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.