ベクトル場とナブラ記号の暗記メモ

正規直交座標系を仮定する．

場 field

$ \nabla = \left( \frac{\partial } {\partial x _ i} {\boldsymbol e}_i \right) = \left( \nabla _ i \right)

勾配 gradient:

$ \nabla f = \left( \nabla _ i f \right) = \left( \frac{ \partial f } {\partial x_i} \right) = \frac{\partial f}{\partial x _ i} {\boldsymbol e } _ i

ベクトル場の勾配：

$ \nabla {\boldsymbol a} = \left( \nabla _ i a _ j \right) = \left( \frac{\partial a _ j } {\partial x _ i} \right)

$ \frac{\partial f}{\partial x_i} = \frac{\partial f}{\partial x'_k} \frac{\partial x'_k}{\partial x _ i} = R _ {ki} \frac{\partial f}{\partial x'_k}

どの関数を微分するかを明示しよう：

$ \nabla _ i \left( f \right) g = \frac{\partial f}{\partial x _ i } g \\ \nabla _ i \left( fg \right) = \frac{\partial f}{\partial x _ i} g + f \frac{\partial g}{\partial x_i}

ヘッセ行列：

$ \nabla \nabla f = \frac{\partial^2 f} {\partial x _ i \partial x _ j}

発散(divergence)：

$ {\rm div} \ {\boldsymbol v} = \nabla \cdot {\boldsymbol v} = \frac{\partial v _ k}{\partial x _ k}

回転(rotation)：

$ {\rm rot} \ {\boldsymbol v} = {\rm curl} \ {\boldsymbol v} = \nabla \times {\boldsymbol v} = \left( \epsilon _ {ijk} \nabla _ j v _ k \right)

スカラー場に対するラプラシアン：

$ \Delta {\boldsymbol f} = \nabla^2 f = \nabla \cdot \nabla f = \frac{ \partial^2 f}{\partial x _ k x _ k} = \frac{\partial^2 f}{\partial x^2_ 0} + \frac{\partial^2 f}{\partial x^2_ 1} + \frac{\partial^2 f}{\partial x^2_ 2}

ベクトル場に対するラプラシアン：

$ \begin{aligned} \Delta {\boldsymbol v} & = \nabla \left( \nabla \cdot {\boldsymbol v} \right) - \nabla \times \left( \nabla \times {\boldsymbol v } \right) \\ & = \nabla^2 {\boldsymbol v} \\ & = \left( \frac{\partial^2 v_i}{\partial x^2_ 0} + \frac{\partial^2 v_i}{\partial x^2_ 1} + \frac{\partial^2 v_i}{\partial x^2_ 2} \right) \end{aligned}

よく使うやつら：

$ \nabla \times \left( \nabla f \right) = \epsilon _ {ijk} \nabla _ j \left( \nabla _ k f \right) = 0

$ \nabla \cdot \left( f {\boldsymbol a} \right) = \frac{ \partial f}{\partial x_k} a_k + f \frac{\partial a_k }{\partial x_k} = ( \nabla f ) \cdot {\boldsymbol a} + f ( \nabla \cdot {\boldsymbol a} )

$ \nabla \cdot \left( \nabla \times {\boldsymbol a} \right) = \nabla _ k \left( \epsilon _ {kmn} \nabla _ m a _ n \right) = \epsilon _ {nkm} \nabla _ k \nabla _ m a _ k = 0

$ \begin{aligned} \nabla \times \left( \nabla \times {\boldsymbol a} \right) & = \epsilon _ {ijk} \nabla _ j \left( \epsilon _ {kmn} \nabla _ m a _ n \right) \\ & = \epsilon _ {ijk} \epsilon _ {mnk} \nabla _ j \nabla _ m a _ n \\ & = \left( \delta _ {im} \delta _ {jn} - \delta _ {in} \delta _ {jm} \right) \nabla _ j \nabla _ m a _ n \\ & = \nabla _ i \nabla _ j a _ j - \nabla _ j \nabla _ j a _ i \\ & = \nabla \left( \nabla \cdot {\boldsymbol a } \right) - \Delta {\boldsymbol a} \end{aligned}

$ \begin{aligned} \nabla \cdot ( {\boldsymbol a} \times {\boldsymbol b} ) & = \frac{ \partial \epsilon_{kmn} a_m b_n}{\partial x_k} \\ & = \epsilon_{nkm} \frac{\partial a_m}{\partial x_k} b_n + \epsilon_{mnk} \frac{\partial b_n}{\partial x_k} a _ m \\ & = \epsilon_{nkm} \frac{\partial a_m}{\partial x_k} b_n - \epsilon_{mkn} \frac{\partial b_n}{\partial x_k} a _ m \\ & = ( \nabla \times {\boldsymbol a} ) \cdot {\boldsymbol b} - ( \nabla \times {\boldsymbol b} ) \cdot {\boldsymbol a} \end{aligned}

$ \begin{aligned} \nabla \times ( {\boldsymbol a} \times {\boldsymbol b} ) & = \epsilon_{ijk} \nabla_j ( \epsilon_{kmn} a_m b_n) \\ & = ( \delta_{im} \delta_{jn} - \delta_{in} \delta_{jm} ) \nabla_j ( a_m b_n) \\ & = \nabla_j (a_i b_j) - \nabla_j (a_j b_i) \\ & = \frac{\partial a_i}{\partial x_j} b_j + a_i \frac{\partial b_j}{\partial x_j} - \frac{\partial a_j }{\partial x_j} b_i - a_j \frac{\partial b_i }{\partial x_j} \\ & = {\boldsymbol b} \cdot ( \nabla {\boldsymbol a} ) + (\nabla \cdot {\boldsymbol b}) {\boldsymbol a} - (\nabla \cdot {\boldsymbol a}) {\boldsymbol b} - {\boldsymbol a} \cdot ( \nabla {\boldsymbol b}) \end{aligned}