The curl of the gradient of a scalar field is always zero. Let's break down the steps to understand why the answer is option D.

Step 1: Find the gradient of the scalar field
The gradient of a scalar field is defined as the vector sum of its partial derivatives with respect to each coordinate direction. In this case, we have:

∇V = (partial derivative of V with respect to x)ax + (partial derivative of V with respect to y)ay + (partial derivative of V with respect to z)az

Taking the partial derivatives, we get:

∇V = (4xy)ax + (4xz)az + (6yz)ay + (3y^2)az + (8zx)ax + (2x^2)ay

Simplifying, we get:

∇V = (4xy + 8zx)ax + (6yz + 2x^2)ay + (4xz + 3y^2)az

Step 2: Find the curl of the gradient
The curl of a vector field is defined as the vector sum of the partial derivatives of its components with respect to each coordinate direction. In this case, we have:

curl(∇V) = (partial derivative of (4xz + 3y^2) with respect to y - partial derivative of (6yz + 2x^2) with respect to z)ax + (partial derivative of (4xy + 8zx) with respect to z - partial derivative of (4xz + 3y^2) with respect to x)ay + (partial derivative of (6yz + 2x^2) with respect to x - partial derivative of (4xy + 8zx) with respect to y)az

Taking the partial derivatives, we get:

curl(∇V) = 0ax + 0ay + 0az

Step 3: Interpret the result
Since the curl of the gradient is zero, we can conclude that the vector field is conservative. This means that it can be expressed as the gradient of a scalar potential function. In other words, there exists a function U such that:

∇U = (4xy + 8zx)ax + (6yz + 2x^2)ay + (4xz + 3y^2)az

However, this function is not unique. In fact, any function U' that differs from U by a constant can also satisfy the above equation. Therefore, we cannot uniquely determine the potential function U from the given information.