       Re: Wolfram, meet Stefan and Boltzmann

• To: mathgroup at smc.vnet.net
• Subject: [mg117486] Re: Wolfram, meet Stefan and Boltzmann
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 20 Mar 2011 04:54:05 -0500 (EST)

```"Anti-derivative" is a somewhat newer term than the very old "indefinite integral" and is used particularly often in complex analysis and in more algebraic contexts (for example differential algebras, where the word "indefinite integral" would definitely sound unsuitable).  For example "anti-derivative" is used in the classic "Complex Analysis" by Lars Ahlfors.

Anti-derivative is also used in textbooks of real analysis where the authors are particularly careful to emphasise concepts rather than computing techniques.
For example, it is also used in Apostol's very popular "Calculus", although it is given there as the alternative (second) choice - the first choice being "the primitive" of a function (the word "indefinite integral" is not even mentioned).

Although the words "antiderivative" and "primitive" are newer than "indefinite integral" they are not really very new - the oldest book I have that uses these terms seems to be "The theory of functions of a real variable" by L.M.Graves, which was published in 1946.

One reason why the words "primitive" and "anti-derivative" are preferred to"indefinite integral" is that they emphasise the non-trivial nature of the fundamental theorem of calculus. To say that you can compute the definite integral by evaluating an indefinite integral at the limits of integration and subtracting sounds almost like a tautology; to say that you can compute the integral of a function by finding and evaluating its primitive, sounds like the profound result that it actually is.

Andrzej Kozlowski

On 19 Mar 2011, at 11:20, AES wrote:

> In article <ilve3r\$emn\$1 at smc.vnet.net>, SigmundV <sigmundv at gmail.com>
> wrote:
>
>> It also astonished me that AES is not familiar with the term
>> 'antiderivative'. The derivative of the antiderivative is the function
>> itself.
>
> Pretty obvious what it means, of course.  But:
>
> 1)  "Antiderivative" doesn't appear in the New Oxford American
> Dictionary; "indefinite integral" does.
>
> 2)  The MIT Math Department's online "Calculus for Beginners" course
> says:
>
>   16.1 The Antiderivative
>
>   The antiderivative is the name we sometimes (rarely) give
>   to the operation that goes backward from the derivative of
>   a function to the function itself . . . The more common name
>   for the antiderivative is the indefinite integral. This is the
>   identical notion, merely a different name for it.
>
> 3)  I'm 250 miles from my home library at the moment, so can't look in
> the indexes of Morse and Feshbach or comparable classics; but amazon.com
> has an online searchable listing for Courant and Hilbert, Methods of
> Mathematical Physics, and "antiderivative" doesn't appear in its index,
> or anywhere else in the book.
>
> And so on . . .
>

```

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