The problem provided asks to show that the vector $\vec{v} = (x + 3y)\hat{i} + (y - 3z)\hat{j} + (x - 2z)\hat{k}$ is solenoidal.

Solution:

To determine if a vector field is solenoidal, we calculate the divergence of the vector field $\vec{v}$. A vector field is solenoidal if its divergence is zero, i.e., $\nabla \cdot \vec{v} = 0$

The divergence of a vector field $\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$ is given by: $\nabla \cdot \vec{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}$

Here, the components of $\vec{v}$ are:

• $v_x = x + 3y$,
• $v_y = y - 3z$,
• $v_z = x - 2z$.

Now, calculate the partial derivatives:

1. $\frac{\partial v_x}{\partial x} = \frac{\partial}{\partial x}(x + 3y) = 1$,
2. $\frac{\partial v_y}{\partial y} = \frac{\partial}{\partial y}(y - 3z) = 1$,
3. $\frac{\partial v_z}{\partial z} = \frac{\partial}{\partial z}(x - 2z) = -2$.

Add these results: $\nabla \cdot \vec{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} = 1 + 1 - 2 = 0$

Conclusion:

Since $\nabla \cdot \vec{v} = 0$, the vector field $\vec{v}$ is solenoidal.

Do you want me to explain any part in more detail or proceed with similar examples?

Related Questions:

1. What is the physical meaning of a solenoidal vector field?
2. How is divergence related to the flux of a vector field?
3. Can you provide an example of a non-solenoidal vector field and calculate its divergence?
4. How does the divergence theorem connect solenoidal fields to closed surfaces?
5. What is the difference between irrotational and solenoidal vector fields?

Tip:

To quickly identify if a vector field is solenoidal, always check if the divergence simplifies to zero by focusing on the coefficients of each unit vector.