Grad-Shafranov 方程为三维空间中具有整体 $\mathrm{SO}\left(2\right)$ 对称性之理想磁流体系统在处处受力平衡时所满足之方程.

假设理想磁流体系统之质量密度场为 $\rho_{\mathrm{m}}$, 速度场为 $\boldsymbol{u}$, 压强场为 $P$, 电流场为 $\boldsymbol{J}$, 磁场为 $\boldsymbol{B}$. 我们写出理想磁流体之动力学方程,

\begin{equation}\label{eq:磁流体之动力学方程}
\begin{aligned}
\rho_{\mathrm{m}} \left(\dfrac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \cdot \boldsymbol{\nabla} \boldsymbol{u}\right) = – \boldsymbol{\nabla} P + \boldsymbol{J} \times \boldsymbol{B}.
\end{aligned}
\end{equation}

并令其处于受力平衡状态, 即速度场 $\boldsymbol{u}$ 之随体导数为零,

\begin{equation}\label{eq:速度场之随体导数为零}
\begin{aligned}
\dfrac{\mathrm{d} \boldsymbol{u}}{\mathrm{d} t} = {} & \dfrac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \cdot \boldsymbol{\nabla} \boldsymbol{u} \\
= {} & 0.
\end{aligned}
\end{equation}

将式 $\eqref{eq:速度场之随体导数为零}$ 代入式 $\eqref{eq:磁流体之动力学方程}$ 即可得到平衡方程,

\begin{equation}\label{eq:磁流体平衡方程}
\begin{aligned}
\boldsymbol{\nabla} P = \boldsymbol{J} \times \boldsymbol{B}.
\end{aligned}
\end{equation}

由麦克斯韦方程可知,

\begin{equation}\label{eq:安培环路定理}
\begin{aligned}
\boldsymbol{\nabla} \times \boldsymbol{B} = \mu_{0} \boldsymbol{J} + \dfrac{1}{c^{2}}\dfrac{\partial \boldsymbol{E}}{\partial t},
\end{aligned}
\end{equation}

其中 $\boldsymbol{E}$ 为电场, $\mu_{0}$ 为真空磁导率, $c$ 为真空中之光速大小. 由于 $c$ 非常大, 因而在位移电流相对于传导电流足够小之情况下, 可忽略式 $\eqref{eq:安培环路定理}$ 等号右边最后一项,

\begin{equation}\label{eq:忽略电场变化项之安培环路定理}
\begin{aligned}
\boldsymbol{\nabla} \times \boldsymbol{B} = \mu_{0} \boldsymbol{J}.
\end{aligned}
\end{equation}

由磁场之高斯定理可知, $\boldsymbol{B}$ 为无源场,

\begin{equation}\label{eq:磁场为无源场}
\begin{aligned}
\boldsymbol{\nabla} \cdot \boldsymbol{B} = 0.
\end{aligned}
\end{equation}

由此可知,

\begin{equation}\label{eq:磁场可表为一矢量场之旋度}
\begin{aligned}
\boldsymbol{B}\left(\boldsymbol{x}\right) = {} & 1^{2} \boldsymbol{B}\left(1 \boldsymbol{x}\right) – 0^{2} \boldsymbol{B}\left(0 \boldsymbol{x}\right) \\
= {} & \left.\left[\lambda^{2} \boldsymbol{B}\left(\lambda \boldsymbol{x}\right)\right]\right|_{\lambda = 0}^{\lambda = 1} \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \dfrac{\mathrm{d}}{\mathrm{d} \lambda} \left[\lambda^{2} \boldsymbol{B}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \left[2 \lambda \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) + \lambda^{2} \dfrac{\mathrm{d}}{\mathrm{d} \lambda} \boldsymbol{B}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \left[2 \lambda \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) + \lambda^{2} \dfrac{\mathrm{d} \left(\lambda \boldsymbol{x}\right)}{\mathrm{d} \lambda} \cdot \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \boldsymbol{B}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \left[2 \lambda \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) + \lambda^{2} \boldsymbol{x} \cdot \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \boldsymbol{B}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \lambda \left[2 \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) + \lambda \boldsymbol{x} \cdot \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \boldsymbol{B}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \lambda \left[3 \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) – \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) + \lambda \boldsymbol{x} \cdot \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) – \lambda \boldsymbol{x} \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \cdot \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) + \lambda \boldsymbol{x} \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \cdot \boldsymbol{B}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \lambda \left[\boldsymbol{B}\left(\lambda \boldsymbol{x}\right) \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \cdot \left(\lambda \boldsymbol{x}\right) – \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) \cdot \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \left(\lambda \boldsymbol{x}\right) + \lambda \boldsymbol{x} \cdot \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) – \lambda \boldsymbol{x} \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \cdot \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) + \lambda \boldsymbol{x} \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \cdot \boldsymbol{B}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \lambda \left\{\boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \times \left[\boldsymbol{B}\left(\lambda \boldsymbol{x}\right) \times \left(\lambda \boldsymbol{x}\right)\right] + \lambda \boldsymbol{x} \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \cdot \boldsymbol{B}\left(\lambda \boldsymbol{x}\right)\right\} \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \lambda \left\{\boldsymbol{\nabla}_{\boldsymbol{x}} \times \left[\boldsymbol{B}\left(\lambda \boldsymbol{x}\right) \times \boldsymbol{x}\right] + 0\right\} \\
= {} & \boldsymbol{\nabla}_{\boldsymbol{x}} \times \left[\int_{0}^{1} \mathrm{d} \lambda \lambda \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) \times \boldsymbol{x}\right].
\end{aligned}
\end{equation}

因此我们可以构造如下矢量场, 称为矢势,

\begin{equation}\label{eq:构造矢势}
\begin{aligned}
\boldsymbol{A}\left(\boldsymbol{x}\right) \equiv \int_{0}^{1}\mathrm{d} \lambda \lambda \boldsymbol{B}\left(\lambda \boldsymbol{x}\right) \times \boldsymbol{x} + \boldsymbol{\nabla}_{\boldsymbol{x}} \varphi\left(\boldsymbol{x}\right).
\end{aligned}
\end{equation}

其中 $\varphi$ 为任意 (在我们所关心之空间区域内处处光滑之) 标量场, 由于 $\boldsymbol{\nabla} \times \boldsymbol{\nabla} \varphi = 0$ 恒成立, 因此由式 $\eqref{eq:磁场可表为一矢量场之旋度}$ 可知, $\boldsymbol{A}$ 之旋度为 $\boldsymbol{B}$,

\begin{equation}\label{eq:矢势之旋度为磁场}
\begin{aligned}
\boldsymbol{\nabla} \times \boldsymbol{A} = \boldsymbol{B}.
\end{aligned}
\end{equation}

由矢势之定义式可知, 相差一个任意标量场梯度之矢势对应同一个物理上之磁场, 因而矢势不唯一, 我们称其具有规范自由度. 此自由度为非物理之自由度, 实为我们所采取之数学描述所带来之冗余自由度. 对应于同一磁场之矢势之间之变换称为规范变换, 由规范变换所联系起来之不同矢势在物理上等价.

假设磁流体系统具有绕某方向 $\boldsymbol{n}$ 之旋转对称性, 亦即是说, 将整个系统绕 $\boldsymbol{n}$ 旋转任意角度后, 与原来相比无任何变化. 这要求所有可观测物理场在旋转后与旋转前完全重合. 假设旋转变换之变换矩阵所对应之旋转张量为 $\boldsymbol{\mathcal{R}}$, 旋转变换即 $\boldsymbol{x} \to \boldsymbol{\mathcal{R}} \cdot \boldsymbol{x}$.

对于标量场 $P$, 其对称性意味着

\begin{equation}\label{eq:标量场之旋转对称性}
\begin{aligned}
P\left(\boldsymbol{x}\right) = P\left(\boldsymbol{\mathcal{R}} \cdot \boldsymbol{x}\right).
\end{aligned}
\end{equation}

即, $\boldsymbol{x}$ 处之 $P$ 跟随旋转到达 $\boldsymbol{\mathcal{R}} \cdot \boldsymbol{x}$ 后, 必须与 $\boldsymbol{\mathcal{R}} \cdot \boldsymbol{x}$ 处之 $P$ 取值相同, 才能确保旋转前后之 $P$ 场无任何变化.

而对于 (赝) 矢量场 $\boldsymbol{B}$, 其对称性意味着

\begin{equation}\label{eq:矢量场之旋转对称性}
\begin{aligned}
\boldsymbol{\mathcal{R}} \cdot \boldsymbol{B}\left(\boldsymbol{x}\right) = \boldsymbol{B}\left(\boldsymbol{\mathcal{R}} \cdot \boldsymbol{x}\right).
\end{aligned}
\end{equation}

即, $\boldsymbol{x}$ 处之 $\boldsymbol{B}$ 跟随旋转到达 $\boldsymbol{\mathcal{R}} \cdot \boldsymbol{x}$ 后, 必须与 $\boldsymbol{\mathcal{R}} \cdot \boldsymbol{x}$ 处之 $\boldsymbol{B}$ 之大小与方向均相同, 才能确保旋转前后之 $\boldsymbol{B}$ 场无任何变化.

我们考虑无穷小旋转, 即绕 $\boldsymbol{n}$ 之旋转角度 $\delta \theta$ 为无穷小量. 则此时旋转张量 $\boldsymbol{\mathcal{R}}$ 用 $\delta \theta$ 展开后保留至一阶项, 可知对任意矢量场 $\boldsymbol{V}$ 均有

\begin{equation}\label{eq:无穷小旋转之旋转张量}
\begin{aligned}
\boldsymbol{\mathcal{R}} \cdot \boldsymbol{V} = \boldsymbol{V} + \delta \theta \boldsymbol{n} \times \boldsymbol{V} + o\left(\delta \theta\right).
\end{aligned}
\end{equation}

因此, 在无穷小旋转下, 式 $\eqref{eq:标量场之旋转对称性}$ 与式 $\eqref{eq:矢量场之旋转对称性}$ 可分别写为

\begin{equation}\label{eq:无穷小旋转下标量场之旋转对称性}
\begin{aligned}
P\left(\boldsymbol{x}\right) = {} & P\left(\boldsymbol{x} + \delta \theta \boldsymbol{n} \times \boldsymbol{x}\right) + o\left(\delta \theta\right) \\
= {} & P\left(\boldsymbol{x}\right) + \left(\delta \theta \boldsymbol{n} \times \boldsymbol{x}\right) \cdot \boldsymbol{\nabla} P\left(\boldsymbol{x}\right) + o\left(\delta \theta\right),
\end{aligned}
\end{equation}

\begin{equation}\label{eq:无穷小旋转下矢量场之旋转对称性}
\begin{aligned}
\boldsymbol{B}\left(\boldsymbol{x}\right) + \delta \theta \boldsymbol{n} \times \boldsymbol{B}\left(\boldsymbol{x}\right) = {} & \boldsymbol{B}\left(\boldsymbol{x} + \delta \theta \boldsymbol{n} \times \boldsymbol{x}\right) + o\left(\delta \theta\right) \\
= {} & \boldsymbol{B}\left(\boldsymbol{x}\right) + \left(\delta \theta \boldsymbol{n} \times \boldsymbol{x}\right) \cdot \boldsymbol{\nabla} \boldsymbol{B}\left(\boldsymbol{x}\right) + o\left(\delta \theta\right).
\end{aligned}
\end{equation}

