To prove that the vector field \(\mathbf{B} = xyz^2 \hat{i} + xyz^2 \hat{j} + xyz^2 \hat{k}\) is solenoidal, we need to show that its divergence is zero.

### Step 1: Write the vector field components
The vector field \(\mathbf{B}\) is given by:
\[
\mathbf{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}
\]
where:
- \(B_x = xyz^2\)
- \(B_y = xyz^2\)
- \(B_z = xyz^2\)

### Step 2: Compute the divergence of \(\mathbf{B}\)
The divergence of a vector field \(\mathbf{B}\) is defined as:
\[
\nabla \cdot \mathbf{B} = \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z}
\]

### Step 3: Differentiate each component
Let's compute each partial derivative:

1. **Partial derivative of \(B_x\) with respect to \(x\):**
   \[
   \frac{\partial B_x}{\partial x} = \frac{\partial}{\partial x} (xyz^2) = yz^2
   \]

2. **Partial derivative of \(B_y\) with respect to \(y\):**
   \[
   \frac{\partial B_y}{\partial y} = \frac{\partial}{\partial y} (xyz^2) = xz^2
   \]

3. **Partial derivative of \(B_z\) with respect to \(z\):**
   \[
   \frac{\partial B_z}{\partial z} = \frac{\partial}{\partial z} (xyz^2) = 2xyz
   \]

### Step 4: Add the partial derivatives
Now, let's sum the partial derivatives to find the divergence:
\[
\nabla \cdot \mathbf{B} = yz^2 + xz^2 + 2xyz
\]

### Step 5: Analyze the result
The expression \(yz^2 + xz^2 + 2xyz\) is generally not zero unless \(x = 0\), \(y = 0\), or \(z = 0\) specifically.

### Conclusion:
The vector field \(\mathbf{B} = xyz^2 \hat{i} + xyz^2 \hat{j} + xyz^2 \hat{k}\) is **not solenoidal** because its divergence is not zero in general.

Would you like to discuss this further or have more details on the implications?

### Related Questions:
1. What conditions must be met for a vector field to be solenoidal?
2. How can we visualize the concept of divergence in three-dimensional space?
3. Can you provide an example of a vector field that is solenoidal?
4. How is the concept of solenoidal fields applied in electromagnetism?
5. What is the relationship between divergence and flux?

### Tip:
When dealing with vector fields, always carefully compute each component's partial derivatives to determine divergence accurately. This will help in determining whether the field is solenoidal.

Learn how to prove that the vector field B = xyz^2 is solenoidal using divergence calculations. This problem involves vector field analysis and the concept of divergence in three-dimensional space. Suitable for undergraduate-level students and those interested in vector calculus.

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