yes but on a higher level. i really want to know for example how to turn an if statement into math. But i'm only still in secondary school and so all i really know currently is +-*/,p
Those are the operations and functions that we use the most, but there's no reason to limit ourselves only to those things and combinations of those things
But, uh, if you have learned about piecewise functions
then "if" statements most directly translate into those
So look up piecewise functions if you haven't learned about them yet @Cosinux
Does the map $\mathbb R^2\to\mathbb R^2$, $(x,y)\mapsto(x^2-y^2,2xy)$ have an inverse when restricted to some open ball? I think no and have it somewhat worked out, but am not sure.
@EsX_Raptor Yes. It's Jacobian matrix is $\begin{pmatrix} 2x&2y\\ -2y&2x\end{pmatrix}$ which has determinant $4(x^2+y^2)$ which is nonzero whenever $(x,y)$ is nonzero. Hence the inverse function theorem says $f$ is a local diffeomorphism at every point not the origin.
This is in fact the map $\Bbb C\to\Bbb C$ that sends $z\to z^2$, and one can in general show that a complex polynomial is a local diffeomorphism (in fact biholomorphism) in any point away from its roots (which are "branched" points).
Huh? 1) Inner products take two inputs, not one. 2) Where did the inner product come from in the first place? That shouldn't be part of the input in "Blah does not converge in the $\infty$-norm" - it didn't show up when you were talking about convergence in norm above.
Definition. Let $X$ denote a set. Whenever $\tau$ is a topology on $X$, write $\tilde{\tau}$ for the collection of subsets of $X$ that are connected from the viewpoint of the space $(X,\tau)$.
In general, we cannot recover $\tau$ from $\tilde{\tau}$. For instance, let $\tau$ denote the usual...
Is someone good here in graph theory? Puzzling on this where trying to find the proper jargon.
Somehow turning graph to tree.
It should be here sufficient to consider undirected graphs so wondering whether block graphs and cliques could be useful (just reading their wikipedia).
@Kari Yeah. I don't know of a concise way to put it (because I don't understand the story yet), but it's like a formalization of vector calculus. The fundamental objects there are vector fields, and you integrate vector fields along paths (line integral) or surfaces (surface integral) and do stuff. But the fundamental objects in this context are, well, differential forms. So turns out what you integrate, differentiate, etc are not vector fields but differential form.
@Kari Nice, anything interesting that you can tell me about?
So far I really haven't seen much as I started reading yesterday, but I've just been looking at equivalent ways of forming initial value problems. There have been a few errors in the lecturers hand-written notes so I haven't been able to make much progress.
We began looking at $(\mathcal{C}([-\tau,\tau];\mathbb{R}^n), \| \cdot \|_{\infty})$ being a complete vector space but the supremum norm he defined looked like: $$\| f \|_{\infty} = \sup_{t\in[-\tau,\tau]} |f(t)|. $$
@Kari Note that not much is lost in assuming that he means $\Bbb R^n$ and Euclidean norm of $f(t)$. Because if he really means $\Bbb R$ and the absolute value of $f(t)$, then that'd just be a special case of what you assumed for $n = 1$!
I find it a bit annoying that I can't explain what's the deal with differential forms in a few words. There is something fundamental missing in my understanding of those, I think, because most of the time what I really understand I can explain in a motivating/easy way.
On a phone, so cant respond much. Does anyone know what an injection of bialgebras requires? Does it do anything to preserve bialg structure, or just two bialgebras and an injection?
a form on a vector space eats parallelpipeds and spits out scalars. a form on a manifold eats a smoothly varying parallelpiped sticking out of each point and spits out a smooth varying scalar, i.e. smooth functions.
@MikeMiller Is it true that given any manifold there exists a vectr filed with finitely many zeros?? For compact manifold it is true (followed from tansversility theorem)
@Semiclassical Sure, but this doesn't actually use anything special about differential forms. You might as well use vector fields and integrate them on surfaces.
The physical picture doesn't tell me why I should use differential forms and not vector fields.
sure. my point is only that the formal aspects don't enter into my picture of a flux integral---the differentials themselves, as 'infinitesimal contributions' to the flux, are what i fall back on
i'm not saying it's internally consistent, mind. just that it illustrates the limitations of my thinking.
I see what you mean, but what I am trying to pin-point is why differential forms is not just a formal thing. I "feel" that it isn't, but I can't find the precise reason.
It seems like an answer to your question would contribute a lot to the understanding, but I haven't thought about it.
to put it a somewhat different way, i tend to think of the basis elements of such a vector field as stuff like unit vectors (possibly position-dependent, if i'm working in spherical coordinates)
rather than having the basis elements be differential operators
though it's funny. i stress that i think of unit vectors rather than differential operators, and yet if someone asked me how my vector transforms under coordinate transformations, i interpret said unit vectors as transforming in precisely the same way as differential operators would naturally
@Semiclassical I think I got an answer to your question. Note that a form eats an oriented parallelpiped (equiv. a bunch of vectors) and spits out it's oriented area (that's what the determinant does), not just an arbitrary scalar. So if I have $f dx_1 \wedge \cdots \wedge dx_n$, and integrate it over some parameterized surface formally by removing the wedges, it shouldn't be too surprising that it should be a natural thing to do, right?
