly in development), but talking about possibilities, sure, that wouldn't ever be a problem.
Well, actually there are a couple of examples of what you said above. At least in the abstract algebra sector, there are a lot of users who occassionally ask abotu whether their mathematical structure make sense and implications http://math.stackexchange.com/questions/1775868/is-this-extension-of-the-real-numbers-a-field-it-involves-a-unit-of-infinity
I invent maths all the time in my spare time. For example just trying to break axioms so that I can get a division by zero I have been doing that for at least 3 years
When I actually share the results to others, however often we all spot some mistakes and it will be sent back into the drawing board
Recently I was studying a linear algebra class which introduce some basic concepts of fields, groups
Once again I tried to divide by zero by constructing an algebraic structure from scratch using just a few axioms because I stumbled across this when browsing the site
We knew that the important ...
However my communication skills is poor thus questions like these people usualy don't understood it. This question hwoever no longer matters since I have checked with my mathematician peer and he point out that I have not completely avoided the axioms to derive $0x=0$
My single most highly rated question is actually from mixing mathematics and philosophy, which my linear algebra professor siad I gone too meta that I might lost my way in the vastness of mathematics
This might be a very stupid, and possibly philosophical question, but attempt to apply mathematics to everything plus inspired by this question caused me to ask this question
Is there any mathematical object that has been proved to exist but cannot be described in words?
If the answer is...
The primary reason why it attract so many upvotes is because there is a alot of dispute about what it actually means. IT also involve some MO answers there
Basically, ever since I become very interested in division by zero and understood from my professors how the do nothing technique works (aka cheating, trick, e.g. 0=x-x, 1=x/x etc.)I have a hackerspace like attitude to maths in addition to the ordinary one
I usually invent new algebraic systems to explore by taking known axioms, and then breaking them
Nah, the $0/0$ on the riemann sphere is not very interesting because it acts like an absorbing element if I recall
In fact, in a lot of systems, $\frac{0}{0}$ are additive absorbers
this include Wheels, probably the cloest thing we have to division by zero
Right now, one of my latest idea is to determine if there is some kind of continuous map between different logic systems so that analogous to how we change basis in linear algebra to make the maths easier, whether we can do something in a more abstract level by pulling the entire proof that "the permenent is #P" into a logic system so that it breaks down, then compute the permenent there, and transform back into the usual logic system to circumvert the proof
However, gut feelign told me that there must be some no-go theorem that prevent me from doing this. My current maths level is however not high enough to explore this idea
Under such hackerspace mindset, one will think of the following besides the usual demand of rigor and meaning of the mathematical objects: 1. The set of all possible ways to prove a given theorem, its optimisation 2. The weak points of the proof and how to break a proof to get something new 3. Theorems and constraints that govern counterexamples as a mathematical object 4. Strong affinity to pathological examples and trying to udnerstand them
This mindset also means any questions that pop up due to misreading of an exercise will be considered seriously, as long it is a geenralisation of what the exericse question is:
This question is inspired from accidentally misreading this question in my tutorial exercise in my linear algebra course because I forgot the 3 in $P_3(\mathbb{R})$
(The actual question can be easily solved by considering the matrix of $T$ and then the eigenvector is found to be the constant p...
I encourage you to share your ideas more with the community. It is true that some of them might be too publishing quality that you might don't want to share directly until they are published on a journal, but for some work in progress ideas, you can choose some of these to shre how you came up your methods and let the community to contribute to your ideas to make it better
I have read previosu chat logs and I have seen you already done some when you explain about the symmetry properties of some integrands with robojohn and others, so keep it up!
@robjohn have you ever tried is one? $$\int_0^1 \left(\frac{\log(1+x)}{1+x}-\frac{\log(1-x)}{1+x}\right) \arctan(x) \textrm{d}x=\frac{\pi^3}{192}+\frac{\log(2)}{2}G$$
I gave it to more students and professors, and no one did it. The idea is to finish it elegantly in a very easy way, such a proof would be required.
Uh, If my memory serves, aren't you already asked robojohn this already around a week ago. I also remember he wrote soemthing about series closed forms afterwards and another user also joined in the discussion?
