Question

There exists a conservative smooth vector field F and G on R^3 such that the divergence of

Answer 1

There does not exist a conservative smooth vector field F such that its divergence is zero.

Step-by-step explanation:

A conservative vector field satisfies the property that its curl is zero.

Let's consider a simple example. Suppose F(x, y, z) = (x^2, y^2, z^2) is a conservative vector field. We can calculate its divergence using the formula div(F) = ∂F/∂x + ∂F/∂y + ∂F/∂z.

div(F) = 2x + 2y + 2z = 2(x + y + z), which is not equal to zero. Therefore, there does not exist a conservative smooth vector field F such that the divergence of F is zero.