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.

Learn how to prove that the vector field B = xyz^2 is solenoidal by calculating its divergence step-by-step. This problem involves vector fields and the divergence theorem in Cartesian coordinates. Suitable for college-level students studying vector calculus.

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