由此可知,

\begin{equation}\label{eq:无穷小旋转下标量场所满足之关系}
\begin{aligned}
\left(\boldsymbol{n} \times \boldsymbol{x}\right) \cdot \boldsymbol{\nabla} P = 0.
\end{aligned}
\end{equation}

\begin{equation}\label{eq:无穷小旋转下矢量场所满足之关系}
\begin{aligned}
\left(\boldsymbol{n} \times \boldsymbol{x}\right) \cdot \boldsymbol{\nabla} \boldsymbol{B} = \boldsymbol{n} \times \boldsymbol{B}.
\end{aligned}
\end{equation}

建立 Euclid 右手坐标系 $\left(x, y, z\right)$, 并将 $z$ 方向 $\boldsymbol{e}_{z}$ 选为 $\boldsymbol{n}$ 之方向. 再由此建立柱坐标系 $\left(R, \phi, Z\right)$, 其与 $\left(x, y, z\right)$ 之坐标变换关系为

\begin{equation}\label{eq:柱坐标系到直角坐标系之变换关系}
\left\{
\begin{aligned}
& x = R \cos \phi, \\
& y = R \sin \phi, \\
& z = Z.
\end{aligned}
\right.
\end{equation}

\begin{equation}\label{eq:直角坐标系到柱坐标系之变换关系}
\left\{
\begin{aligned}
& R = \sqrt{x^{2} + y^{2}}, \\
& \phi = \arctan \dfrac{y}{x}, \\
& Z = z.
\end{aligned}
\right.
\end{equation}

单位基矢 $\left[\boldsymbol{e}_{R}\ \boldsymbol{e}_{\phi}\ \boldsymbol{e}_{Z}\right]$ 与 $\left[\boldsymbol{e}_{x}\ \boldsymbol{e}_{y}\ \boldsymbol{e}_{z}\right]$ 之变换关系为

\begin{equation}\label{eq:柱坐标系基矢到直角坐标系基矢之变换关系}
\begin{aligned}
\left[\boldsymbol{e}_{x}\ \boldsymbol{e}_{y}\ \boldsymbol{e}_{z}\right] = \left[\boldsymbol{e}_{R}\ \boldsymbol{e}_{\phi}\ \boldsymbol{e}_{Z}\right]
\begin{pmatrix}
\cos \phi & \sin \phi & 0 \\
– \sin \phi & \cos \phi & 0 \\
0 & 0 & 1
\end{pmatrix}.
\end{aligned}
\end{equation}

\begin{equation}\label{eq:直角坐标系基矢到柱坐标系基矢之变换关系}
\begin{aligned}
\left[\boldsymbol{e}_{R}\ \boldsymbol{e}_{\phi}\ \boldsymbol{e}_{Z}\right] = \left[\boldsymbol{e}_{x}\ \boldsymbol{e}_{y}\ \boldsymbol{e}_{z}\right]
\begin{pmatrix}
\dfrac{x}{\sqrt{x^{2} + y^{2}}} & – \dfrac{y}{\sqrt{x^{2} + y^{2}}} & 0 \\
\dfrac{y}{\sqrt{x^{2} + y^{2}}} & \dfrac{x}{\sqrt{x^{2} + y^{2}}} & 0 \\
0 & 0 & 1
\end{pmatrix}.
\end{aligned}
\end{equation}

由此可计算柱坐标系之各基矢对各坐标之偏微商为

\begin{equation}\label{eq:柱坐标系之各基矢对坐标之偏微商}
\begin{aligned}
\begin{pmatrix}
\dfrac{\partial \boldsymbol{e}_{R}}{\partial R} & \dfrac{\partial \boldsymbol{e}_{\phi}}{\partial R} & \dfrac{\partial \boldsymbol{e}_{Z}}{\partial R} \\
\dfrac{\partial \boldsymbol{e}_{R}}{\partial \phi} & \dfrac{\partial \boldsymbol{e}_{\phi}}{\partial \phi} & \dfrac{\partial \boldsymbol{e}_{Z}}{\partial \phi} \\
\dfrac{\partial \boldsymbol{e}_{R}}{\partial Z} & \dfrac{\partial \boldsymbol{e}_{\phi}}{\partial Z} & \dfrac{\partial \boldsymbol{e}_{Z}}{\partial Z}
\end{pmatrix}
=
\begin{pmatrix}
0 & 0 & 0 \\
\boldsymbol{e}_{\phi} & – \boldsymbol{e}_{R} & 0 \\
0 & 0 & 0
\end{pmatrix}.
\end{aligned}
\end{equation}

还可计算得到

\begin{equation}\label{eq:柱坐标系各坐标之梯度}
\left\{
\begin{aligned}
& \boldsymbol{\nabla} R = \boldsymbol{e}_{R}, \\
& \boldsymbol{\nabla} \phi = \dfrac{\boldsymbol{e}_{\phi}}{R}, \\
& \boldsymbol{\nabla} Z = \boldsymbol{e}_{Z}.
\end{aligned}
\right.
\end{equation}

空间点之位置 $\boldsymbol{x}$ 可用单位基矢展开,

\begin{equation}\label{eq:空间点之位置可用单位基矢展开}
\begin{aligned}
\boldsymbol{x} = {} & x \boldsymbol{e}_{x} + y \boldsymbol{e}_{y} + z \boldsymbol{e}_{z} \\
= {} & R \cos \phi \left(\boldsymbol{e}_{R} \cos \phi – \boldsymbol{e}_{\phi} \sin \phi\right) \\
& + R \sin \phi \left(\boldsymbol{e}_{R} \sin \phi + \boldsymbol{e}_{\phi} \cos \phi\right) \\
& + Z \boldsymbol{e}_{Z} \\
= {} & R \boldsymbol{e}_{R} + Z \boldsymbol{e}_{Z}.
\end{aligned}
\end{equation}

因而

\begin{equation}\label{eq:旋转对称轴叉乘空间位置矢量}
\begin{aligned}
\boldsymbol{n} \times \boldsymbol{x} = {} & \boldsymbol{e}_{z} \times \boldsymbol{x} \\
= {} & \boldsymbol{e}_{Z} \times \boldsymbol{x} \\
= {} & \boldsymbol{e}_{Z} \times \left(R \boldsymbol{e}_{R} + Z \boldsymbol{e}_{Z}\right) \\
= {} & R \boldsymbol{e}_{\phi}.
\end{aligned}
\end{equation}

考虑式 $\eqref{eq:柱坐标系各坐标之梯度}$ 与式 $\eqref{eq:旋转对称轴叉乘空间位置矢量}$, 可得

\begin{equation}\label{eq:旋转对称轴叉乘空间位置矢量点乘梯度算符}
\begin{aligned}
\left(\boldsymbol{n} \times \boldsymbol{x}\right) \cdot \boldsymbol{\nabla} = {} & \left(\boldsymbol{n} \times \boldsymbol{x}\right) \cdot \left(\boldsymbol{\nabla} R \dfrac{\partial}{\partial R} + \boldsymbol{\nabla} \phi \dfrac{\partial}{\partial \phi} + \boldsymbol{\nabla} Z \dfrac{\partial}{\partial Z}\right) \\
= {} & R \boldsymbol{e}_{\phi} \cdot \left(\boldsymbol{e}_{R} \dfrac{\partial}{\partial R} + \dfrac{\boldsymbol{e}_{\phi}}{R} \dfrac{\partial}{\partial \phi} + \boldsymbol{e}_{Z} \dfrac{\partial}{\partial Z}\right) \\
= {} & \dfrac{\partial}{\partial \phi}.
\end{aligned}
\end{equation}

另外, 还可计算得

\begin{equation}\label{eq:旋转对称轴叉乘磁场}
\begin{aligned}
\boldsymbol{n} \times \boldsymbol{B} = {} & \boldsymbol{e}_{z} \times \boldsymbol{B} \\
= {} & \boldsymbol{e}_{Z} \times \boldsymbol{B} \\
= {} & \boldsymbol{e}_{Z} \times \left(B_{R} \boldsymbol{e}_{R} + B_{\phi} \boldsymbol{e}_{\phi} + B_{Z} \boldsymbol{e}_{Z}\right) \\
= {} & B_{R} \boldsymbol{e}_{\phi} – B_{\phi} \boldsymbol{e}_{R}.
\end{aligned}
\end{equation}

将式 $\eqref{eq:柱坐标系之各基矢对坐标之偏微商}$, 式 $\eqref{eq:旋转对称轴叉乘空间位置矢量点乘梯度算符}$, 式 $\eqref{eq:旋转对称轴叉乘磁场}$ 代入式 $\eqref{eq:无穷小旋转下标量场所满足之关系}$ 与式 $\eqref{eq:无穷小旋转下矢量场所满足之关系}$, 即可得,

\begin{equation}\label{eq:环向对称性对压强场之数学体现}
\begin{aligned}
0 = {} & \left(\boldsymbol{n} \times \boldsymbol{x}\right) \cdot \boldsymbol{\nabla} P \\
= {} & \dfrac{\partial P}{\partial \phi}.
\end{aligned}
\end{equation}

\begin{equation}\label{eq:环向对称性对磁场之数学体现}
\begin{aligned}
0 = {} & \left(\boldsymbol{n} \times \boldsymbol{x}\right) \cdot \boldsymbol{\nabla} \boldsymbol{B} – \boldsymbol{n} \times \boldsymbol{B} \\
= {} & \dfrac{\partial \boldsymbol{B}}{\partial \phi} – \left(B_{R} \boldsymbol{e}_{\phi} – B_{\phi} \boldsymbol{e}_{R}\right) \\
= {} & \dfrac{\partial}{\partial \phi} \left(B_{R} \boldsymbol{e}_{R} + B_{\phi} \boldsymbol{e}_{\phi} + B_{Z} \boldsymbol{e}_{Z}\right) – \left(B_{R} \boldsymbol{e}_{\phi} – B_{\phi} \boldsymbol{e}_{R}\right) \\
= {} & \dfrac{\partial B_{R}}{\partial \phi} \boldsymbol{e}_{R} + B_{R} \dfrac{\partial \boldsymbol{e}_{R}}{\partial \phi} \\
& + \dfrac{\partial B_{\phi}}{\partial \phi} \boldsymbol{e}_{\phi} + B_{\phi} \dfrac{\partial \boldsymbol{e}_{\phi}}{\partial \phi} \\
& + \dfrac{\partial B_{Z}}{\partial \phi} \boldsymbol{e}_{Z} + B_{Z} \dfrac{\partial \boldsymbol{e}_{Z}}{\partial \phi} \\
& – \left(B_{R} \boldsymbol{e}_{\phi} – B_{\phi} \boldsymbol{e}_{R}\right)\\
= {} & \dfrac{\partial B_{R}}{\partial \phi} \boldsymbol{e}_{R} + B_{R} \boldsymbol{e}_{\phi} \\
& + \dfrac{\partial B_{\phi}}{\partial \phi} \boldsymbol{e}_{\phi} – B_{\phi} \boldsymbol{e}_{R} \\
& + \dfrac{\partial B_{Z}}{\partial \phi} \boldsymbol{e}_{Z} \\
& – \left(B_{R} \boldsymbol{e}_{\phi} – B_{\phi} \boldsymbol{e}_{R}\right) \\
= {} & \dfrac{\partial B_{R}}{\partial \phi} \boldsymbol{e}_{R} + \dfrac{\partial B_{\phi}}{\partial \phi} \boldsymbol{e}_{\phi} + \dfrac{\partial B_{Z}}{\partial \phi} \boldsymbol{e}_{Z},
\end{aligned}
\end{equation}

