📝 题目
例 2 设有微分形式
$$ \omega = a\mathrm{\;d}x + b\mathrm{\;d}y + c\mathrm{\;d}z, $$
$$ \theta = A\mathrm{\;d}x + B\mathrm{\;d}y + C\mathrm{\;d}z, $$
试计算 $\omega \land \theta$ .
💡 答案与解析
解 我们有
$$ \omega \land \theta = \left( {{aB} - {bA}}\right) \mathrm{d}x \land \mathrm{d}y $$
$$ + \left( {{bC} - {cB}}\right) \mathrm{d}y \land \mathrm{d}z $$
$$ + \left( {{cA} - {aC}}\right) \mathrm{d}z \land \mathrm{d}x $$
$$ = \left| \begin{array}{ll} a & b \\ A & B \end{array}\right| \mathrm{d}x \land \mathrm{d}y $$
$$ + \left| \begin{array}{ll} b & c \\ B & C \end{array}\right| \mathrm{d}y \land \mathrm{d}z $$
$$ + \left| \begin{array}{ll} c & a \\ C & A \end{array}\right| \mathrm{d}z \land \mathrm{d}x. $$