Use the definitions and equality of mixed partial derivatives.
curl(▽f) = curl<∂f/∂x, ∂f/∂y, ∂f/∂z> =
|..i........j........k..|
|∂/∂x...∂/∂y...∂/∂z| = <∂²f/∂y∂z - ∂²f/∂z∂y, -(∂²f/∂x∂z - ∂²f/∂z∂x), ∂²f/∂x∂y - ∂²f/∂y∂x> = <0, 0, 0>.
|∂f/∂x..∂f/∂y..∂f/∂z|
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To fix notation, let's write F = <f, g, h>.
Since curl F =
|∂/∂x...∂/∂y...∂/∂z| = <∂h/∂y - ∂g/∂z, ∂f/∂z - ∂h/∂x, ∂g/∂x - ∂f/∂y>,
|..f........g.......h..|
div(curl F)
= div <∂h/∂y - ∂g/∂z, ∂f/∂z - ∂h/∂x, ∂g/∂x - ∂f/∂y>
= ∂/∂x (∂h/∂y - ∂g/∂z) + ∂/∂y (∂f/∂z - ∂h/∂x) + ∂/∂z (∂g/∂x - ∂f/∂y)
= (∂²h/∂x∂y - ∂²g/∂x∂z) + (∂²f/∂y∂z - ∂²h/∂y∂x) + (∂²g/∂z∂x - ∂²f/∂z∂y)
= 0, by equality of mixed partials.
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I hope this helps!
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Verified answer
Use the definitions and equality of mixed partial derivatives.
curl(▽f) = curl<∂f/∂x, ∂f/∂y, ∂f/∂z> =
|..i........j........k..|
|∂/∂x...∂/∂y...∂/∂z| = <∂²f/∂y∂z - ∂²f/∂z∂y, -(∂²f/∂x∂z - ∂²f/∂z∂x), ∂²f/∂x∂y - ∂²f/∂y∂x> = <0, 0, 0>.
|∂f/∂x..∂f/∂y..∂f/∂z|
-------------------------
To fix notation, let's write F = <f, g, h>.
Since curl F =
|..i........j........k..|
|∂/∂x...∂/∂y...∂/∂z| = <∂h/∂y - ∂g/∂z, ∂f/∂z - ∂h/∂x, ∂g/∂x - ∂f/∂y>,
|..f........g.......h..|
div(curl F)
= div <∂h/∂y - ∂g/∂z, ∂f/∂z - ∂h/∂x, ∂g/∂x - ∂f/∂y>
= ∂/∂x (∂h/∂y - ∂g/∂z) + ∂/∂y (∂f/∂z - ∂h/∂x) + ∂/∂z (∂g/∂x - ∂f/∂y)
= (∂²h/∂x∂y - ∂²g/∂x∂z) + (∂²f/∂y∂z - ∂²h/∂y∂x) + (∂²g/∂z∂x - ∂²f/∂z∂y)
= 0, by equality of mixed partials.
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I hope this helps!