即

\begin{equation}\label{eq:环向对称性对压强场与磁场之数学体现}
\left\{
\begin{aligned}
& \dfrac{\partial P}{\partial \phi} = 0, \\
& \dfrac{\partial B_{R}}{\partial \phi} = \dfrac{\partial B_{\phi}}{\partial \phi} = \dfrac{\partial B_{Z}}{\partial \phi} = 0.
\end{aligned}
\right.
\end{equation}

式 $\eqref{eq:环向对称性对压强场与磁场之数学体现}$ 即为系统环向对称性之体现.

式 $\eqref{eq:无穷小旋转下标量场所满足之关系}$ 与式 $\eqref{eq:无穷小旋转下矢量场所满足之关系}$ 实际上为微分几何中李导数在该情景下之应用, 在微分流形上, 张量场
$$
T = {T^{\mu_{1} \cdots \mu_{k}}}_{\nu_{1} \cdots \nu_{l}} \dfrac{\partial}{\partial x^{\mu_{1}}} \otimes \cdots \otimes \dfrac{\partial}{\partial x^{\mu_{k}}} \otimes \mathrm{d} x^{\nu_{1}} \otimes \cdots \otimes \mathrm{d} x^{\nu_{l}}
$$
沿矢量场
$$
v = v^{\mu} \dfrac{\partial}{\partial x^{\mu}}
$$
方向之李导数计算方法如下,

\begin{equation}\label{eq:李导数计算方法}
\begin{aligned}
\mathcal{L}_{v}{T^{\mu_{1} \cdots \mu_{k}}}_{\nu_{1} \cdots \nu_{l}} = {} & v^{\rho} \dfrac{\partial}{\partial x^{\rho}} {T^{\mu_{1} \cdots \mu_{k}}}_{\nu_{1} \cdots \nu_{l}} \\
& – {T^{\rho \cdots \mu_{k}}}_{\nu_{1} \cdots \nu_{l}} \dfrac{\partial v^{\mu_{1}}}{\partial x^{\rho}} \\
& – \cdots \\
& – {T^{\mu_{1} \cdots \rho}}_{\nu_{1} \cdots \nu_{l}} \dfrac{\partial v^{\mu_{k}}}{\partial x^{\rho}} \\
& + {T^{\mu_{1} \cdots \mu_{k}}}_{\rho \cdots \nu_{l}} \dfrac{\partial v^{\rho}}{\partial x^{\nu_{1}}} \\
& + \cdots \\
& + {T^{\mu_{1} \cdots \mu_{k}}}_{\nu_{1} \cdots \rho} \dfrac{\partial v^{\rho}}{\partial x^{\nu_{l}}}.
\end{aligned}
\end{equation}

式 $\eqref{eq:无穷小旋转下标量场所满足之关系}$ 与式 $\eqref{eq:无穷小旋转下矢量场所满足之关系}$ 正是压强场与磁场沿矢量 $\dfrac{\partial}{\partial \phi} = R \boldsymbol{e}_{\phi}$ 方向之李导数为零所导致之结果,

\begin{equation}\label{eq:压强场沿环向坐标基矢之李导数}
\begin{aligned}
0 = {} & \mathcal{L}_{\frac{\partial}{\partial \phi}} P = \left(0 \dfrac{\partial}{\partial R} + 1 \dfrac{\partial}{\partial \phi} + 0 \dfrac{\partial}{\partial Z}\right) P \\
= {} & \dfrac{\partial P}{\partial \phi},
\end{aligned}
\end{equation}

\begin{equation}\label{eq:磁场沿环向坐标基矢之李导数}
\begin{aligned}
0 = {} & \mathcal{L}_{\frac{\partial}{\partial \phi}} \boldsymbol{B} \\
= {} & \mathcal{L}_{\frac{\partial}{\partial \phi}} \left(B_{R} \boldsymbol{e}_{R} + B_{\phi} \boldsymbol{e}_{\phi} + B_{Z} \boldsymbol{e}_{Z}\right) \\
= {} & \mathcal{L}_{\frac{\partial}{\partial \phi}} \left(B_{R} \dfrac{\partial}{\partial R} + \dfrac{B_{\phi}}{R} \dfrac{\partial}{\partial \phi} + B_{Z} \dfrac{\partial}{\partial Z}\right) \\
= {} & \left[\left(0 \dfrac{\partial}{\partial R} + 1 \dfrac{\partial}{\partial \phi} + 0 \dfrac{\partial}{\partial Z}\right) B_{R} – \left(B_{R} \dfrac{\partial}{\partial R} + \dfrac{B_{\phi}}{R} \dfrac{\partial}{\partial \phi} + B_{Z} \dfrac{\partial}{\partial Z}\right) 0\right] \dfrac{\partial}{\partial R} \\
& + \left[\left(0 \dfrac{\partial}{\partial R} + 1 \dfrac{\partial}{\partial \phi} + 0 \dfrac{\partial}{\partial Z}\right) \dfrac{B_{\phi}}{R} – \left(B_{R} \dfrac{\partial}{\partial R} + \dfrac{B_{\phi}}{R} \dfrac{\partial}{\partial \phi} + B_{Z} \dfrac{\partial}{\partial Z}\right) 1\right] \dfrac{\partial}{\partial \phi} \\
& + \left[\left(0 \dfrac{\partial}{\partial R} + 1 \dfrac{\partial}{\partial \phi} + 0 \dfrac{\partial}{\partial Z}\right) B_{Z} – \left(B_{R} \dfrac{\partial}{\partial R} + \dfrac{B_{\phi}}{R} \dfrac{\partial}{\partial \phi} + B_{Z} \dfrac{\partial}{\partial Z}\right) 0\right] \dfrac{\partial}{\partial Z} \\
= {} & \dfrac{\partial B_{R}}{\partial \phi} \dfrac{\partial}{\partial R} + \dfrac{1}{R} \dfrac{\partial B_{\phi}}{\partial \phi} \dfrac{\partial}{\partial \phi} + \dfrac{\partial B_{Z}}{\partial \phi} \dfrac{\partial}{\partial Z},
\end{aligned}
\end{equation}

由式 $\eqref{eq:压强场沿环向坐标基矢之李导数}$ 与式 $\eqref{eq:磁场沿环向坐标基矢之李导数}$ 即可直接得到式 $\eqref{eq:环向对称性对压强场与磁场之数学体现}$, 与由式 $\eqref{eq:无穷小旋转下标量场所满足之关系}$ 与式 $\eqref{eq:无穷小旋转下矢量场所满足之关系}$ 所得到之结果相同.

