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The specified current density is $\stackrel{鈬赌}{J}$.

To prove,

$鈭�\left[\left(\stackrel{鈬赌}{鈭�}脳\stackrel{鈬赌}{U}\right).\left(\stackrel{鈬赌}{鈭�}脳\stackrel{鈬赌}{V}\right)-\stackrel{鈬赌}{U}.\left(\stackrel{鈬赌}{鈭�}脳\stackrel{鈬赌}{鈭�}脳\stackrel{鈬赌}{V}\right)\right]d蟿=鈭�\left(\stackrel{鈬赌}{U}脳\stackrel{鈬赌}{鈭�}脳\stackrel{鈬赌}{V}\right).d\stackrel{鈬赌}{a}$

Vector product rule of two arbitrary vector functions $\stackrel{鈬赌}{U}$and $\stackrel{鈬赌}{V}$is

localid="1658559820207" $\stackrel{\mathbf{鈬赌}}{\mathbf{鈭�}}\mathbf{.}\mathbf{(}\stackrel{\mathbf{鈬赌}}{\mathbf{U}}\mathbf{脳}\stackrel{\mathbf{鈬赌}}{\mathbf{鈭�}}\mathbf{脳}\stackrel{\mathbf{鈬赌}}{\mathbf{V}}\mathbf{)}\mathbf{=}\mathbf{(}\stackrel{\mathbf{鈬赌}}{\mathbf{鈭�}}\mathbf{脳}\stackrel{\mathbf{鈬赌}}{\mathbf{V}}\mathbf{)}\mathbf{.}\mathbf{(}\stackrel{\mathbf{鈬赌}}{\mathbf{鈭�}}\mathbf{脳}\stackrel{\mathbf{鈬赌}}{\mathbf{U}}\mathbf{)}\mathbf{.}\mathit{U}\mathbf{.}\mathbf{(}\stackrel{\mathbf{鈬赌}}{\mathbf{鈭�}}\mathbf{脳}\stackrel{\mathbf{鈬赌}}{\mathbf{鈭�}}\mathbf{脳}\stackrel{\mathbf{鈬赌}}{\mathbf{V}}\mathbf{)}$

According to divergence theorem, the volume integral of the divergence of a vector function is

localid="1658559830383" $\mathbf{鈭�}\mathbf{鈭�}\mathbf{.}\stackrel{\mathbf{鈬赌}}{\mathbf{U}}\mathit{d}\mathit{蟿}\mathbf{=}\mathbf{鈭�}\stackrel{\mathbf{鈬赌}}{\mathbf{U}}\mathbf{.}\mathit{d}\stackrel{\mathbf{鈬赌}}{\mathbf{a}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{3}\mathbf{\right)}$

The curl of the magnetic field is

localid="1658559839318" $\stackrel{\mathbf{鈬赌}}{\mathbf{鈭�}}\mathbf{脳}\stackrel{\mathbf{鈬赌}}{\mathbf{B}}\mathbf{=}{\mathit{渭}}_{\mathbf{0}}\stackrel{\mathbf{鈬赌}}{\mathbf{J}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{4}\mathbf{\right)}$

Here, ${渭}_0$is the permeability of free space.

Take volume integral of both sides of equation (2) and use equation (3) on the left hand side to get,

$鈭�\left(\stackrel{鈬赌}{U}脳\stackrel{鈬赌}{鈭�}脳\stackrel{鈬赌}{V}\right).d\stackrel{鈬赌}{a}=鈭�\left(\stackrel{鈬赌}{鈭�}脳\stackrel{鈬赌}{V}\right).\left(\stackrel{鈬赌}{鈭�}脳\stackrel{鈬赌}{U}\right)-\stackrel{鈬赌}{U}\left(\stackrel{鈬赌}{鈭�}脳\stackrel{鈬赌}{鈭�}脳\stackrel{鈬赌}{V}\right)d蟿$

Assume that there are two values of magnetic fields ${\stackrel{鈬赌}{B}}_1$ and role="math" localid="1657773504274" ${\stackrel{鈬赌}{\mathrm{B}}}_2$ and two corresponding magnetic vector potentials ${\stackrel{鈬赌}{A}}_1$ and ${\stackrel{鈬赌}{A}}_2$. The difference between the values is defined as

$$\begin{gathered} {\stackrel{鈬赌}{B}}_3={\stackrel{鈬赌}{B}}_1-{\stackrel{鈬赌}{B}}_2 \\ {\stackrel{鈬赌}{A}}_3={\stackrel{鈬赌}{A}}_1-{\stackrel{鈬赌}{A}}_2 \\ {\stackrel{鈬赌}{B}}_3=\stackrel{鈬赌}{鈭�}脳{\stackrel{鈬赌}{A}}_3 \end{gathered}$$

Since the current density is uniquely specified, from equation (4),

$$\begin{gathered} \stackrel{鈬赌}{鈭�}脳{\stackrel{鈬赌}{B}}_3=\stackrel{鈬赌}{鈭�}脳{\stackrel{鈬赌}{B}}_1-\stackrel{鈬赌}{鈭�}脳{\stackrel{鈬赌}{B}}_2 \\ ={渭}_0\stackrel{鈬赌}{J}-{渭}_0\stackrel{鈬赌}{J} \\ =0 \end{gathered}$$

Set $\stackrel{鈬赌}{U},\stackrel{鈬赌}{V}=\stackrel{鈬赌}{A_3}$in equation (1), to get,

$$\begin{gathered} 鈭�\left[\left(\stackrel{鈬赌}{鈭�}脳{\stackrel{鈬赌}{A}}_3\right).\left(\stackrel{鈬赌}{鈭�}脳{\stackrel{鈬赌}{A}}_3\right)-{\stackrel{鈬赌}{A}}_3\left(\stackrel{鈬赌}{鈭�}脳\stackrel{鈫�}{鈭�}脳{\stackrel{鈬赌}{A}}_3\right)\right]d蟿=鈭�\left({\stackrel{鈬赌}{A}}_3脳\stackrel{鈫�}{鈭�}脳{\stackrel{鈬赌}{A}}_3\right).d\stackrel{鈫�}{a} \\ 鈭�{\stackrel{鈬赌}{B}}_3.{\stackrel{鈬赌}{B}}_3-{\stackrel{鈬赌}{A}}_3.\left(\stackrel{鈫�}{鈭�}脳{\stackrel{鈬赌}{B}}_3\right)d蟿=鈭�\left({\stackrel{鈬赌}{A}}_3脳{\stackrel{鈬赌}{B}}_3\right).da \end{gathered}$$

Use equation (5) to get,

$鈭�B_3^{2}d蟿=鈭�\left({\stackrel{鈬赌}{A}}_3脳{\stackrel{鈬赌}{B}}_3\right).d\stackrel{鈬赌}{a}$

If either magnetic field or the potential is uniquely specified on the surface then either ${\stackrel{鈬赌}{A}}_3=0$ or ${\stackrel{鈬赌}{B}}_3=0$. In both cases the right hand side of the previous equation becomes zero. Hence $B_3=0$ in the volume.

Thus, the magnetic field is uniquely determined in the volume.