Why start at x, y, z - User contributions [en-gb]

Function application without parentheses

2022-10-21T15:01:40Z

Spagety:

[[Category:Ambiguities]]

It's sometimes OK to write a function name followed by its argument, without any parentheses. Because there's [[There is no function application symbol|there is no function application symbol]], this can look like a [[Missing multiplication symbol|multiplication]].

This is an example of [[juxtaposition means combine in the obvious way]].

This is most usually done with the trigonometric functions and logarithms, e.g.

\[ \sin \theta, \ln x \]

So is it only OK to omit the parentheses for well-known functions? Or is it for any function whose name is longer than one letter? Christian Lawson-Perfect asked<ref>[https://twitter.com/christianp/status/1412411447552917515 Tweet by Christian Lawson-Perfect], [https://mathstodon.xyz/@christianp/106534017984381320 toot by Christian Lawson-Perfect]</ref><ref>[https://twitter.com/christianp/status/1036895727442644992 Earlier tweet by Christian Lawson-Perfect]</ref>, and got mixed responses.

Some people interpret \(\sin(x)^2\) to mean \(\sin(x^2)\) rather than \((\sin x)^2\), even though \(f(x)^2\) normally means \((f(x))^2\).

Deyan Ginev searched for occurrences of <code>f x</code> in arXiv papers<ref>[https://gist.github.com/dginev/0587ce0264f03f2787503b6e4c14a342 A report on the direct uses of "f x" (without parentheses) in arXiv] by Deyan Ginev</ref> and found about 7,000 papers containing that pattern, containing a wide range of exotic uses. In a handful of which, "f x" was indeed a function application.

[[File:F-x-for-brevity.png|thumb|alt="To simplify expressions we often omit parentheses writing f x = f (x) etc." [math/0201098] |Clear statement of the "f x" shorthand, from arXiv:math/0201098]]
==References==
<references/>

Field

2022-10-21T14:55:34Z

Spagety:

[[Category:Ambiguities]]
[[Category:Language]]
[[Category:Local variations]]

The word "field" has two unrelated meanings in mathematics. In algebra, it refers to a nontrivial commutative ring in which every nonzero element is a unit, while in differential geometry and theoretical physics, it refers to a function from a manifold to a vector space. This ambiguity does not exist in some languages like french and german, where fields in the former sense are referred to as "bodies" (french: corps, german: Körper).

Inverse functions and preimages

2022-10-20T20:07:14Z

Spagety: Created page with "Category:Ambiguities Category:Inconsistencies Category:Unpleasantness Category:Conflicting definitions The notation \(f^{-1}\) usually denotes the composition..."

[[Category:Ambiguities]]
[[Category:Inconsistencies]]
[[Category:Unpleasantness]]
[[Category:Conflicting definitions]]

The notation \(f^{-1}\) usually denotes the compositional inverse function of \(f\), that is, a function \(g\) such that, for all \(x\) in the domain of \(f\), \(g(f(x)) = x\), and, for all \(x\) in the domain of \(g\), \(f(g(x))=x\). This is a special case of the convention that \(f^n\) denotes the \(n\)-fold composition of \(f\) with itself, but, confusingly, it is also used in some contexts where \(f^n\) normally denotes \(n\)-fold ''pointwise multiplication'' of \(f\) with itself; see [[Powers of trigonometric functions]]. The inverse function does not exist unless \(f\) is injective.

In some contexts, \(f^{-1}\) denotes the preimage function from the power set of the codomain of \(f\) to the power set of the domain of \(f\), defined by \(f^{-1}(Y) = \{x : f(x) \in Y\}\). This exists for all functions, whether or not they are injective.

In other contexts, like when \(f\) is an element of a ring of functions on a space, \(f^{-1}\) can denote the ''multiplicative'' inverse of \(f\). This only exists if \(f(x)\) is invertible for all \(x\).

==References==
[https://math.stackexchange.com/questions/1024280/most-ambiguous-and-inconsistent-phrases-and-notations-in-maths Most ambiguous and inconsistent phrases and notations in maths] on math.stackexchange
<references/>

Field

2022-10-20T17:16:30Z

Spagety: Created page with "Category:Ambiguities Category:Language The word "field" has two unrelated meanings in mathematics. In algebra, it refers to a nontrivial commutative ring in which ev..."

[[Category:Ambiguities]]
[[Category:Language]]

The word "field" has two unrelated meanings in mathematics. In algebra, it refers to a nontrivial commutative ring in which every nonzero element is a unit, while in differential geometry and theoretical physics, it refers to a function from a manifold to a vector space. This ambiguity does not exist in some languages like french and german, where fields in the former sense are referred to as "bodies" (french: corps, german: Körper).