将式 $\eqref{eq:矢势之旋度为磁场}$ 在柱坐标系中展开,

\begin{equation}\label{eq:在柱坐标系中用矢势表示磁场}
\begin{aligned}
\boldsymbol{B} = {} & \boldsymbol{\nabla} \times \boldsymbol{A} \\
= {} & \boldsymbol{\nabla} R \times \dfrac{\partial \boldsymbol{A}}{\partial R} + \boldsymbol{\nabla} \phi \times \dfrac{\partial \boldsymbol{A}}{\partial \phi} + \boldsymbol{\nabla} Z \times \dfrac{\partial \boldsymbol{A}}{\partial Z} \\
= {} & \boldsymbol{e}_{R} \times \dfrac{\partial}{\partial R} \left(A_{R} \boldsymbol{e}_{R} + A_{\phi} \boldsymbol{e}_{\phi} + A_{Z} \boldsymbol{e}_{Z}\right) \\
& + \dfrac{\boldsymbol{e}_{\phi}}{R} \times \dfrac{\partial}{\partial \phi} \left(A_{R} \boldsymbol{e}_{R} + A_{\phi} \boldsymbol{e}_{\phi} + A_{Z} \boldsymbol{e}_{Z}\right) \\
& + \boldsymbol{e}_{Z} \times \dfrac{\partial}{\partial Z} \left(A_{R} \boldsymbol{e}_{R} + A_{\phi} \boldsymbol{e}_{\phi} + A_{Z} \boldsymbol{e}_{Z}\right) \\
= {} & \boldsymbol{e}_{R} \times \left(\dfrac{\partial A_{R}}{\partial R} \boldsymbol{e}_{R} + A_{R} \dfrac{\partial \boldsymbol{e}_{R}}{\partial R} + \dfrac{\partial A_{\phi}}{\partial R} \boldsymbol{e}_{\phi} + A_{\phi} \dfrac{\partial \boldsymbol{e}_{\phi}}{\partial R} + \dfrac{\partial A_{Z}}{\partial R} \boldsymbol{e}_{Z} + A_{Z} \dfrac{\partial \boldsymbol{e}_{Z}}{\partial R}\right) \\
& + \dfrac{\boldsymbol{e}_{\phi}}{R} \times \left(\dfrac{\partial A_{R}}{\partial \phi} \boldsymbol{e}_{R} + A_{R} \dfrac{\partial \boldsymbol{e}_{R}}{\partial \phi} + \dfrac{\partial A_{\phi}}{\partial \phi} \boldsymbol{e}_{\phi} + A_{\phi} \dfrac{\partial \boldsymbol{e}_{\phi}}{\partial \phi} + \dfrac{\partial A_{Z}}{\partial \phi} \boldsymbol{e}_{Z} + A_{Z} \dfrac{\partial \boldsymbol{e}_{Z}}{\partial \phi}\right) \\
& + \boldsymbol{e}_{Z} \times \left(\dfrac{\partial A_{R}}{\partial Z} \boldsymbol{e}_{R} + A_{R} \dfrac{\partial \boldsymbol{e}_{R}}{\partial Z} + \dfrac{\partial A_{\phi}}{\partial Z} \boldsymbol{e}_{\phi} + A_{\phi} \dfrac{\partial \boldsymbol{e}_{\phi}}{\partial Z} + \dfrac{\partial A_{Z}}{\partial Z} \boldsymbol{e}_{Z} + A_{Z} \dfrac{\partial \boldsymbol{e}_{Z}}{\partial Z}\right) \\
= {} & \boldsymbol{e}_{R} \times \left(\dfrac{\partial A_{R}}{\partial R} \boldsymbol{e}_{R} + \dfrac{\partial A_{\phi}}{\partial R} \boldsymbol{e}_{\phi} + \dfrac{\partial A_{Z}}{\partial R} \boldsymbol{e}_{Z}\right) \\
& + \dfrac{\boldsymbol{e}_{\phi}}{R} \times \left(\dfrac{\partial A_{R}}{\partial \phi} \boldsymbol{e}_{R} + A_{R} \boldsymbol{e}_{\phi} + \dfrac{\partial A_{\phi}}{\partial \phi} \boldsymbol{e}_{\phi} – A_{\phi} \boldsymbol{e}_{R} + \dfrac{\partial A_{Z}}{\partial \phi} \boldsymbol{e}_{Z}\right) \\
& + \boldsymbol{e}_{Z} \times \left(\dfrac{\partial A_{R}}{\partial Z} \boldsymbol{e}_{R} + \dfrac{\partial A_{\phi}}{\partial Z} \boldsymbol{e}_{\phi} + \dfrac{\partial A_{Z}}{\partial Z} \boldsymbol{e}_{Z}\right) \\
= {} & \left(\dfrac{\partial A_{\phi}}{\partial R} \boldsymbol{e}_{Z} – \dfrac{\partial A_{Z}}{\partial R} \boldsymbol{e}_{\phi}\right) \\
& + \dfrac{1}{R} \left(- \dfrac{\partial A_{R}}{\partial \phi} \boldsymbol{e}_{Z} + A_{\phi} \boldsymbol{e}_{Z} + \dfrac{\partial A_{Z}}{\partial \phi} \boldsymbol{e}_{R}\right) \\
& + \left(\dfrac{\partial A_{R}}{\partial Z} \boldsymbol{e}_{\phi} – \dfrac{\partial A_{\phi}}{\partial Z} \boldsymbol{e}_{R}\right) \\
= {} & \left(\dfrac{1}{R} \dfrac{\partial A_{Z}}{\partial \phi} – \dfrac{\partial A_{\phi}}{\partial Z}\right) \boldsymbol{e}_{R} + \left(\dfrac{\partial A_{R}}{\partial Z} – \dfrac{\partial A_{Z}}{\partial R}\right) \boldsymbol{e}_{\phi} + \left(- \dfrac{1}{R} \dfrac{\partial A_{R}}{\partial \phi} + \dfrac{A_{\phi}}{R} + \dfrac{\partial A_{\phi}}{\partial R}\right) \boldsymbol{e}_{Z} \\
= {} & \left[\dfrac{\partial A_{Z}}{\partial \phi} – \dfrac{\partial \left(R A_{\phi}\right)}{\partial Z}\right] \dfrac{\boldsymbol{e}_{R}}{R} + \left(\dfrac{\partial A_{R}}{\partial Z} – \dfrac{\partial A_{Z}}{\partial R}\right) \boldsymbol{e}_{\phi} + \left[\dfrac{\partial \left(R A_{\phi}\right)}{\partial R} – \dfrac{\partial A_{R}}{\partial \phi}\right] \dfrac{\boldsymbol{e}_{Z}}{R}.
\end{aligned}
\end{equation}

由此可知,

\begin{equation}\label{eq:在柱坐标系中用矢势表示磁场各分量}
\left\{
\begin{aligned}
& B_{R} = \dfrac{1}{R} \left[\dfrac{\partial A_{Z}}{\partial \phi} – \dfrac{\partial \left(R A_{\phi}\right)}{\partial Z}\right], \\
& B_{\phi} = \dfrac{\partial A_{R}}{\partial Z} – \dfrac{\partial A_{Z}}{\partial R}, \\
& B_{Z} = \dfrac{1}{R} \left[\dfrac{\partial \left(R A_{\phi}\right)}{\partial R} – \dfrac{\partial A_{R}}{\partial \phi}\right].
\end{aligned}
\right.
\end{equation}

将式 $\eqref{eq:环向对称性对压强场与磁场之数学体现}$ 代入式 $\eqref{eq:在柱坐标系中用矢势表示磁场各分量}$,

\begin{equation}\label{eq:环向对称性对矢势之影响}
\left\{
\begin{aligned}
& 0 = \dfrac{\partial B_{R}}{\partial \phi} = \dfrac{1}{R} \left[\dfrac{\partial^{2} A_{Z}}{\partial \phi^{2}} – \dfrac{\partial^{2} \left(R A_{\phi}\right)}{\partial \phi \partial Z}\right] = \dfrac{1}{R} \left[\dfrac{\partial}{\partial \phi} \dfrac{\partial A_{Z}}{\partial \phi} – \dfrac{\partial}{\partial Z} \left(R \dfrac{\partial A_{\phi}}{\partial \phi}\right)\right], \\
& 0 = \dfrac{\partial B_{\phi}}{\partial \phi} = \dfrac{\partial^{2} A_{R}}{\partial \phi \partial Z} – \dfrac{\partial^{2} A_{Z}}{\partial \phi \partial R} = \dfrac{\partial}{\partial Z} \dfrac{\partial A_{R}}{\partial \phi} – \dfrac{\partial}{\partial R} \dfrac{\partial A_{Z}}{\partial \phi}, \\
& 0 = \dfrac{\partial B_{Z}}{\partial \phi} = \dfrac{1}{R} \left[\dfrac{\partial^{2} \left(R A_{\phi}\right)}{\partial \phi \partial R} – \dfrac{\partial^{2} A_{R}}{\partial \phi^{2}}\right] = \dfrac{1}{R} \left[\dfrac{\partial}{\partial R} \left(R \dfrac{\partial A_{\phi}}{\partial \phi}\right) – \dfrac{\partial}{\partial \phi} \dfrac{\partial A_{R}}{\partial \phi}\right].
\end{aligned}
\right.
\end{equation}

构造一个新矢量场 $\boldsymbol{\mathcal{A}} = \mathcal{A}_{R} \boldsymbol{e}_{R} + \mathcal{A}_{\phi} \boldsymbol{e}_{\phi} + \mathcal{A}_{Z} \boldsymbol{e}_{Z}$, 其各分量定义为

\begin{equation}\label{eq:由矢势构造新矢量场}
\left\{
\begin{aligned}
& \mathcal{A}_{R} \equiv \dfrac{\partial A_{R}}{\partial \phi}, \\
& \mathcal{A}_{\phi} \equiv \dfrac{\partial A_{\phi}}{\partial \phi}, \\
& \mathcal{A}_{Z} \equiv \dfrac{\partial A_{Z}}{\partial \phi}.
\end{aligned}
\right.
\end{equation}

从而式 $\eqref{eq:环向对称性对矢势之影响}$ 可写为

\begin{equation}\label{eq:用新矢量场表述磁场对称性}
\left\{
\begin{aligned}
& \dfrac{1}{R} \left[\dfrac{\partial \mathcal{A}_{Z}}{\partial \phi} – \dfrac{\partial \left(R \mathcal{A}_{\phi}\right)}{\partial Z}\right] = 0, \\
& \dfrac{\partial \mathcal{A}_{R}}{\partial Z} – \dfrac{\partial \mathcal{A}_{Z}}{\partial R} = 0, \\
&\dfrac{1}{R} \left[\dfrac{\partial \left(R \mathcal{A}_{\phi}\right)}{\partial R} – \dfrac{\partial \mathcal{A}_{R}}{\partial \phi}\right] = 0.
\end{aligned}
\right.
\end{equation}

即

\begin{equation}\label{eq:新矢量场各分量为零之条件组合起来}
\begin{aligned}
\left[\dfrac{\partial \mathcal{A}_{Z}}{\partial \phi} – \dfrac{\partial \left(R \mathcal{A}_{\phi}\right)}{\partial Z}\right] \dfrac{\boldsymbol{e}_{R}}{R} + \left(\dfrac{\partial \mathcal{A}_{R}}{\partial Z} – \dfrac{\partial \mathcal{A}_{Z}}{\partial R}\right) \boldsymbol{e}_{\phi} + \left[\dfrac{\partial \left(R \mathcal{A}_{\phi}\right)}{\partial R} – \dfrac{\partial \mathcal{A}_{R}}{\partial \phi}\right] \dfrac{\boldsymbol{e}_{Z}}{R} = 0.
\end{aligned}
\end{equation}

式 $\eqref{eq:新矢量场各分量为零之条件组合起来}$ 与式 $\eqref{eq:在柱坐标系中用矢势表示磁场}$ 等号右边之展开式具有相同之形式, 因此其事实上即为

\begin{equation}\label{eq:新矢量场无旋}
\begin{aligned}
\boldsymbol{\nabla} \times \boldsymbol{\mathcal{A}} = 0.
\end{aligned}
\end{equation}

即 $\boldsymbol{\mathcal{A}}$ 为无旋场. 由此可知,

\begin{equation}\label{eq:新矢量场可表为一标量场之梯度}
\begin{aligned}
\boldsymbol{\mathcal{A}}\left(\boldsymbol{x}\right) = {} & 1 \boldsymbol{\mathcal{A}}\left(1 \boldsymbol{x}\right) – 0 \boldsymbol{\mathcal{A}}\left(0 \boldsymbol{x}\right) \\
= {} & \left.\left[\lambda \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right]\right|_{\lambda = 0}^{\lambda = 1} \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \dfrac{\mathrm{d}}{\mathrm{d} \lambda} \left[\lambda \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \left[\boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right) + \lambda \dfrac{\mathrm{d}}{\mathrm{d} \lambda} \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \left[\boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right) + \lambda \dfrac{\mathrm{d} \left(\lambda \boldsymbol{x}\right)}{\mathrm{d} \lambda} \cdot \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \left[\boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right) + \lambda \boldsymbol{x} \cdot \boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right] \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \left\{\boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right) + \left[\boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right] \cdot \left(\lambda \boldsymbol{x}\right) – \lambda \boldsymbol{x} \times \left[\boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \times \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right]\right\} \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \left\{\left[\boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \left(\lambda \boldsymbol{x}\right)\right] \cdot \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right) + \left[\boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right] \cdot \left(\lambda \boldsymbol{x}\right) – \lambda \boldsymbol{x} \times \left[\boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \times \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right]\right\} \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \left\{\boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \left[\lambda \boldsymbol{x} \cdot \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right] – \lambda \boldsymbol{x} \times \left[\boldsymbol{\nabla}_{\lambda \boldsymbol{x}} \times \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right]\right\} \\
= {} & \int_{0}^{1} \mathrm{d} \lambda \left\{\boldsymbol{\nabla}_{\boldsymbol{x}} \left[\boldsymbol{x} \cdot \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right] – 0\right\} \\
= {} & \boldsymbol{\nabla}_{\boldsymbol{x}} \left[\int_{0}^{1} \mathrm{d} \lambda \boldsymbol{x} \cdot \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right)\right].
\end{aligned}
\end{equation}

