内積や外積の偏微分 - biochem

簡単だが、後の参照用にまとめておこう。

$\frac{\partial \mathbf{x}\cdot\mathbf{y}}{\partial t} = \frac{\partial}{\partial t} (x_1y_1 + x_2y_2 + x_3y_3) = \frac{\partial x_2}{\partial t}y_2 + x_2 \frac{\partial y_2}{\partial t} + \frac{\partial x_3}{\partial t}y_3 + x_3 \frac{\partial y_3}{\partial t} = \frac{\partial \mathbf{x}}{\partial t} \cdot\mathbf{y} + \mathbf{x} \cdot \frac{\partial \mathbf{y}}{\partial t}$

つまり、積の微分の公式が成り立つ。

$\begin{eqnarray} \frac{\partial \mathbf{x}\times\mathbf{y}}{\partial t} &=& \frac{\partial }{\partial t}(x_2y_3 - x_3y_2, x_3y_1 - x_1y_3, x_1y_2 - x_2y_1) \\ &=& (\frac{\partial x_2}{\partial t}y_3 - \frac{\partial x_3}{\partial t}y_2 + x_2 \frac{\partial y_3}{\partial t} - x_3 \frac{\partial y_2}{\partial t}, \\ &&\, \frac{\partial x_3}{\partial t}y_1 - \frac{\partial x_1}{\partial t}y_3 + x_3 \frac{\partial y_1}{\partial t} - x_1 \frac{\partial y_3}{\partial t}, \\ &&\, \frac{\partial x_1}{\partial t}y_2 - \frac{\partial x_2}{\partial t}y_1 + x_1 \frac{\partial y_2}{\partial t} - x_2 \frac{\partial y_1}{\partial t}) \\&=& \frac{\partial \mathbf{x}}{\partial t} \times \mathbf{y} + \mathbf{x} \times \frac{\partial \mathbf{y}}{\partial t}\end{eqnarray}$

つまり、こちらも積の微分公式が成り立つ。