Definition talk:Regular Value

The problem is that it is tedious and confusing for others to define pushforward without a rigorous definition of tangent spaces, tangent bundles and vector fields, which do not exist yet. The symbol $f_* \vert_x$ is a notation for "the pushforward of $f$ at $x$". After thinking a while, one could take $f_* \left(x\right)$ or $f_* \restriction_x$, but the latter produces a thick space and it is not really $f_*$ restricted to $x$, but $f_*$ evaluated at $x$ acting on tangent vectors. I read the house style page and tried sticking to it, which points did I not follow? I will try following it next time.--Geometry dude (talk) 20:00, 15 September 2014 (UTC)

Sorry, forgot to remove the "tidy" template (I've removed it now), but here are a few pointers:

1. Within a left/right pair, we insist on a pair of braces: \left({x}\right) as opposed to \left(x\right), for a few reasons, one of which is portability.

2. No need for \colon or \lbrace, \rbrace -- we use : and \{, \} consistently as they're simpler. In fact, for the latter it's \left\{ {, }\right\} to keep with the consistent left/right style.

3. Please, no underlines in internal links: it makes it much more difficult to do a global search on entities. Thus, please use e.g. [[Definition:Smooth Manifold]] and not [[Definition:Smooth_Manifold]].

Every page is designed on $\mathsf{Pr} \infty \mathsf{fWiki}$ to be usable as an entry page. That is, anyone should be able to come to it with no other knowledge at all, and be able (through dint of linking back and filling in the gaps of their knowledge until they reach a level they *do* understand) to make sense of the page.

In this context, the reader has no chance of doing that until we have links to:

a) Definition:Pushforward to define what $f_* \left({x}\right)$ means

b) Definition:Tangent Space to define what $T_x$ means

c) whatever $|_x$ actually means in this context.

We already have Definition:Pushforward Measure which is adequately defined -- it should be straightforward to define the concept in the topological context without too much trouble.

It is, however, noted that in the Definition:Tangent Space page we still need to define what the derivative is of a manifold, which is a long way off yet -- we've barely got started on Complex Analysis.

(Incidentally, if you put \restriction into braces: {\restriction} it seems to lose the thick space before it. Don't know why.)

In a perfect world, the groundwork for *all* pages on this site would be covered before jumping into something as heavy as this -- but a few early contributors did not share the yet-to-evolve $\mathsf{Pr} \infty \mathsf{fWiki}$ philosophy of providing a rigorous framework into which to build up the library of proofs; they preferred to post up deep and interesting proofs of things which could not be understood without a solid background in the particular subject.

As has been stated a few times on this site: it's easy to just post up proofs -- anyone can do that. It is far more challenging to construct the rigorous framework and house style into which the entirety of the mathematical structure underpinning these proofs is consistent and easily comprehended. --prime mover (talk) 21:02, 15 September 2014 (UTC)

Let me note some thoughts. $f_* \vert_x$ is not a restriction.

I think, $f_* \vert_x$ is usually written as $d_x f$ in the literature.

If $X_x \in T_x X$ defined as Definition:Tangent Vector/Definition 2 by:

$\forall \phi \in \map {C^\infty} {V, \R} : \map {X_x} \phi := \map {\dfrac \d {\d \tau} {\restriction_0} } {\map {\phi \circ \gamma} \tau}$

then $\map {d_x f} {X_x} \in T_{\map f x} Y$ is e.g. to be defined by:

$\forall \psi \in \map {C^\infty} {\map f V, \R} : \map {\map {d_x f} {X_x} } \psi := \map {\dfrac \d {\d \tau} {\restriction_0} } {\map {\psi \circ f \circ \gamma} \tau} $

--Usagiop (talk) 20:57, 9 November 2022 (UTC)