Final answer:
The vector f=(xy, y, z) is not the gradient of a function because its curl does not equal zero. Also, f is not the curl of a vector field because its divergence is not zero. The electric field E can be found by taking the negative gradient of the potential.
Step-by-step explanation:
To decide if the vector f = (xy, y, z) is the gradient of a function, we must first remember that if f is the gradient of a scalar field ϕ (phi), then f must be conservative. This means that the curl of f (denoted as ∇ x f) must be equal to zero. However, when we calculate the curl of the vector field f, we find that it does not equal zero, indicating that f is not the gradient of a function.
As for whether f is the curl of a vector field, we use a similar criterion. For a vector field F, if f is the curl of F, then the divergence of f (denoted as ∇ ⋅ f) must be zero. To check this, we need to compute the divergence of f. If the result is not zero, then f cannot be the curl of any vector field. For the given vector f, the divergence is not zero, so f cannot be the curl of a vector field either.
We should also recall the relationship between the electric field and the potential. The electric field E can be found by taking the negative gradient of the potential (∇V). The components Ex, Ey, Ez would correspond to the partial derivatives of the potential with respect to x, y, and z, respectively.