[texhax] derivatives and integrals: math operators

Toby Cubitt tsc25 at cantab.net
Fri Aug 7 14:12:45 CEST 2009

Eduardo M KALINOWSKI wrote:
> On Sex, 07 Ago 2009, Toby Cubitt wrote:
>> Phil Parker wrote:
>>> Another physics quirk is to write \int dx\,f(x) instead of \int
>>> f(x)\,dx as we
>>> do. That just creates unnecessary confusion, especially in elementary
>>> (as in Calculus III) double and triple integrals.
>> I have to disagree here. I think people do this precisely to avoid
>> confusion in multiple definite integrals. Writing
>>   \int_0^1 dx \int_{-1}^1 dy f(x,y)
> How does one know when the integral ends with this notation? Especially
> if something comes after the f(x, y), such as
> \int_0^1 dx \int_{-1}^1 dy f(x,y) + \pi
> Does one integrate f(x, y) + \pi, or do we add \pi to the final result?

In your example, I think the usual convention would apply: products bind
more strongly than addition, so the "+ \pi" isn't integrated over. But in
general, brackets might be necessary to avoid ambiguity.

>> makes it clear which limits belong with which variable, whereas
>>   \int_0^1 \int_{-1}^1 f(x,y) dx dy
>> can easily lead to mistakes (at least amongst my Calculus I students!).
> To mean the same as the really weird notation of your first equation
> (which I had never seen), and the complete formula below, it should be
> \int_0^1 \int_{-1}^1 f(x,y) dy dx

True. Which only illustrates my point :)

> And I don't see how that's ambiguous. The \int is the start of the
> operator, the dx (or similar) is its end.

I never said it was ambiguous, just that it's liable to lead to mistakes
in calculations. Which is why some physicists (who, after all, are
generally the ones who have to *compute* definite integrals) sometimes
prefer to place the measure next to the integral.

The problem stems from having to scan all the way to the end of the
expression and parse the ordering, just to figure out which limit applies
to which variable, whilst the integration itself is being carried out left
to right. This seems to confuse the human brain, and be prone to errors.

If NASA engineers can crash a Mars orbiter by forgetting which units
they're supposed to be using, physicists can get integrals wrong by
applying the wrong limits to the wrong variable. It seems to me that
there's a valid argument for notational help in avoiding mistakes like
this, and that it shouldn't be dismissed out of hand as "creating
unnecessary confusion".

> And just like with parenthesis, brackets, etc, the first dx ends the
> last integral, and so on.

The notation for integration is rather unique and unusual in mathematics.
The vast majority of operators don't get split up in this way. Opening and
closing delimiters are often confusing for human brains to parse (ever
tried to read a large block of unindented Lisp code? :)

I'm not advocating religiously putting the measure next to the integral in
all cases. I just think the notation has its place, and it seemed worth
attempting to explain why I believe it's used occasionally.


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