Are curl and divergence local properties?
Example: curl is 0 and/or divergence is non-zero for a field. Does that mean that it has this property at some point in that field or all the points in that field?
Like, I am thinking of a simple electric field lines due to a stationary charge. The divergence is non-zero at the point where the charge is located but if we move sufficiently far away then the lines are basically parallel and we have a situation where divergence is almost zero.
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$\begingroup$Classically both, as well as the gradient, are local operators. Applied to any smooth function they depend only on the values in the vicinity of a point. Generalizing to weak derivatives things are a bit different, with these derivatives being defined in terms of functionals, typically as integrations of test functions over a region with specific boundary behavior.
ADDED
But as you guess, and guess correctly you did, the classical derivative, curl, divergence can assign different values at different points.
$\endgroup$ $\begingroup$The curl and divergence operators, $\nabla \times$ and $\nabla \cdot$, are operators which send scalar functions, say $f(x,y)$ to vector functions ($\nabla \times f)$ and scalar functions ($\nabla \cdot f$) respectively. So the curl and divergence are operators, which result in new functions, and therefore are global rather than local. So at two points in the field $f$, the divergence and curl will generally take two different values.
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