正交曲面坐标系

坐标系的三个单位方向向量正交的坐标系称为正交曲面坐标系，例如柱坐标系和球坐标系都是正交曲面坐标系。

度规张量(Metric Tensor)

记直角坐标 $(x, y, z)$ 和新坐标 $(q_{1}, q_{2}, q_{3})$ ，有

d x = \frac{\partial x}{\partial q_{1}} d q_{1} + \frac{\partial x}{\partial q_{2}} d q_{2} + \frac{\partial x}{\partial q_{3}} d q_{3}

则

(d x)^{2} = \sum_{i, j = 1, 2, 3} \frac{\partial x}{\partial q_{i}} \frac{\partial x}{\partial q_{j}} d q_{i} d q_{j}

同理

(d y)^{2} = \sum_{i, j = 1, 2, 3} \frac{\partial y}{\partial q_{i}} \frac{\partial y}{\partial q_{j}} d q_{i} d q_{j}

(d z)^{2} = \sum_{i, j = 1, 2, 3} \frac{\partial z}{\partial q_{i}} \frac{\partial z}{\partial q_{j}} d q_{i} d q_{j}

进而有

(d s)^{2} = (d x)^{2} + (d y)^{2} + (d z)^{2} = \sum_{i, j = 1, 2, 3} (\frac{\partial x}{\partial q_{i}} \frac{\partial x}{\partial q_{j}} + \frac{\partial y}{\partial q_{i}} \frac{\partial y}{\partial q_{j}} + \frac{\partial z}{\partial q_{i}} \frac{\partial z}{\partial q_{j}}) d q_{i} d q_{j}

度规

g_{i j} = g_{j i} = \frac{\partial x}{\partial q_{i}} \frac{\partial x}{\partial q_{j}} + \frac{\partial y}{\partial q_{i}} \frac{\partial y}{\partial q_{j}} + \frac{\partial z}{\partial q_{i}} \frac{\partial z}{\partial q_{j}}

对于正交曲面坐标系，有 $g_{i j} = 0, \forall i \neq j$ 。

(d s)^{2} = \sum_{i = 1}^{3} g_{i i} (d q_{i})^{2} = \sum_{i = 1}^{3} (h_{i} d q_{i})^{2}

度规系数/拉梅系数

h_{i} = \sqrt{g_{i i}} = \sqrt{{(\frac{\partial x}{\partial q_{i}})}^{2} + {(\frac{\partial y}{\partial q_{i}})}^{2} + {(\frac{\partial z}{\partial q_{i}})}^{2}} (i = 1, 2, 3)

线微分

d \vec{s} = h_{1} d q_{1} \hat{q_{1}} + h_{2} d q_{2} \hat{q_{2}} + h_{3} d q_{3} \hat{q_{3}}

面微分

d \vec{σ} = h_{2} h_{3} d q_{2} d q_{3} \hat{q_{1}} + h_{1} h_{3} d q_{1} d q_{3} \hat{q_{2}} + h_{1} h_{2} d q_{1} d q_{2} \hat{q_{3}}

体微分

d τ = h_{1} h_{2} h_{3} d q_{1} d q_{2} d q_{3}

线积分

\int_{C} \vec{V} d \vec{s} = \sum_{i = 1}^{3} \int_{C} V_{i} h_{i} d q_{i}

面积分

\iint_{S} \vec{V} d \vec{σ} = \iint_{S} V_{1} h_{2} h_{3} d q_{2} d q_{3} + \iint_{S} V_{2} h_{1} h_{3} d q_{1} d q_{3} + \iint_{S} V_{3} h_{1} h_{2} d q_{1} d q_{2}

体积分

∭_{V} ϕ d τ = ∭_{V} ϕ h_{1} h_{2} h_{3} d q_{1} d q_{2} d q_{3}

梯度(grad)

\nabla f = \frac{1}{h_{1}} \frac{\partial f}{\partial q_{1}} \hat{q_{1}} + \frac{1}{h_{2}} \frac{\partial f}{\partial q_{2}} \hat{q_{2}} + \frac{1}{h_{3}} \frac{\partial f}{\partial q_{3}} \hat{q_{3}}

散度(div)

\nabla \cdot \vec{V} = \frac{1}{h_{1} h_{2} h_{3}} [\frac{\partial}{\partial q_{1}} (h_{2} h_{3} v_{1}) + \frac{\partial}{\partial q_{2}} (h_{1} h_{3} v_{2}) + \frac{\partial}{\partial q_{3}} (h_{1} h_{2} v_{3})]

拉普拉斯算符(div grad) $\nabla^{2} ϕ \equiv \nabla \cdot (\nabla ϕ)$

\nabla^{2} ϕ = \frac{1}{h_{1} h_{2} h_{3}} [\frac{\partial}{\partial q_{1}} (\frac{h_{2} h_{3}}{h_{1}} \frac{\partial ϕ}{\partial q_{1}}) + \frac{\partial}{\partial q_{2}} (\frac{h_{1} h_{3}}{h_{2}} \frac{\partial ϕ}{\partial q_{2}}) + \frac{\partial}{\partial q_{3}} (\frac{h_{1} h_{2}}{h_{3}} \frac{\partial ϕ}{\partial q_{3}})]

旋度(curl)

\nabla \times \vec{B} = \frac{1}{h_{1} h_{2} h_{3}} | \begin{array}{ccc} h_{1} \hat{q_{1}} & h_{2} \hat{q_{2}} & h_{3} \hat{q_{3}} \\ \frac{\partial}{\partial q_{1}} & \frac{\partial}{\partial q_{2}} & \frac{\partial}{\partial q_{3}} \\ h_{1} B_{1} & h_{2} B_{2} & h_{3} B_{3} \end{array} |

对矢量的拉普拉斯算符

\nabla^{2} \vec{V} = \nabla \cdot (\nabla \cdot \vec{V}) - \nabla \times (\nabla \times \vec{V})