17

Interior multiplication from John Lee's Smooth Manifolds

See the above excerpt from John Lee's Introduction to Smooth Manifolds. In differential topology and linear algebra, there's an operator called the interior product or contraction operator, denoted $\iota_X(\omega)$ or alternatively with some other "half box" like operator $X_|\omega$. Does anyone know how to markup that operator in LaTeX?

Google has turned up nothing except for this thread: Symbol for contraction / interior product?

That thread did receive an accepted answer, which was \invneg from the MnSymbol package, but the suggestion given there does not seem correct to me. Or at least, it does not match what I'm familiar with. That answer results in an inverted negation symbol, whereas the contraction operator should have a stroke down at the floor of the line, and a long stroke going up almost the full height of the line. It can be found in many texts, but why can I not find the correct LaTeX markup?

ziggurism
  • 373
  • 4
    Looks like http://tex.stackexchange.com/questions/14/how-to-look-up-a-math-symbol to me – Joseph Wright Dec 19 '11 at 21:44
  • Is the following link a good reference that shows the notation you're after: The Interior Product? We're after an image to see whether this could be contained within an existing font or command (and therefore perhaps be solved by @JosephWright's link), or perhaps requires a larger construct. – Werner Dec 19 '11 at 21:45
  • The rendering of the character in Asana Math and in the STIX fonts is just like a "reversed \lnot". – egreg Dec 19 '11 at 21:50
  • Detexify lets you draw the symbol and shows you matches from all the various LaTeX packages. Pretty awesome: http://detexify.kirelabs.org/classify.html. – Justin Bailey Dec 20 '11 at 00:00
  • 1
    detexify doesn't recognize it – ziggurism Dec 20 '11 at 10:46
  • But the "Comprehensive Latex Symbols" list included one called \righthalfcup which looks like it might be right. – ziggurism Dec 20 '11 at 10:48
  • Are there issues with using the MnSymbol package? I found this thread http://tex.stackexchange.com/questions/17370/how-to-use-mnsymbol-along-with-other-packages Should I try to find the same symbol in another package? – ziggurism Dec 20 '11 at 11:18
  • @Werner: That link does describe the mathematical operation I'm seeking. It shows one notation with no typesetting difficulties, which is a subscript on an \iota or i. There is an alternate notation which is what I'm looking for, which is not mentioned in that link. – ziggurism Dec 20 '11 at 13:16
  • Could you find an adequate picture of this symbol, so we know what we are looking for? – Seamus Dec 20 '11 at 13:17
  • @egreg: Ok, I think you're right. Except it's an inverted, not reversed, negation symbol. I originally said that the inverted negation was incorrect, but some further googling suggests that some people actually do use that notation for interior product. However it remains true that several texts I'm familiar with use a somewhat different notation with longer strokes than a negation symbol. So there are at least two alternative symbol choices for this operator. I prefer the longer one, and am still seeking the correct markup (it may be \righthalfcup from MnSymbol though) – ziggurism Dec 20 '11 at 13:20
  • @Seamus: I'm not sure how to find symbols on the web, but if you go to the amazon page for "Smooth Manifolds" by "John Lee" at http://www.amazon.com/dp/0387954953/ then choose "Search inside another version of this book" and then search for "interior multiplication", you will see the symbol near the top of page 335 in the first result. – ziggurism Dec 20 '11 at 19:12
  • 1
    I'm reading a paper which uses \rfloor (it's on arxiv so you can read the source), but it looks terrible and nonstandard. – ziggurism Dec 23 '11 at 19:27
  • Possible duplicate of https://tex.stackexchange.com/q/51125/9285 – ziggurism Dec 07 '17 at 16:42

3 Answers3

20

I stumbled upon the correct markup used in Wikipedia's article on geometric algebra. I looked at the code and found the symbol is given by \lrcorner. \llcorner is the same thing but flipped horizontally.

Alex Eftimiades
  • 316
9

You can import the symbol from MnSymbol by saying

\DeclareFontFamily{U}{MnSymbolC}{}
\DeclareSymbolFont{MnSyC}{U}{MnSymbolC}{m}{n}
\DeclareFontShape{U}{MnSymbolC}{m}{n}{
    <-6>  MnSymbolC5
   <6-7>  MnSymbolC6
   <7-8>  MnSymbolC7
   <8-9>  MnSymbolC8
   <9-10> MnSymbolC9
  <10-12> MnSymbolC10
  <12->   MnSymbolC12}{}
\DeclareMathSymbol{\intprod}{\mathbin}{MnSyC}{'270}

This way you won't override your math fonts with MnSymbol.

The alternative way with reversing the \lnot symbol is

\usepackage{graphicx}
\newcommand{\intprod}{\mathbin{\raisebox{\depth}{\scalebox{1}[-1]{$\lnot$}}}}

If the symbol is needed also in other sizes, then

\newcommand{\intprod}{\mathbin{\mathpalette\dointprod\relax}}
\newcommand{\dointprod}[2]{%
  \raisebox{\depth}{\scalebox{1}[-1]{$#1\lnot$}}}

Here's a picture of the output in the two ways.

enter image description here

egreg
  • 1,121,712
5

Inspired in the answers by @egreg and @alex-eftimiades my solution was to define the \lrcorner symbol as a mathematical binary operator, i.e.

\newcommand{\iprod}{\mathbin{\lrcorner}}

So, putting it at work

  \begin{equation*}
  \begin{split}
  X \iprod (\omega_1 \we \omega_2) & = (X \iprod \omega_1) \we \omega_2 + (-1)^r \omega_1 \we (X \iprod \omega_2)
  \\
  X(\omega) & = X \iprod \de{\omega} + \de{(X \iprod \omega)}
  \\
  X(Y \iprod \omega) & = [X,Y] \iprod \omega + Y \iprod X(\omega)
  \end{split}
  \end{equation*}

yields

Result of the definition

Dox
  • 5,729