因此我们可以构造如下标量场 $\chi$,

\begin{equation}\label{eq:构造新标量场}
\begin{aligned}
\chi\left(\boldsymbol{x}\right) \equiv \int_{0}^{1} \mathrm{d} \lambda \boldsymbol{x} \cdot \boldsymbol{\mathcal{A}}\left(\lambda \boldsymbol{x}\right) + C,
\end{aligned}
\end{equation}

其中 $C$ 为任意常数, 由于 $\boldsymbol{\nabla} C = 0$ 恒成立, 因此由式 $\eqref{eq:新矢量场可表为一标量场之梯度}$ 可知, $\chi$ 之梯度为 $\boldsymbol{\mathcal{A}}$,

\begin{equation}\label{eq:新标量场之梯度为新矢量场}
\begin{aligned}
\boldsymbol{\nabla} \chi = \boldsymbol{\mathcal{A}},
\end{aligned}
\end{equation}

即

\begin{equation}\label{eq:分量展开版新标量场之梯度为新矢量场}
\begin{aligned}
\dfrac{\partial \chi}{\partial R} \boldsymbol{e}_{R} + \dfrac{\partial \chi}{\partial \phi} \dfrac{\boldsymbol{e}_{\phi}}{R} + \dfrac{\partial \chi}{\partial Z} \boldsymbol{e}_{Z} = \mathcal{A}_{R} \boldsymbol{e}_{R} + \mathcal{A}_{\phi} \boldsymbol{e}_{\phi} + \mathcal{A} \boldsymbol{e}_{Z},
\end{aligned}
\end{equation}

从而

\begin{equation}\label{eq:分量各自相等版新标量场之梯度为新矢量场}
\left\{
\begin{aligned}
& \dfrac{\partial \chi}{\partial R} = \mathcal{A}_{R} = \dfrac{\partial A_{R}}{\partial \phi}, \\
& \dfrac{1}{R} \dfrac{\partial \chi}{\partial \phi} = \mathcal{A}_{\phi} = \dfrac{\partial A_{\phi}}{\partial \phi}, \\
& \dfrac{\partial \chi}{\partial Z} = \mathcal{A}_{Z} = \dfrac{\partial A_{Z}}{\partial \phi}.
\end{aligned}
\right.
\end{equation}

将式 $\eqref{eq:分量各自相等版新标量场之梯度为新矢量场}$ 代入式 $\eqref{eq:在柱坐标系中用矢势表示磁场}$ 即可得,

\begin{equation}\label{eq:将新标量场与矢势之关系代入磁场之矢势表达式}
\begin{aligned}
\boldsymbol{B} = {} & \left[\dfrac{\partial A_{Z}}{\partial \phi} – \dfrac{\partial \left(R A_{\phi}\right)}{\partial Z}\right] \dfrac{\boldsymbol{e}_{R}}{R} + \left(\dfrac{\partial A_{R}}{\partial Z} – \dfrac{\partial A_{Z}}{\partial R}\right) \boldsymbol{e}_{\phi} + \left[\dfrac{\partial \left(R A_{\phi}\right)}{\partial R} – \dfrac{\partial A_{R}}{\partial \phi}\right] \dfrac{\boldsymbol{e}_{Z}}{R} \\
= {} & \left[\dfrac{\partial \chi}{\partial Z} – \dfrac{\partial \left(R A_{\phi}\right)}{\partial Z}\right] \dfrac{\boldsymbol{e}_{R}}{R} + B_{\phi} \boldsymbol{e}_{\phi} + \left[\dfrac{\partial \left(R A_{\phi}\right)}{\partial R} – \dfrac{\partial \chi}{\partial R}\right] \dfrac{\boldsymbol{e}_{Z}}{R} \\
= {} & \dfrac{\boldsymbol{e}_{Z} \times \boldsymbol{e}_{\phi}}{R} \dfrac{\partial}{\partial Z} \left(R A_{\phi} – \chi\right) + B_{\phi} \boldsymbol{e}_{\phi} + \dfrac{\boldsymbol{e}_{R} \times \boldsymbol{e}_{\phi}}{R} \dfrac{\partial}{\partial R} \left(R A_{\phi} – \chi\right) \\
= {} & \left[\boldsymbol{e}_{R} \dfrac{\partial}{\partial R} \left(R A_{\phi} – \chi\right) + \boldsymbol{e}_{Z} \dfrac{\partial}{\partial Z} \left(R A_{\phi} – \chi\right)\right] \times \boldsymbol{\nabla} \phi + 0 \dfrac{\partial}{\partial \phi} \left(R A_{\phi} – \chi\right) + R B_{\phi} \boldsymbol{\nabla} \phi \\
= {} & \left[\boldsymbol{e}_{R} \dfrac{\partial}{\partial R} \left(R A_{\phi} – \chi\right) + \boldsymbol{e}_{Z} \dfrac{\partial}{\partial Z} \left(R A_{\phi} – \chi\right)\right] \times \boldsymbol{\nabla} \phi + \dfrac{\boldsymbol{e}_{\phi} \times \boldsymbol{e}_{\phi}}{R^{2}} \dfrac{\partial}{\partial \phi} \left(R A_{\phi} – \chi\right) + R B_{\phi} \boldsymbol{\nabla} \phi \\
= {} & \left[\boldsymbol{e}_{R} \dfrac{\partial}{\partial R} \left(R A_{\phi} – \chi\right) + \dfrac{\boldsymbol{e}_{\phi}}{R} \dfrac{\partial}{\partial \phi} \left(R A_{\phi} – \chi\right) + \boldsymbol{e}_{Z} \dfrac{\partial}{\partial Z} \left(R A_{\phi} – \chi\right)\right] \times \boldsymbol{\nabla} \phi + R B_{\phi} \boldsymbol{\nabla} \phi \\
= {} & \boldsymbol{\nabla} \left(R A_{\phi} – \chi\right) \times \boldsymbol{\nabla} \phi + R B_{\phi} \boldsymbol{\nabla} \phi.
\end{aligned}
\end{equation}

由此我们可定义两个标量场,

\begin{equation}\label{eq:定义磁面函数}
\begin{aligned}
\psi \equiv R A_{\phi} – \chi,
\end{aligned}
\end{equation}

\begin{equation}\label{eq:定义极向电流函数}
\begin{aligned}
F \equiv R B_{\phi}.
\end{aligned}
\end{equation}

则由式 $\eqref{eq:将新标量场与矢势之关系代入磁场之矢势表达式}$, 式 $\eqref{eq:定义磁面函数}$ 与式 $\eqref{eq:定义极向电流函数}$, 可将磁场表示为

\begin{equation}\label{eq:用磁面函数与极向电流函数表示磁场}
\begin{aligned}
\boldsymbol{B} = \boldsymbol{\nabla} \psi \times \boldsymbol{\nabla} \phi + F \boldsymbol{\nabla} \phi.
\end{aligned}
\end{equation}

由式 $\eqref{eq:环向对称性对压强场与磁场之数学体现}$ 与式 $\eqref{eq:分量各自相等版新标量场之梯度为新矢量场}$ 可知,

\begin{equation}\label{eq:磁面函数之环向对称性}
\begin{aligned}
\dfrac{\partial \psi}{\partial \phi} = {} & \dfrac{\partial}{\partial \phi} \left(R A_{\phi} – \chi\right) \\
= {} & R \left(\dfrac{\partial A_{\phi}}{\partial \phi} – \dfrac{1}{R} \dfrac{\partial \chi}{\partial \phi}\right) \\
= {} & 0,
\end{aligned}
\end{equation}

\begin{equation}\label{eq:极向电流函数之环向对称性}
\begin{aligned}
\dfrac{\partial F}{\partial \phi} = {} & \dfrac{\partial}{\partial \phi} \left(R B_{\phi}\right) \\
= {} & R \dfrac{\partial B_{\phi}}{\partial \phi} \\
= {} & 0.
\end{aligned}
\end{equation}

由式 $\eqref{eq:忽略电场变化项之安培环路定理}$ 与式 $\eqref{eq:用磁面函数与极向电流函数表示磁场}$, 并考虑式 $\eqref{eq:磁面函数之环向对称性}$ 与式 $\eqref{eq:极向电流函数之环向对称性}$, 可得

