C^1 or C^2? Investigating Vector Identity

[madachi]

One of the basic vector identities is

$\nabla \cdot (\nabla f \times \nabla g) = 0$

Is this true if $f$ and $g$ are $C^{1}$ ? (Or they must be $C^{2}$ functions?

Thanks!

 

[phyzguy]

I think this follows from the equality of mixed partial derivatives. So I think that f and g must be al least twice differentiable for this to hold.