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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.

What is the value of p for the vector V = (2x^2)î - (py - 4z^3)j + 5z^4k to be solenoidal at point (1, -1, 0)?

Determine the value of p such that the vector field V = (2x^2)î - (py - 4z^3)j + 5z^4k is solenoidal at the point (1, -1, 0). Learn the use of divergence in vector calculus and how solenoidal fields are computed. This problem is ideal for undergraduate students in physics or engineering.

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