\begin{equation}\label{eq:用磁面函数与极向电流函数表示电流}
\begin{aligned}
\mu_{0} \boldsymbol{J} = {} & \boldsymbol{\nabla} \times \boldsymbol{B} \\
= {} & \boldsymbol{\nabla} \times \left(\boldsymbol{\nabla} \psi \times \boldsymbol{\nabla} \phi + F \boldsymbol{\nabla} \phi\right) \\
= {} & \boldsymbol{\nabla} \phi \cdot \boldsymbol{\nabla} \boldsymbol{\nabla} \psi – \nabla^{2} \psi \boldsymbol{\nabla} \phi + \nabla^{2} \phi \boldsymbol{\nabla} \psi – \boldsymbol{\nabla} \psi \cdot \boldsymbol{\nabla} \boldsymbol{\nabla} \phi + \boldsymbol{\nabla} F \times \boldsymbol{\nabla} \phi \\
= {} & \boldsymbol{\nabla} \phi \cdot \left(\boldsymbol{\nabla} R \dfrac{\partial}{\partial R} + \boldsymbol{\nabla} \phi \dfrac{\partial}{\partial \phi} + \boldsymbol{\nabla} Z \dfrac{\partial}{\partial Z}\right) \left(\boldsymbol{\nabla} R \dfrac{\partial \psi}{\partial R} + \boldsymbol{\nabla} \phi \dfrac{\partial \psi}{\partial \phi} + \boldsymbol{\nabla} Z \dfrac{\partial \psi}{\partial Z}\right) \\
& – \left(\boldsymbol{\nabla} R \dfrac{\partial}{\partial R} + \boldsymbol{\nabla} \phi \dfrac{\partial}{\partial \phi} + \boldsymbol{\nabla} Z \dfrac{\partial}{\partial Z}\right) \cdot \left(\boldsymbol{\nabla} R \dfrac{\partial \psi}{\partial R} + \boldsymbol{\nabla} \phi \dfrac{\partial \psi}{\partial \phi} + \boldsymbol{\nabla} Z \dfrac{\partial \psi}{\partial Z}\right) \boldsymbol{\nabla} \phi \\
& + \left(\boldsymbol{\nabla} R \dfrac{\partial}{\partial R} + \boldsymbol{\nabla} \phi \dfrac{\partial}{\partial \phi} + \boldsymbol{\nabla} Z \dfrac{\partial}{\partial Z}\right) \cdot \left(\boldsymbol{\nabla} R \dfrac{\partial \phi}{\partial R} + \boldsymbol{\nabla} \phi \dfrac{\partial \phi}{\partial \phi} + \boldsymbol{\nabla} Z \dfrac{\partial \phi}{\partial Z}\right) \boldsymbol{\nabla} \psi \\
& – \left(\boldsymbol{\nabla} R \dfrac{\partial \psi}{\partial R} + \boldsymbol{\nabla} \phi \dfrac{\partial \psi}{\partial \phi} + \boldsymbol{\nabla} Z \dfrac{\partial \psi}{\partial Z}\right) \cdot \left(\boldsymbol{\nabla} R \dfrac{\partial}{\partial R} + \boldsymbol{\nabla} \phi \dfrac{\partial}{\partial \phi} + \boldsymbol{\nabla} Z \dfrac{\partial}{\partial Z}\right) \left(\boldsymbol{\nabla} R \dfrac{\partial \phi}{\partial R} + \boldsymbol{\nabla} \phi \dfrac{\partial \phi}{\partial \phi} + \boldsymbol{\nabla} Z \dfrac{\partial \phi}{\partial Z}\right) \\
& + \boldsymbol{\nabla} F \times \boldsymbol{\nabla} \phi \\
= {} & \dfrac{\boldsymbol{e}_{\phi}}{R} \cdot \left(\boldsymbol{e}_{R} \dfrac{\partial}{\partial R} + \dfrac{\boldsymbol{e}_{\phi}}{R} \dfrac{\partial}{\partial \phi} + \boldsymbol{e}_{Z} \dfrac{\partial}{\partial Z}\right) \left(\boldsymbol{e}_{R} \dfrac{\partial \psi}{\partial R} + \boldsymbol{e}_{Z} \dfrac{\partial \psi}{\partial Z}\right) \\
& – \left(\boldsymbol{e}_{R} \dfrac{\partial}{\partial R} + \dfrac{\boldsymbol{e}_{\phi}}{R} \dfrac{\partial}{\partial \phi} + \boldsymbol{e}_{Z} \dfrac{\partial}{\partial Z}\right) \cdot \left(\boldsymbol{e}_{R} \dfrac{\partial \psi}{\partial R} + \boldsymbol{e}_{Z} \dfrac{\partial \psi}{\partial Z}\right) \boldsymbol{\nabla} \phi \\
& + \left(\boldsymbol{e}_{R} \dfrac{\partial}{\partial R} + \dfrac{\boldsymbol{e}_{\phi}}{R} \dfrac{\partial}{\partial \phi} + \boldsymbol{e}_{Z} \dfrac{\partial}{\partial Z}\right) \cdot \dfrac{\boldsymbol{e}_{\phi}}{R} \boldsymbol{\nabla} \psi \\
& – \left(\boldsymbol{e}_{R} \dfrac{\partial \psi}{\partial R} + \dfrac{\boldsymbol{e}_{\phi}}{R} \dfrac{\partial \psi}{\partial \phi} + \boldsymbol{e}_{Z} \dfrac{\partial \psi}{\partial Z}\right) \cdot \left(\boldsymbol{e}_{R} \dfrac{\partial}{\partial R} + \dfrac{\boldsymbol{e}_{\phi}}{R} \dfrac{\partial}{\partial \phi} + \boldsymbol{e}_{Z} \dfrac{\partial}{\partial Z}\right) \dfrac{\boldsymbol{e}_{\phi}}{R} \\
& + \boldsymbol{\nabla} F \times \boldsymbol{\nabla} \phi \\
= {} & \dfrac{1}{R^{2}} \dfrac{\partial}{\partial \phi} \left(\boldsymbol{e}_{R} \dfrac{\partial \psi}{\partial R} + \boldsymbol{e}_{Z} \dfrac{\partial \psi}{\partial Z}\right) \\
& – \left(\dfrac{\partial^{2} \psi}{\partial R^{2}} + \dfrac{1}{R} \dfrac{\partial \psi}{\partial R} + \dfrac{\partial^{2} \psi}{\partial Z^{2}}\right) \boldsymbol{\nabla} \phi \\
& + 0 \\
& – \left(\boldsymbol{e}_{R} \dfrac{\partial \psi}{\partial R} + \dfrac{\boldsymbol{e}_{\phi}}{R} \dfrac{\partial \psi}{\partial \phi} + \boldsymbol{e}_{Z} \dfrac{\partial \psi}{\partial Z}\right) \cdot \left(- \dfrac{\boldsymbol{e}_{R} \boldsymbol{e}_{\phi}}{R^{2}} – \dfrac{\boldsymbol{e}_{\phi} \boldsymbol{e}_{R}}{R^{2}}\right) \\
& + \boldsymbol{\nabla} F \times \boldsymbol{\nabla} \phi \\
= {} & \dfrac{1}{R^{2}} \boldsymbol{e}_{\phi} \dfrac{\partial \psi}{\partial R} \\
& – \left(\dfrac{\partial^{2} \psi}{\partial R^{2}} + \dfrac{1}{R} \dfrac{\partial \psi}{\partial R} + \dfrac{\partial^{2} \psi}{\partial Z^{2}}\right) \boldsymbol{\nabla} \phi \\
& + \dfrac{\boldsymbol{e}_{\phi}}{R^{2}} \dfrac{\partial \psi}{\partial R} \\
& + \boldsymbol{\nabla} F \times \boldsymbol{\nabla} \phi \\
= {} & \boldsymbol{\nabla} F \times \boldsymbol{\nabla} \phi – \left(\dfrac{\partial^{2} \psi}{\partial R^{2}} – \dfrac{1}{R} \dfrac{\partial \psi}{\partial R} + \dfrac{\partial^{2} \psi}{\partial Z^{2}}\right) \boldsymbol{\nabla} \phi \\
= {} & \boldsymbol{\nabla} F \times \boldsymbol{\nabla} \phi – \left[R \dfrac{\partial}{\partial R} \left(\dfrac{1}{R} \dfrac{\partial \psi}{\partial R}\right) + \dfrac{\partial^{2} \psi}{\partial Z^{2}}\right] \boldsymbol{\nabla} \phi.
\end{aligned}
\end{equation}

定义 Grad-Shafranov 算子 $\Delta^{\ast}$,

\begin{equation}\label{eq:Grad-Shafranov算子之定义}
\begin{aligned}
\Delta^{\ast} \equiv R \dfrac{\partial}{\partial R}\left(\dfrac{1}{R} \dfrac{\partial}{\partial R}\right) + \dfrac{\partial^{2}}{\partial Z^{2}},
\end{aligned}
\end{equation}

则式 $\eqref{eq:用磁面函数与极向电流函数表示电流}$ 可表示为

\begin{equation}\label{eq:用Grad-Shafranov算子简化电流表达式}
\begin{aligned}
\mu_{0} \boldsymbol{J} = \boldsymbol{\nabla} F \times \boldsymbol{\nabla} \phi – \Delta^{\ast} \psi \boldsymbol{\nabla} \phi.
\end{aligned}
\end{equation}

将式 $\eqref{eq:用磁面函数与极向电流函数表示磁场}$ 与式 $\eqref{eq:用Grad-Shafranov算子简化电流表达式}$ 代入式 $\eqref{eq:磁流体平衡方程}$, 并考虑式 $\eqref{eq:磁面函数之环向对称性}$ 与式 $\eqref{eq:极向电流函数之环向对称性}$, 可得

\begin{equation}\label{eq:平衡方程用磁面函数与极向电流函数表示}
\begin{aligned}
\mu_{0} \boldsymbol{\nabla} P = {} & \mu_{0} \boldsymbol{J} \times \boldsymbol{B} \\
= {} & \left(\boldsymbol{\nabla} F \times \boldsymbol{\nabla} \phi – \Delta^{\ast} \psi \boldsymbol{\nabla} \phi\right) \times \left(\boldsymbol{\nabla} \psi \times \boldsymbol{\nabla} \phi + F \boldsymbol{\nabla} \phi\right) \\
= {} & -\boldsymbol{\nabla} \psi \cdot \left(\boldsymbol{\nabla} F \times \boldsymbol{\nabla} \phi\right) \boldsymbol{\nabla} \phi – \Delta^{\ast} \psi \left|\boldsymbol{\nabla} \phi\right|^{2} \boldsymbol{\nabla} \psi + \Delta^{\ast} \psi \boldsymbol{\nabla} \phi \cdot \boldsymbol{\nabla} \psi \boldsymbol{\nabla} \phi + F \boldsymbol{\nabla} \phi \boldsymbol{\nabla} \phi \cdot \boldsymbol{\nabla} F – F \left|\boldsymbol{\nabla} \phi\right|^{2} \boldsymbol{\nabla} F \\
= {} & \boldsymbol{\nabla} \phi \boldsymbol{\nabla} \phi \cdot \left(\boldsymbol{\nabla} F \times \boldsymbol{\nabla} \psi\right) – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi \boldsymbol{\nabla} \psi + F \boldsymbol{\nabla} F\right) \\
= {} & – \dfrac{\boldsymbol{e}_{\phi}}{R^{2}} \left(\dfrac{\partial F}{\partial R} \dfrac{\partial \psi}{\partial Z} – \dfrac{\partial F}{\partial Z} \dfrac{\partial \psi}{\partial R}\right) – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi \boldsymbol{\nabla} \psi + F \boldsymbol{\nabla} F\right) \\
= {} & – \dfrac{\boldsymbol{e}_{\phi}}{R^{2}} \left(\dfrac{\partial F}{\partial R} \dfrac{\partial \psi}{\partial Z} – \dfrac{\partial F}{\partial Z} \dfrac{\partial \psi}{\partial R}\right) – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi \dfrac{\partial \psi}{\partial R} + F \dfrac{\partial F}{\partial R}\right) \boldsymbol{\nabla} R – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi \dfrac{\partial \psi}{\partial Z} + F \dfrac{\partial F}{\partial Z}\right) \boldsymbol{\nabla} Z \\
= {} & – \dfrac{\boldsymbol{e}_{R}}{R^{2}} \left(\Delta^{\ast} \psi \dfrac{\partial \psi}{\partial R} + F \dfrac{\partial F}{\partial R}\right) – \dfrac{\boldsymbol{e}_{\phi}}{R^{2}} \left(\dfrac{\partial F}{\partial R} \dfrac{\partial \psi}{\partial Z} – \dfrac{\partial F}{\partial Z} \dfrac{\partial \psi}{\partial R}\right) – \dfrac{\boldsymbol{e}_{Z}}{R^{2}} \left(\Delta^{\ast} \psi \dfrac{\partial \psi}{\partial Z} + F \dfrac{\partial F}{\partial Z}\right) \\
= {} & – \dfrac{\boldsymbol{e}_{R}}{R^{2}} \left(\Delta^{\ast} \psi \dfrac{\partial \psi}{\partial R} + F \dfrac{\partial F}{\partial R}\right) – \dfrac{\boldsymbol{e}_{\phi}}{R^{2}} \dfrac{\partial \left(F, \psi\right)}{\partial \left(R, Z\right)} – \dfrac{\boldsymbol{e}_{Z}}{R^{2}} \left(\Delta^{\ast} \psi \dfrac{\partial \psi}{\partial Z} + F \dfrac{\partial F}{\partial Z}\right),
\end{aligned}
\end{equation}

