= Curl of a gradient
{title2=$\nabla\times\nabla f=0$}
For a twice continuously differentiable <function>, the <curl> of its <gradient> vanishes. In components, $\epsilon_{ijk}\partial_j\partial_kf=0$ because the second derivatives are symmetric in $j,k$ while the <Levi-Civita symbol> is antisymmetric. Thus adding a gradient to a <vector potential> preserves the represented curl. No topological assumption is needed for this identity, unlike the converse assertion that every <curl-free vector field> has a global potential.
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