日: 2014年4月15日

Gradient, divergence, curl and Laplacian in orthogonal curvilinear coordinates

If \Phi is a scalar function and \bold{A} = A_1\bold{e_1} + A_2\bold{e_2} + A_3\bold{e_3} a vector function of orthogonal curvilinear coordinates u_1, u_2, u_3, we have the following results.

1. \displaystyle \nabla\Phi = grad\Phi = \frac{1}{h_1}\frac{\partial\Phi}{\partial u_1}\bold{e_1} + \frac{1}{h_2}\frac{\partial\Phi}{\partial u_2}\bold{e_2} + \frac{1}{h_3}\frac{\partial\Phi}{\partial u_3}\bold{e_3}

2. \displaystyle \nabla\cdot\bold{A} = div\bold{A} = \frac{1}{h_1h_2h_3}\left[ \frac{\partial}{\partial u_1}(h_2h_3A_1) + \frac{\partial}{\partial u_2}(h_3h_1A_2) + \frac{\partial}{\partial u_3}(h_1h_2A_3) \right]

3. \displaystyle \nabla\times\bold{A} = curl\bold{A} = \frac{1}{h_1h_2h_3}\left| \begin{array}{ccc} h_1\bold{e_1} & h_2\bold{e_2} & h_3\bold{e_3} \\ \frac{\partial}{\partial u_1} & \frac{\partial}{\partial u_2} & \frac{\partial}{\partial u_3} \\ h_1A_1 & h_2A_2 & h_3A_3 \end{array} \right|

4. \displaystyle \nabla^2\Phi = Laplacian\ of\ \Phi\\ = \frac{1}{h_1h_2h_3}\left[ \frac{\partial}{\partial u_1}\left( \frac{h_2h_3}{h_1}\frac{\partial\Phi}{\partial u_1} \right) + \frac{\partial}{\partial u_2}\left( \frac{h_3h_1}{h_2}\frac{\partial\Phi}{\partial u_2} \right) + \frac{\partial}{\partial u_3}\left( \frac{h_1h_2}{h_3}\frac{\partial\Phi}{\partial u_3} \right) \right]

These reduce to the usual expressions in rectangular coordinates if we replace (u_1, u_2, u_3) by (x, y, z), in which case \bold{e_1}, \bold{e_2} and \bold{e_3} are replaced by \bold{i}, \bold{j} and \bold{k} and h_1 = h_2 = h_3 = 1.

直交曲線座標における勾配，発散，回転およびラプラシアン

　仮に \Phi が一つのスカラー関数であり，また \bold{A} = A_1\bold{e_1} + A_2\bold{e_2} + A_3\bold{e_3} が直交曲線座標 u_1, u_2, u_3 のベクトル関数の時，下記の結果を得ます．

1. \displaystyle \nabla\Phi = grad\Phi = \frac{1}{h_1}\frac{\partial\Phi}{\partial u_1}\bold{e_1} + \frac{1}{h_2}\frac{\partial\Phi}{\partial u_2}\bold{e_2} + \frac{1}{h_3}\frac{\partial\Phi}{\partial u_3}\bold{e_3}

2. \displaystyle \nabla\cdot\bold{A} = div\bold{A} = \frac{1}{h_1h_2h_3}\left[ \frac{\partial}{\partial u_1}(h_2h_3A_1) + \frac{\partial}{\partial u_2}(h_3h_1A_2) + \frac{\partial}{\partial u_3}(h_1h_2A_3) \right]

3. \displaystyle \nabla\times\bold{A} = curl\bold{A} = \frac{1}{h_1h_2h_3}\left| \begin{array}{ccc} h_1\bold{e_1} & h_2\bold{e_2} & h_3\bold{e_3} \\ \frac{\partial}{\partial u_1} & \frac{\partial}{\partial u_2} & \frac{\partial}{\partial u_3} \\ h_1A_1 & h_2A_2 & h_3A_3 \end{array} \right|

4. \displaystyle \nabla^2\Phi = Laplacian\ of\ \Phi\\ = \frac{1}{h_1h_2h_3}\left[ \frac{\partial}{\partial u_1}\left( \frac{h_2h_3}{h_1}\frac{\partial\Phi}{\partial u_1} \right) + \frac{\partial}{\partial u_2}\left( \frac{h_3h_1}{h_2}\frac{\partial\Phi}{\partial u_2} \right) + \frac{\partial}{\partial u_3}\left( \frac{h_1h_2}{h_3}\frac{\partial\Phi}{\partial u_3} \right) \right]

　仮に (u_1, u_2, u_3) を (x, y, z) で置換すると，以下の場合，つまり \bold{e_1}, \bold{e_2} および \bold{e_3} が \bold{i}, \bold{j} および \bold{k} で置換され， h_1 = h_2 = h_3 = 1 で置換されるような場合などには，これらの結果は直交座標系の通常の式に短縮されます．