Re: Wolfram, meet Stefan and Boltzmann

• To: mathgroup at smc.vnet.net
• Subject: [mg117507] Re: Wolfram, meet Stefan and Boltzmann
• From: Robert Rosenbaum <robertr at math.uh.edu>
• Date: Mon, 21 Mar 2011 06:14:51 -0500 (EST)

```I should probably let this tangent topic die off, but I thought I'd give my two cents.

"Anti-derivative" is a useful and frequently used term.  I use it often when teaching calculus and it's also used in textbooks.

However, the phrase "the anti-derivative" is meaningless since anti-derivatives are never unique.  When teaching, I always say "an anti-derivative," lest the students become confused.

Best,
Robert

On Mar 20, 2011, at 4:54 AM, Andrzej Kozlowski wrote:

> "Anti-derivative" is a somewhat newer term than the very old "indefinite
integral" and is used particularly often in complex analysis and in more al=
gebraic contexts (for example differential algebras, where the word "indefi=
nite integral" would definitely sound unsuitable).  For example "anti-deriv=
ative" is used in the classic "Complex Analysis" by Lars Ahlfors.
>
> Anti-derivative is also used in textbooks of real analysis where the auth=
ors are particularly careful to emphasise concepts rather than computing te=
chniques.
> For example, it is also used in Apostol's very popular "Calculus", althou=
gh 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 "indef=
inite 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" b=
y 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 th=
e 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 comput=
e the integral of a function by finding and evaluating its primitive, sound=
s 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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