If $∇ · F = 0$ and $∇ ·G = 0$, would $F + G$ or $F \times G$ have zero divergence guaranteed? I know divergence is closed under addition, and $\operatorname*{div}(F\times G)=G· \operatorname*{curl} F − F · \operatorname*{curl} G$.
Zero divergence in vector field
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calculus
vector-fields
divergence
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0$\nabla$ is a linear operator so $\nabla\cdot (F+G) = \nabla\cdot F + \nabla\cdot G$. Try to find a counter-example to the last part, i.e. try to create some vectors such that $\nabla\cdot F = 0 $,$\nabla\cdot G = 0 $, but $\nabla\cdot (F\times G) \not= 0$. For example a simple example like $F = (y,0,0)$ for some suitable $G = (0,*,*)$ should do the trick. – 2017-02-09