Product notation in partial differentiation

Question

For a function $f:\mathbb{R}^n\to \mathbb{R}$, is it correct to write, for any $n\in \mathbb{N}$, the expression $$ \frac{\partial^n f}{\partial x_1\cdots \partial x_n}=\frac{\partial^n f}{\prod_{i=1}^n\partial x_i} $$ Context: I am not against the notation on the left, but it would be interesting to write it like that since in my original expression, the partial derivative is multiplied by the term $\prod_{i=1}^n x_i$, and it would "look" better. Any thoughts?

Answer 1

I never saw such a notation. And the left side expression is sort of symbolic and what is in its "denominator" cannot be considered as a product. It is rather the whole symbol that is a "product" in the sense of an operator composition (the i-th operator in the "product" being that of partial derivation rel. to the i-th variable). So your notation (right side) might not communicate well to people not aware of your convention ...