[texhax] derivatives and integrals: math operators

Eduardo M KALINOWSKI eduardo at kalinowski.com.br
Fri Aug 7 13:40:07 CEST 2009

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?

> 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

And I don't see how that's ambiguous. The \int is the start of the  
operator, the dx (or similar) is its end. And just like with  
parenthesis, brackets, etc, the first dx ends the last integral, and  
so on.

> Especially if f(x,y) is a long and complicated expression, pushing the
> measure far away from the limits. An alternative is to indicate the
> variables explicitly in the limits,
>   \int_{x=0}^{x=1} \int_{y=-1}^{y=1} f(x,y) dx dy
> but this seems to be too long-winded for most people.

eduardo at kalinowski.com.br

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