Well, not quite a very satisfying answer yet, but it's intuitively understandable that a form should have something to do with volume.
E.g., if I take a parametrized n-manifold $M$ inside $\Bbb R^m$, and I have the form $f dx_1 \wedge \cdots \wedge dx_n$, then evaluating that at a point $x$ in $M$, it spits out $f(x)$ times the volume of the parallelpiped on $T_xM$ spanned by a bunch of tangent vectors $v_1, \cdots, v_n$ I have fed it to.
there i have two ways of thinking about $\int_C \mathbf{F}\cdot d\mathbf{x}$: either as a riemann sum $\sum_k \mathbf{F}_k\cdot (\Delta \mathbf{x})_k$, or via the FTC using some locally-defined antiderivative
in neither case do i seem to be appealing to the nature of $\mathbf{F}\cdot d\mathbf{x}$ as an object which acts on vector fields.
(i should have some kind of limit in front of the Riemann sum, but w/e)
the first one is just direct summation of infinitesimal contributions, and the latter makes use of some 0-form in order to compute the result.
now, i suspect that what i'm missing is that i'm not treating $C$ with sufficient respect (whatever that means)
@balarka does that make any sense? not sure it does.
yes, it does. I think there should be a way to interpret the integral using the fact that $\mathbb{F} \cdot d\mathbb{x}$ eats the tangent vectors of $C$.
@SemiC If I feed in a tangent vector $\mathbf{T}$ in $T_x \mathcal{C}$ to $\mathbb{F} \cdot d\mathbf{x}$, then $\mathbb{F} \cdot d\mathbf{x} (\mathbf{T}) = \mathbb{F} \cdot \mathbf{T}$. I am pretty sure this should somehow mean that the integral is really $\int_C \mathbb{F} \cdot \mathbf{T} ds$ where $ds$ means an infinitsimal patch of the curve $\mathcal{C}$, and that is exactly it. But maybe I am thinking too hard.
This is a good conceptual question I don't really know the answer to.
we were discussing how the interpretation of differential forms as something which eats k vector fields and spits smooth function links with something which can be integrated over parametrized submanifolds.
btw. where i was going was that, if one uses the arc length parametrization, then $\int_C \mathbf{F}\cdot d\mathbf{x}=\int_0^L \mathbf{F}\cdot \hat{\mathbf{T}}\,ds$
I don't know of a rigorous way to fill that bit in. One can think of $ds$ as an infinitisimal patch on my curve $C$, and the intuition is more or less what it should be.
I guess a good question could be "if it's not a 1-form, what the hell is it?". That is, how to make the expression $\int F \cdot T ds$ rigorous. It's not clear to me how one would do that.
@Semiclassical I looked again and essentially the guy is saying what I wrote above. Pulling back say the form $fdx$ by some smooth function $g$ gives $f(g(x))g'(x)dx$, and $f(g(x)) g'(x)$ is precisely what you get when you feed in the vector $g'(x)$ to $f dx$.
@robjohn Thanks. Remembering this event is OK, not matter the day.
People should think (once in a while) of a simple fact if talking about God: what is the mathematical probability that all 12 apostols were highly gifted and spread the gospels in such an amazing way with so poor knowledge they had, and they also did many outstanding miracles.
This is for those that said Jesus didn't even existed, or he was just an ordinary person.
It's not so surprising, but admittedly the formal aspect is not clear to me. That is, it's not clear to me how to do the integration and things in the setting of symmetric tensors, formally.
As the starting point for my experiment I assumed that the imaginary parts of the Riemann zeta zeros are of the form:
$$\Im \{ \rho_n \} = \frac{2\pi}{\log x_n}$$
where $x_n$ is unknown. Therefore I solved for $x_n$ and got this integer sequence:
$$\Large a_1(n) = (n+1) \left(\left[\frac{1}{e^...
One last question on religion: is the modern world today able to turn 12 simple, uneducated people into persons like the apostles in the time of Jesus such that they are able to change the world? There are 12, each one very special and able of doing amazing things!
Definition. Let $X$ denote a set. Whenever $\tau$ is a topology on $X$, write $\tilde{\tau}$ for the collection of subsets of $X$ that are connected from the viewpoint of the space $(X,\tau)$.
In general, we cannot recover $\tau$ from $\tilde{\tau}$. For instance, let $\tau$ denote the usual...
As the starting point for my experiment I assumed that the imaginary parts of the Riemann zeta zeros are of the form:
$$\Im \{ \rho_n \} = \frac{2\pi}{\log x_n}$$
where $x_n$ is unknown. Therefore I solved for $x_n$ and got this integer sequence:
$$\Large a_1(n) = (n+1) \left(\left[\frac{1}{e^...
