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$\begingroup$

I've always wanted to know what the name of the vertical bar in these examples was:

$f(x)=(x^2+1)\vert_{x = 4}$ (I know this means evaluate $x$ at $4$)

$\int_0^4 (x^2+1) \,dx = \left.\left(\frac{x^3}{3}+x+c\right) \right\vert_0^4$ (and I know this means that you would then evaluate at $x=0$ and $x=4$, then subtract $F(4)-F(0)$ if finding the net signed area)

I know it seems trivial, but it's something I can't really seem to find when I go googling and the question came up in my calc class last night and no one seemed to know.

Also, for bonus internets; What is the name of the horizontal bar in $\frac{x^3}{3}$? Is that called an obelus?

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    I don't know if it has a name, and if it does, I don't know how useful that information would be to you for talking about mathematics (since I am pretty sure most people would not recognize it).2011-07-20
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    "bonus internets" - ??? Are these the internets that Al Gore *didn't invent* ?2011-07-20
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    As Qiaochu Y. notes, knowing a Latinate name would not help in actual communication, since it is pronounced "evaluated at" in math. Maybe "pipe" in computer science, but that's unrelated. The _double_ vertical line _does_ have a Latinate name, "vel", which is just "or" in Latin. Horizontal lines for grouping (as in large fractions, but also in radicals) are "vinculi" (singular "vinculum"). But no one calls them that. Latin's former status as prestige language seems fading, the way the Sumerian language's prestige for Egypt and Mesopotamia seems to be long gone.2011-07-20
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    @paul garrett I know it wouldn't help in communication, but I just want to know for the sake of knowing.2011-07-20
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    @OghmaOsiris Ok! Do I get a bonus internet, for "vinculum"? :)2011-07-20
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    it's called \mid, at least if you are a TeXnitian. :)2011-07-20
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    Why the vote to close for too localized? This question is about the terminology of a widely used symbol... How is that localized?2011-07-20
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    @BBischof: No, the symbol in question is (best typeset as) `\right|`, `\mid` has the wrong spacing properties.2011-07-20
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    @Zhen Lin, I would say that depends on the usage, I have no spacing issues for using it in set-builder notation or restrictions. Moreover you forgot to mention the \left. However we are deviating quite far from the point of the question. My comment was meant as a joke.2011-07-20
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    I found this question trying to find out how to typeset this in Mathematica, and now that I found out, I just wanted to add it's "\[RightBracketingBar]" for anyone with the same question2017-04-28

4 Answers 4

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In the wikipedia article for the symbol no name for this particular use of it is mentioned, just that it is read as, simply, "evaluated at". It has a number of suggested names for the symbol from different situations though:

verti-bar, vbar, stick, vertical line, vertical slash, or bar, think colon, poley or divider line

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    I saw that, which is another reason why I asked my question.2011-07-20
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Jeff Miller calls it "bar notation" in his Earliest Uses of Symbols of Calculus (see below). The bar denotes an evaluation functional, a concept whose importance comes to the fore when one studies duality of vector spaces (e.g. such duality plays a key role in the Umbral Calculus).

The bar notation to indicate evaluation of an antiderivative at the two limits of integration was first used by Pierre Frederic Sarrus (1798-1861) in 1823 in Gergonne’s Annales, Vol. XIV. The notation was used later by Moigno and Cauchy (Cajori vol. 2, page 250).

Below is the cited passage from Cajori

enter image description here enter image description here

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This may be called Evaluation bar. See, in particular, here (Evaluation Bar Notation:).

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    Example 1: Page 2, third item from the end, here: http://www.math.uaa.alaska.edu/~afmaf/classes/math201/notes/IntParts/IntegByParts.pdf2011-07-20
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    Example 2: Page 2, item 5, here: http://www.princeton.edu/~slynch/bayesbook/errata.pdf2011-07-20
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    Example 3: You can the sentence "and I couldn't figure out how to make the "evaluation bar" immediately to the right of the uv terms any bigger", here: http://www.physicsforums.com/showthread.php?t=3761902011-07-20
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    I like this name. +1 just for that XD2011-07-20
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    @Patrick: Thanks. Apparently, this is standard terminology.2011-07-20
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    Then perhaps I like it because many do.2011-07-20
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    the link doesn't work...2013-02-20
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    @draks An alternate post regarding the same topic may be found [**here**](http://math.stackexchange.com/questions/126868/function-vertical-line-and-two-values-notation-for-evaluation-at-1-point-or-2/130010#130010).2013-07-17
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In my calculus book, the vertical bar is called the "evaluation symbol", and this phrase is bolded when first mentioned. It makes sense, I suppose.

Copy paste from wikipedia: Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a vinculum or fraction bar, between them.