Show that the vector field

F(x, y, z) = xz²i + x³yz²j y⁵k

cannot be written as the curl of another vector field, that is,

F curl(G).

SolutionIn a previous example we showed that

div(F) =

and therefore

div(F) --?-- = 0.

If it were true that

F = curl(G),

then the theorem that statesif

F = Pi + Qj + Rk

is a vector field on ³ and P, Q, and R have continuous second-order partial derivatives, then

div(curl(F)) = 0

would give

div(F) = div(curl(G)) = 0

which contradicts

div(F) --?-- = 0.

Therefore F is not the curl of another vector field.