其中 $\dfrac{\partial \left(F, \psi\right)}{\partial \left(R, Z\right)}$ 为雅可比行列式,

\begin{equation}\label{eq:极向电流函数与磁面函数之雅可比行列式}
\begin{aligned}
\dfrac{\partial \left(F, \psi\right)}{\partial \left(R, Z\right)} \equiv {} &
\begin{vmatrix}
\dfrac{\partial F}{\partial R} & \dfrac{\partial \psi}{\partial R} \\
\dfrac{\partial F}{\partial Z} & \dfrac{\partial \psi}{\partial Z}
\end{vmatrix} \\
= {} & \dfrac{\partial F}{\partial R} \dfrac{\partial \psi}{\partial Z} – \dfrac{\partial F}{\partial Z} \dfrac{\partial \psi}{\partial R}.
\end{aligned}
\end{equation}

因此有

\begin{equation}\label{eq:柱坐标系中压强场对各坐标之偏微商}
\left\{
\begin{aligned}
& \mu_{0} \dfrac{\partial P}{\partial R} = – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi \dfrac{\partial \psi}{\partial R} + F \dfrac{\partial F}{\partial R}\right), \\
& \mu_{0} \dfrac{\partial P}{\partial \phi} = – \dfrac{1}{R} \dfrac{\partial \left(F, \psi\right)}{\partial \left(R, Z\right)}, \\
& \mu_{0} \dfrac{\partial P}{\partial Z} = – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi \dfrac{\partial \psi}{\partial Z} + F \dfrac{\partial F}{\partial Z}\right).
\end{aligned}
\right.
\end{equation}

由式 $\eqref{eq:环向对称性对压强场与磁场之数学体现}$ 与式 $\eqref{eq:柱坐标系中压强场对各坐标之偏微商}$ 可知,

\begin{equation}\label{eq:压强场之环向对称性导致极向电流函数与磁面函数之雅可比行列式为零}
\begin{aligned}
0 = {} & \dfrac{1}{\mu_{0}} \dfrac{\partial P}{\partial \phi} \\
= {} & – \dfrac{1}{R} \dfrac{\partial \left(F, \psi\right)}{\partial \left(R, Z\right)}
\end{aligned}
\end{equation}

即雅可比行列式 $\dfrac{\partial \left(F, \psi\right)}{\partial \left(R, Z\right)}$ 为零. 这意味着 $F$ 与 $\psi$ 这两个变量不独立, 即它们之间存在某种约束关系,

\begin{equation}\label{eq:极向电流函数与磁面函数之间存在约束关系}
\begin{aligned}
\exists f \ \mathrm{s.t.} \ f\left(F, \psi\right) = 0.
\end{aligned}
\end{equation}

可以由式 $\eqref{eq:极向电流函数与磁面函数之间存在约束关系}$ 反解出 $F$ 作为 $\psi$ 之函数,

\begin{equation}\label{eq:极向电流函数为磁面之函数}
\begin{aligned}
F = F\left(\psi\right).
\end{aligned}
\end{equation}

式 $\eqref{eq:极向电流函数为磁面之函数}$ 表明, 当将 $F$ 表示为 $\psi$ 之函数时, 其不显含 $\left(R, Z\right)$, 而是通过 $\psi$ 间接依赖于 $\left(R, Z\right)$. 因此有

\begin{equation}\label{eq:极向电流函数之梯度用磁面函数之梯度表示}
\begin{aligned}
\boldsymbol{\nabla} F = \dfrac{\mathrm{d} F}{\mathrm{d} \psi} \boldsymbol{\nabla} \psi.
\end{aligned}
\end{equation}

将式 $\eqref{eq:极向电流函数之梯度用磁面函数之梯度表示}$ 代入式 $\eqref{eq:平衡方程用磁面函数与极向电流函数表示}$ 可得,

\begin{equation}\label{eq:压强场之梯度用磁面函数值梯度表示}
\begin{aligned}
\mu_{0} \boldsymbol{\nabla} P = {} & – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi \boldsymbol{\nabla} \psi + F \boldsymbol{\nabla} F\right) \\
= {} & – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi + F \dfrac{\mathrm{d} F}{\mathrm{d} \psi}\right) \boldsymbol{\nabla} \psi.
\end{aligned}
\end{equation}

即

\begin{equation}\label{eq:柱坐标系中压强场对各坐标之偏微商用磁面函数对各坐标之偏微商表示}
\left\{
\begin{aligned}
& \mu_{0} \dfrac{\partial P}{\partial R} = – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi + F \dfrac{\mathrm{d} F}{\mathrm{d} \psi}\right) \dfrac{\partial \psi}{\partial R}, \\
& \mu_{0} \dfrac{\partial P}{\partial \phi} = – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi + F \dfrac{\mathrm{d} F}{\mathrm{d} \psi}\right) \dfrac{\partial \psi}{\partial \phi} \ \left(= 0\right), \\
& \mu_{0} \dfrac{\partial P}{\partial Z} = – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi + F \dfrac{\mathrm{d} F}{\mathrm{d} \psi}\right) \dfrac{\partial \psi}{\partial Z}.
\end{aligned}
\right.
\end{equation}

由式 $\eqref{eq:柱坐标系中压强场对各坐标之偏微商用磁面函数对各坐标之偏微商表示}$ 可计算 $P$ 与 $\psi$ 之雅可比行列式,

\begin{equation}\label{eq:压强场与磁面函数之雅可比行列式}
\begin{aligned}
\dfrac{\partial \left(P, \psi\right)}{\partial \left(R, Z\right)} \equiv {} &
\begin{vmatrix}
\dfrac{\partial P}{\partial R} & \dfrac{\partial \psi}{\partial R} \\
\dfrac{\partial P}{\partial Z} & \dfrac{\partial \psi}{\partial Z}
\end{vmatrix} \\
= {} & \dfrac{\partial P}{\partial R} \dfrac{\partial \psi}{\partial Z} – \dfrac{\partial P}{\partial Z} \dfrac{\partial \psi}{\partial R} \\
= {} & – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi + F \dfrac{\mathrm{d} F}{\mathrm{d} \psi}\right) \left(\dfrac{\partial \psi}{\partial R} \dfrac{\partial \psi}{\partial Z} – \dfrac{\partial \psi}{\partial Z} \dfrac{\partial \psi}{\partial R}\right) \\
= {} & 0.
\end{aligned}
\end{equation}

即 $P$ 与 $\psi$ 这两个变量亦不独立,仿照式 $\eqref{eq:极向电流函数与磁面函数之间存在约束关系}$, 式 $\eqref{eq:极向电流函数为磁面之函数}$ 与式 $\eqref{eq:极向电流函数之梯度用磁面函数之梯度表示}$ 之讨论, 可以同样得到,

\begin{equation}\label{eq:压强场为磁面之函数}
\begin{aligned}
P = P\left(\psi\right).
\end{aligned}
\end{equation}

\begin{equation}\label{eq:压强场之梯度用磁面函数之梯度表示}
\begin{aligned}
\boldsymbol{\nabla} P = \dfrac{\mathrm{d} P}{\mathrm{d} \psi} \boldsymbol{\nabla} \psi.
\end{aligned}
\end{equation}

将式 $\eqref{eq:压强场之梯度用磁面函数之梯度表示}$ 代入式 $\eqref{eq:压强场之梯度用磁面函数值梯度表示}$ 可得,

\begin{equation}\label{eq:压强场之梯度中包含磁面函数之梯度}
\begin{aligned}
– \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi + F \dfrac{\mathrm{d} F}{\mathrm{d} \psi}\right) \boldsymbol{\nabla} \psi = {} & \mu_{0} \boldsymbol{\nabla} P \\
= {} & \mu_{0} \dfrac{\mathrm{d} P}{\mathrm{d} \psi} \boldsymbol{\nabla} \psi,
\end{aligned}
\end{equation}

即

\begin{equation}\label{eq:压强场对磁面函数之微商}
\begin{aligned}
\mu_{0} \dfrac{\mathrm{d} P}{\mathrm{d} \psi} = – \dfrac{1}{R^{2}} \left(\Delta^{\ast} \psi + F \dfrac{\mathrm{d} F}{\mathrm{d} \psi}\right).
\end{aligned}
\end{equation}

将式 $\eqref{eq:压强场对磁面函数之微商}$ 进行整理, 即可得

\begin{equation}\label{eq:Grad-Shafranov方程}
\begin{aligned}
\Delta^{\ast} \psi = – \mu_{0} R^{2} \dfrac{\mathrm{d} P}{\mathrm{d} \psi} – F \dfrac{\mathrm{d} F}{\mathrm{d} \psi}.
\end{aligned}
\end{equation}

式 $\eqref{eq:Grad-Shafranov方程}$ 即为 Grad-Shafranov 方程.

考察空间中两点 $\boldsymbol{x_{\mathrm{i}}} = R_{\mathrm{i}} \boldsymbol{e}_{R} + Z_{\mathrm{i}} \boldsymbol{e}_{Z}$ 与 $\boldsymbol{x_{\mathrm{f}}} = R_{\mathrm{f}} \boldsymbol{e}_{R} + Z_{\mathrm{f}} \boldsymbol{e}_{Z}$. 在它们之间通过一条 (分段光滑之) 曲线相连,

\begin{equation}\label{eq:空间两点通过分段光滑之曲线相连}
\begin{aligned}
\boldsymbol{x}\left(s\right) = {} & \boldsymbol{e}_{R}\left(s\right) R\left(s\right) + \boldsymbol{e}_{Z}\left(s\right) Z\left(s\right) \\
= {} & \boldsymbol{e}_{x} R\left(s\right) \cos\left[\phi\left(s\right)\right] + \boldsymbol{e}_{y} R\left(s\right) \sin\left[\phi\left(s\right)\right] + \boldsymbol{e}_{z} Z\left(s\right), \boldsymbol{x}\left(s_{\mathrm{i}}\right) = \boldsymbol{x}_{\mathrm{i}}, \boldsymbol{x}\left(s_{\mathrm{f}}\right) = \boldsymbol{x}_{\mathrm{f}}.
\end{aligned}
\end{equation}

将该曲线绕 $Z$ 轴旋转某一角度 $\Delta \theta$, 形成一个二维曲面 $\Sigma$, 其法向量处处与 $\boldsymbol{e}_{\phi}$ 垂直, 与极向平面相切. $\Sigma$ 上任意点可由两个参数 $\left(s, \theta\right)$ 描述. 则 $\left(s, \theta\right)$ 处之 (有向) 无穷小面积元可写为

\begin{equation}\label{eq:有向面积元表达式}
\begin{aligned}
\mathrm{d} \boldsymbol{S} = {} & \left[\boldsymbol{x}\left(s + \mathrm{d} s, \theta\right) – \boldsymbol{x}\left(s, \theta\right)\right] \times \left[\boldsymbol{x}\left(s, \theta + \mathrm{d} \theta\right) – \boldsymbol{x}\left(s, \theta\right)\right] \\
= {} & \left(\dfrac{\partial \boldsymbol{x}}{\partial s} \mathrm{d} s\right) \times \left(\dfrac{\partial \boldsymbol{x}}{\partial \theta} \mathrm{d} \theta\right).
\end{aligned}
\end{equation}

