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What are concepts that are generally notated by $\partial$?

I know two things that are denoted by $\partial$. Those are "the boundary of a set in a topological space" and "partial derivatives". What are other concepts generally notated by $\partial$?

The reason why I am asking this:

I'm now studying differential calculus in the context of normable topological vector spaces. Let $V_1,...,V_n,W$ be nonzero normed spaces and $E$ be open in $\prod_{j=1}^n V_j$ and $f:E\rightarrow W$ be a function. Then, when $p\in E$, $\frac{\partial f}{\partial x_i}(p)$ means a bounded linear transformation from $V_i$ to $\prod_{j=1}^n V_j$ which is the Fréchet-derivative of $f$ along axis $V_i$ at $p_i$. When $V_j$'s are $\mathbb{R}$, $\frac{\partial f}{\partial x_i}(p)$ can be viewed as a single element in $W$. To be very precise, I decided to denote $\frac{\partial f}{\partial x_i}(p)(1)$ by $\partial_{x_i} f(p)$. However, I wonder if there is a case when $\partial_{x_i} f(p)$ is a standard notation for some other concepts.

Rubertos
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1 Answers1

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An example use is for divided difference operators on polynomial rings with Coxeter group actions. If $p$ is a (multivariate) polynomial and $s$ is a simple reflection with corresponding simple root $x_s$, then $$\partial_s(p)=\frac{p-s(p)}{x_s}$$ If $w=s_1s_2\cdots s_n$ and $\ell(w)=n$, then $$\partial_w=\partial_{s_1}\partial_{s_2}\cdots\partial_{s_n}$$ You can do a Google search for something like "divided difference operators Schubert calculus" or you could look at my dissertation (linked in my profile).

Matt Samuel
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