If two functions have the same divergence and curl in some region, then are the functions equal?
2026-04-07 01:51:48.1775526708
If two functions have the same divergence and curl in some region, then are they equal?
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Then their difference will have the same divergence and curl. So you are asking whether a vector field of zero divergence and curl is necessarily zero.
There a trivial counterexamples to this; for instance a constant vector field.
There are less trivial examples too. If $f$ is a harmonic function, its gradient has divergence zero (essentially the definition of harmonic function) while every gradient has curl zero. So you can take the gradient of a harmonic function. There are lots of harmonic functions...