My questions is relatively simple and that's why I don't want to create an independent post for it. Basically, for proving in maths, can observation be used? Once the same step has been shown to work for many numbers, and it's evidently clear why it happens, can it thus said to be proved? Thanks
@ThePointer Your approach is fine, but as mentioned, you should write down $\lim_{x \to 0} \sin(x)/x$ and $\lim_{x \to 0} \sin(x)/(2x)$ explicitly than just writing down "$0/0$" which is not really an answer.
Suppose that we have the function $f(x,y)=\left\{\begin{matrix} \frac{xy^2}{x^2+y^4} &, (x,y) \neq (0,0) \\ 0 &, (x,y)=(0,0) \end{matrix}\right.$.
Could it hold that $D_{\overline{a}} f(0,0)$ exists for each $\overline{a}=(a_1, a_2) \in \mathbb{R}^2$ with $||\overline{a}||=1$ or $||\overline{a}||=2$ and that f is not continuous in (0,0) ?
Can anyone show me why this is true: Let $a,b,c$ and $d$ be real numberssuch that $a^4+b^4+c^4+d^4=16$. Then $a^5+b^5+c^5+d^5=32$ if and only if one of $a,b,c,d$ is $2$ and others are zero.
The function written by Evinda might be related to the "windmill saddle function" $\frac{x^2}{x^2+y^2}$,, which is known to have discontinous first derivatives at the orign
@Evinda Why don't you try working it out? $f$ is not continuous at the origin is easy to see (approach by different curves). It is possible that every directional derivative exists but that the function is not continuous.
ok I definitely suck at mutivariable limits because of the lack of experience to see the relevent directions (I often start with y=mx, but it almost always fail for the "windmill saddle" and its relatives)
These multivariable calculus continunity exercises are excellant is demonstrating how vast the number of possible paths one can approach any point in the function, which is why multivariable limits are a pain to calculate without experience or sandwich theorem
I use google
If you type any functions of x and y, google will plot them
The issue is that all these paths are located in the space of functions, one of these being $\mathcal{C}^{\infty}(\mathbb{R})$ and there are simply uncountably too infinitely many elements to test them all
$\nabla f \cdot \vec{a}$ give zero unless I must have done something wrong (I might need to read my notes again on how to take the gradient in piemeal functions
@BalarkaSen @Idomathart How do you guys know where the ridges are when just staring at those expressions without graphs? (It's just like the magic when figuring ou how to integrate something)
--- (NB, actually that thing is not a saddle, better just call it the Windmill surface)
Some time later I might ask or experiment whether you can make a function of the form $\frac{P(x,y)}{Q(x,y)}$, where $P,Q$ are polynomials and get a ridge that is not a simple power of x or y
Otherwise, letting $y=x^n$ for $n \in \mathbb{R}$ will guareentee a solution for multivarable limits for these types of functions
I wonder about a closed form for
$ ? = \sqrt[3] {1 + \sqrt[3] {1 + 2 \sqrt[3] {1 + 3 \sqrt[3] {1 + 4 \sqrt[3] {1 + 5 \sqrt[3] \cdots}}}}} $
I tried solving the related equation $ f(x) ^3 = 1 + (x+y) f(x+1) $ for various fixed values integer $y$ , but I failed.
It appears
$$ ? = \sqrt[3] {1 +...
Recently I was studying a linear algebra class which introduce some basic concepts of fields, groups
Once again I tried to divide by zero by constructing an algebraic structure from scratch using just a few axioms because I stumbled across this when browsing the site
We knew that the important ...
I wonder about a closed form for
$ ? = \sqrt[3] {1 + \sqrt[3] {1 + 2 \sqrt[3] {1 + 3 \sqrt[3] {1 + 4 \sqrt[3] {1 + 5 \sqrt[3] \cdots}}}}} $
I tried solving the related equation $ f(x) ^3 = 1 + (x+y) f(x+1) $ for various fixed values integer $y$ , but I failed.
It appears
$$ ? = \sqrt[3] {1 +...
