Question

How to Verify this question?
Suppose a function f and a vector field F defined on R^3 are differentiable. Verify that curl(gradient f) = 0 and div(curl F) = 0.

Answer 1

\[\nabla f = <f_x~,~f_y ~ ,f_z>\\ \nabla \times (\nabla f)=?\]

Answer 2

just do the the cross product,
and use the property \(f_{xy}=f_{yx} ; ~~f_{yz}=f_{zy} \) etc