由于

\begin{equation}\label{eq:曲面上点之坐标}
\begin{aligned}
\boldsymbol{x}\left(s, \theta\right) = \boldsymbol{e}_{x} R\left(s\right) \cos\left[\phi\left(s\right) + \theta\right] + \boldsymbol{e}_{y} R\left(s\right) \sin\left[\phi\left(s\right) + \theta\right] + \boldsymbol{e}_{z} Z\left(s\right),
\end{aligned}
\end{equation}

因此有

\begin{equation}\label{eq:计算曲面上点对曲线参数之偏微商}
\begin{aligned}
\dfrac{\partial \boldsymbol{x}\left(s, \theta\right)}{\partial s} = {} & \boldsymbol{e}_{x} \left[\dfrac{\mathrm{d} R\left(s\right)}{\mathrm{d} s} \cos\left[\phi\left(s\right) + \theta\right] – R\left(s\right) \dfrac{\mathrm{d} \phi\left(s\right)}{\mathrm{d} s} \sin\left[\phi\left(s\right) + \theta\right]\right] \\
& + \boldsymbol{e}_{y} \left[\dfrac{\mathrm{d} R\left(s\right)}{\mathrm{d} s} \sin\left[\phi\left(s\right) + \theta\right] + R\left(s\right) \dfrac{\mathrm{d} \phi\left(s\right)}{\mathrm{d} s} \cos\left[\phi\left(s\right) + \theta\right]\right] \\
& + \boldsymbol{e}_{z} \dfrac{\mathrm{d} Z\left(s\right)}{\mathrm{d} s} \\
= {} & \boldsymbol{e}_{R}\left(s, \theta\right) \dfrac{\mathrm{d} R\left(s\right)}{\mathrm{d} s} + \boldsymbol{e}_{\phi}\left(s, \theta\right) R\left(s\right) \dfrac{\mathrm{d} \phi\left(s\right)}{\mathrm{d} s} + \boldsymbol{e}_{Z}\left(s, \theta\right) \dfrac{\mathrm{d} Z\left(s\right)}{\mathrm{d} s},
\end{aligned}
\end{equation}

以及

\begin{equation}\label{eq:计算曲面上点对角度参数之偏微商}
\begin{aligned}
\dfrac{\partial \boldsymbol{x}\left(s, \theta\right)}{\partial \theta} = {} & – \boldsymbol{e}_{x} R\left(s\right) \sin\left[\phi\left(s\right) + \theta\right] \\
& + \boldsymbol{e}_{y} R\left(s\right) \cos\left[\phi\left(s\right) + \theta\right] \\
= {} & \boldsymbol{e}_{\phi}\left(s, \theta\right) R\left(s\right).
\end{aligned}
\end{equation}

由式 $\eqref{eq:计算曲面上点对曲线参数之偏微商}$ 与式 $\eqref{eq:计算曲面上点对角度参数之偏微商}$, 可得

\begin{equation}\label{eq:曲面上单位参数间隔之面积元}
\begin{aligned}
\dfrac{\partial \boldsymbol{x}\left(s, \theta\right)}{\partial s} \times \dfrac{\partial \boldsymbol{x}\left(s, \theta\right)}{\partial \theta} = R\left(s\right) \left[- \boldsymbol{e}_{R}\left(s, \theta\right) \dfrac{\mathrm{d} Z\left(s\right)}{\mathrm{d} s} + \boldsymbol{e}_{Z}\left(s, \theta\right) \dfrac{\mathrm{d} R\left(s\right)}{\mathrm{d} s}\right].
\end{aligned}
\end{equation}

由式 $\eqref{eq:用磁面函数与极向电流函数表示磁场}$, 式 $\eqref{eq:有向面积元表达式}$ 与式 $\eqref{eq:曲面上单位参数间隔之面积元}$, 可得

\begin{equation}\label{eq:曲面上之面积元与磁场点乘}
\begin{aligned}
\mathrm{d} \boldsymbol{S} \cdot \boldsymbol{B} = {} & \mathrm{d} s \mathrm{d} \theta \left(\left.\dfrac{\partial \psi\left(R, Z\right)}{\partial R}\right|_{\left(R, Z\right) = \left(R\left(s\right), Z\left(s\right)\right)} \dfrac{\mathrm{d} R\left(s\right)}{\mathrm{d} s} + \left.\dfrac{\partial \psi\left(R, Z\right)}{\partial Z}\right|_{\left(R, Z\right) = \left(R\left(s\right), Z\left(s\right)\right)} \dfrac{\mathrm{d} Z\left(s\right)}{\mathrm{d} s}\right) \\
= {} & \mathrm{d} s \mathrm{d} \theta \dfrac{\mathrm{d}}{\mathrm{d} s} \psi\left(R\left(s\right), Z\left(s\right)\right).
\end{aligned}
\end{equation}

由式 $\eqref{eq:用Grad-Shafranov算子简化电流表达式}$, 式 $\eqref{eq:有向面积元表达式}$ 与式 $\eqref{eq:曲面上单位参数间隔之面积元}$, 可得

\begin{equation}\label{eq:曲面上之面积元与电流密度点乘}
\begin{aligned}
\mathrm{d} \boldsymbol{S} \cdot \boldsymbol{J} = {} & \mathrm{d} s \mathrm{d} \theta \dfrac{1}{\mu_{0}} \left(\left.\dfrac{\partial F\left(R, Z\right)}{\partial R}\right|_{\left(R, Z\right) = \left(R\left(s\right), Z\left(s\right)\right)} \dfrac{\mathrm{d} R\left(s\right)}{\mathrm{d} s} + \left.\dfrac{\partial F\left(R, Z\right)}{\partial Z}\right|_{\left(R, Z\right) = \left(R\left(s\right), Z\left(s\right)\right)} \dfrac{\mathrm{d} Z\left(s\right)}{\mathrm{d} s}\right) \\
= {} & \mathrm{d} s \mathrm{d} \theta \dfrac{1}{\mu_{0}} \dfrac{\mathrm{d}}{\mathrm{d} s} F\left(R\left(s\right), Z\left(s\right)\right).
\end{aligned}
\end{equation}

因此, 穿过曲面 $\Sigma$ 之磁通量为

\begin{equation}\label{eq:穿过曲面之磁通量}
\begin{aligned}
\Delta \Psi_{\mathrm{p}} = {} & \int_{\Sigma} \mathrm{d} \boldsymbol{S} \cdot \boldsymbol{B} \\
= {} & \int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}} \mathrm{d} s \int_{0}^{\Delta \theta} \mathrm{d} \theta \dfrac{\mathrm{d}}{\mathrm{d} s} \psi\left(R\left(s\right), Z\left(s\right)\right) \\
= {} & \Delta \theta \left[\psi\left(R_{\mathrm{f}}, Z_{\mathrm{f}}\right) – \psi\left(R_{\mathrm{i}}, Z_{\mathrm{i}}\right)\right],
\end{aligned}
\end{equation}

从而单位环向角度所对应之极向磁通量为

\begin{equation}\label{eq:单位环向角度之极向磁通量}
\begin{aligned}
\Psi_{\mathrm{p, n}} = {} & \lim_{\Delta \theta \to 0} \dfrac{\Delta \Psi_{\mathrm{p}}}{\Delta \theta} \\
= {} & \left[\psi\left(R_{\mathrm{f}}, Z_{\mathrm{f}}\right) – \psi\left(R_{\mathrm{i}}, Z_{\mathrm{i}}\right)\right].
\end{aligned}
\end{equation}

用与计算极向磁通量相同之方法计算流过曲面 $\Sigma$ 之电流 (即穿过曲面 $\Sigma$ 之电流密度通量) 为

\begin{equation}\label{eq:穿过曲面之电流}
\begin{aligned}
\Delta I_{\mathrm{p}} = {} & \int_{\Sigma} \mathrm{d} \boldsymbol{S} \cdot \boldsymbol{J} \\
= {} & \dfrac{1}{\mu_{0}} \int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}} \mathrm{d} s \int_{0}^{\Delta \theta} \mathrm{d} \theta \dfrac{\mathrm{d}}{\mathrm{d} s} F\left(R\left(s\right), Z\left(s\right)\right) \\
= {} & \dfrac{\Delta \theta}{\mu_{0}} \left[F\left(R_{\mathrm{f}}, Z_{\mathrm{f}}\right) – F\left(R_{\mathrm{i}}, Z_{\mathrm{i}}\right)\right],
\end{aligned}
\end{equation}

从而单位环向角度所对应之极向电流为

\begin{equation}\label{eq:单位环向角度之极向电流}
\begin{aligned}
I_{\mathrm{p, n}} = {} & \lim_{\Delta \theta \to 0} \dfrac{\Delta I_{\mathrm{p}}}{\Delta \theta} \\
= {} & \dfrac{1}{\mu_{0}}\left[F\left(R_{\mathrm{f}}, Z_{\mathrm{f}}\right) – F\left(R_{\mathrm{i}}, Z_{\mathrm{i}}\right)\right].
\end{aligned}
\end{equation}

若旋转曲面 $\Sigma$ 自相交, 则应将式 $\eqref{eq:穿过曲面之磁通量}$ 与式 $\eqref{eq:穿过曲面之电流}$ 理解为按曲面上之参数域 $\left(s, \theta\right)$ 计代数重数或分段处理以去重; 若曲线经过 $Z$ 轴, 则应将曲线于轴上进行分段处理, 在 $R > 0$ 之区域使用柱坐标之表达式, 并在物理场具有适当正则性之条件下取 $R \to 0^{+}$ 之极限.

由于式 $\eqref{eq:单位环向角度之极向磁通量}$ 与式 $\eqref{eq:单位环向角度之极向电流}$ 之物理意义, 变量 $\psi$ 通常被称为极向磁通函数 (Poloidal Flux Function) 或通量函数 (Flux Function); 变量 $F$ 通常被称为极向电流函数 (Poloidal Current Function). 因此 Grad-Shafranov 方程式 $\eqref{eq:Grad-Shafranov方程}$ 为关于极向磁通函数之二维偏微分方程. 由极向磁通函数之定义式 $\eqref{eq:定义磁面函数}$ 以及变量 $\chi$ 之定义式 $\eqref{eq:构造新标量场}$ 可以看出, 相差一个常数之极向磁通函数对应相同之物理系统. 即极向磁通函数存在一个由数学描述方式所带来之冗余自由度, 该冗余自由度不具有物理上之可观测效应.