The divergence of a magnetic vector field $\vec{B}$ must be zero everywhere. Which of the following vector fields cannot be a magnetic vector field?

$$\vec{B} (x, y, z) = −z\vec{i} + y\vec{j} + x\vec{k}$$

Given that B = curl A, use the divergence theorem to show that $\oint \mathbf{B} \cdot \mathbf{n} d \sigma$ over any closed surface is zero.

Prove that the divergence of a curl is always zero.

Are the statements true or false? Give reasons for your answer.

If a vector field $\vec{F}$ in $3$-space has zero divergence then $\vec{F} = a\vec{i} + b\vec{j} + c\vec{k}$ where $a, b$ and $c$ are constants.

Suppose that $\nabla \cdot \mathbf{F}=0$ and $\nabla \cdot \mathbf{G}=0$. Does $\mathbf{F} \times \mathbf{G}$ necessarily have zero divergence?

Suppose that $\nabla \cdot \mathbf{F}=0$ and $\nabla \cdot \mathbf{G}=0$. Does $\mathbf{F}+\mathbf{G}$ necessarily have zero divergence?