Divergence and Curl: Can a Non-Constant Function Have Both 0?

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ehrenfest
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I am trying to think of a non-constant function whose divergence and curl is 0. It seems like this is impossible to me. Any hints?
 

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Well, if the curl is zero, you can write the field as a gradient of some scalar function. So, if the divergence is also zero, what does that tell you? (Hint: Laplace equation)
 
Potential theory would be dreadfully boring to study if the Laplace equation had only constant solutions..
 

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