(a) The time-dependent vector field $\mathbf{F}$ is related to the vector field $\mathbf{B}$ by

$$\mathbf{F}(\mathbf{x}, t)=\mathbf{B}(\mathbf{z})$$

where $\mathbf{z}=t \mathbf{x}$. Show that

$$(\mathbf{x} \cdot \nabla) \mathbf{F}=t \frac{\partial \mathbf{F}}{\partial t} \text {. }$$

(b) The vector fields $\mathbf{B}$ and $\mathbf{A}$ satisfy $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that $\boldsymbol{\nabla} \cdot \mathbf{B}=0$.

(c) The vector field $\mathbf{B}$ satisfies $\boldsymbol{\nabla} \cdot \mathbf{B}=0$. Show that

$$\mathbf{B}(\mathbf{x})=\nabla \times(\mathbf{D}(\mathbf{x}) \times \mathbf{x})$$

where

$$\mathbf{D}(\mathbf{x})=\int_{0}^{1} t \mathbf{B}(t \mathbf{x}) d t$$