To prove that the vector field $\mathbf{B} = x y z^2 \hat{i} + 0 \hat{j} + 0 \hat{k}$ is solenoidal, we need to show that its divergence is zero.

Step-by-step proof:

1. Definition of a solenoidal vector field: A vector field $\mathbf{B}$ is solenoidal if its divergence is zero everywhere, i.e., $\nabla \cdot \mathbf{B} = 0$

2. Given vector field: The vector field is $\mathbf{B} = x y z^2 \hat{i} + 0 \hat{j} + 0 \hat{k}$.

   In component form, we can express $\mathbf{B}$ as: $\mathbf{B} = \left( B_x, B_y, B_z \right)$ where: $B_x = x y z^2, \quad B_y = 0, \quad B_z = 0$

3. Formula for the divergence: The divergence of a vector field $\mathbf{B} = (B_x, B_y, B_z)$ in Cartesian coordinates is: $\nabla \cdot \mathbf{B} = \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z}$

4. Compute the partial derivatives:

   • For $B_x = x y z^2$: $\frac{\partial B_x}{\partial x} = \frac{\partial}{\partial x}(x y z^2) = y z^2$
   • For $B_y = 0$: $\frac{\partial B_y}{\partial y} = \frac{\partial}{\partial y}(0) = 0$
   • For $B_z = 0$: $\frac{\partial B_z}{\partial z} = \frac{\partial}{\partial z}(0) = 0$

5. Substitute into the divergence formula: $\nabla \cdot \mathbf{B} = y z^2 + 0 + 0 = y z^2$

6. Conclusion: The divergence of $\mathbf{B}$ is: $\nabla \cdot \mathbf{B} = y z^2$ Since $\nabla \cdot \mathbf{B} \neq 0$, the vector field $\mathbf{B} = x y z^2 \hat{i} + 0 \hat{j} + 0 \hat{k}$ is not solenoidal.

Therefore, $\mathbf{B}$ is not solenoidal, as its divergence is not zero.

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Do you want further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What are the physical implications of a vector field being solenoidal?
2. How is the solenoidal condition used in fluid dynamics?
3. Can a vector field that is not solenoidal be transformed into a solenoidal one?
4. How does the divergence theorem relate to solenoidal fields?
5. What are examples of solenoidal vector fields in electromagnetism?

Tip: The divergence of a vector field helps measure how much the field spreads out from a point. Solenoidal fields represent incompressible flows, where there is no net outflow from any point.