The given vector field $\mathbf{F}$ is:

$\mathbf{F} = x^2y \hat{i} + (z^3 - 3x) \hat{j} + 4y^2 \hat{k}$

We are asked to compute the divergence of $\mathbf{F}$, denoted by $\text{div} \, \mathbf{F}$, which is defined as the dot product of the del operator $\nabla$ with the vector field $\mathbf{F}$. In three dimensions, the divergence is given by:

$\text{div} \, \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}$

Here, $F_1 = x^2y$, $F_2 = z^3 - 3x$, and $F_3 = 4y^2$.

Now, we calculate each partial derivative:

1. $\frac{\partial F_1}{\partial x} = \frac{\partial}{\partial x}(x^2y) = 2xy$
2. $\frac{\partial F_2}{\partial y} = \frac{\partial}{\partial y}(z^3 - 3x) = 0$ (since there is no $y$-dependence in $F_2$)
3. $\frac{\partial F_3}{\partial z} = \frac{\partial}{\partial z}(4y^2) = 0$ (since there is no $z$-dependence in $F_3$)

Thus, the divergence of $\mathbf{F}$ is:

$\text{div} \, \mathbf{F} = 2xy + 0 + 0 = 2xy$

So, the value of $\text{div} \, \mathbf{F}$ is $2xy$.

Would you like further details or clarifications?

Here are five related questions for further practice:

1. How would you compute the curl of the vector field $\mathbf{F}$?
2. What is the divergence of $\mathbf{F}$ if $\mathbf{F} = xz\hat{i} + y^2\hat{j} + 3z\hat{k}$?
3. How would the divergence change if $\mathbf{F}$ contained exponential terms?
4. What is the physical significance of the divergence of a vector field in fluid dynamics?
5. How would you calculate the divergence of $\mathbf{F}$ in cylindrical coordinates?

Tip: The divergence of a vector field gives insight into whether there is a net "source" or "sink" at any point in the field.