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Occasionally, I see a closed integral - often with relation to Green's Theorem - where the curve or region is denoted as: $\partial R$, and once I even saw: $\partial \text{Rect}$. Here is an example: $$ \text{circ} = \oint _{\partial R} \vec{F}\cdot d\vec{r} $$ I am wondering why this notation is used for the curve? most of the time we just use $C$ or $R$ for 'curve' or 'region'. I wondered if this had anything to do with the theorem of line integrals, with the curve being the line integrated on some gradient field, but that doesn't make much sense for a couple reasons. Thank you for reading, all answers are appreciated :)

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This use of "$\partial$" means "boundary of". Several such uses appear in this question and its answers. So $\partial$Rect is the boundary of a rectangle and $\partial R$ is the boundary of the region $R$. (Typically, one imagines the boundary positively oriented, but the path integrals you mention are oblivious to whether the boundary is parametrized in the positive or negative direction -- or even in a wandering back and forth manner -- as long as the path encloses the region exactly once.)

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  • $\begingroup$ thank you :), I was confused, because I thought it had something to do with partial derivatives. $\endgroup$
    – Poobean
    Aug 24 '21 at 2:13
  • $\begingroup$ So, ..., there is a connection with partial derivatives. See Greens theorem. One can integrate the "plain integrand" on the boundary or the partial derivative of the integrand on the region. There's a partial either applied to the integrand or to the region. But this is more a suggestive notational hack than anything like partial differentiation of the region. $\endgroup$ Aug 24 '21 at 2:17

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