-1

I am currently studying PDE's where the integration of partial derivative is commonplace. I have seen that in a lot of literature the integral of a partial derivative is just written as

$$\int f(x,y)dx$$

But I find that if you were to write something like $\int f(x,y)\partial x$ it would be much clearer that the integral will have an arbitrary function rather than just a constant. In my head this makes sense but I was wondering if there reason this convention is not widely adopted or if it is even correct to write an integral like this at all?

Diaz.Mrn
  • 136
  • 1
    @LL3.14 I think that your comment would be fine if posted as an answer. – Mark S. Feb 12 '23 at 01:03
  • What notation is widely adopted in contrast to the one you might prefer has to do with taste, democracy, tradition but little to do with mathematics itself. Personally I follow the crowd and most importantly I spend a few words to explain if necessary what my notation means. – Kurt G. Feb 12 '23 at 10:24
  • There is more than taste, democracy and tradition. One should still try for example to be at least coherent ... here are other advantages one can think about when choosing a notation https://mathoverflow.net/questions/366070/what-are-the-benefits-of-writing-vector-inner-products-as-langle-u-v-rangle – LL 3.14 Feb 12 '23 at 10:41

1 Answers1

1

You are not integrating a derivative or a partial derivative of $f$, but just the function $f$ itself with respect to one variable. You can see $\mathrm dx$ (the Lebesgue measure) as the derivative of the function $x$, but then it only depends on one variable, so there is no reason to write a $∂$ symbol. See also here for more about the notation $\mathrm dx$.

The notation $\mathrm d f$ on the contrary was already chosen to be the (total) differential of $f$, which is why one then has to denote with a $\partial f$ the partial derivatives.

LL 3.14
  • 12,457