The question asks for the value of $p$ such that the given vector field $\mathbf{V}$ is solenoidal at the point $(1, -1, 0)$.

The vector field $\mathbf{V}$ is provided as: $\mathbf{V} = \left( 2x^2 \right) \hat{i} - \left( py - 4z^3 \right) \hat{j} + 5z^4 \hat{k}$ A vector field is solenoidal if its divergence is zero, i.e.: $\nabla \cdot \mathbf{V} = 0$

Let's compute the divergence of the vector field $\mathbf{V}$:

$\nabla \cdot \mathbf{V} = \frac{\partial}{\partial x} (2x^2) + \frac{\partial}{\partial y} \left( -(py - 4z^3) \right) + \frac{\partial}{\partial z} (5z^4)$

• The partial derivative with respect to $x$ of $2x^2$ is: $\frac{\partial}{\partial x} (2x^2) = 4x$
• The partial derivative with respect to $y$ of $-(py - 4z^3)$ is: $\frac{\partial}{\partial y} (-(py - 4z^3)) = -p$
• The partial derivative with respect to $z$ of $5z^4$ is: $\frac{\partial}{\partial z} (5z^4) = 20z^3$

Thus, the divergence of the vector field is: $\nabla \cdot \mathbf{V} = 4x - p + 20z^3$

At the point $(1, -1, 0)$, substituting $x = 1$, $y = -1$, and $z = 0$, we get: $\nabla \cdot \mathbf{V} = 4(1) - p + 20(0)^3 = 4 - p$

Since the vector field is solenoidal at this point, we set the divergence to zero: $4 - p = 0$

Solving for $p$: $p = 4$

Therefore, the value of $p$ is $\boxed{4}$.

Would you like further clarification or more details? Here are some related questions you might explore:

1. What does it mean for a vector field to be solenoidal in general terms?
2. How would the solution change if the point were different from $(1, -1, 0)$?
3. What is the physical significance of a solenoidal vector field?
4. Could the vector field be solenoidal at all points for a different value of $p$?
5. What are other examples of solenoidal vector fields in physics?

Tip: Solenoidal vector fields often represent incompressible fluid flows, where the divergence of the velocity field is zero, implying constant density.