Vektorfeld/3/Rotation/Definition

Rotation eines Vektorfelds

Zu einem partiell differenzierbaren Vektorfeld

$G\colon U\longrightarrow \mathbb {R} ^{3}$

auf einer offenen Teilmenge ${}U\subseteq \mathbb {R} ^{3}$ nennt man

${}\operatorname {rot} _{}^{}{\left(G\right)}(P):={\begin{pmatrix}{\frac {\partial G_{3}}{\partial x_{2}}}(P)-{\frac {\partial G_{2}}{\partial x_{3}}}(P)\\{\frac {\partial G_{1}}{\partial x_{3}}}(P)-{\frac {\partial G_{3}}{\partial x_{1}}}(P)\\{\frac {\partial G_{2}}{\partial x_{1}}}(P)-{\frac {\partial G_{1}}{\partial x_{2}}}(P)\end{pmatrix}}\,$

die Rotation von ${}G$.